ForMotherMayIholdyouinmemory

SACREDGEOMETRYFORARTISTS,DREAMERS,ANDPHILOSOPHERS

“Fromadropofwater to theexpanseof the solar systemandbeyond,thisbookrevealsanunseenharmonypervadingthenaturalworld.”

MARKVIDLERANDCATHERINEYOUNG,AUTHORSOFSACREDGEOMETRYOFTHEEARTH

“JohnOscarLieben has developed a simple frameworkwithwhich toviewthewholeworldofgeometricformationandproportion.Hisvisuallanguage maintains wholeness while giving detailed examples inarchitecture,musicalproportion, thehumanform,cosmictime,andtheelements. This art and craft can form sacred images which, liketraditionalmandalasandyantras,findabalancedorderintheworld.”

RICHARDHEATH,AUTHOROFTHEHARMONICORIGINSOFTHEWORLD,SACREDNUMBERANDTHEORIGINSOFCIVILIZATION,ANDMATRIXOF

CREATION

CONTENTS

CoverImage

TitlePage

Dedication

Epigraph

Introduction.AHarmonyThatStructurestheUniverse

Chapter1.TheFieldofHarmony

THERINGFORMTHEVESICA-TONECONSTRUCTIONTONE

Chapter2.TheGeometryofTime

THECENTEROFTIMETHETRITONEHARMONIOUSRECTANGLES

Chapter3.MoonandEarth

MEASURESOFTIMEANDSPACETHETHREEMASTERS(PART1)CONSTRUCTINGASEVEN-TONESCALETHETHREEMASTERS(PART2)EARTHANDMOONTHEINNEREARTH

Chapter4.TheTempleofWater

LANDANDWATERTHESHAPEOFRAINDROPSTHETEMPLEGARDENATEMPLEOFWATER

Chapter5.MusicalVessels

PLANETARYPOTTERYTHEWORLDINAGRAINOFSAND

Chapter6.TheSpiralingOctave

THESKYSPEAKSOCTAVESPIRALSMUSICALSPACES

Chapter7.HarmonicStructures

THEHARMONICSERIESTHEMEANTHERECIPROCALMUSICALRECTANGLESTHEVESICA-HARMONICCONSTRUCTIONTHEMINORTHIRD

Chapter8.TheCircleofResonance

RESONANCETWOFROMONEONEFROMTWOTHEBIRTHOFTHEMOONCROSSINGSHARMONICSCIENCE

Chapter9.TheMusicalUniverse

ASABOVE,SOBELOWTHEWATEROFLIFETHEGREATCENTRALSUNTHEGALACTICCORE

Chapter10.TheTreeofTime

ASPIRALOFTIMETHEVESICAOFTIMETHEERABEFORETIMETHETREEOFTIMEWITHINTHEROOTOFTIME

Chapter11.TheCosmicFlower

THESYMBOLINTHESKYSACREDCOSMOLOGYTHESYMBOLSPEAKSCEREMONY

Appendix1.TheVesselofCeremony

CEREMONYWITHINGEOMETRICSPACECLASSICALGEOMETRYCREATINGSACREDSPACEANEARTH-HONORINGCEREMONYABIRTHOFTHEMOONCEREMONYAWATER-HONORINGCEREMONYABIRTHOFTHECOSMOSCEREMONYASUN-HONORINGCEREMONY

Appendix2.AMiscellanyofNumbersandForms

Endnotes

Bibliography

IllustrationCredits

AbouttheAuthor

AboutInnerTraditions•Bear&Company

BooksofRelatedInterest

Copyright&Permissions

Index

INTRODUCTION

AHarmonyThatStructurestheUniverse

WEAREINTHEMIDSTOFAREVIVALofanancientwayoflookingattheworldasaninterlockingsystemofharmoniousnumbers.Therevelationsthatresultfromthisperceptionawakenfeelingsofwonderinthosewhocareaboutsuchthings.Why should it surprise us that this is the way things are? Marie-Louise vonFranzwritesinNumberandTime:

Wehavelongknownthatnumbersdidnotalwayspossessthedry-as-dust abstract quality that characterizes them today. Formerly theywerenotonlysomething“godlike”butpossessed—curiouslyenough—anall-embracingsignificance.Theydidnotdividebutunitedtwoworlds.1

When we begin to see in this way it becomes clear that the concept of ameaninglessuniverseinwhichevolutionproceedsblindlyfromrandomeventtorandom event is a delusion. This newway of seeing is really very old. It hasbeen called harmonics, which includes the disciplines of number, geometry,music,andcosmology—thequadriviumoftheancientmysteryschools.Ofgreatimportance to theancientswas thediscoveryof thenumerical ratios thatwerethebasisofmusical intervals.Whilemusicianshad intuitivelyemployed theseratios since the beginnings of their art, the practitioners of the harmonic artsrealizedthatsuchintervalscouldalsobeobtainedthroughtheintegraldivisionsofthemusicalstring.Thisdiscoveryofanumericalbasisforsensoryexperiencewas thebeginningof exact science.This scienceof sound as embodied in themusical scales became the foundation of a musical-numerical conception of

nature,whichincludedideasasdiverseasastronomicalcyclesandthenatureofthehumansoul.Theartofgeometrywaspartofthisexplorationsinceitcouldvisuallyportraynumbers,justasmusiccouldportraythemthroughsound.Intheprocessoftheirgeometricworkitbecameapparenttotheharmoniciststhatthegenerated forms had symbolic qualities—that is, they could somehow awakenperceptions about higher realities. For example, the circle seemed to have anaffinity with universal cycles and with the infinite, and the square with theearthly and the finite.Thus the circle-in-the-square could represent the humanbeing who was seen as an earthly vessel containing a divine essence. Thissymbolic character of geometry provides a simple way of contemplatingcomplexideasandprocesses,andassuchitbecameanindispensablefoundationinthetrainingofbothartistsandphilosophersintheancientworld.

Today,asinthepast,whenarchitectsanddesignersofspaceswhoareworkingwithsacredintentionwishtoplaceafirmfoundationundertheirworktheymustturntothelawsofcreation,whichincludethelawsthatdeterminecosmictimecycles and the shapes that arise from natural forces.While these laws can becomplex, there are simple ways of making use of them, and this is whatharmonics offers. The study of harmonics is an entry into a world of cosmicmeaning.Thisancientartisnowrisingfromthedarkseaoftheunconsciouslikealostcontinentwhereonceagreat,goldencivilizationflourished.Asweunearththe artifacts,wewonderwhy they should appear now. Is there a reason? Isn’teverything random andmeaningless as somany of us have been taught? Theharmonicist knows that this is untrue, and the reason themountains are risingfromthedarkseaisthat—itistime.Thewheelhasturned,theflowerisopening,andtheLawmustbefulfilled.

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This book demonstrates specific ways of creating harmonious forms usinggeometryandnumbersderivedfromthemusicalscale.Inthisapproachweareworking with tools that are as ancient as time yet perfectly suited to beimpeccableinstrumentsforthecreationofnewforms.Thereaderwillfindthatthedisciplineofharmonicsnaturallyweavestogethersymbolicform(geometric,numerical,andmusical)andthepractical.Thepracticalapplications,thehumancreations, will be seen to arise naturally from the energy contained withinprimordial number andgeometry.This process resonateswith nature’swayofcreation in which complex forms arise from the simple geometric shapes ofatomicstructures.

Ihavefoundthatthereisacreativematrixinwhichtheseenergiescaninteract.Ihavecalledthisthevesicaconstruction.Itscentralfigureisthevesica,theleaf-shaped product of the intersection of two circles.Most geometers are familiarwith thevesicapiscis, thevesica that is formedwhen the twocircles intersecteachother’scenters.Thevesicapiscishasbeenanimportantformgeneratorinsacred geometry. An example of its power is demonstrated in chapter 11 inwhich it generates the foundational polygons. I have explored some othervesicas, which can take infinite forms ranging from a thin sliver, when itsintersectingcirclesarebarelypenetratingoneanother,towiderandwiderfiguresresembling fish and eyes and then to broad figures shaped like fruit until itbecomes a single circle as the intersecting circles join and become congruent.While this figuremay seem simplistic, it actually has profound symbolic andgenerativeproperties.Theprocessof intersectionof thevesica-formingcirclesexactly corresponds to the intersections ofwaves,which are perhaps themostuniversalofdynamicprocesses innature.The infinitevarietyofvesica shapescorrespondstotheinnumerableheight-to-widthratiosofcreatedbeings.Andthevesica ratios thatwewill use are of great interest in their intimate associationwithtonalratios.Thiswillbediscussedfurtherinchapter1.

Theestablishmentofthevesica-constructionmatrixisalsodescribedinthefirstchapter.We should note that almost all the constructions in the book can bemadeusingclassicalgeometry,which isgeometrycreatedusingonlycompassand straightedge—that is, without the need for measurement. While thissimplifies the drawing process, there are deeper reasons for this restriction. Itrequiresonetoproceedonestepatatime,buildingeachnewpartonapreviouspart. This is the way of nature: from seed to sprout to leaf to flower. Thesequencemustbefollowed.Onecannothurrythrough,andinthisprocessthereisroomforquietcontemplationthatinvitestheunknown,thenew,toenter.

Alsointhefirstchapterweintroducetheseedformofourvesicaconstructions,thecircle-in-the-square,asimpleformwithgreatgenerativepotential.Thisisthefield,thematrixwithinwhichourvesicageometrywillbegenerated.Webrieflymentionaninvaluabletechniqueofgeneratingtonalratiosfromtheintersectionof the many kinds of diagonals that the square generates. This method isdescribed in appendix 2, drawing A2.2. This technique makes it possible tocreate a direct correspondence between the vesica and the ratios found on themusicalstring.Simplyput,wehavecreatedaharmonicworkshopinwhichwe

willworkfortheremainderofthebook.

Inchapter2wejourneyfurtherintotherealmoftone-geometryandhowitcanbe used to reveal connections between theMayan calendar, an ancient Greektemple of Apollo (whose ground-plan ratio will occur in different contextsthroughout thebook),and the lunationcycle.Twoother themes thatwill recurthroughout the book are introduced: the golden proportion and the tritoneinterval, themysteriouscenterofthemusicalscale.Inthischapterwebegintogetasenseofthenonrandomnatureofcosmicspace-time—thatthereseemstobeamusical-geometricharmonythatstructuresthecosmos.

Chapter3explores,amongotherthings,theharmonyoftheEarth-Moonsystem.Thesignificanceofthe366rotationsoftheEarthonitsyearlyvoyagearoundtheSunisnotedaswellasitsrelationtotheelusivehuangchungnoteoftheancientChinese.Thisnumber366willplayanimportantroleinfuturechapters.Next,asimpleexplanationoftheformationofadiatonicmusicalscaleisjoinedwithanimaginary dialogue among three ancientmasterswho are selecting the tonallycharged ratios that form the basis of our foot-and-milemeasuring system.Wewill realize that thissystemreflects themusicalstructureofcosmicspace-timeand takes an honored place alongside the ubiquitous decimal metric systems,whichhavetheirownvirtuesbutseemsomewhatdrywhenusedtodescribeanenchanted cosmos. Next we use our vesica-tone construction to accuratelygenerateamodeloftheinnerEarth.

Chapter4explores thesubstanceofwater.Surprisingly, thewater-to-landratiooftheEarthgeneratesthesamevesicaconstructionasthatoftheApollotemplepreviously mentioned. The shape of the waterdrop is considered andgeometricallyconstructedandcomparedtocertainNeolithicflattenedstanding-stonecirclesinBritain.Theintersectingvesicalpatternformedbyraindropsonthe surface of a pond is related to the Flower of Life, a symbol of universalcreation. An accurate model of the molecule of water is generated from thevesica construction at the interval of themajor third.Thismodel becomes thecenterofadesignforawatertemplegarden.

Chapter5demonstrateshowthevesica-toneconstructioncanbeusedtogeneratethe sizes of the inner planets and theMoon in relation toEarth.We take thisfurtherandgeneratepotteryshapesfromtheseplanetarycircleswhosesizesarethen compared, resulting in several interesting ratios.We briefly look at three

leafforms—perhapsthemostubiquitousvesicalshapeinnature—anddiscovertheirtones.ThenumericalharmoniesbetweenEarthandSunarenoted,andthisleadstothedrawingofaneedle-likevesica,whichdescribestheirsizerelation.

Chapter 6 explores more of the numerical relations in the Earth-Moon-Sunsystem. We gain a new appreciation for the numbers of the Pythagoreanintervalsthatseemtomostaccuratelygeneratethesynchronicitiesofthesystem.Thenwe lookathowthevesicageneratesoctaveconstructionswithwhichwecan drawmore harmonious forms.Andwe demonstrate how tonal spirals canalsobegeneratedfromthevesicaconstruction,oneofwhichistheancientspiralof fifths,which includesall the tonesof thePythagoreanscale.Andanautilusspiralwillbeanalyzedusingavesicaconstruction.

Chapter7describes theharmonicseriesandcoverssomeessential informationsuchas thearithmetic,geometric,andharmonicmeanpositionsonthemusicalstring that correspond to important ratios within the vesica construction. WebrieflyconsidersometonallybasedroomdimensionsusedbyarchitectsofbothRenaissance buildings and nineteenth-and twentieth-century concert halls. Thelayout of a ground plan of a building is generated from a vesica-harmonicconstruction at the minor-third interval. We show how this intervalsymmetricallydividesthescaleandhowit is intimatelyinvolvedwithphi(ϕ),thegoldenratio.

Chapter8 considers the resonanceprinciple andhow it canbe included in thevesicaconstruction.Thetwo-from-oneandtheone-from-twocreationprinciplesarecompared.Ageometric-mythiccreationstoryabouttheBirthoftheMoonisconstructedfromthirteendrawings.Weexaminetheprincipleoftheintersectionof energies using a crossed-vesica construction overlaid on a Greek sculptureand amedieval cathedral. A ground plan of a building is constructed using asimilarmethod.Weanalyzeacrossed-polygondrawingbythesixteenth-centurymasterGiordanoBruno.Aharmoniouspotteryfigure is thenconstructedusingthecrossed-polygonprinciple.

Chapter 9 explains how the principle of correspondence when validated byharmoniousnumberbecomesawelcomesimplificationofcomplexknowledge.Examples of this principle are discovered in the Sun-Earth-Moon system.Wenoteabiblicalvalidationfortheassociationoftheintervalofthethirdandwater,whosemoleculewehavealreadygenerated from the thirdvesicaconstruction.

Thisinformationwillbeofuseinchapter10.WeassignacosmicnumbertotheGreatCentralSunofesotericloreandproceedtogenerateharmoniesthatrelatethe ring shape of the inner Earth to the orbit of the solar system around thegalactic core. The ground plan for a Galactic Temple is created using anasymmetricalcrossed-vesicaconstruction.

Chapter10isastudyofcosmictimedrawnfromCarlCalleman’selucidationoftheMayancalendar.ThenumbersoftheEarth-Moonsystemwhencorrelatedtothe16.4-billion-yearMayancosmichistoryarefoundtoagreewiththefindingsof modern science. A pre-big-bang period also results from this analysis; itsexistenceishypothesizedbymoderncosmologyandispartofaboriginalmyth.SomeofthenumbersthatdescribethiscosmichistoryarethesameaswehaveseeninourEarth-MoonstudiesaswellasinouranalysisoftheApollotempleinGreece.Weconstructseveraldrawingsshowingthemusicalandlunaraspectsofthecalendar.AnotherdrawingshowsaTreeofTimewith its roots in thepre-big-bang, orDreamtime, period of cosmic history.Using the particular vesicashape,which embodies the progression of the calendar,we design the groundplanofaShipofTimeTemple.Wefindcorrespondencesbetweenthecalendarasawhole, the5,125-yearLongCountsectionof thecalendar,and the tzolkinritualyearof260days.

Chapter11takesusonajourneyintosacredcosmology,intoregionswheretheonly guides are number, geometry, tone, and imagination.We realize that thetoolsoftheancientharmonicistsarestillfreshandnew.Weseehow—bymeansofoursymbolicpolygonalimagery—spirit,consciousness,life,andtimeemergefrom the universal field. We further explore the Dreamtime, mentioned inchapter10.Weintroducetheideaofceremony,whichissimplyanothermeansofexplorationoftheregionsofexperienceclosedtoreductionistscience.

Appendix1followstheexampleoftheancientswhowerenaturallyledbytheirstudyofharmonicstocreatesacredspaces,thetemplesinwhichceremoniesofinvocationand thanksgivingwereconducted.Fiveexamplesofceremoniesarethe subjects of this appendix. Our simplified approach is to generate, usingcirclesandlines,symbolsofEarth,Moon,water,Sun,andcosmic time.Thesearelargedrawingsmadeonthegroundorinalargeroom.Thedimensionsofthedrawingsaccuratelycorrespondtotherealitiestheysymbolize.Withinthesefivedrawings,whoseconstructionwedemonstratestep-by-step,ceremoniescantakeplace. We include instructions for creating sacred space and suggestions for

guided journeys, and so forth. These are ceremonies of honoring, ofthanksgiving, and of purification, and while the geometry itself is fixed, theconduct of each ceremony is open to the interpretation of facilitators andparticipants.

Appendix2coversavarietyofarchitecturaltechniquesandhistoricalpractices,atechniqueforlayingoutabookpageanddesigningtypefaces,adiscussionofthe tonal content of the Fibonacci series, the construction of ellipses fromvesicas, harmonic analyses of ancient sculpture and temples, tonal qualities ofthemoon’sphases,andothermaterialrelatedtoharmonicformandthevesica.

1

THEFIELDOFHARMONY

OURINQUIRYBEGINSWITHTHECONSTRUCTIONofaprimalform:thecircle-in-the-square.Thissimpleshapeisasourceofinnumerableforms,anditstwoelementscanrepresentthepolaroppositesofmindandspiritorthefiniteandtheinfinite.The circle, symbolizing spirit, with its unknowable (because irrational) πcircumference, is the first of the two forms to appear.We can imagine that itradiates from its centerpointof zerodimension, recalling themoderncreationmythoftheoriginoftheUniversefromaninfinitesimalsingularity.

Fromthecircle,symbolofprimalunity,theenclosingsquareisgenerated.Withitsrationalandthusmeasurablesides,eachofwhichequalsthecirclediameter,the square is symbolic of reason and understanding. In enclosing the circlewithinthesquarewearesymbolicallyattemptingtounderstand,tobringwithinthe realm of reason, that which can never be fully known. This attempt isamplifiedwhenwe try to “square the circle”—that is, to construct, usingonlycompassandstraightedge,acircleandasquarewithequalperimetersorequalareas.Theancientpractitionerswhoattemptedcirclesquaringknewtheresonantpowerofsymbols,thetruelanguageoftheunconscious.Theybelievedthatthesymbolofthesquaredcircle,whichrepresentsacongruenceofmindandspirit,might, when perceived by the unconscious, help to manifest the symbolizedunity in ordinary reality. The geometers were able to come very close tosquaring the circle. But an exact circle squaring has, in modern times, beenshowntobeimpossible—ameaningfuldiscovery,whichdemonstratesourlimitsin our quest for unity with the divine: there will always be realms that areunreachable.

The present approach to geometric construction can beginwith a straight linethat is bisected using compass, straightedge, and scriber (drawing 1.1 [1] and

[2]). The process also generates, as a kind of side effect, the vesica piscismentionedintheintroduction.Ourpurpose,however,istoconstructthecircle-in-the-squareusing thesesameclassical toolsofcompassandstraightedgeandscriber.Thosepersonswhowould ratheruseacomputerdrawingprogramcanmaintainthepurityoftheclassicalmethodbyproceedingonestepatatimewithcirclesandstraightlinesandlettingeachstepbeguidedbythepreviousstep—thatis,bynotsimplymeasuringoutfiguresbutlettingtheinherentpropertiesofonestageleadtothenextstagesuchaswedemonstrateindrawings1and2.

Drawing1.1. (1)The first steps ingenerating theprimal formof thecircle-in-the-square.Drawalinethatwillbethediameterofthecircle.(2)Bisectthelinewith a vesica piscis, establishing a circle center and also the length of theradius.

Drawing1.2.(1)Afterdrawingtheinitialcircle,draw(2)fourcircleswiththesame diameter as the circle in (1) with centers at the ends of the crossingdiameters.Theintersectingpointsofthesefourcirclesgiveusthecornersoftheenclosingsquare.Drawlinesfromthesepointstomakethesurroundingsquare.

THERINGFORM

The circle within a circle—the primal wave pattern symbolizingrhythmic temporal expansion or, in space, a vessel holding anessence—is the ring form, which exists throughout material andnonmaterialnature.Itisthecellwithitsnucleus,theouterlayersandcoreofcelestialbodies,theorbitsofplanetsaroundtheSun,thefleshandstoneofafruit.Theringsymbolconnectsdiversephenomena—ofallshapes.Theclassictemple,thoughgenerallyarectangle,isalsoa ring with its outer corridors protecting the inner sanctum in amanner similar to theway theouterbodiesof animalsprotect theirvitalinnerorgans.

Figure 1A. “The voice is moved in an infinite number of undulating circlessimilar to thosegenerated instandingwater ifastone iscast into it,whenweseeinnumerableringsspreadingforth fromthecenterandtravelingoutas faras they possibly can. . . .”wrote Vitruvius.1Here two stones cast into a pondgeneratewiththeirripplestheformofthevesica,atwo-dimensionalsymboloftheintersectionofradiantenergiesinthethree-dimensionalworld.

Innatureitiscommonfortheringshapetobeexpandedtoincludeseveralrings—anestingofcirclesorothershapes.Thus,theinnerEarthandtheatmosphereconsist of regions arranged in concentric layers. The human body consists ofskin, muscle, fascia, and so forth, which protect and nourish the organs.Furthermore,itisbelievedthatthereareenergeticringssurroundingthephysicalbody—such as the subtle etheric andmental bodies—which correspond to thelayersofatmosphereandmagnetismsurroundingtheEarth.Figure1Bshowsafewexamplesoftheringforminnatureandartthat,again,areareminderoftheuniversal occurrence of this form whose simplest expression are the ringsspreadingoutfromaraindropfallingintowater.Thegeometerworkswiththisidea of rings but in a context of harmony unaffected by the environmentalfactorsofthephysicalworldthattendtodistortthepurityofprimalform.Aswe

will see below, there are ring structures thatwemay generatewith the vesicaconstructioninwhichtherelationsamongtheringsarehighlyharmonious.

THEVESICA-TONECONSTRUCTION

Asanintroductiontothevesicaform,Ipresentanexamplethatshowshowthisformrevealsthewayweseetheworld.Whenapersonstandsontheedgeofthesea,thehorizonisaboutthreemilesaway.Imagineasecondpersonstandinginarowboat that is floating on the first person’s horizon.We can affirm that thecircle of perception of both persons combined is about sixmiles in diameter.Furthermoretheircirclesofperceptionintersectoneanother,formingacommonvesica-shapedspace(seedrawing1.3below).Thenameforthisparticularvesicais the vesica piscis, formed when two circles intersect one another at theircenters.Nowletusimaginethatthepersonintherowboatbeginsrowingtowardthepersonontheshore.Thesizeoftheircirclesofperceptionremainthesame,buttheybegintooverlaponeanothermoreandmoredeeply.Thevesicagrowswiderandwideras its shape traverses innumerableheight-to-width ratiosuntiltheboat reaches theshoreand thepersonscome together.Their twocirclesofperception have joinedwith the vesica, which has expanded into a circle; thethreeelementshavebecomeasinglecircle.

So,generallyspeaking,whenwe lookatanotherpersonweare joined to themwithinaperceptualvesicawhosesizeandshapedependsonthedistancewearefromthem.(Thisappliesalsotoobjectsthatwecanimaginebeingsentientandhavingcirclesofperception.)Butwhatisinterestingforthepresentstudyisthetonalaspectofthisphenomenon.Ifweimagineastraightlinebetweenourshorepersonandtherowboatperson,thislinecanrepresentastretchedmusicalstring.Now, as the rowboat person rows toward the shore person, tones are beingsounded along the musical string. This is the basic idea behind the vesicaconstructionsshownindrawings1.4and1.5onpages13and14,respectively.

We begin with our circle-in-the-square. The horizontal diameter of the circlebecomesourmusicalstring.Wenowneedtolocatethepointsonthestringthatcorrespondtomusical intervals.Wenotehere therequirementforanumber tobeusableasamusicaltone:itmustbeafractioncomposedofwholenumbers.Anirrationalnumbercannotbeusedasatonalnumberforthesimplereasonthatitcannotbeexactlylocatedonthemusicalstring.Tofindatoneonthestringwe

decideonaratio,say2/3,andmeasureoutaone-thirdsegment leavinga two-thirdssegment.Thenweplaceastop(suchasafinger)atthe2/3pointandpluckthe2/3length.Inthiscasewehavesoundedtheintervalofthemusicalfifthinrelationtotheopenstring,consideredtobethefundamental(ifwehadpluckedthe 1/3 length, an octave fifth would have sounded). See drawing A2.2 for adescriptionofthetechniqueofdiagonalcrossings.

Figure1B.Fiveringsbothman-madeandnatural:(a)Azteccalendarstone;(b)rosewindowofNotreDamedeParis;(c)Chinesepidisc;(d)crosssectionofabeet;(e)waterripplesinatidepool.

Drawing 1.3. As described in the text the vesical shape encloses the sharedvisually perceived space between a person standing on the shore and anotherpersonstandinginarowboatonthehorizon(1).In(2)and(3)theothercanbeimagined to be rowing a boat toward the shore and in so doing altering theshape of the shared vesica. Thus we see a geometric connection between thepsychic-numeric tools that determine the way we perceive and the vesica. Inotherwordsweliveandmoveincircularspaceborderedbythehorizon.Otherbeingsmoveinandoutofourcircle,andourspatialrelationshiptothemcanbeexpressedinidealterms—thatis,asvesicalgeometrythatrevealsthatthevisualspace we have in common has a vesica shape that widens or narrows as weapproachorrecedefromoneanother.

Drawing1.4.The intervalof the fifthon the string/diameter isgenerated fromcrosseddiagonals.

Yetwewouldprefer,atleastinourgeometricaldrawings,tolocatetonalnodeswithoutresorting tomeasurementsorasfewaspossible.One importantuseofmeasurementsthatisemployedinthisbookistoidentifytheratioofthepoints

on the string/diameter that are generated from the crossed diagonals. Thesepoints create rational divisions of the string/diameter,which, interestingly, aregeneratedfromirrationaldiagonals thatare themselvesgeneratedfromrationaldivisionsofthesidesofthesquare.Wemightsaythattherationalintheformofsimplenumbersappliedtothesidesofthesquareistransmutedinthecauldronof the irrational, in the formof crossed diagonals, into a newkind of rationalnumber—namely,thenumbersofthemusicalintervals.Withthepresentmethodwecanuse thenumerouscrossingsof thegreatvarietyofdiagonalswithin thesquareas tools to locateour tonalnodeson thestring/diameterof thecircle inthe square. The crossing of two diagonals locates a point, which, when aperpendicularfromitisdroppedtothestring,locatesatonalnode(seedrawing1.4).Thediagrams indrawing1.5 locatesomeof the tonesofacommon just-tunedscaleusingthismethod.

Let us now consider how the vesica itself is generated. It is instructive toconsiderthisgeometricconstructionasasymbolicenactmentofcosmicprocess.Thisimaginativewayofseeingisactuallyoneoftheprincipalvirtuesofsacredgeometry.Thus, theoriginalcirclecanbeseenas the fieldofcreationwaitingfortheinitiatingimpulse.Atthecenterofthecircleistheprimordialpointfromwhich the circular field was generated. Being at the halfway point of thestring/diameter,itistonallyattheoctavepositionofthewholestring—thatis,atthe first octave. On either side, from center to diameter end, are innumerabletonalpositionswhosepitchesbecomehigherastheyapproachtheends.

Figure 1C. Cells doubling during mitosis. The implied vesica between thedaughter cells is symbolically moving through all the tones from thefundamentalat the single cell position to the final separationwhen theoctavewillbesounded.

Drawing1.5.Theintersectionoftwodiagonallinesthatoriginatefromintegraldivision points of the containing square result in crossing nodes that, when aperpendicular is dropped to the main diameter or musical string, divide thestring intorationalormusicalsegments.When thisdiameterpoint ismirroredontheoppositesideoftheverticalaxisandcirclesaredrawnthroughthesetwopoints,a vesica is formed that can thenbeassociatedwith the tonegeneratedfrom the original intersectionof diagonals. In this drawing twelve vesica-toneconstructions created from various diagonals within the square result in atwelve-tonescale.Thegeometrictritone,thecenterofthescale,isanexception:being an irrational quantity, it is generated by a circle arc. This crossing ofdiagonalswithinthesquareisthebeginninggesturethatwewillusetogeneratethevesicaconstructions throughout thebook. (For thevesica ratiosandotherpropertiesseedrawingA2.1.)

Drawing 1.6.The creation of formas symbolized by the intersection ofwavesandtheresultingvesica,thesymbolofform.Thisdrawingmightbeconsideredasapreviewoftheseparatingcellsinfigure1C:themaleandfemalepronucleijointoformthefirstcell,then,asin1C,theybeginseparatingintotwo.

Now let us imagine a very high vibration beginning at each end that causescircularwavestobeginripplingthroughthemediumofthecircularfield.Whenthewaves impinge on the string/diameter they cause a tone to issue from thestring. So, as thewave front grows andmoves along the string, a descendingglissandooftonesissounded(orreallyaunisonoftheglissando,becausethereare twowavefronts,eachmovingfromitspole toward thecenter).Thewaveshaveadestinationrepresentedbythecentralverticaldiameter.Thisistheportalinto the 2/1 ratio, the first octave, the placewhere the intersection of the twowaves takes place and the series of vesicas begin.During this processwe can

accurately state that all tones are being produced.But nowwedecide thatweonly need a few of these tones for the scalewe are building. Let us stop theprocess at, for example, 2/3 of thewhole string length.We have noted abovehowgeometricallywecanfindthe2/3nodeonthestring—withdiagonals thatconnectintegernodesontheperimeterofthesquare.Wenowhaveanexampleofthesymbolicpurposeofthesquarethatwehaveconstructedaroundthecircle.It is a workshop, a laboratory within which the transcendent circle may bestudied and transmuted into a source of harmonious form. The tools that areemployedarethediagonalsandtheirgeneratedpointsandnumbers.

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As we have noted, the crossings of these various diagonals of the squaregeneratetonalnodes,placeswherethewavefronts,expandingfromtheequatorends, can be stopped and where the musical string can then be plucked toproduce our geometrically derived tones. The node, in this case the 2/3 ormusicalintervalofthefifth,isnowsounded,andthevibrationalwaveofsoundexpandsthroughthecircularfielduntilitisstoppedatthelimitofthefield.Thisprocess takes place simultaneously on both sides of the vertical axis andgeneratesacentralvesica,anewformcomposedofcirclearcsarisingwithinanenvironment formerly dominated by circles and straight lines. Each vesica isuniquewith its own height-to-width ratio, its own ring structure, and its owntone,itssong.Avesicacanassumeaninnumerablevarietyofshapes,fromtheslenderestofgrasslikeformstoformsofnear-circularfullness.Theshapeofthevesicadependsonitstone,whichcorrespondstothepointonthestring/diameterwherewehavestoppedthestring.

Figure1D.Theancientharmonicistsusedadevicecalledamonochordthatconsistedofoneormorestringsstretchedacrossasoundingboardonwhichmeasurementscouldbemade.Onthemonochordtheycouldlocatenumericalratiosandhearthembystoppingthestringatthe

measuredpointandpluckingittosoundthetone.Above,athree-stringmonochord.

*

Thefirstoctave,theplacewherethevesicatakesform,couldbecalledtherealmofphysicalreality,ofthree-dimensionalspace,twodimensionsofwhich,heightandwidth,containthegeometryofthevesica.Thethird,depth,issuggestedtotheimaginationbuthasnoplaceinthissymbolicgeometry,becausetoincludeitwould defeat this geometry’s purpose: the simplification of the overwhelmingcomplexityoftheworld.

We recall that outside the central octave are higher octaves—that is, tonalpositionsoneachsideofthecentraloctave.Thecirclesofthesehigheroctaves,inourpresentconception,havenotyet“takenform”byenteringtherealmofthevesica by joining with their mirrors—that is, intersecting with their

correspondingcirclesontheothersideoftheverticalaxisrepresentingthemainoctave. They exist as pure moving energy until they cross the threshold intoform. They are represented by the growing circles that emanate from thediameterendsofthecontainingcircle.

Thereisanotherinterestingfeatureofthevesicaconstruction.Wenotethatthereisanimpliedinnercirclewithinthevesicaandanimpliedcirclethatenclosesthevesica(seedrawing1.7).TheinnercirclewewillcallC4,theoutercircleC3,andtheenclosingcircle that iscircumscribedby thecontainingsquarewewillcallC1.Thetwovesica-formingcircleswewillcallC2.Astrikingpropertyofvesicaconstructionsisthegeometricproportionthatexistsamongthediametersofthenestedcircles:

C1:C3::C3:C4.

Drawing1.7.Thecirclesofthevesicaconstruction:C1,theenclosingcircle;C2,thetwovesica-formingcircles;C3,thecirclethatenclosesthevesica;C4,the

circlethatisenclosedbythevesica.

Indemonstrationofthisfactletuslookmorecloselyatthedetailsofourvesicaconstructionof themusical fifth (drawing1.4).Givenourcircle in the square,whose construction is shown in drawings 1.1 and 1.2,we use the crossing ofquarter-squareanddouble-squarediagonals to locatea tonalnodeon thecirclediameter.Webisectthedistancebetweenthisnodeandthefartherdiameterendtogiveusacenterforourvesica-formingcircle(C2).Thiscircleismirroredontheoppositesideresultinginavesica,inthiscaseavesicapiscis,generatedfromtwocircles, eachofwhich intersects theother’scenter.With thisvesicapiscisgenerated by the vesica construction at the interval of the fifth the diameterproportionoftheimpliedvesicaringsis

C1:C3::C3:C4=theirrationalvalueof√3,or1.732...,whichistheratiooftheinnerdiagonalbetweenoppositecornersofacubetoitssideandalsothedistancebetweenparallelsidesofthehexagon.

It becomes apparent that the limitless variety of vesica constructions makeavailable to us awealth of proportional ring structures that are expressions ofgeometricandtonalrealities.Theseidealstructuresdonotexistinavacuumbutinsteadcorrespondtotheubiquitousringstructuresoforganicandcosmicformsthatunderliethenaturalworld.

TONE

Tonalquantitiesareexpressedasratioscomposedofwhole-numberfractionsofthestringlength,suchas2/3forthefifthintervalor4/5forthethird.Theinverseof these fractions—that is, 3/2 and 5/4—are the frequency ratios, the rate ofvibrations of the string in relation to the vibration rate of the fundamental oropen string,which is designated as 1/1.Theoctave,with frequency2/1 at thecenter of the string, is the beginning point from which the series of vesicasunfold. The octave represents the principle of doubling that we encounter inmostifnotalllifeprocesses.Theseries1,2,4,8,16,32,64,128,256,512,andso forth, is an octave or doubling progression embedded in natural growthprocesses.Thesingleeggorcellwhenactivateddividesintotwo;thenthetwobecome four, and so on. So the octave, the doubling of the fundamental tone,

correspondstothisuniversaldoublinggesture.

While it has been noted that the octave, fifth, and third have been present inmusical scales throughout time and across cultures, there is an even firmerfoundationfortheseandfortheotherintervalstobepresentintheharmonicists’toolbox. This foundation is contained in the harmonic series in which all theintervalsofthescalearepresent.Followingisasimplifieddescription.

An object when struck will vibrate in many different frequencies.Themusical string is themodel for thisphenomenonbecauseof itssimplicityandminimumofdistortingfactorssuchasshape,size,andmaterialcomposition.Anopenmusicalstringwillvibrateasasinglewhole but also simultaneously in two parts, three parts, four parts,five parts, and so on, in theory, to infinity. While this complexvibration is going on all the harmonics are being sounded withdecreasing volume in higher and higher octaves. To draw on thetones within the harmonic series when constructing a scale orgenerating a vesica construction is thus a “lawful”way to proceed,sincewearedrawingonratioswhichareembeddedinthevibratorynatureofexistence.

InthefollowingtablewechoosethefirstharmonictobeinthekeyofC—thatis,thefundamentalistheCtonethat,becauseitisthetonesoundedbyouropenmusicalstring,hasa ratioof1/1.Thesecondharmonic,withratio1/2,dividesthe string in two, sounding an octave of the fundamental. The third harmonicdividesthestringintothirds,soundingafifth(1/3is theoctaveofthe2/3fifthintervalandismultipliedby2tobringitintothemainoctave),andsoon.Thefirstnineteenharmonicsgeneratethetones(lessone)ofthestandardscale.Someareexactjust-intonationtones,whileothersarecloseapproximations.Wehaveto jump to the twenty-seventhharmonic to find the final toneof thechromaticscale, themajor sixth interval. This happens to be the Pythagorean sixthwithratio16/27(.5925),quiteclosetothejustsixthof3/5(.60)ratio.

For clarity we have, through successive multiplications by 2, reduced theincreasinglyhigherharmonics into themainoctave—that is,between½and1.Forexample,1/14×2=1/7;1/7×2=2/7;2/7×2=4/7,theminorseventhinthemainoctave.Becausetheharmonicsdonotalwaysexactlycorrespondtothe

ideal ratios the little arrows next to some of the octave tics indicate suchdiscrepancies: up arrowmeans the tone is sharp of exact tuning; down arrowmeansthetoneisflat.

2

THEGEOMETRYOFTIME

ITISDIFFICULTTOESTABLISHANDMAINTAINastandardforspatialmeasureinthematerialworld.Even if aphysical standard is established, sculpted into shape,andenshrinedinagoldenboxthereisalwaysthechance(orinevitability)thatitwill be lost in the clash of civilizations or disintegrated by time. Yet time ismeasuredwithnumbersthataremoreorlessfixedandthusoffersanonarbitrarysourceofnumbersand,asweshallsee,highlymeaningfulwaystoapproachthemeasurementofspace.Forexample,thelunardiameterof2,160milesmayhavebeen assigned its length by unknown astronomers because of the fact that thelengthofaprecessionalageis2,160years.

A supporting synchronicity is the fact that the lunation cycle of newmoon tonewmoonis29.53days,andthelunarorbitalperiod,whichistheperiodofoneorbit of theMoon in relation to the fixed stars, is 27.32days.Thequotientofthese twocycles,29.53÷27.32, is1.08,whichalso refers to2,160, theMoondiameter,since2,160÷2=1,080, theradiusof theMooninmiles.Thesefewexamplessuggest that thedimensionsandthetimecyclesof theMoonmaybemeaningful—thatis,theydonotconsistofrandomnumbersbutinsteadarepartofaninterconnectednumericalsystem.

*

Continuingwiththelunar-timepropertieswenotethatthereare12.36lunationswithin a year of 365 days. Here we encounter the golden number, phi, orϕ(1.618...),since2/ϕ=1.236.Thusthereareexactly20/ϕlunationsof29.53daysinayearof365days:

20/ϕ×lunationof29.53days=365days,since20÷1.618=12.36,and12.36×29.53=365days.

Furthermore, we note that the diameter relation inmiles of Earth toMoon is7,920/2,160,whichequals3.66(11/3).(WehereusetheancientcanonicalvalueoftheEarthdiameterasopposedtothemodernmeasureof7,926miles.)SotheMoon diameter is exactly 3/11 (1/3.666) that of Earth’s diameter.And 366 isalsoanimportanttimenumber,beingthenumberofrevolutionsofEarthonitsaxis in one year. (Herewe have a congruence of a space number and a timenumber,asynchronicitythatshouldalertustoitsimportance.Laterwewillseethis number in its roles as both a tonal frequency and a number embedded incosmicevolution.)

*

Letusbrieflylookatthesevaluesinatonalcontext.Theratio3/11(.2727)istheoctave above the tone 6/11 (.5455), known as the neutral seventh.We rarelyencounter11-basedorundecimal tones in theWestern scale except inmodernavant-gardeworksuchasthatofHarryPartch,althoughArabiccomposershavetraditionally used these ratios. In the harmonic series 11 is the number of theundecimal tritone of 8/11 (.7273), a ratio that is the third octave of the 1/11(.0909)harmonicnode.Elevenisakeynumberinancientcanonsofproportion.For example, it divides an even 720 (an important highly composite number)timesinto7,920,thevalueofEarth’sdiameter.ElevenisthenumberthatrelatesthediametersizesofEarthandJupiter(seedrawing2.1below).Alongwiththenumber 7, also rare inWestern music, the number 11 and its multiples wereimportantcalculatingtoolsfortheancientbuilders.Theratio22/7,forexample,isagoodapproximationofπ.Theratio11/14approximatestheareaofacirclewithdiameterequalto1.Theratio21/11(1.91)isthevolumeratioofacubetoits inscribed sphere,whose diameter equals 1. The 7/11 (.6364, an undecimalaugmentedfifth)tonegeneratesavesicaconstructionthataccuratelyrevealstherelative sizes of the moon (the C4 circle), the Earth circle (C1), and anapproximation of the size of Mars (the C3 circle) as shown in drawing 2.2(below). Another coincidence involving the number 11 is that the differencebetweenthe365-dayyearandtwelvelunations(354days)equalselevendays—asifpointingustowardthiskeynumberinEarth-Moonrelations.

Owingtothedifferencebetweensolarandsiderealtime,theEarth

rotatesuponitsaxis366timesinayearof365days.Earthcompletesarevolutiononitsaxisin23hoursand56minutes.Thisisthesiderealday,thetimethatthestarsappeartocompleteonerevolutionaroundtheEarth.Atthesametime,theEarthadvancesaboutadegreeeastwardonitsorbitaroundtheSunasitturnseastwardarounditsaxis.Thesamemeridianthatwasfoundatnoonononedayonthenextdayis1/365behind.ThustheEarthperformsonecompleterotationonitsaxisplusthe365thpartofarotationeveryday.Thereforein365daystheEarthturns366timesonitsaxis.Andsothereare366siderealdaysinayear.

Drawing2.1.(a)Crosseddiagonalsfromtwocornerstoadjacent1/3pointsonthe perimeter of the square generate the 6/11 tone. This is the octave of theMoontoneof3/11,whosediameterinrelationtotheEarthdiameteris3:11.(b)ThistonegeneratesaC4circlewhosediameteris1/11ofthatoftheenclosingcircle.Theratio1/11isthediameterratioofEarthtoJupiter.

Drawing 2.2. (a)The 7/11 tone is generated from diagonals originating frompointsthatdividetheperimeterlinesofthesquareintofiveparts.(b)TheareaoftheinnervesicacircleinrelationtotheC1enclosingcircleareaisinthepreciseMoon-Earthratioof3/11.Thecirclethatenclosesthevesicaisquiteclosetotherelative size of Mars, whose diameter equals 4,221 miles. The circle in thisdrawinghasarelativediameterof4,136miles.Thus4136/4221=.98—thatis,the Mars circle is 98 percent accurate. The ratio of the vesica = 1.9149 =√3.666,with3.666=11/3.

THECENTEROFTIME

Ancientcalendarmakerswerefacedwiththeobstacleoftheincommensurabilityofthemonthandtheyear.But,aswehaveseenwiththe20/ϕrelationoflunarto solar year, there is an underlying pattern ormeaning in this discrepancy. Itmay be that the Maya discovered this pattern, and it was a factor in theirdevelopmentoftheirvigesimalsystemofnumerationbasedonthenumber20asopposedtoourbase10decimalsystem.

OneofthemostimportanttimeperiodsfortheMayawastheuinal,aperiodof20days.Thirteenuinalsequaledatzolkin,theMayanritualyearof260days.Forourpresentdiscussion,thetzolkinisofinterestinthatitmarkstheverycenterof

theyear-as-scale.

LikemanyancientculturestheMayawereconcernedwithlocatingthecenterofthecosmos.Itisbelievedthatthenorthernpeoples,fromwhomtheMayaweredescended,consideredtheNorthStar tobethecosmiccenter.Ancestorsof theMayacontinuedthisbeliefevenastheymigratedsouth.YetinsouthernMexicoand Central America where they settled this belief was no longer relevant,because the polestar had long since moved from its place overhead to aninsignificantplacenearthehorizon.AccordingtoJohnMajorJenkins,theMayasomehowlocatedtheGalacticCenterintheregionoftheDarkRiftoftheMilkyWay and realized, longbefore our culturemade the samediscovery, that theyhadfoundthetruecenterofthecosmos.

TheMayantzolkinof260daysalsodesignatedacenter—thetonalcenteroftheyear.Weknowthatthecenterofthemusicalscaleisnotthearithmeticmeanoraveragebetween fundamentalandoctavebut rather is locatedat thegeometricmean.Thearithmeticmeanof thestring lengthwouldbe thefundamentalplustheoctavedividedby2:

1+½÷2=3/4,

which is the interval of the fourth.The tone that resides at the tonal center iscalled the tritone. The formula for locating this tonal center at the geometricmean is to take the square root of the fundamental times the octave stringlengths,or

√(1×½)=.707...,

whichisanirrationalquantity.

“TheuseinIndianmusicoftheaugmentedfourth,ortritone,atthecriticaltimesofmidnightandmiddayremindsusofthemagicalimportanceattachedtothosehours,andoftheuseofthetritone(diabolusinmusica)byWesternmusiciansfortherepresentationofmagic,whichisnothingotherthanthepossibleintersection,atcertaincriticalhours,ofworldsthatcannotnormallycommunicate.”

—ALAINDANIÉLOU1

Aswenotedearlier, tohaveanactual tone that canbe locatedona stringwemust have a rational fraction. There aremany tritones that have been used asapproximationsto.707...;5/7isoneoftheseandhasadecimalvalueof.7142,which,whenmultipliedby365days,equals2605/7,quiteclosetothe260-daytzolkin.Anearlyidenticaltritonevalueof52/73(.712)producesanexacttzolkinperiod:

52/73×365days=260days,

andmayhavebeenusedbytheMayabecauseoftheimportanttimenumbersofwhichitiscomposed.

TheMayanCalendarRoundof18,980daysistheperiodinwhichtheseasonalyearof365days(thehaab)andthetzolkinaresynchronized:

18,980days÷52=365days(haab);18,980days÷73=260days(tzolkin).

This52/73ratioalsorelatesthediametersoftheMoonandMercury:

2,160miles(Moon)3,032miles(Mercury)=5273.

THETRITONE

Strictlyspeaking, thetritoneinterval iscomposedof threewholetones.Thusathree-tone interval between any two tones is a tritone. For simplicity wewillconsider the tritone to be the interval that joins the fundamental and the tonalcenterofthescale—forexample,CandF♯.Thetritoneisconsideredtobethemostdissonantintervalandhasoftenbeenshunnedbycomposers,mostnotablybycertainmedievalauthoritieswhocalleditthediabolusinmusica,the“devilinmusic.” These circumstances somehow reflect the fact that a true tritone, theactualtonalcenter, isunreachablebecauseofits irrationalnature.Andbecauseof this, an even integral bisectionof the scale and thusof thewhole tone thatstraddles the center is impossible. This situation is said to have disturbed the

ancient Greek harmonicists for whom the scale was an essential tool of theirhighlyrationalharmonicscience.Wecancomparethistotheequallyperplexingsituation of the irrational diagonal of the square. TheKabbalist Leonora Leetoffersaperspectiveonthisproblem.

Itwastheincommensurabilityofthediagonaltothesidethatwasthegreat secret the Pythagorean initiate was vowed to keep, not, asgenerally taught, because this fact destroys the rationality ofPythagoreangeometry,butpreciselybecauseitisarevelationofthevital necessity of the suprarational, infinite element in all finitestructures that the square root contains a powerwhosenature is, intruth, unutterable in terms of finite measure and that points to thedeepestofcosmicmysteries.2

The same idea can be applied to the incommensurable tritone that lies at theheartofthemysteryembodiedbythescaleoftonesthatjoinstwoworlds—therationalworld of number (quantity) and the unconscious realmwherein soundgivesbirthtofeeling(quality).Thetritoneseemstopointustowardthesacredcenterwherethegodsreside.Inthiscontextwerecallthepreviouslydiscussedringstructureinwhichanouterringholdsaninneressence.Thecentraltritonecouldbeconsideredtheunknowablecoreofthescalesurroundedbyconcentricringsoftones.

Theregionofthetritone,situatedbetweentheharmoniousintervalsofthefourthandthefifth,hasbeenthesourceofvariousrationalapproximationsthroughouthistorysince,aswehavenoted,theactualtonalcenteratstringlength.707...(√2 ÷ 2) cannot, because of its irrational nature, be exactly located on themusicalstringbutmustbeapproximatedbyarationalfraction.Clusteredaroundthe irrational .707. . . are 8/11 (.7273), the tritone derived from the harmonicseries; 5/7 (.7143), an ancient tritone (Ptolemy,Mersenne); 729/1,024 (.7119),Arabic; 32/45 (.7111) (Ptolemy), the most commonly used tritone in moderntuning;45/64(.7031),theinversionof32/45;17/24(.7083),Arabic;7/10(.70),Euler’stritone;52/73(.7123),theMayantritone,discussedabove;and512/729(.7023), the Pythagorean tritone. Though seemingly unwieldy the Pythagoreantritonesimplyconsistsofthreewholetonesaddedtogether:

Drawing2.3.(a)Weinscribeasquareinacircle.Thequartersquarediagonal=.7071...,whichgeneratesacircle(b)thatintersectsthemusicalstringatthetritone.In(c)weseehowthetritoneanddiagonalofthesquarearerelated:thetritone-diagonal is thecatalystof theoctave-doublingprocesssimilar tospacedoubling: the large square area is twice that of the small square: as thefundamentalistwicethestring-lengthoftheoctave.

Figure2A.ThetempleofApolloEpikouriosatBassai,Greece,fifthcenturyBCE

8/9×8/9×8/9=512/729(.7023).

(Toaddintervalsmultiplytheirratios;tosubtractonefromanotherdividethem.)

HARMONIOUSRECTANGLES

The Pythagorean tritone vesica construction may have been the initiatinggeometry of the ground plan of the temple of Apollo Epikourios at Bassai,Greece,thoughttohavebeendesignedbyIktinos,thearchitectoftheParthenon.Wehaveadualpurposeinchoosingthistempleasoursubject.

Thefirstpurposeistoshowapracticaluseofthetone-vesicaconstruction;thesecondistoexplainseveraloftheharmoniescontainedintheplanofthistemplethatarerevealedinthenumbersandthegeometricanalysisofthegroundplan.TheApollotemplehasanoveralldimensionof130.8×52.919feetmeasuredattheeuthynteria,thefirstlevelingcoursethatslightlyprotrudesfrombeneaththebottomstepofthetemple.Thelength-to-widthratioisthus2.472,whichequals4/ϕ,aratiowehavelookedatpreviouslyinrelationtoitstimeproperties,whichinclude the fact that thereare12.36 lunations inayear (that is,365÷29.53=12.36)and2.472÷2=1.236.

The 4/ϕ ground plan of the Apollo temple thus embodies a double yearlylunationcyclealongwiththeotherinterestingpropertiesthatthisratiogenerates.

After thePythagoreantritoneis locatedonthehorizontalaxisof thesquarebymeans of crossed diagonals (see drawing 2.4), we proceed with the vesicaconstructionthatresultsinthegroundplanofthetemple.Inthedetail(drawing2.5)wecanbetterunderstandtheadvantageofthegeometricmethodofbuildinglayout in creating a unity that would have been impossible if we had simplymeasured out a “harmonious rectangle.” The construction lines themselvesindicatethelengthandwidthofthetempleaswellascolumnplacementandthecellaborders.

Drawing2.4.Theground-planconstructionof the templeofApolloEpikouriosat Bassai, Greece. (1) Using diagonals we locate the Pythagorean tritone =512/729 = .7023 on the string/diameter. (2) A vesica with ratio of 1.572(√2.472)isconstructed.(3)Tangentstothevesicaareextendedtotheenclosingsquareproducinganoverallground-planrectangleof2.472(4/ϕ)ratio.

Anothertritonewewillnowconsideristheirrationalsharpenedtritone(.691...)that,thoughitmaybeapproximatedwith9/13(.692)or38/55(.6909),wewillcontinue to express as .691 because of certain important associations that thisnumberhas.Tobeginwith, .691 is the reciprocalof1.4472—that is, 1/.691=1.4472,and

1.4472=1+(1÷√5).

Thesquarerootof5,or2.236,istheratioofthediagonalofthedoublesquaretoitssideandisintimatelyinvolvedwithϕ:

(√5+1)÷2=ϕ.

Drawing2.5.Thisdetailofdrawing2.4showshowthelinesoftheconstructioncoincidewiththetemplestructures.

Drawing2.6.Theϕtritonevesicaconstruction.(1)Theestablishmentofthe.691nodeatBisshownindrawing2.8(below).ThelineABisbisectedandthecircleisdrawnandmirrored(2)formingavesicawithheight-to-widthratioof1.618,orϕ.In(3)theinterior(C4)andexterior(C3)vesicacirclesaredrawn.TheirdiametersareC1=1;C2=.691;C3=.618(1/ϕ);C4=.382(1/ϕ2);andtheirratioisC1/C3=C3/C4=1.618,orϕ.

Drawing2.7.AnotherapproachtodrawingtheApollotemplerectangle.Theϕtritonevesicaconstruction(left)perfectlyholdsthedoublesquarethat,whenitsbaseisextendedtothecircles(right),producesthe2.472(4/ϕ)rectangle.

There is anotherGreek templewhose ground plan is based on this sharpenedtritonenumber.691,andthatisthefamousParthenon,alsodesignedbyIktinosalong with the architect Callicrates. The euthynteria of the Parthenon hasdimensionsof238×111.31feet,resultinginaratioof2.138,and

2.138=1.4472+.691.

Thegroundplan is thusbasedon thefrequency(1.4472)plus thestring length(.691)ofoursharpenedtritone.JayHambidgesaysofthis1.4472ratio:“ThisisthekeyratiofortheParthenonplanandappearsmanytimesinGreekdesign.”3The .691 string-length ratio generates a vesicawith a length-to-width ratio of1.618,orϕ,andforthisreasonwewillnowreferto.691astheϕtritone.Thecirclesofitsvesicaconstructionhavethefollowingdiameters:

C1=1;C2=.691;C3=.618(1/ϕ);C4=.382(1/ϕ2).

The construction of the .691 tonal node is shown in drawing 2.8, and theϕvesica construction is shown in drawing 2.6 (2) and (3). We note that thisconstructionalsogeneratestherelativesizesofEarthandMercury,theC1circlerepresentingEarth,andtheC4,Mercury.

It is interesting how theϕ tritone vesica perfectly holds the double square, asshownindrawing2.7.Infact,thewholeϕtritonevesicaconstructioncouldbegeneratedfromthedoublesquare,whosegeometricpotencywasrecognizedbythe ancients. For example, the King’s Chamber of the Great Pyramid has adouble-square floorplanasdoes thehypostylehallofSeti IatKarnak,Egypt.TheGreek temple ofArtemis atEphesus has a floor-plan ratio of 2.09.Manyotherancienttemplesareclosetoadouble-squaregroundplan,whichsuggeststhatthebuildersmayhavebegunwithadoublesquareandthenmodifieditforvariousreasons.

Drawing 2.8. A construction of the Parthenon euthynteria, or first levelingcourse.In(a)and(b)weconstructthe.691ϕtritonethatintersectsthestring/diameteratapointfromwhicharectangle(c)isdrawn.In(d)aperpendicularto the (c) rectangle generates a reciprocal rectangle. The two conjoinedrectangles form rectangle (e), the completed Parthenon euthynteria. Theimportantprincipleofratioandreciprocalisthusdemonstrated.Thisprinciplefindsexpressioninmusicaltone,thestringlengthbeinginreciprocalrelationtothefrequency:1/stringlength=frequency.

Drawing2.9.Aninternaltimepropertyofthedoublesquarebecomesapparentwhenwe scale its long side to equal 12.Then thehalf diagonalof thedoublesquareequals12.3693(10×2/ϕ),whichisthenumberoflunationsinasolaryear.Thesolaryearequals365.2422days,andthelunationcycleequals29.53days.So,365.2422÷29.53=12.3683,accurate towithina thousandthof thehalfdiagonalofthedoublesquare.(AfterRobinHeath’sdrawing)

*

Wehaveseenthatthedoublesquarealsogeneratestheground-planratioofthe

templeofApolloEpikouriosatBassai(drawing2.7).PerhapsIktinoslaidoutadoublesquareonthegroundandsimplyuseditsdiagonalsasradiitosweepouttwoarcsthat,intheprocessofgeneratingthe4/ϕrectangle,createdavesicaofϕratio.Whilethisisthesimplestapproach,thePythagoreantritoneconstructionshownindrawings2.4and2.5morepreciselyrevealstheinnerstructureofthetemple.PerhapsIktinosusedbothapproaches—orneither.

3

MOONANDEARTH

INTHISCHAPTERWEWILLCONTINUETOEXPLOREthedimensionsandtimeperiodsoftheMoon-Earthsysteminthecontextofthetonalandcanonicalnumbersthatreveal the deepermeanings of their relationship. There is an intimate relationbetween Earth and her life-forms and the Moon. Among some indigenouspeoplestheMoonisknownasGrandmotherMoon.Sheisthematureelder,thewisdomkeeperwhoteachesthemysteriesoftimeandchange,whodrawsusintothemysterywithhermaturebeauty,wholightsourwaythroughthedarkplacesorwithholdsherlightsowemaylearntowalkinthedark.Sheistheonewhoinitiates youngwomen into the path of femininewisdom through themonthlybloodritual.Sheismistressof thewatersmovingtheseaandthewaterwithinlife-forms in daily and monthly rhythms through the great binding force ofgravity. She clearly signals the changing intensities of her energy through thevaryingphasesofherlightsothatallmayprepareforandcomfortablyadjusttothechangingmonthlyrhythms.MirceaEliadewritesoftheMoon:

It is through the moon’s phases—that is, its birth, death, andresurrection—that men came to know at once their own mode ofbeinginthecosmosandthechancesfortheirsurvivalorrebirth.Itisthroughlunarsymbolismthatreligiousmanwasledtocomparevastmassesofapparentlyunrelatedfactsandfinallytointegratetheminasinglesystem.Itisevenprobablethatthereligiousvalorizationofthelunar rhythms made possible the first great anthropo-cosmicsynthesesoftheprimitives.Itwaslunarsymbolismthatenabledmantorelateandconnectsuchheterogeneousthingsas:birth,becoming,death, and resurrection; the waters, plants, woman, fecundity, and

immortality; the cosmic darkness, prenatal existence, and life afterdeath, followed by a rebirth of lunar type (“light coming out ofdarkness”); weaving, the symbol of the “thread of life,” fate,temporality,anddeath;andyetothers. Ingeneral,mostof the ideasof cycle, dualism, polarity, opposition, conflict, but also ofreconciliationofcontraries,ofcoincidentiaoppositorum,wereeitherdiscoveredorclarifiedbyvirtueof lunar symbolism.Wemayevenspeak of a metaphysics of the moon, in the sense of a consistentsystem of “truths” relating to themode of being peculiar to livingcreatures, to everything in the cosmos that shares in life, that is, inbecoming,growthandwaning,deathandresurrection.Forwemustnotforgetthatwhatthemoonrevealstoreligiousmanisnotonlythatdeath is indissolubly linked with life but also, and above all, thatdeathisnotfinal,thatitisalwaysfollowedbyanewbirth.1

AndwhenshedisappearsatthedarkoftheMoonourcommonlineageisfullyrevealed:thestarsofourgalaxy,theGreatMotherfromwhichtheMoonandallthingsinthiscorneroftheUniversewerebornfromexplodingstars.

YetGrandmotherMoonismorethananinspirerofthemythicimagination.Herpresencehas shapedEarth inmanyways. It is believed that she stabilized theorbitoftheerraticearlyEarthandhelpedcreatetheconditionsfortherisingoflife. TheMoon also reveals through her dimensions and cycles mathematicalwisdom that helps us to understand our world. We have mentioned the 11/3(3.66)diameterrelationofEarthandMoon.Wecanlookatthisanotherway:

Figure3A.EarthandMoonintheirpropersizerelation

Earthdiameter+Moondiameter=1+.272(3/11)=1.272=14/11.

The number 1.272 is a key number or numerical region in our study. Itapproximatesthearearatioofasquaretoitsinscribedcircle(1.27323)andthe√ϕfunction(1.27202);and14/11(1.27273)givesusarationalvaluewithwhichto do harmonicwork inmusic and design. This numberwas enshrined in theGreat Pyramid, whose cross section (drawing 3.1, below), in referencing thisimportantratio,revealssignificantrelationshipsintheEarth-Moonstructure.

Drawing3.1.TheEarthandMooninpropersizerelationareembodied in thecross section of the Great Pyramid. A circle squaring is implied in thisconstruction:thesquaresurroundingEarthhasthesameperimeter(within.002)asthecircleintersectingthecenteroftheMoon.(AfterJohnMichell’sdrawing)

Theoveralleffectofthisenchantedconstruction,basedonJohnMichell’swork,suggeststhepossibilityofaharmoniouslydesignedcosmos.

MEASURESOFTIMEANDSPACE

There is evidence that the musical scale was one of the principal tools thatcertain ancient civilizations used to investigate theworkingsof nature and thesoul.WehaveheardthatthePythagoreansconsiderednumbersandtonalratiosto be the foundation of the created world. The Chinese also recognized thecosmicsignificanceoftone.IntheShihChiitwaswrittenthatthesixlu(tones)are “the root stock of the myriad things.”2 Joseph Needham writes that theChinesemade practical application of this understanding in their use of pitchintervalsas thebasisof theirmeasuringsystem.Hestates thatoriginallygrainscoopswereusedasprimitivepercussiveinstrumentsorbellsandthatlaterthesize and tone of these vessel/tone-makerswere calibrated to correspond to anexactgrain-holdingcapacity.Needhamnotessomefurtherdevelopmentsofthisconcept.

The simple grain scoop having evolved into a bell, and the simplebell into a standardized measure or chung of fixed dimensions,capacity andweight, aswell asmusical pitch, itwas natural,whenpipes became the standard pitch-givers, that they should inherit themeasuring functions which had one time belonged to bells.Consequentlywe read of the number of grains ofmilletwhich theHuangchung[thefundamentaltone]pipeoughtproperlytocontain....Thestandardmeasure,chung,hadtoemittheHuangchungnote.3

Thehuangchungwasaparticularmusicalpitchthatwasbelievedtoexistwithinnature.As timepassed, elaboratedivination ritualswere conducted to find thehuangchungsothatmusical instrumentscouldbe tunedto itandthesystemofmeasurecouldbecorrelatedtoit.AstheChinesesoughtthehuangchungtonetoconstructtheirscaleandtodeterminetheothermeasurementsthatarosefromthescale, so other ancient cultures sought a natural length that could serve as anincorruptiblemodulefortheirmeasuringsystems.Itmaybethattheylookedtotheeternalheavens to find it. Itwasnatural,whenyou thinkabout it, that theMoon was chosen. Easy to see, full of visual cues in the form of maria andcraters, it had long been used as a timekeeper by humans. And its regularadvancingandretreatingborderbetweendarkandlightevokesasenseofprecisespatialregularity,asifnotjusttimebutalsospaceisbeingmeasuredout.

THETHREEMASTERS(PART1)

AsAristarchusdemonstrated,agoodapproximationoftherelativesizesofEarthandMooncanbedeterminedbyobservingtheshapeandsizeofEarth’sshadowcast on the Moon during eclipses. The ancients seemed to have determinedaccurate values for both the Earth’s and theMoon’s diameters. JohnMichellwritesthattheseearlyharmonicists,inselectingtheMoonfortheirstandard,hadassigned the number 127, or 35,831,808, as the number of units in thecircumference of the Moon. This standard was then divided by an ancientapproximationofπ, 864/275, togive11,404,800units for theMoondiameter.Buttherewereotherfactorstobeconsidered.Thenumericaldescriptionoftheworkingsofanenchantedcosmoscouldnotsimplybedrydecimalnumbersbutmusthavedeepermeanings thatwould take the formofmusical and temporalcorrespondences.

Let us now imagine an ancient scenario whose actors are three masterharmonicists who are seeking an alternate system of measure to the decimal-basedsystemwithwhichtodesignatethediametersoftheEarthandMoon.TheMoondiameterischosenastheonetobedetermined,andfromthistheknownMoon/Earthratioof3/11willdeterminetheEarth’sdiameter.Theharmonicistsare Maestro, an elder from central Asia; Maestra, a wisewoman from amegalithic building clan of the British Isles; and the Traveler, a scholar ofancientarchitecture.

Maestro:ItwouldbegoodtohaveauniversalnumberfortheMoondiametersothatitmightbeeasilydiscoveredbyourdescendants.

Maestra:Anumberwithtimeresonanceandwithmusicalmeaning.

Traveler:...arichnumberbefittingGrandmotherMoon.

Maestro:IfeelweneedtoenclosetheMooninacubesothatwehavearationalstructurewithwhichtoapproachher.Andthenwemustsubdividethecubeintomanysmallercubes.Thenumberofsmallcubesacrossafaceofthegreatcubewillthenbeappliedtothetwo-dimensionalcircleoftheMoon.Thequestionis:Howmanysmallercubesshallwefillthegreatcubewith?

Traveler: Let us make use of our revered decimal system that can give ussimpleestimates.Howabouttenmillionlittlecubes?

Maestra:(calculating)...Thenthelargecubehasasidewhoselengthis215.4smallcubes.

Maestro:Weneedsomethinglonger.Let’sgowithtenbillionlittlecubesintheMooncube.

Maestra:TheMooncubenowhassidesof2,154units.

Traveler:That’s close to the 2,160-year precessional age number; just a littlecommaofdifference.

Maestra:It’s1/360tobeexact,alittleoctavetritone.Anotherinterestingtimefact is that the lunationof29.53daysdividedby theMoon’sorbitalperiodof27.32daysequals1.08,orhalfof2.16.

Maestro:And2,160seemsresonantwithtimenumbers.

Traveler:Ithasamusicalpedigreealso.Since216isamajorsixthwithinour128–256octave.

Maestra:Right;27/16×128=216.Bythewayhere’sanothermajorsixthfact:16/27×365daysinayearequals216daysplusafewhours.

Maestro:SoourMoondiameterhas2,160units.AndmultiplyingbytheknownEarth/Moonratioof11/3theEarthdiameterwouldbe?

Maestra: Would be 7,920 units. This is also a good number. Dividing it byelevengivesus720,oneofourmostdivisiblenumbers:7,920isalsodivisibleby2,3,4,5,6,8,9,10,11,12,15,16,and18.Thisisanumberwithwhichwecanwork.

Traveler:And so the Earth’s circumference is 7,920 times our π of 864/275,giving...

Maestra:Ananswerof24,883.2units...whichwecancallmiles.

We pause the imaginary conversation here to give a brief introduction to the

subject ofmusical scales and, in particular, a simpleway to construct a basicPythagoreanscaleanditsrelationtojustintonation.

CONSTRUCTINGASEVEN-TONESCALE

WeknowthattheintervalsoftheoctaveandthefifthhavebeenpresentinthemusicofmostculturesacrossEarthandthroughtime.Musicianscanlocatethemwith intuitiveaccuracyon themusical string.From these two intervals a scalemay be generated.We can approach our primordial scale from either of twodirections.Wecansoundthefundamentalfirstandthensoundthefifth.Orwecan sound theoctave first and then sound a fifthdown. (Note: on themusicalstring theoctavedivides thestring in two, the fifth into2/3and1/3segments,andthefourthinto3/4and1/4segments.)Butnowwehavethreeintervals:theoctave,thefifth,andthe“fourth,”sinceafifthdownfromtheoctavelandsusonthe same tone as a fourth up from the fundamental, the fourth being anotherharmonious-soundinginterval.

Ourscaleisnow

fundamental——————fourth–fifth——————

Atthispointwehaveascale,butitisnotofmuchuseforcreatingrich-soundingmusicuntilwecanfill in theblankplaces.Howdowedothat?Onewayis totaketheintervalbetweenthefourthandthefifth,whichwecalla“wholetone”(WT)(8/9of thestringlength),andaddit to thefundamentalwithinthelowerpart of the scale and to the fifth in the upper part of the scale.We can keepadding whole tones to the upper and lower parts of the scale giving us thefollowingtonalintervals.

Wenotice twogaps:betweenthe thirdandthefourthandbetweentheseventhandtheoctave.The8/9wholetoneistoolargetofitbetweentheseintervals.Tofind an interval that does fit we subtract the third from the fourth throughdivision:3/4÷64/81=243/256.Andwesubtracttheseventhfromtheoctaveinthe sameway: 1/2 ÷128/243= 243/256.These are the Pythagorean semitoneswithstring-lengthof.9364(roughly15/16)ofthevalueofthewholetone,whichwasslightlytoobigtofitinthespacesbetweenthirdandfourthandseventhandoctave.WehavejustcompletedtheconstructionofthePythagoreanseven-tonescale.Thecalculationsabove the scale showhow the intervalswerearrivedatthroughmultiplication,suchas

3/4×8/9=2/3

It happens that all these tones are composed of the factors 2 and 3 or theirmultiples.Ifweintroducethenumber5anditsmultiplesintothissystemwecansimplifythemorecomplexfractions—namely,64/81,16/27,128/243—withoutvaryingtheirvaluesbytoomuch,asdemonstratedbytheirdecimalequivalents,listedinthefollowingtable.

PythagoreanTone Adjusted64/81(.7901) 4/5(.8000)

16/27(.5926) 3/5(.6000)

128/243(.5267) 8/15(.5333)

These adjustments are simplymade by adding the “syntonic comma,” a smallinterval with frequency ratio 81/80, to the Pythagorean ratios. The result is aseven-tonejust-intonationscalewiththefollowingintervalsbetweenthetones:

Theintervalsarenow8/9,wholetone;9/10,minorwholetone;15/16,semitone.

Further divisions can be made, and, of course, innumerable tones of widelyvarying ratios may be inserted into this basic structure. Now back to theconversation.

THETHREEMASTERS(PART2)

Maestro:Wehaveourbasic longunit—themile—ofwhich thereare2,160 intheMoondiameterand7,920intheEarthdiameter—andnowweneedsmallerunits.

Maestra:Letusfirstestablishaneasilydecipherablenumberofsmallerunitsforthelunarcircumference.

Maestro: A large number is needed. It must be of eternal vintage such as amusicalnumber.

Traveler:Possiblytwelve,thenumberoftonesinthefullscaleraisedtosomepower.

Maestro:Suchasseven,thetonesinthebasicscale.

Maestra: So, 127 smaller units in the Moon circumference: that would be35,831,808units.

Traveler:Dividedbyourπnumberof864/275, results in11,404,800units intheMoondiameter.

Maestro:So,11,404,800dividedbythemilecountof2,160equals . . .5,280,thenumberofsmallerunits...feet...inourmile.

Traveler:Well,with528we’rejustbarelyintothe512–1024octave.

Maestra:And512/528givesusanintervalof32/33.Soundsfamiliar.

Traveler:That’sanotherbeautifulcomma.

Maestro:Itstwelvelunationsdividedbytheyearlydaycount.

Maestra:Thatis,354/365=32/33.

Maestro:Sowehaveourfoot,5,280ofwhichgointoamile.And2,160milesintheMoon’sdiameterand7,920inEarth’s.Now,howdowedividethefoot?

Traveler:Thedecimalmetrologistsalwaysdividebyten.Itwouldbebetter, Ithink,nottomixsystems.

Maestro:Consistencyisimportant.Let’slookatournumbersandwhatdividesevenlyintothem.

Maestra:Well,2,160,1,080,7,920,5,280arealldivisiblebymanynumbers.

Traveler:Isuggestweusethenoblestnumberofthemall.

Maestro:That,again,wouldbe12.

Maestra:So,2,160÷12=180;7,920÷12=660;1,080÷12=90;5,280÷12=440;and12itself,ofcourse,ishighlydivisible—by1,2,3,4,and6.

Traveler:Andrelatestoourtwelvezodiacalsigns...andtherearetwelvetonesinthescale....Icouldgoon.

Maestro:Areweallagreed?

Traveler:Twelveitis.

Maestra:Twelve...inches...toournewfoot.

Traveler:Whatoftheinch?Ittooneedsdivisionforfinermeasurements.

Maestro:Thetemptationistousetenths.But,again,letusnotmixsystems.

Maestra:Whatisthesimplestdivisionofthemall?

Traveler:Divisionintotwo.

Maestro:Ofcourse.Anditsreciprocal,doubling.Theseareeternal.

Traveler:Tworesonateswiththeoctave,withtheprincipleofduality,withthelife generation principle of one from two. In geometry section into two is themostprimalgestureofthatart.Itisgoodtoincludethisinoursystem.

Maestra:Theinch,then,woulddivideintohalves,quarters,eighths,andsoon.

Maestro:Sobeit.

EARTHANDMOON

ThecentralcorrespondencethatemergesfromourdiscussionisthatthediameterratioofEarth toMoon=7,920/2,160=3.66,andEarthrevolvesabout itsaxis366timesinitsyearlyorbitaroundtheSun.Asatonalfrequency,366iswithinthemainoctaveof256–512.Itsfrequencyratiois366/256,anditsstring-lengthratiois256/366=.699—closetothe7/10tritone.Aswehavenoted,theancientChinesesearchedforatoneembeddedincosmicstructurethatwouldserveasabeginningoftheirmusicalscale—theelusivehuangchung.LevarieandLevyin

theirbookTonewrote:

IntraditionalChinesemusic,thepitchnormwasatone,calledhuangchung(“yellowbell”)whichsoundslikeourf-sharpandcorresponds,as subsequent measurement revealed, to a frequency of 366. Thistoneandthepipemeasuresthatdefineditwerederivedfromcertainsacred cosmic proportions, which recur throughout Chinesephilosophyandreligion.4

Our.699stringlengthismultipliedbytheyearlyrevolutionsoftheEarthof366days:

.699×365days=255.83days

within99percentofthe256traditionalfrequencyofmiddleC.Thetritoneoftheyearthusestablishesthebeginningofthescale.

There are other results that the 366-frequency number generates. Itsestablishment of theC-256 frequency as the beginning of the scale introduceslunarandsolarnumbersintothescalewhenthePythagorean-sixthfrequencyof27/16(A-432)issubstitutedforthestandard5/3(A-440)oftheC-264scale,

27/16×256=432,

givingthemile-basedmeasuresof

4,320miles÷2=2,160,theMoondiameterand432,000=Sunradius.

Another instance of the 366number occurswhenwe calculate the area of theMooncircle:

Moonradius1,0802miles×π=3,664,353squaremiles.

Inchapter10wewilldiscoverhowthenumber366playsan important role incosmichistory.

From the previous discussion a question arises: Are we meant to draw ourmusicalscales fromnumbersembedded in thecosmos?WedoknowthatboththeGreeksandtheChineseproscribedthealteringofthescales.Becausemusicwas considered to be of great importance to the harmonious functioning ofsociety, to allow changes to its carefully crafted structure would inviteinharmonious elements to enter into and corrupt the social order. This mightindicate that their musical systems were drawn from imperishable andunchanging laws of nature that could nomore be altered than could the timecyclesofSunandMoon.

Certainly the octave system of numbers, with its doubling and halvingproperties,isfundamentaltouniversallaw.Thisismostevidentinthetwo-from-oneprocessofmitosisandtheone-from-twoprocessofsexualreproduction.Thenumbers2,4,8,16,32,64,128,256,512,andsoforth,arethusobviouschoicesforearlymusicaltheoristslookingforrawmaterialsforascale.TheGreeksusedthesepowersof2andaddedthepowersof3(3,6,9,27,54,108,andsoforth)togenerate intervalsfor theirscale.Three isanothernumber thathasuniversalcorrespondences, but with a different quality than 2: past, present, future;microcosm,mesocosm,macrocosm;birth,life,death;threedimensionsofspace;andsoon.Laterothernumbersworthyof forming intervalswereadded to themusicalandmeasurementcanons—suchasthepowersof5seriesandϕ-basednumbersthathadbeendiscoveredintimecyclessuchas,asnotedabove,intheMoon-Earth-Sunrelation.

THEINNEREARTHThroughthediligentworkofearthscientistswenowknowagreatdealaboutthesizesofthelayersoftheinnerEarth.Muchofthisworkhasbeendonethroughthe tracking of seismic waves as they ripple through the Earth reflecting andrefractingastheyencounterdifferentmaterialsanddifferentdensities.Wenowhavefairlyaccuratemeasuresofthedepthsandthicknessesofthemainlayersofcrust,mantle, andcoreaswell as their secondary layers.Butasgeometerswewonder if we can create a model of the ring structure of the Earth withoutresorting to measurement. We find that this is possible using the geometricmethodswehavealreadydiscussed.

TheMoon,whichwehave learnedhowtoconstruct fromacirclerepresentingEarth, seems tohave ageometricpresence insideEarth.As shown indrawing

8.3, the revolving barycenter, or center ofmass of the Earth-Moon system, islocated at a distance from the surface of the Earth of oneMoon radius. TheMoon revolves around this barycenter. The Earth does also, although itsmovement is more of a slight wobble. There is another way in which EarthcontainstheMoonthatwewilllookatinthefollowingdrawings.Drawing3.2(below) uses crossed diagonals to locate a node that produces the diameter ofVenusrelativetoEarth.Whenmirrored,thisnodegivesusthethicknessofthecombined uppermantle and crust of theEarth. In drawing 3.3we construct adouble-Moon circlewhose diameter happens to be the size of the diameter ofEarth’soutercore.WhenplacedwithintheEarthdrawingitalsodefinestheringofthelowermantle.Finally,themajor-sixthconstructionlocatestheinnercore(drawing3.4,below).

Drawing3.2.(a)ThefirsttwoofaseriesofdrawingsinwhichthelayersoftheinnerEarthareconstructedusingtonalratiosthatgeneratediagonalcrossings.Heretheintersectionofthetwodiagonalsdrawnfromquarter-squarepointsonthe enclosing square locates a point on the string/diameter towhich an innercircle isconstructed.Thispoint isat19/20of themaindiameter length that isconsidered the diameter ofEarth (7,920miles). The circle drawn to the pointhas a diameter of 7,524 miles, the diameter of Venus. (b) Next, the point ismirrored and a new circle drawnwith a diameter of 9/10 (7,128miles). (The

Venusnodeof 19/20 is the ratioof anancient semitone citedbyEratosthenesandPtolemy.Theratio9/10istheminorwholetone.)TheringthatisformedisproportionallyaboutthesizeofthecrustanduppermantleoftheEarth.

Drawing 3.3. (a) Quarter-square diagonals are crossed to locate the 6/11(.5455)nodeonthestring/diameter.(b)Thisgeneratesacircleofproportionally4,320 miles in diameter, which is double the Moon diameter of 2,160 miles.When centered on the construction this circle defines the perimeter of theEarth’soutercoreandtheinnerperimeterofthelowermantle.

Drawing 3.4. The inner-core circle (with diameter one-fifth that of Earth) issimplylocatedwiththediagonalconstructionforthemajor-sixthinterval,againusingquarter-squarediagonals.Thediameterratioofoutercoretoinnercoreis2.72 (4,320/1,584), which, in feet, is the length of the megalithic yard (MY)discovered by Alexander Thom, who believed that theMY was the basic unitusedinthelayoutofthemegalithicmonumentsofBritain.Thenumber2.72alsoreferstotheMoon/Earthdiameterratioof3/11(.272).

Drawing 3.5. The rings of the inner Earth. The layers of the inner Earthdescribedinthepreviousdrawingscombinetoformagrandringshapeandanobjectofcontemplation.

4

THETEMPLEOFWATER

IN THIS CHAPTER WE WILL CONSIDER THE SUBSTANCE OF WATER in a harmoniccontext.Weknowthatwaterconstitutesamajorityofourphysicalbodiesandofthe surfaceofourworld,but,ofcourse, it ismore than that. Itspowerand itscharacterhas,overtheeons,integrateditselfwithinourpsychicstructure,whereitresidesasasymbolfortheunconsciousitself—thegreatDeepwithitshiddencontentsthat,likeunknownseacreatures,occasionallysurfacewearingbeautifulormysteriousormonstrousforms.

Watertakesonalltheformsofhumanfeeling:theperfectmirrorofastilllakesuggests the feeling that can arise in deep meditation; the wild thunderstormreflectschaoticemotion;themonotonousrhythmofthecontinuousdripremindsusofthepassingoftime;therushingmountainstreaminlatespringisanimageof passion; the gentle sun-shower on a hot day can soothe us like the softglissandoofaharp.Waterhas,inoldertraditions,beenassociatedwith,orevenidentified with, the soul, and the reasons are evident. No life-forms can existwithoutwaterorwithoutananimatingsoul.But,likethesoul,watercanbecomeclouded, can become impure. Yet to cleanse it, often all that is required isstillness. Thewater becomes clean as the sediment sinks to the bottomof thepond;thesoulinmeditationreturnstoitssourceandbecomeswhole.

Yet, in themidstofsuch intuitivecorrespondencesweallowourselves toseeknumerical relationships and geometrical symbols sowemay better understandthis substance. Perhaps seeingwater in a context that is filledwith numericalsynchronicities, as we are finding with theMoon and the Earth, will help usapproachit inasacredway.Numberscanoftenrevealaunitiveenergybehind

theseeminglyrandomformsandprocessesoftheworld.

LANDANDWATER

The proportion of surface land and water areas on Earth as determined bymodernsurveyingis,approximately:

Land:148,300,000km2=29%Water:361,800,000km2=71%

Total:510,100,000km2

We can visually express this ratio of 71 percent water to 29 percent landarea(which happens to be about the water-to-solid ratio within the adulthumanbody) with a vesica construction (see drawing 1.7) in which the innercircle(C4), enclosed by the vesica, represents the land area and the circleinscribingthe vesica (C3) represents the water area. It is interesting that thePythagoreantritone vesica construction, which we have used to lay out thegroundplanoftheApollotemple(drawings2.4and2.5),generatesanearlyexact71/29arearatiobetweenitsouter(C3)andinner(C4)vesicacircles(seedrawings4.1and4.2below).

C3area(water):.3176=71.2%C4area(land):.1285=28.8%

and71.2÷28.8=2.472(4/ϕ)

TheApollo templeground-plan ratio, alongwith its previouslydiscussed timeaffinities,takesonanewmeaningwhetherornotthebuilderswereawareofit.

I pause here for a daydream. Because the water-area circle in this vesicaconstructionisconsideredtobebeneaththeland-areacircle,theimageIseehereisthatofanislandfloatingontheworldoceanlikethehuman-builtfloatingreedislandsoftheUrospeopleoutonLakeTiticaca.Thedaydreamthenremembersmodern plate-tectonic theory inwhich the crust of theEarth is believed to be“floating”andgraduallymovingontheviscousmantlematerialremindingmeof

the primordial continent of Pangaea before it separated into our presentcontinents. I imagine the Apollo temple superimposed on this drawing, withwhich it aligns perfectly, itswidth being that of the land circle and its lengthextending to the containing circle. It thus symbolically holds the land and itsinhabitantssafelywithinitselfasitnavigatesacrossthemysteriousdarkseaofexistence.

Drawing 4.1. The water and land construction. We begin with thePythagoreantritonedrawingthatweusedinchapter2tolayoutthegroundplanoftheApollotemple.

Drawing 4.2.Thewater and land construction. The inner circle of the vesica(C4)representsthelandsurfaceareaoftheEarth.Theoutervesicacircle(C3)representsthewatersurfaceareaoftheEarth.Theratiois

Water=71.2%÷Land=28.8%=2.472(4/ϕ).

THESHAPEOFRAINDROPS

Adropofwatersuspendedinacloudhasasphericalshape.Asitfallstoearthitfirstassumestheshapeofaspherethatisflattenedalongthetop.Duringitslongjourneytheflattenedsideoscillatesbetweenthetopandbottomastheraindroptries to maintain its structural integrity. The shape of a drop dripping from afaucet, shown in figure 4A, has the same top-flattened spherical shape as theraindropinitsinitialfallingstage,asshowninfigure4B.Wecanapproximatethisformbyfirstdrawingacircleinasquareanddoingsomesimplegeometry(drawing4.3).Asimilarformcanbemadewithinavesicaconstruction(drawing4.5).

Figure4A.Thewaterdroptakingform

Figure4B.Theelegantshapeofthefallingwaterdrop

Drawing4.3.Aconstructionofthewaterdropshape.In(1)drawacircle-in-the-square.Drawacircle(a)withcenteratthelowercornerofthesquareandwithradius equal to the side of the square to intersect the vertical diameter of thecircle.Drawanothercircle(b)withcenteratthebottomoftheverticalaxiswithradiusextendingtotheintersectionofcircle(a)andtheverticalaxis.(2)Deletetheconstructionlinesexceptforthearcofcircle(b)lyingwithinthemaincircle.

(3)Deletetheremainingarcofthemaincircleandthesquare,leavingthedropform. Properties of the construction: diameter of circle = 1; area of drop =.699;perimeterofdrop=3.001.Thediameterofthecontainingcircleandtheperimeterofthedropareinanear-integralrelationof1to3.Thereisasimilarintent between this exercise and the circle-squaring process: to bring theunknowableintoarationalframework.

The.699ratioofdroptosquarealsorecallsourdiscussioninchapter2ofthehuangchungfrequencyof366withitsstring-lengthratioof.699,atonethatsomehowinheresinthiswaterdropfigure.

Otherharmonies include thecrescentshapeabove thedrop thathasanareaof.0864(864beingtheSunnumber);thedrop/crescentarearatiois.699÷.0864=8.09=10×ϕ/2;andthearearelationofdroptocircleis.699÷.7854=.8899=8/9,awholetone.

Thewaterdropshapeissimilar to thestanding-stoneflattened-circlerings built by Neolithic peoples in Britain. Many of these weresurveyedbyAlexanderThomduring the twentieth century, suchasthe one shown in drawing 4.4. Thom believed that the Neolithicbuildersconsciouslysoughttoestablishathree-to-oneratiobetweencircumferenceanddiameter.Hewrotethat“megalithicmanliked...togetasmanyaspossibleof thedimensionsofhisconstructionstobemultiplesofhisbasicunit.”1ThisfactenabledThomtodiscovertheunitusedbytheNeolithicbuildersofBritain,themegalithicyard,which equals 2.72 feet. Thom continues: “The reason for hisobsessionwithintegersisnotentirelyclear,butundoubtedlytheunitwasuniversallyused,perhapsuniversallysacred.”2

Perhaps the flattened-circle space with its integral proportions wasconsidered to have an energetic power thatmade it suitable for thecosmological and ceremonial functions that took place within thering.Wehaveseenthatthemusicalintervals,tobeintune,mustbecomposedof exact integral fractionsof string length.Perhaps theseflattened circles of the ancientBritonswere an attempt at “tuning”theirritualspacestoaone-to-threeharmony—onebeingthediameterand three being the perimeter of the rings. The creation of the

flattened circles seems to be generated from the same quest thatproduced the circle-squaring efforts: to somehow transmute the π-generated and therefore unknowable circle into a rational figure, tobringtheheavenlydowntoEarth.

Drawing 4.4. In (1) is Professor Alexander Thom’s drawing of the standing-stoneflattenedcircleknownasLongMegandHerDaughters.In(2)and(3)areThom’sdrawingsoftwotypesofflattenedcirclesthathavenearintegralratiosbetweenperimeteranddiameter.In(2),theratiois3.059,andin(3)theratiois2.957. Further discussions of the geometry can be found in Thom’s bookMegalithicSitesinBritain.

Drawing 4.5. A waterdrop shape generated from the ϕ tritone vesicaconstruction. The drop is in Pythagorean tritone relation to the surroundingsquare=1.Theareaofthecircle=.7854.Thecircle/dropratio=1.118,halfof√5 (2.236). The area of the crescent above the drop is .0831. Therefore, thecrescent-to-dropratiois.118=1÷2ϕ3.Thereisanearϕrelationbetweenthecirclecircumference (π)and thecrescentperimeter that=1.936:π÷1.936=1.622,whichiswithin99percent.

ItmayalsobethatZalzal,theArabicmusicianandmakeroftones,playedwiththisideawhenhediscoveredhis3/8tonewithratio21/22(.9545),which,whenmultipliedbyπ,is

21/22×π=2.999.

So ifwewantacirclewithacircumferenceofascloseaspossible toaneventhreefeetwecouldbeginwithadiameterof21/22feet.Zalzalwasundoubtedlyawareof the ancientpractical valueofπof22/7,which equals3.1428,within99percentofπ,whichhesimplytripledtogethisfrequency.

It is interesting that our waterdrop geometry has the same goal—theestablishment of rationality. Perhaps the waterdrops themselves seek rational

forms.Astheydescendfromskytoearththeyrealizethattoberationalistobesomehowreal,justastonesmustbecomposedofrationalfractionsinordertobelocatedandsoundedonamusicalstring.Eachraindropassumesaglissandoofformsas it falls to earth. It strikes the solid earth and is absorbedbecomingalife-nourishing resource.Or it strikeswaterwhere it generates a circularwavethat interacts with other drop-generated circular waves to form a pattern ofintersectingvesicas(seefigure4C).

Drawing4.6.The“Mayantritone”of52/73(.7123)stringlength(left)generatesa waterdrop formwith an area of 32/45 (.711) in relation to the area of thecontaining square. This ratio is the standard just-intonation tritone. Theperimeter of the drop is 3.022 (center),maintaining the near-integral relationwith the diameter. The .0744 (2/27) crescent area is in octave relation to thePythagorean sixth (16/27). The drop-to-crescent area ratio of .711/.0744 =9.556,whichisthesquareof3.09=6.18÷2.Ontherightisaconceptionofanenclosedpoolofwateroraritualspacewiththeformofthisharmonicraindrop.

Drawing 4.7.The Flower of Life is an ancient symbol that can represent thecomplex intersections of energies that constitute phenomena as diverse asmolecular bonds, social networks, and the elements of the individual creativeprocess.BeingacomplexwaveinterferencepatternitcouldalsorepresentwhatLaszlohascalled theAkashicField,orA-field,auniversal informationmatrixexistingthroughoutspaceandholdingenergyandinformationinthemediumofwaveinterference.

Figure 4C is a photo of raindrops falling into water and the resultingintersectionsofwaves,ametaphoricalimageofthiswave-interferencepropertyofthefieldorofthecreativeprocessitselfinwhichsenseimpressions,dreamsandfantasies,memoriesofwordsand images fromtheworkofothers,visionsduringtimesofhighemotion,intuitions,andsoon,intersectinmyriadways.

Figure4C.Raindropsinpool.

“Thephysicalbodyismadeupofintersecting,pulsatingenergyfields.Whatwecalla‘physicalbody’—flesh,bonesandblood—rapidlydisappearswhenhighlymagnified.Aphysicalbody,therefore,oranypieceofmatter,canbeviewedasaninterferencepatternofelectromagneticfieldsthatchangewiththepassageoftime.”

ITZHAKBENTOV3

Another example of such interconnectedness is the root system of a redwoodgrove.Aredwoodtreecangrowupto350feet(107meters)inheightandweighamillionpounds(450,000kilos).Yetitsrootsgoonly6to12feetdeepintotheearth.Theredwoodcompensatesbyspreadingout isrootsforhundredsoffeetand, in theprocess, interweaveswithother redwoodrootsystems.Thusall thetrees together, using a similar pattern as the multiple intersecting raindrops,create a synergetic strength that allows the trees to remain standing for athousandyearsormore.

Themultipleintersectionsofcirclesasasymbolforthesynergyofcreativeworkfindsgeometricexpressionin theFlowerofLifesymbol(seedrawing4.7).As

notedinthecaption,thissymbolcouldalsorepresentwhatErvinLaszlocallstheAkashicField,namedaftertheancientIndianconceptoftheakasha,oruniversalfieldinwhichallknowledgeiscontainedandfromwhichcreationarises.Laszloconsiders theA-field to be an information field containing, in themedium ofwaveinterference,arecordofeverythingthathashappenedintheUniverse.Heoffers a parable to give a sense of how information can be held in interferingwaves.

Whenaship travelson thesea’ssurface,wavesspread in itswake.These affect the motion of all other ships in that part of the sea.Everyship—andeveryfish,whale,orobjectinthatpartofthesea—is exposed to thesewaves and its path is in a sense “informed” bythem.All vessels and objects “makewaves,” and theirwave frontsintersectandcreateinterferencepatterns.

Ifmany thingsmove simultaneously in awavingmedium, thatmedium becomes modulated: full of waves that intersect andinterfere. This is what happens when several ships ply the sea’ssurface.Whenwe view the sea from a height—a coastal hill or anairplane—we can see traces of all the ships that passed over thatstretchofwater.Wecanalsoseehowthetracesintersectandcreatecomplexpatterns.Themodulationof the sea’s surface by the shipsthatplyitcarriesinformationontheshipsthemselves.It ispossibletodeducethelocation,speed,andeventhetonnageofthevesselsbyanalyzingtheinterferencepatternsofthewavestheyhavecreated.4

Another watery symbol of an information field comes from Peruvian shamandon Antonio in a conversation with Alberto Villoldo. He describes, in ametaphorofa lagoon, thenatureof theenergetic realmthatseers tune in to inordertoseethepast.

Ifyouchangeyourperspectiveonthewater,seewherethesunshineson its surface and look at it from that point of view, from above,where the eagle flies, then you can see into its depths, see whatsupportsthesurface,seewhatliesbelow....Thatperspectiveallowsyoutoseenotonlythepresentconditionofthelagoon,butmuchofits history, everything that has touched its surface and sunk to thebottom.Youcanevenseetheeffectthatwhateverhaspenetratedthe

surface has had on the life of the lagoon: a sunken log that plantshavegrownonandfishmustswimaround.Everythingthathaseverfallenintothelagoonhassomehowaltereditscharacter.Somethingsare deeply embedded and are no longer distinguishable, but all ofthemarevisible.5

Theideaofthephenomenonofwaveinterferenceasarepositoryofinformationcanbesymbolized ifwedraw twosmall intersectingcirclearcs,whichdonotseem,at first, topossessanyusable information.Butby finding thecentersofthearcsandconstructingthevesicaandtheothercirclesoftheconstructionwebegin to see the richness of the information contained in the initial seeminglysimplisticfigure(drawing4.8,below).

Theinterpenetrationofformswithinoneanother,symbolizedbythevesica,isanaturalmechanismthroughwhichspaceissharedamongdifferentbeings.

Drawing4.8.Beginningwithafragmentofawave-interferencepattern(1),wefindthecirclecentersofthearcsbydrawingvesicapiscesatanytwopointsoneacharc, bisecting the vesicasand intersecting thebisectors to find the circlecenters (2). In (3)wedraw the full intersecting circles, and in (4) and (5)wecompletethevesicaconstruction.Thisexercisehasaholographicqualitytoit—that is, it shows how the whole is somehow contained in the part. When weunfold the information contained in the original crossed arcs we produce avesica-piscisconstructiongeneratedfromthetonalfifthintervalthatcontainsallthepotentialnumericalandgeometricalinformationofthisimportantfigure.

*

In theplantworld the intersectionofdifferent rootsystemscanbe representedby the same symbol that can represent the intersecting gravitational fields of

binary star systems.Thebringing togetherof twoormore things is an ideaofgreatimportanceinourinternalandexternalcreativeprocesses:thebirthofnewlife from egg and seed; the joining of two tones to produce an interval; thejoiningofpaintandcanvastomakeapainting;thecreationofametaphorthat,in joining two images or ideas, creates a poetic unity. The following exampleshowsacreativeintersectionthatbringstogethertwodisciplines:gardeningandsacredgeometry.

THETEMPLEGARDEN

Our example is an actual garden thatwas created from an arrangement ofϕ-based circles and rectangles derived from an enclosing rectangle with a ratioequal to 2.236, or the square root of 5. This figure can also be seen as theintersectionoftwoϕrectangles.Theideaoftheintersectionofenergiesisverymuchathemeofthegarden.Thelargecentralvesicaisitselftheintersectionoftwocirclesthatcouldrepresentthemasculineandfeminine,which,whenjoined,represent the integrated self. Or the two implied vesica-forming circles couldrepresentspiritandmatterjoinedtogeneratelife,orwaterandsoil,whichjointonourishplants.Thevesicasymbolisopentowhateverinterpretationarisesinthemindof theviewer.Thegarden thusbecomesmore thanaplace togrowfoodandflowers. It isnowaplaceofcontemplationfilledwithsymbols interactingwith living, growingbeings. It is also aplaceof learning.The constructionofsuch a garden, ofwhich there are endless variations, can educate the buildersabout theϕproportionandothersystemsofproportionandwillbeapracticalintroductiontothesubjectofsacredgeometry.

In ancient times the making of architectural drawings on the earth was anadvanced skillpracticedby theharpedonaptae as theGreekscalled the“rope-stretchers.”Theywouldheat-treattheirropessothattheirlengthsremainstable.Agoodropewithstakesateachendmadeanearthcompass.Chalklines,thenasnow,wereusedtomakestraightlines.Thesesametoolscanbeusedtolayoutagarden.Sometimesmeasurementiscalledforbutitisgoodtolimititandletthegeometryunfoldinanaturalway.

ATEMPLEOFWATER

InthesenextdrawingsweagainemploythemagnitudeoftheMoondiameter—thistimeinconjunctionwiththeEarthdiameter—toassistusincreatingasmalltemplegardeninwhichthesubstanceofwaterwillbehonored.Forthisprojectwechooseasacredform—thewatermolecule,H2O,withitstwohydrogenandoneoxygenatoms.WewillusethismoleculeasthecenterpieceofagroundplanforaTempleofWater.SurroundingthetemplewewillplacefourMooncirclesrepresentingthemainlunationphases.Thesewillsymbolicallyholdandregulatethemoleculeofwaterwithinthetemplejustas,inthephysicalworld,theMoonstructures thewater of the seas andof all living things.Within the temple arethree circles—the large oxygen circle and the two smaller hydrogen circlesconnectedwithtwolinesthatrepresentthebondenergythatholdsthemoleculetogether.

Drawing4.9.Ageometricgardendesignwhosethemeisthegoldenproportion.Thecentralvesicaisconstructedaccordingtothedirectionsshownindrawing2.6.Adescriptionoftheelementsareshowninthefollowingdrawings.Figures4D,4E,and4Farephotosofthegardenunderconstructionandfinished.

Figure4D.

Figure4E.

Figure4F.

Drawing4.10.Theoverallshapeofthegardenisarectanglewitharatioequalto√5(2.236).Theinternalstructureconsistsoftwooverlappingϕrectangles.

Drawing4.11.Otherϕrectanglesinthegarden

Drawing4.12.Somecosmic relationships in thegarden.Thediameter ratioofEarth toMercury=2.618 (ϕ2); thediameter ratioofEarth toMoon=3.666(11/3);thesmallthree-sectionrectangleintheupperrightcornerhasthesameratioastheoverallrectangle—2.236(√5)—andthusreferstotheancientdictum“Asabove,sobelow.”Thiswasapractical formulaofsacredsciencethroughwhichtheunknowncouldbeapproachedthroughtheknown,thegreaterthroughthelesserorviceversa;forexample,thecorrespondencesbetweenringshapesondifferentlevelsofbeingasdiscussedinchapter1.

Drawing4.13.Some.-familydimensionswithinthegarden:

.618=1/ϕ .191=1/2ϕ2

.382=1/ϕ2 .543=.691/√ϕ .118=1/2ϕ3 .264=.691/ϕ2

.236=1/ϕ3 .427=.691/ϕ .691=1/(1+1/√5)

OurfirsttaskistogeneratetheMooninitspropersizerelationshiptotheEarththatisgeneratedbyourbasicformofthecirclecontainedwithinthesquare.Thisis done through the agency of the golden rectangle, which is constructed asshownindrawings4.14and4.15.Werecallfromchapter1howtonalnodescanbe locatedon thestring/diameterofourmaincircleusingvariousdiagonalsofour main circle. In drawing 4.18 (below) we make a vesica constructiongeneratedfromthemajor-thirdintervalbymeansofdiagonals.Thelinesdrawnbetweenthepolesandequatorendsofthevesicawillbethebondlengthsforthewatermolecule. The internal angle of thismusical-third vesica happens to be104.5°,whichisthebondangleofthemoleculeofwater.

Drawing4.14.TheTempleofWaterconstructionbeginswith thecircle-in-the-square. (a) The circle representsEarth. (b)Half-square diagonals are drawn,thebaselineofthesquareisextended,andagoldenrectangleisdrawn.

Drawing 4.15. (a) An arc with a radius the length of the golden rectangleintersectstheoppositeside(top)atthe√ϕpoint.Anotherrectangleisdrawn.(b)Themaindiameter isextendedto thenewrectangle.Theextensionisbisected,giving a center point fromwhich to draw aMoon circle. Its size is in properproportiontotheEarthcircle.

Thetwopolesandoneoftheequatorendsofthevesicawillbethecentersofthethreeatomiccirclesthatconstitutethewatermolecule.Thesecirclesmustbein

scale to thebondlengthsconnectingthepolesandequator.Amazingly,weareable to accomplish this withoutmeasurement by using the same construction,with a slight variation, that we used to generate theMoon from the Earth indrawing4.15.The resultingmodelof thewatermolecule can thus serve as anaccuratesymbolinourTempleofWater.AroundtheEarthsquarewithitswatermoleculeweconstructachacana(drawing4.19),astructurebasedonasymboloftheAndeanpeoples.Inthiscaseit ismodifiedsothatitholdsanimprintofthewatermolecule.

Drawing4.16.(a)Alargercirclecenteredontheaxiscrossingisdrawnthroughthe center of the Moon circle. This locates the center points for the otherlunation phases (b). In the processwe have squared the circle since this newcircle and the Earth-containing square have the same perimeters (towithin afewthousandths).

Drawing 4.17. (a) A diagonal crossing locates the 4/5 node (the major-thirdpoint) on the string/diameter. (b) This point is mirrored, the halfway centerpointsarelocated,andavesicaconstructionismade.

Drawing 4.18. (a) The inner angle of the major-third vesica is drawn. It is104.5°, which equals the bond angle of liquid water. (b) Now, the sameconstruction that produced the Moon circle in drawings 4.14 and 4.15 isreplicated in miniature. The Moon is considered to be the Earth and the ϕrectangleand√ϕ rectangleareconstructedgeneratingasmallMoon. In (c)acircleisdrawnaroundthesmallEarth-Moonpair.Thiscirclewillbetheoxygenatom in our water model. (d) Half of the oxygen-circle diameter will be thehydrogen-atomdiameter.

Drawing4.19.(a)Theoxygenatomiscenteredatthevertexofthebondangle,and thehydrogenatomsareplacedon theends. (b)Next,achacana isdrawnaroundthewatermoleculeasshown.

Drawing4.20.AsimpledesignforaTempleofWaterwithentryandexit

Drawing 4.21. Thewatermoleculewith its scientifically derived bond lengthsand atomic sizes, both of which come very close to our scaled-up andgeometricallyderivedTempleofWatermolecularcomponents

Drawing 4.22.Here the chacana is doubled, providing a raised garden bedsurrounding the interior of the water temple garden. In this version we haveoriented the temple to face the new moon: to be in the center of the watermoleculeistobefacingeast.ThedirectionscorrespondtotheidealrisingtimesoftheMoonphases.Thenewmoonrisesatdawn,thedirectionofeast.Thefirstquarter rises at noon; the full moon at sundown; and the last quarter atmidnight. Integral measurements were used in this particular temple garden.Theoveralllengthoftheinnerchacanais11feet,andtheouterchacana14feet.ThediameteroftheMoonis3feet,andthewidthoftheoutercircularpathis2feet.Theseandtheratiosamongthedimensionsareshownindrawing4.23.

Drawing4.23.AconstructionoftheTempleofWatergardenusingintegralandϕ-basedmeasurements:

Hydrogendiameter=1.909=21/11(3×√ϕ/2)Oxygendiameter=3.818=42/11(3√ϕ)Oxygen/Moon=3.818÷3.0=14/11(√ϕ)Moon/Hydrogen=3.0÷1.909=11/7(√4/ϕ)Outerchacana/Innerchacana=14/11(√ϕ)

5

MUSICALVESSELS

WE ARE BEGINNING TO CONSIDER THE MUSICAL SCALE in a different way: as arepository of numerical tools that can aid us in finding a simpler way toapproachcreation—whetheritbethephysicalformsofthenaturalworldorthecreationsofhumanimagination.Itoftenseemsthat thingsexistasmembersofscales—sometimes within a single species and sometimes across differentspecies—and can be understood as intervals whose relationships can beperceived in terms ofmusical harmony. This perceptual practice can serve torelate things with no apparent connection and can sometimes result in newinsights. Furthermore, we have seen in our constructions of the land-to-waterratio of Earth’s surface (in chapter 3) how tonal numbers can be useful ingeneratingformsthatcloselyapproximatephysicalreality.TherelativesizesoftheinnerplanetsofoursystemaswellastheMooncanalsobederivedinsuchamanner—thatis,usingtonalratiostogeneratevesicaconstructionsinwhichtheinnermost vesica circle (C4 in drawing 1.7), representing a planet, will be inaccurate proportion to the enclosing circle, representing Earth (C1). Theaccompanyingdrawings(5.1to5.4)illustratethisapproach.Theresultingtonesarelistedinthefollowingtable.

Planet TonalRatio IntervalNameEarth 1 fundamental

Venus 19/20(.95) semitone

Mars 8/15(.5333) seventh

Mercury ±9/13(.691) sharpenedtritone

Moon 7/11(.636) minorsixth

Drawing5.1.Theϕtritoneconstructionresultsinanaccuraterepresentationofthe relative sizes of Earth andMercury. Aϕ section construction of the halfdiameterisusedhereresultinginthe1to.382,Earth/Mercuryratio;.382=1/ϕ2.

PLANETARYPOTTERY

In the spirit of play, let us use this odd collection of planetary intervals to

generate forms in the shape of pottery vessels. Drawings 5.6 to 5.11 (below)show the planetary pots generated by a single construction—the crosseddiagonalsofthesquareandthehalfsquare(thesameconstructionthatwehaveused to generate the fifth interval).Wewill consider the areas of the pots inrelationtothefundamentalastheirtones.(Thepotareashavebeendeterminedwith a computer drawing program.) The fundamental tone is the area of theenclosingsquarethatisequalto1.Allofthetonesofthepotshavebeenbroughtintoasingleoctave.Ournexttaskistocomparethetones/areasoftheplanetaryvesselstorevealtheirintervallicrelations.Asurprisingratio—thatbetweenthevesselsofVenusandMars—isthesquareofϕ,or2.618.Thisisagoodexampleof the hidden harmonies that might be discovered using such transmutationalgeometry.Curiously,2.618isthesameratiothatexistsbetweenthediametersoftheEarthandMercury:7,926miles÷3,027miles=2.618.

Drawing5.2.TheconstructionofVenusinrelationtoEarth.In(1)werepeattheconstructionwemadeindrawing3.2oftheVenusdiameterpointat19/20ofthediameter of Earth. In (2) the Venus circle is drawn to the 19/20 point. In(3)theVenuscircleiscenteredwithinthedrawing,andavesicaisconstructedaroundit.Thetonegeneratingthisvesicahastheratio39/40(.975),anancientquartertonecitedbyAvicennaandEratosthenes.

Drawing5.3.Thediagonalconstructionofthe7/11(.636)tonethatgeneratesavesicawhoseenclosedcirclehastheratioof3/11inrelationtotheEarthcircle.The actual Moon diameter is 2,160 miles—an accuracy of 99 percent. (Seedrawing2.2forfurtherpropertiesofthisconstruction.)

Drawing 5.4. The construction of Mars in relation to Earth. In (1), using

diagonals, a circle is constructedwith diameter of 8/15, the tonal ratio of themajor seventh. In (2) this circle is centered within the Earth circle. In (3) avesicaisdrawntoenclosetheMarscircle.Thegeneratingtoneofthevesicais23/30(.7667),atonelyingbetweenthethirdandthefourth.

Drawing5.5.(1)AstoragejarfromSusa,thirdmillenniumBCE,fitsnicelyintoa major-third vesica with tonal ratio of 4/5 (.80). (2) The Sanctuary Rhyton,Crete, 1600 BCE. It is surrounded by a vesica at the interval of the majorseventhwithstring-lengthratioof8/15(.5333).Theratioofthevesica=3.872=√15.

Drawing5.6.TheconstructionoftheMercuryvesselusingcrossedfullandhalf-square diagonals. This vessel occupies .1379 of the area of the square thatequals1.Broughtintothemainoctave,thevesselequals.5516,orabout11/20,aminorseventh.

Drawing 5.7.TheVenus vessel occupies .6214, or about 5/8 of the square—aminorsixth.Thenumber.6214isalsoveryclose(within99percent)to1/ϕ.

Drawing5.8.TheEarthvesseloccupies27/40(.675)oftheareaofthesquare.Thisintervaliscalleda“gravefifth.”

Drawing5.9.TheMoonvesselwitharea=.0786occupiesatenthofthesquare.Broughtintothemainoctavethisequals4/5,amajorthird.

Drawing 5.10. TheMars vessel has an area = .2373. Brought into the mainoctavethisequals.9492,or243/256,thelimma,orPythagoreansemitone.

Drawing5.11.Theratiosamongtheareas(asnotedindrawings5.6-5.10)oftheplanetary vessels along with descriptions and historical references. OurtranslationsoftheinnerplanetsandtheMoonintopotteryformsandthefurthertranslations of their area ratios intomusical ratios hopefully will allow us toappreciate, in a differentway, the relations among the planets. The followingintervalshavebeenbroughtintoasingleoctave.

A/B=13/12,athree-quartertone(IbnSina)A/C=1.414,octaveof√2,thegeometrictritoneA/D=octaveof11/9,aone-and-three-quartertone(Partch)A/E=octaveof15/14,asemitone(Ptolemy,AlFarabi)B/C=2.618=ϕ2,broughtintooctaveitequals.764,aflattedfourthB/D=9/8,thejustwholetoneB/E=160/81,theinverseofthesyntoniccommaof81/80ratioC/D=12/7,aseptimalsixthC/E=3/2,theperfectfifthD/E=7/4,ajustminorseventh(Ptolemy,Mersenne,Partch)(SeedrawingA2.15formorepotteryforms.)

Perhapsthissortofexercisecanserveasamodelforinvestigationsofspacebydesignerswhowishtohaveananchorinlaw,inthiscasethelawsofgeometryandoftonalintervals,yetwishtoplaywithnumberandform.Suchanapproachmight also appeal to musical composers who would like to experiment withtonesderivedfromgeometry.

THEWORLDINAGRAINOFSAND

Throughout the bookwe have seen the power of the vesica as a tool of form

generation. Now let us do another exercise with those most common vesicaformsof all: leaves. It is interesting to look at leaves and other vesica-shapedformssuchaseyesandfishinaharmonicway—thatis,astonalphenomenon.Infigure5Awehavetakenthreeleavesandappliedtonalvesicastotheirborders.Perhapstheseareinsomewaytheprimal“tones”oftheleaves,whicheachleafinaspeciesplaysinitsownway,someabithigherandsomelower.Couldweexpandthisideatoincludedifferentkindsofbeingssuchasafishwhoselength-to-widthratioequalsthatofachosenleaf?Isthere,then,someharmonicrelationbetween them? And what of non-vesica-shaped beings that simply share thesamelength-to-widthratio?

Figure5A.Ascaleofleaves.Fromleft,southerncatalpa,purplebeech,ash.

Drawing5.12.Thesoutherncatalpa leaf surroundedby themajor-thirdvesicageneratedat4/5(.80)ofthestring/diameterlength.Thevesicahasalength-to-widthratioof1.291,thesquarerootof1⅔.

Drawing5.13.Thepurplebeech leaf is enclosed ina fifth vesicageneratedat2/3(.667)ofthestring/diameter.Thevesicahasaratioof1.732=√3,makingitavesicapiscis.

Drawing 5.14. The ash leaf vesica has a generating tone of 8/15 (.5333), amajor-seventhinterval.Thevesicaratio=3.873=√15.

Letushaveacloser lookatall theformsavesicamight takewithinanoctavestructure.Wemay say that the vesica of the fundamental is a circle, and thevesicaoftheoctaveisastraightline.Thustheinnumerableformsbetweenthetworepresentallpossibleratiosofheighttowidthandsonumericallyrepresentallcreatedbeings.Eachofthesevesicascan,ofcourse,beconsideredasbeinggeneratedfromaparticulartone,andeachcreaturecanbeassociatedwithatone.

Asthevesicasapproachtheoctavewehaveseenthattheybecomesmallerandthinner.Yetat thesametimetheirheight-to-widthratio increases.Theyare, inessence, moving toward a point of infinitesimal thinness and infinite height-towidth ratio (drawing 5.15). There is an implied form that might help toilluminatethepresentsubject.Werecallthestructureofthevesica,howitsarcsenclosetheoctave,thecenterofthestring/diameter.

Drawing5.15.The vesicas of the twelve tones.Between each vesica there areimplied intermediate vesicas that have their own tones. The area between theseventh(noteB,8/15)andtheoctave,forexample,isaregioninwhichvesicasof increasingly larger height-to-width ratios take form even as the vesicasthemselves shrink in absolute height. Among those vesicas are the following(none ofwhich are pictured here): AnArabic tone of ratio 33/64 (.5156), forexample, generates a vesica of 5.656 ratio. The vesica with ratio = 7 isgenerated by the double septimal tritone of ratio 25/49 (.5102). The vesicasbegin to appear like blades of grass as they approach the octave. The vesicawithratio=11,whichrepresentsthediameterratioofJupitertoEarth—thatis,Earth is the circle enclosed by the vesica, and the Jupiter circle encloses thevesica—isgeneratedbyatoneof61/121(.5041).Thevesicawithratio=109isgeneratedby the5,940/11,879 tone(.500042084).This is theratio thatrelatesthediametersoftheSunandEarth—theEarthbeingthecirclewithinthevesicaand the Sun the circumcircle around the vesica. We begin to see the infinitespaciousnessintheintervalsbetweentones.Inthislastsectionofthescale,thevesicasbecomemicroscopicasthetonesbecomemoreandmoreimpossibletodistinguish with human ears. Their numbers, however, reveal that the vesicaratios approach the infinite even as their physical heights approach the

infinitesimal.Forexample,wecanimagineavesicainwhichtheinnercircleisthesizeofanatomandtheoutercirclethesizeoftheEarth.Yetstill,withintheoctave,spaceremains.

Thus,whenweconsiderallthetonesasawhole,theresultisanestedgroupingofvesicasenclosedbyacircle.Again,wehavetheuniversalringshapeinwhichaninneressenceisheldwithinaprotectingvessel,aformsharedbyallbeingsfromplanets to cells to atoms.Here it consists of concentric layers of vesicasthat symbolize the length-to-width ratiosofbeingsof all sizes (drawing5.16).The needle-like vesica generated from the Earth/Sun diameter relation indrawing5.17(below)vesicawouldbecomealmostinvisibleifitwereshrunktoitsproperheightwithindrawing5.16.An interesting fact is revealedwhenweconsider the actualdimensionsof thisvesica:TheSundiameter (the lengthofthe vesica) equals 864,000 miles, and the Earth diameter (the width of thevesica)equals7,926miles.Their ratioequals109.Now, ifwemultiply109×864,000 we get the actual size of the C1 enclosing circle in the vesicaconstruction,864,000×109=94,176,000miles,

which, surprisingly, happens to be the apheliondistance from theEarth to theSun,

Sun/Earth=Aphelion/Sun=109.

Thus thereare109Earths in theSundiameterand109SunsfromtheEarth totheSun.

Drawing 5.16.The nested vesicas of a standard twelve-tone scale.Within thecentral vesica is space for an infinity of vesicas whose ratios becomeprogressivelylargerastheyshrinkinsizetowardtheinfinitesimal.

Drawing 5.17. A computer-generated measured conception of a vesicaconstructiongivingasenseoftherelativesizesofEarthandSun.TheC2vesica-generating circles are very close to the octave (.50) having the string-lengthvaluesof .500042,whichgeneratesthelongthinEarth-Sunvesicawithlength-to-widthratioof109.Inthedrawingontheleft theC3circle thatcontains thevesicaistootinytobeseen.Wescaleitup,andthedrawingontherightrevealsthe tinyEarthwithin its needle-like vesica, both containedwithin the giganticSun.BesidestherealizationofhowsmalltheEarthisinrelationtotheSunthereis another interesting fact brought out by the geometry—namely, that thecontaining circle (C1) (shown on the left) for the C2 vesica circles has adimensionthattranslatesintotheSun-Earthdistance,specificallytheaphelion,or farthest away distance, of around 94millionmiles (see accompanying textdescription of this phenomenon and the proportion it generates). A simplerclassical construction, of similar accuracy, of the Earth-Sun size relationshipcan be done using an octave-reduction technique that is demonstrated in thefollowingchapter.

6

THESPIRALINGOCTAVE

ONEPURPOSEOFTHISBOOKISTOENCOURAGEreaderstobecomefamiliarwiththeancienttoolsoftone,number,andgeometryinordertousethemtocreatenewforms—formsthatwillreflecttheharmoniesinherentinthenumericalstructuresoftheworld.Wehavepreviouslyseenafewexamplesofhowsuchtoolsmighthave been used by the ancient temple builders. And, in our exercises withplanetaryandlunardimensionswehavediscoveredpossibleharmonicprocessesbehindtheirformation.

In this chapter we will use our tools to create harmonious forms using twoessentialfiguresoftheharmonicrealm—namely,theoctaveandthespiral.Thenumbersthatemergefromtheseexperimentswillbeexpressedgeometricallyinorderthatwemayvisuallyexperiencethem.LetusbeginbyagainlookingatourEarth-Sun construction, yet this time in the context of an octave-halvingprogression of the Earth diameter. We consider the Sun of 864,000 milesdiameter equal to1 and theEarthof7,926milesdiameter equal to1/109.Weneedtobringthisfractionintothemainoctave,whichwecandobydoublingit6times:

2/109,4/109,8/109,16/109,32/109,64/109.

The sixth double, 64/109 (.5872), can be located on the string/diameter of theSuncircleusingcrosseddiagonalsasshownindrawing6.1(below).AcircleisdrawnthatintersectsthegeneratedpointonthehorizontaldiameterandextendstothecircumferenceoftheenclosingSuncircle.Locatingthecenteroftheinitial64/109 circle gives us the first octave point at 32/109, and we draw a circle

throughitaswedidwiththemainoctavepoint.Wecontinueuntilwereachthesixth octave, whose circle is the size of the Earth circle in relation to thecontainingSuncircle.

TheNumber108isimportantinspiritualandcosmiccontexts.TheZenpriestwearsajuzu,abraceletwith108beads.TheHinduhasajapamalarosarywith108beads.TheTibetanBuddhistsalsohavea108-beadrosary.Somesaythereare108gopis,thefemaledisciplesofLordKrishna.BothAyurvedicandChinesemedicinerecognize108pressurepointsonthebody.Yang-styletaich’ihas108moves.Thereare108danceformsintheIndiantradition.Theratio3/11,theMoon/Earthdiameterratio,multipliedby108isequalto29.455,whichiswithin.997of29.53,thedaysinalunation.And108dividedby4equals27,within.988ofthelunarsiderealperiodof27.32days.Thenumber108timesthe2,160-milediameteroftheMoonequals233,280miles,whichisabouttheaveragedistancefromEarthtoMoon.(Oddlyenough,aboutthesamedistance,i.e.,234,054,resultsfrommultiplyingEarth’sdiameterof7,926bythelunationcycleof29.53days.)And108timesEarth’sdiameterof7,926milesequals856,008miles(theSundiameterwithin99percent).Substitutingour“idealEarth”of8,000miles,asdescribedinthetext,makestheSundiameterexactat864,000miles;and108timesthisSundiameterequals93,312,000miles,whichistheaverageoftheEarth-Sundistance.

Drawing6.1.TheEarth-Sunoctaveconstruction,clockwisefrombottomleft,(1)shows us the construction of the 1/5 point on the string/diameter thatwe canthentransfertoourmaindrawing,(2)inwhicha64/109diameterC2circleisconstructed.Througha sixfoldoctave reduction, (3) a circle is found that hasthe size of the Earth diameter in relation to the main enclosing circle thatrepresentstheSun.In(4)theEarthcircleisenlargedandshowninrelationtoacircumferencearcoftheSun.

The 64/109 tone (.5872) is very close (within 99 percent) to the Pythagoreanmajor sixth of 16/27 (.5925) ratio. Their difference is 108/109. Ifwewere tosubstitute the Pythagorean ratio into our geometry, our octave halving wouldresultinanEarthdiameterofexactly8,000milesandaSun/Earthratioof108:

864,000miles÷8,000miles=108.

Withtheseroundnumbersweseemtohaveenteredtherealmofthearchetypal,of Plato’s world of perfect Forms symbolized by simple integral ratios, puretones,andstraightforwardgeometry.

THESKYSPEAKS

TheSunnumberof864istheoctaveof432,whichiswithinthemaindoubling-generatedoctave256–512.Inthisoctave,432,whichisatthePythagorean-sixthintervalpositionof16/27—thatis,256/432=16/27—wecangetaninsightintotheharmonicnatureoftheimperial(mile/foot)systemofmeasure.

The Pythagorean-sixth interval position of 432 is also the octave of 216(2 × 108), the Moon number, the Moon diameter being 2,160 miles, and216×2=432.Thus,our“idealEarth”of8,000miles,ourSunof864,000miles,and our Moon of 2,160 miles all fit into a seamless harmonic system. It isinterestingthatthemeasuredEarthof7,926milesmustbeperfectedoridealizedsoastofitintothiscosmicsystem.Perhapsthisistheheavenlymessagetous—thatwe,asco-creators,mustbring theEarth toa stateof cosmicperfection inorder to harmonize with the Universe. All these small discrepancies, or“commas,” that arise in our harmonic studies remind me of the refrain fromLeonardCohen’ssong“Anthem”:“There isacrack ineverything / that’showthelightgetsin.”

Drawing 6.2.The octave construction of the “ideal Earth” on the face of theSun.In(1–2)thePythagorean-sixthvesicaconstructionismade.In(3–4)octavecirclesarelocatedresultinginanEarthcircle108ththesizeoftheSuncircle,or,inactualmeasurement,8,000miles.

Let us look more closely at this discrepancy, or “comma.” The differencebetweenthemeasuredEarth(7,926)andtheidealEarth(8,000)is74miles.Aninterestingnumberresultswhenwedivide8,000by74:

8,000miles÷74miles=108.108108,

which mysteriously yet obviously emphasizes the importance of this number108.

Thereciprocalof74equals1/74,which,whenbroughtintothemainoctave,is1/74,1/37,2/37,4/37,8/37,16/37,32/37—and

32÷37=.864864,theSunnumberrepeated.

Also,letusconsiderthereciprocaloftheideal108Sun/EarthratiothatwecansuperimposeontheidealEarth:

1/108×8,000miles=74+miles(74.074).

The physical result of this experiment would be a circular area of 74 milesdiameter, representing the Sun, located on Earth. Interestingly, the measuredEarth equals the ideal Earth minus the Sun reciprocal: 8,000miles − 74miles = 7,926miles, as if to say that theway to bring Earth intoperfectionsomehowlieswith theSun—perhaps thehonoringof theSunin themannerofancientcivilizationsorchangingourenergysystemstosolarpower.

Let us apply our Sun/Earth reciprocal to theMoon. Our Earth/Sun reciprocalcircleof74milesfitswithinthediameteroftheMoon29.19times—veryclosetothe29.53numberofdaysinalunation:2,160miles÷74miles=29.19miles.

Againweencounterasmalldiscrepancy,orcomma,betweenthecalculatedandthereal.Inthiscasethedifferencebetween29.53and29.19isaboutasyntoniccomma of 80/81, the same little interval that we have used to transformPythagorean intervals to just-intonation intervals.Thus81/80×29.19=29.55,whichiswithin99percentofthe29.53lunationperiod.

*

Butwehavedigressedfromourthemesofoctaveandspiral.Becausetheuseofϕ-family ratios as generating tones in our constructions generally result inharmoniousfigures,wehaveusedϕ-basedoctaveconstructionsindrawings6.3and6.4toconstructfiguresrelatedtoournowfamiliar4/ϕnumberthatwefirstencounteredintheApollotempleratio.

Drawing6.3.Usingtheoctaveconstructionattheϕtritone,asubtleellipsewithlength-to-width ratio of 17/18 is produced within a context filled withharmoniousrelationships.(1)Thevesica-generatingcirclesarehalved—thatis,theyareat theoctaveof thegenerating toneof .691since .691/2= .3455. (2)Thecentersoftheoctavecirclesarejoinedwiththevesicapolestoproduceanellipse.In(3)weseetheellipseinrelationtothecontainingcircle.Theirarearelation is 4/ϕ, our now familiar Apollo temple ratio. In (4) the ellipse isenclosedinarectanglewithratioof17/18,thestring-lengthratioofanArabic

semitone.Thisdesigncouldbetheplanforaharmonicgarden.

Drawing 6.4. (1) At the .809 (ϕ/2) point on the string/ diameter a vesica isconstructed. (2)Then twooctave circlesaredrawn. (3) and (4)The rectanglejoining them has a ratio of 4/ϕ or 2.472, the ground-plan ratio of theApollotempledescribedinchapter2.

OCTAVESPIRALS

Wehaveseenhowthevesica-constructionmatrixisabletogenerateoctavesofthe fundamental tone. There is another way we can geometrically expressoctaveswiththevesicaconstruction.Thisisbyusingthevesicaspiralasshownindrawing6.5.Theheight-to-widthratioofthevesicadeterminestheratiooftheprogressionofthisstraight-linespiral.Thus,inourexamplewehaveconstructeda vesica with the ratio of 2 from our minor sixth tone. The segments of thestraight-line spiral are in 2-to-1 ratio, the ratio of the octave. In our secondexample(drawing6.6)theratioofthevesicais1.5,whichisthefrequencyratioof the perfect fifthwith string length of 2/3 (.667) of themusical string. Thespiralthatwethenconstructcontainsaprogressioninwhicheachtonesegmentisfollowedbyatonesegmentthatisinafifthrelationtoit.ThisistheancientspiraloffifthsthatestablishesthetonesofthePythagoreanscale.Intheprocessof unfolding twelve tones we traverse seven octaves. There is, however, adiscrepancy,acomma,whichbecomesapparentwhenwetransposeanumberofthetonesbacktothemainoctave.Amongtheseisthefinaltonejustsharpoftheoctavebythefraction73/74—thePythagoreancomma.

Drawing 6.5. (1) The minor sixth tone with string-length ratio of 5/8 isestablished.(2)Thevesicaconstructionattheminorsixthgeneratesavesicaofheight-to-width ratio of 2. A straight-line spiral is then constructed in whichadjacentsegmentlengthsareintheratioof2.Thusbyconsideringthesegmentsas string lengths we can visually get a sense of the pitch relationships of thesuccessiveoctaves.Wecando thisprocedurewithanyvesica: thevesicaratiowilldeterminetheprogressionoftones.

Drawing6.6.Thespiralof fifths. In (a) the13/18 tritone (the inverseof theϕtritoneof9/13)isconstructedgeneratingavesicawithlength-to-widthratioof3/2, the frequency ratioof the fifth.Thisallowsus todrawa spiral (b)whosesuccessivesegments,consideredasmusicalstrings,areinafifthrelationtooneanother. In the accompanying table are the segment-length ratios and theirratioswhen transposedandadjustedwith thePythagoreancommaof74/73 tothemainoctave.TheresultisthetonesofthePythagoreanscale:

MUSICALSPACES

Aswehaveseen,anacceptanceofthefoot-milesystemasvalidcosmicmeasureis important for theunderstandingof thiswork.While it is true that the ratiosbetween space and timemeasures, such as the oneswe have been looking at,would be meaningful no matter what the measuring system, the presence of“canonical” mile measures, many of which are related to musical numbers,weavesthesystemsofharmonicscienceintoameaningfultapestry—atapestrythatlendscolorandlifetotheotherwiseasceticrealmofdrynumber.TheSunnumber864(864,000miles=Sundiameter)relatestotheMoonnumberof216(2,160 miles = Moon diameter) through 4. This would be the same in anysystem, yet 864 is also a musical number, being an octave up from 432, anumberprizedbymanymusiciansasanalternativetotheA=440Hzstandard.The 432 A relates to its fundamental of 256 through the Pythagorean-sixthintervalofa16/27string length—that is,256/432=16/27. Itmakessense that432, theSun radius number,would set the tone in ameaningful solar system.

Althoughwerelatethesenumbersnowtothemoderncycles-per-secondsystem,thenumbersthemselvesservedtheharmonicartsinancienttimes.Thedoublinggeometricprogressionseries1,2,4,8,16,32,64,128,256,512waswellknowntotheancients,andtheyusedittoformtheirmusicalintervals.Theycombinedthedoublednumberswiththetriplinggeometricseries1,3,9,27,81,243,729to define the tones of the Pythagorean scale. As we previously noted, whenmultiples of 5 were included in the numerical toolbox, certain musical ratiosbecamesimpler—thatis, theycouldbecomposedofsmallernumbers: thusthePythagorean 16/27 becomes 3/5 in just intonation. The “5-limit” numbers,thoughwelcome simplificationsof interval ratios,unfortunatelyhaveobscuredthecosmicassociationsofthePythagoreanratiossuchaswehaveseenwiththePythagoreansixth.

HereisaninterestingaspectofourSunnumber864,whichJohnMichellcallsthe foundation number. He writes: “In the language of symbolic number 864clearly pertains to a center of radiant energy, the sun in the solar system,Jerusalemonearth,theinnersanctuaryofthetemple,thealtarwithinit,andthecornerstoneonwhichthewholeedificeisfounded.”1

Thefrequencynumber864maybeusedtocreateaPythagoreanscale.Webeginbyreducing864alongwithtwoofitspermutations—namely,486and648—totheir tones: 864 is within the 512–1,024 octave. Thus 512/864 = 16/27, thePythagorean sixth. The number 486 is within the 256–512 octave:256/486=128/243,thePythagoreanseventh.And648,inthe512–1,024octave,becomes64/81,thePythagoreanthird.Nextwedividethethirdbytheseventh,whichequals2/3, thefifth;andwedividethethirdbythesixthtogiveus3/4,the fourth. Then the seventh is divided by the sixth to give us 8/9, thewholetone.Theresultisaseven-tonePythagoreanscale:1,8/9,64/81,3/4,2/3,16/27,128/243.

Other important tones that are rare in practical music making include, as wenotedearlier, theundecimal tones.Wehaveseeninchapter2howthenumber11 had an importance to the ancients rarely found in modern approaches toharmonic thought. In ancient times there were many undecimal tones in use,particularlyintheArabicworld:an11/16(3¼)tone;an11/18(4¼)tone;a6/11Arabicneutralsevenththatweusedindrawing2.1togeneratethe11/1Jupiter-to-Earthdiameter ratio;an11/12 toneusedon theArabic lute,which isa tonethatreferstotheEarth/Sunratio:792/864=11/12;a10/11wholetonecitedby

Ptolemy; a 22/27 tone of the eighth-century musician Zalzal, whose octaves11/27,11/54,11/108,11/216,11/432arecomposedofMoonandSunnumbers(seedrawing6.7).

Drawing6.7.Theconstructionofaspiralwhoseratioequalsthetoneknownasthe“WostaofZalzal.”(1)Aminor3rddiagonalconstructiongeneratesavesica

with the ratio (within99%)of27/22 (1.2272). (2)Astraight line spiralwhoseadjacentsegmentsare in theratioof22/27 isconstructed.A three-pointcirclearc spiral is constructed on the straight-line spiral. (3) The finished spiralshowing the tonalrelationship.TheWostaofZalzal isnamedafter itscreator,MansurZalzal,anArabicmusicianoftheearlyninthcentury.Thestring-lengthratio,22/27(.8148),iscalleda“neutralthird.”

The spiral form that we have been considering is the gesture of the octaveprogressionasitstonespiralsinward,soundinghigherandhigherpitchesduringits journey within the realm of vibration.We can imagine this form in threedimensionslikeaspiralstaircaseasitrecedesfromsightintotheregionswherethe tonescanonlybedetectedby themost sensitive instruments.Andstill thespiralkeepswindingintotherealmofunimaginablyhighpitches.

Drawing 6.8. (1) The vesica construction at the major third uses crosseddiagonals to establish the 4/5 (.80) tonal point. (2) A straight-line spiral isconstructed. The segments are in 1.291 ratio. (3)A spiral composed of three-pointcirclearcsisdrawn.(4)Thisspiralapproximatesthelogarithmicspiralofthenautilus.

Figure6A.Picturedhereare(a)Nautilusshell;(b)Begonia‘Duartei’leaf;and(c)spiralgalaxyM74.Regardingthespiral,A.S.Eddingtonwrote:“Theformofthearms—alogarithmicspiral—hasnotyetgivenanycluetothedynamicsofthespiralnebulae.Butthoughwedonotunderstandthecauseweseethatthereisawidespreadlawcompellingmattertoflowintheseforms.”A.SEddington2

Thereisonespiralthatcanbeconstructedwithcirclearcsratherthanspiralvearcs. This is the golden spiral, an archetypal logarithmic spiral that can beconstructedwithclassicalgeometry—thatis,withlinesandcircles.Drawing6.9showsthegoldenspiralalongsideaspiraldrawnfrompointsontheborderofanautilusshell.

Drawing 6.9. Two spirals, (1) one produced from points along the edge of anautilusshell,(2)theotherthegoldenspiralconstructedfromcirclearcs.Theseareexamplesofgnomonicgrowth.Thegnomonisafigurethat,whenaddedtoan original figure, generates a new figure with the same proportion as theoriginal.Withthegoldenrectangle,shownontheright,thegnomonisasquarethat, when added to a golden rectangle, results in another golden rectangle.

Thustheshapeofthegoldenrectangleconsistsofaspiralofsquares.Thesamegnomonic principle applies to all logarithmic spirals (such as the nautilusspiral):eachspiralcanbeanalyzedbyaspiralingsuccessionof rectanglesofequalratio.

8

THECIRCLEOFRESONANCE

AMONG OUR HARMONIC TOOLS IN THIS STUDY the most important is the vesicaconstruction. The vesica itself in its innumerable shapes could be called asymbolforformresidingwithinamatrixofconnectedcircles,thewholecreatinga unity composed of number, tone, geometry, and proportion. To review: thecircles of the vesica construction are the C1 containing circlewhose diameterrepresents themusical string; the twoC2 circles that form thevesica (eachC2circlesectionsthestring/diameteratatonalposition);theC3circlethatenclosesthe vesica; and the C4 circle that is enclosed by the vesica. The diameters ofthese circles are closely related, and the following equationsdemonstrate theirrelationship.

C1/C3=C3/C4=vesicaratio(C4+1)÷2=C2

C3=√C4C4=2C2−1

1/C3=vesicaratio

Wecanaffirmthateachvesicaalongwithitsmatrixofcirclesisgeneratedbyaparticulartonalratioofwhichthereareinfinitelymanywithintheoctave.

Drawing8.1.Areviewof the vesicaconstruction showing its circlesand theirrelationshipstoeachother.TheC1circlethatresideswithinitsenclosingsquarehas a diameter equal to 1. This is the string/diameter on which the musicalratios are located. Both of the C2 circles have diameters equal to the string-length of the tone that is being translated into geometry. Their intersectionproducesavesica.TheC4circleisinscribedwithinthevesica,andtheC3circlecircumscribesthevesica.

Thus the creative process as symbolized by the vesica construction can beexpressedasnumbergeneratestone,whichgeneratesform.

There is another circle thatwe can add to the vesica construction. This is thecirclethatconnectsthecentersofthetwoC2vesica-formingcircles.WewillcallthistheC5circle(seedrawing8.2).Sinceitconnectsthecentersofthetwotonalcircles,theC5circlesymbolizestheprincipleofresonance.

RESONANCE

When a musical string is tuned to a certain frequency and plucked, a nearbystring of the same frequency will begin vibrating. We call this phenomenonresonance.Thisappliesnotjusttomusicalstringsbuttoanytwoobjectsofthesame or nearly the same frequency. Hans Kayser gives a broad view of theresonanceprinciple:

Whatstrikesusmostforciblyisthefactthatthereare“variousbodiesofthesamenaturalfrequency.”Thisalsoseemsobviousatfirst,butbecomes remarkablewhenwe grasp the phenomenon of resonanceabsolutely, and observe its autonomous realization, wherever andhowever itmaybe.Then resonancebecomes a value-formof greatamplitude. In physics and its applications to technology we seeresonance used copiously. In chemistry, resonance becomescatalysis: themysterious catalytic processes are scarcely explicablewithouttheanalogyofresonance.Thesamegoesforenzymesinthebodies of plants and animals: they can also only be interpreted asbeing-resonances in the material life cycle, as strengthenings orweakeningsofresonance,heldinequilibriumbytheharmonyofthecollectiveorganism.Buttherearealsoresonancesinthepsychicandspiritual realms. In the sympathies and antipathies of humanrelationships, in the simultaneous emergence of identical ideas anddiscoveries in different minds and nations and even in the greathistoricalresonances,oftenwithgiganticeffectsbothinthepositiveandthenegativesense,weseetheresonanceprincipleelevatedtoavirtuallyuniversalprinciple.1

Resonance is a power that joins things together. It is a binding force thatoperatesintheuniversalrealmofvibration.

TWOFROMONE

LetuscontinuewithourC5resonancecircle,whichcanbeconsideredtobethelink between elements of a similar nature—that is, in direct tonal relation(includingoctavesofthetoneinquestionthatarealsoresonatedbythesounding

of the tone). Thus in our constructionmodelwe can imagine a circularwavebeing formed at the center of our C1 containing circle that, as it widens,intersects the string/diameter at all tonalpoints.Eachof thesepoints,when sostimulated, creates a sound wave (C2) that expands until it is stopped at thecontainingcircle.

Thissound-basedsymbolhasaffinitieswiththemodernscientificcreationstory,whichresonateswiththeoldcreationmyths.Themodernstoryhypothesizesthatsoundwasaprimordialcreativeenergythatorganizedgascloudsintostarsandgalaxiesaftertheexpansionofspacecalledthebigbang.Someoftheoldmythsalsosuggestthatsoundwasaprimarycreativeforceforcreation.

Ourgeometricsymbolshowstheoriginoftheresonancewavetobeatthecenterofthefield.AversefromtheTaoTeChing,translatedbyGiaFu-FengandJaneEnglish,seemstodescribetheensuingprocess:

TaobegotOne;Onebegottwo;Twobegotthree;

Threebegotthetenthousandthings.

TheTao is the unknownor unknowableCause, a dimensionless point or seedfilledwithlimitlesspotency.TheOneistheoriginalwave(C5)thatripplesoutfromthispointgenerating thefield.The twoisembodied in the twoC2circle/wavesthatarebroughtintoexistencebythemovementoftheoneandthatmovethrough the tonal spectrum generating sound. The three is the vesica that isformedfromtheintersectionofthetwowaveswithinthemainoctave.Thetenthousand things are the infinite shapes that the vesica may take from theslenderest slivers representing tones near the octave to the almost circularvesicasattheotherendofthescalenearthefundamental.

Drawing 8.2. The C5 circle connects the centers of the two vesica-formingcircles.Thesecentersareattheoctavepositionsofthetwotones.TheC5circlecan thus be considered a resonance circle since the two identical tones arejoinedtogetherbyresonance.

ONEFROMTWO

There are variousways of using the dynamics of the vesica construction as a

model of creation. As we have just seen, the C5 circle can be seen as theoriginator of the creative process—a two-from-one model like the mitosisprocess of a living cell. Another approach is the one-from-two model asembodiedinthehumanfertilizationcycleinwhichtwoelements,themaleandfemale carriers of genetic material, join to produce the first cell. Thiscorresponds to the phenomenon cited earlier of two stones simultaneouslydroppedintoapoolofwater thatcreatetwowavesthatgraduallyintersectoneanother. The C5 circle is also present in this model and connects the twoelements.Inthiscaseitisshrinkingorcondensingasthetwowave-circlesgrowinsize.

Withintheovumthemaleandfemalepronucleiareveryclosetothesamesizeastheymovetowardoneanother.Theysoonmergetheirgeneticmaterial,whichis transported on a vesica-shaped “mitotic spindle” composed ofmicrotubulesthatemergefromtwincentrioleslocatedatthepolesofthevesica.Azygote,theprimordialcell, isnowformed thatcontains thechromosomesofbothparents.Wecanusethisveryprocesstoimagineastorycalled“TheBirthoftheMoon.”

THEBIRTHOFTHEMOON

While it is generally acknowledged that none of theMoon-origin theories areadequatetoexplainallofthefacts,wedoknowthatEarth’scrustandthesurfaceoftheMoonarecomposedofthesamematerials.Wecanthereforehypothesizethat theMoonwas somehowgenerated fromEarth.Themostwidely acceptedtheoryisthatduringtheearlyhistoryofthesolarsystemawanderingplanet-sizeobjectaboutthesizeofMars,knownasTheia,strucktheyoungEarth,sendingtons ofmatter into orbit. Through gravitational attraction thematter graduallycoalesced into the Moon. The accompanying drawings are a mythical-geometricalinterpretationoftheeventsleadinguptothebirthevent.

Webeginwithtwocirclesoneitherendofthemaindiameter,drawing8.3(2).Eachofthesehasadiameterthatis1/11ofthemaindiameter.Wewillconsiderthe1/11circlestobethemasculineandfemininepronucleigeneratedfromEarthandJupiterand thecondensing (C5) resonancecircle tobe thegestatingMoonitself as itmoves from idea to reality, from conception to birth.We can nowwatchouropposingcirclesas theymovetowardconjunction.Theratio1/11 isthe thirdoctaveof the8/11 tritonederived from theharmonicseries,and8/11

willbetheintervalatwhichtheMoonisborn.Theratio1/11alsoconnectstheplanet Jupiter andEarth since the averagediameter ratioof Jupiter toEarth is11/1.

Drawing 8.3. The Birth of the Moon, parts 1–6. (1) The Earth awaits theinseminationoftheMoonseed.At(2)theseedoftheMoonhasbeenplantedbyJupiter. Its size is 1/11, which is the reciprocal of the Jupiter/Earth diameterratioof11/1.ItisconnectedbytheC5resonancecircletoitstwinattheotherendof thediameter.The resonancecircle represents thecondensingMoon. In(3) the condensing proto-Moon is 6,480 miles in diameter, 3 times its finaldiameterof2,160miles.At(4)theEarth/Moonbarycentercirclewithdiameterof5,760milesisestablished.Thebarycenteristhecenterofgravity,withinthe

Earth,of theEarth-Moonsystem.At (5) the resonancecircle is the sizeof thecombined radii of Earth andMoon. At (6) the resonance circle × 100 = theradiusoftheSun.

Drawing8.4.TheBirthoftheMoon,parts7–9.In(7)theMoonat3,600milesindiameter embodies the universal standard used in measuring time and spacesuchas360degreesinacircle,ortheMayantuntimeperiod.In(8)atemplateof the finalMoonsizecanbe foundencircledby thevesicaas if to inform theMoon-resonancecircleofitsfinalsize.Attheendofthisperiodofperhaps100million years after the forming of the Earth the Wandering Planet (WP),propelled toward Earth by Jupiter, strikes Earth. In (9) the Moon is born,emergingfromthewomboftheEarth.

RichardHeathnotessomesurprisingconnectionsbetweentheMoonandJupiter:“EarthresonateswithJupiterthroughtheMoon.TheslingshoteffectofJupiter’sgravitationalfieldpropelledthemeteorthatcollidedwithEarthandcreatedtheMoon.”Jupiterhas,inmythicterms,impregnatedMotherEarth.Heathgoesontoexplainsomeoftheseconnections:“JupiterissynchronizedtotheMoonusingunits based on the number twenty-five (five squared),”2 and he gives thefollowingexample:

Drawing 8.5. The Birth of the Moon. (1) The moment of birth as shown indrawing8.4 (9). (2)An8/11 to1 rectangle isdrawnenclosing the vesica, themoon,andthelefthalfofthesquare.Thisrectanglehasadiagonalof2/ϕ.Itscenterpointat1/ϕcoincideswith thecrossingof themoonand thehorizontalaxis.(3)Thebaseofthe8/11rectanglebecomesasideofapentagon.Thevertexof the next side intersects themain circle. The next side intersects the centralaxis of the rectangle. (4) The pentagon containing the newly bornmoon. Theperimeter of the pentagon = 11/3, the same ratio as that of the Earth/Moondiameter.

25ofJupiter’ssynodicperiodof398.88days=27.321practicalyearsof365days.

Now,27.321daysisthesiderealorbitalperiodoftheMoon.Thusthereare365oftheseperiodsin25Jupitersynods.Heathnotesthattheotherimportantlunarperiod—thelunationbetweennewmoonsof29.531days—relatestobothϕandthenumber25,theJupiternumber.

Thereare309lunationsin25practicalyears,and309=618÷2=1,000÷2ϕ.

Wenowcontinueourgeometriccreationmyth,whichisillustratedindrawings8.3 to 8.5. We have established our undecimal theme that will find its finalexpressionatthecataclysmicbirthoftheMoonwithitsdiameterequalto3/11that of Earth. The drawings illustrate the process of gestation as the (C5)resonance circle that represents the gestating Moon condenses in undecimalincrements within the womb of the Earth. It is interesting, as described indrawing 8.5, that at the time of birth a pentagon, the vessel of the goldenproportionandsymbolof life,formsaroundtheMoon—andthat theperimeterofthepentagonequals11/3(3.66), theEarth/Moondiameterratio.Theseseemtobeauspicioussignsforthebirthofanentitythatwillsoprofoundlyinfluencelife on Earth through its stabilizing effects on the Earth’s orbit and its poweroverthesubstanceoflife,water.

CROSSINGS

InconsideringtheC5resonancecircleweshouldnotethatbesidesmovingalongthe horizontal diameter of the C1 containing circle it also moves along thevertical diameter. It thus intersects four nodes at a time. A resulting vesicaconstructionwould thenconsistof twocrossedvesicas.This constructionmayhavebeenusedbyancient architects and sculptors togenerate regulating linesfortheircreations,suchasweseeindrawing8.7.Thisconstructionbeginswithone of the simplest beginning gestures in geometry: the arc generated by thequarter-square diagonal, shown in drawing 8.6.Whenmirrored, it generates avesica. In terms of tonewe are in the realm of √2 (1.414 . . .), the irrationalregion of the geometric tritone, the center of the scale.Within the octave itsnumerical expression is .707 . . .,which equals√2/2.Another identical vesicadrawnat rightangles to thisvesicageneratesacentral region inwhichwecandraw an intersecting circle and a square whose area measures happen to be

extremely close.Thecircle-squaringpropertyof this figuremaybeoneof thereasonswhyitwasanaturalsettingoutconstructionforancientarchitects.

Drawing8.6.(1)Anarcwithradiusofthequarter-squarediagonalisdrawnandmirrored and a vesica is formed. (2) The same construction is done on thehorizontal axis resulting in crossed vesicas. (3) A square and a circle areinscribedwithinthecommonareaofthecrossedvesicas.Theirareasarewithin99 percent of being equal. (4) A succession of circles and squares areconstructed around both figures. This space-dividing technique, calledquadrature,wasusedasagroundplanlayoutinancientarchitecture.The.707point of the arcs on the diameters, considered as musical strings, is at thegeometric tritone sectionpoint, the centerof the scale.Twomoreexamplesof

quadratureinarchitectureareshowninappendixdrawingsA2.29andA2.30.

Drawing 8.7. Examples of crossed-theta vesica constructions as possiblesettingoutpatterns forancientsculptureandmedievalarchitecture.On the left(1)istheDoryphorosofPolycleitos,aRomanmarblecopyoftheGreekbronzeoriginal, circa 440 BCE. The center of the body-as-monochord is the octaveposition at the genitals. Like the octave this is the place of generation. Thevesica arc intersects the string/diameter at the tritone, the tonal center of thescale. The vital lung-heart area is defined by the vesica arc and quadraturecircle. Mirroring this on the other side of the octave are the knees, thosemechanisms that allow humans an upright and hands-free stance and all thatthis implies.The sizeof thehead isdeterminedby thecontainingcircleandaquadrature circle that circumscribes the squareand circle generated from thecrossingofthevesicas.Thepatternsquare-circle-square-circle,andsoon,isthepattern of the quadrature technique. The upraised hand, originally holding aspear, lies at an intersection of the crossed vesicas and a quadrature circle,perhapsacluefromtheartistabouthisdesigncanon.

Ontheright(2)isthegroundplanofAmiensCathedral(begun1220CE).Thecrossed thetavesicascreatenotonlyawidthbutalsoan inner space thatcanserveasthestartingpointforquadrature.Otherelementssuchasthebaywidthandthebeginningoftheapsecanthenbedetermined:thespacebetweenthetwosquaresisthebaywidth.Thesemicircleabovetheoutersquareistheapse.Thevertical vesica also establishes connections between distant points of the planandthusprovidesanoverallunifyingstructure.

Drawing 8.8. (1) A crossed-vesica construction composed of the vesicaconstruction at theminor third (5/6 string length) and its inverse—themajor-sixth construction (3/5 string length). (2) The vesica circles are drawn andenclosedinsquares.(3)Afteranumberofdeletions,theresultisagroundplanofabuilding.

Figure8A.Bruno’sFiguraMentis.

The crossed vesicas each have a length-to-width ratio of 2.414, a numbersometimes called the silver number, or theta (θ), a reference to the goldennumber,ϕ. The θ ratio has a similar property toϕ: its fractional part is thereciprocaloftheratio

2.414=1÷.414,likeϕinwhich1.618=1÷.618.

We have seen in drawing 6.9 how the square is the gnomon of the goldenrectangle.Inasimilarwaythegnomonoftheθrectangleis thedoublesquare,whichis tosay thatadoublesquareaddedtoaθrectangleproducesanotherθrectangle.

Drawing8.9.AnanalysisofGiordanoBruno’sFiguraMentis (Drawingof theMind)showninfigure8A.Thegeometricalthemeisthejoiningof5and6.(1)Avesicapisciswithaninscribedrhombusisdrawn.(2)Twointersectinghexagonsare drawn using the sides of the rhombus as the initial two sides. (3) Thedoubled-hexagon diagonals are drawn. (4) Two circles with centers on thevesicapolesandradiiextendingtothehexagonverticesareconstructed.Theselocate the points on the vertical axis fromwhich two irregular pentagons aredrawn.(5)Theinnerirregularstarhexagonisdrawntoexistingpoints,andthecomposite hexagon-pentagon figure is drawn. (6) A section of the drawing isisolated showing the hex-pent composite. The joining of 5 with 6 has severalpossiblemeanings.Thehexagon,whichembodiesthenumber6,hastheperfectradial symmetry often found in nonorganic matter such as crystals. Thepentagon, which embodies the number 5, has the offset symmetry found inorganic life-forms. The joining of the two produces a unifying symbol of thecreatedworld that includesbothprinciples. JohnAnthonyWestnotes that“tothePythagoreans...Five[was]thenumberofLife.WemaycallSixthenumberoftheworld....Five,inbecomingSix,engendersorcreatestimeandspace.”3

Tonally,5/6,theminorthird,istheratiothattransformsπintoϕ:5/6×π=ϕ2,soϕ=√5/6π.

Continuing the theme of the crossing as a creative foundation gesture weconsider Giordano Bruno’s Figura Mentis (figure 8A) in which the hexagonisjoinedwiththepentagon(drawing8.9).Wecanconsidersuchintersectionsassymbolsof the joiningof twoprinciples.Abriefdescriptionof thisconcept isdiscussedinthecaptionaccompanyingthedrawing.Afurtherapplicationofthisideaisshownindrawing8.10inwhichapotteryformiscreatedfromthejoiningofthehexagonandoctagon.

HARMONICSCIENCE

InthisbookIhaveassumedtheexistenceofanancientscienceofnumber,tone,and geometry. My approach toward this science has consisted of using thenumbersandshapesofgeometryandofthemusicalscaleastoolsfordiscovery—asIimaginethattheancientsmusthavedone.Whathasresultedisawayofapproachingthecreationofharmoniousformusingtechniquesbasedonthefirmfoundationofeternalnumber.IhaveusedasmycentraldiagramafigureIcallthevesicaconstruction,whichcanbegeneratedbyanyratio.ForthemostpartIhavechosenratiosfrommusicalscales.Thereasonsforthisarethecompellingconstructions that these ratios generate. For example, we have seen how theharmonious interval of the fifth generates the useful vesica piscis with its √3ratio.Similarlytheintervalofthefourthgeneratesavesicawiththeimportant√2ratio.

Whilewehavenotlimitedourselvestomusicalnumbersasgeneratorsofvesicaconstructions,itisinterestingthat,intheprocessofcreatingtheseconstructions,onecangetinthehabitofthinkingofallnumbersastones.Throughtheoctavedoublingandhalvingprocessanynumbercanbebrought into themainoctavethat residesbetween1and2or,moreprecisely,on themusicalstringbetween1/2and1.Weneedonlytakethereciprocalofthenumberanddoubleitasmanytimesasrequiredtobringitintothemainoctave.Forexample,thereciprocalof27is1/27so:

1/27,2/27,4/27,8/27,16/27,whichisthePythagoreansixth.

Anotherwaytoproceedistodeterminewhichoctavethenumberinquestioniswithinanddivideitintothefundamentalofthatoctavetofindthestring-lengthratio.Thus27isintheoctave16–32,and16/27istheratio.Or,letustakethenumber84,which is in theoctave64–128 and64/84 reduces to 16/21, a tonecalledtheseptimalfourth.

Drawing8.10.Acompositeconstructionofthehexagonandoctagongeneratesapotteryshape.(1)Avesicapiscisisconstructed.Acirclewiththesameradiusasthe vesica piscis circles is centered on the upper pole of the vesica. Twoconstruction linesaredrawnwithin this upper circle as shown. (2)Theuppercirclenowhaspointstowhichahexagonmaybedrawn.Alargecircleisdrawnthroughcircle-hexagonintersectionsasshown.Thisistheoctagoncircle.(3)A

largercirclewithradiusequal to thequarter-circlediagonalof thepreviouslyconstructed circle is drawn. The axis ends of this circle are connectedwith asquare. (4)Thecornersof the squareare the centerpointsof four circlearcswith radii intersecting the center. These arcs divide the square at pointsallowingustodrawtheoctagon.(5)Partsof theoctagonaredeletedandit isjoined to thehexagon. (6)Three-point circlearcs join the verticesof hexagonand octagon resulting in a pottery shape. The ratio of the pot is 16/15, thefrequencyratioofthemajorsemitonemusicalinterval.Theperimeterofthepotisequal(within99percent)tothecircumferenceoftheoriginalcircle.

The hexagon, symbol of crystalline elementary matter, is the first shape toemergefromtheprimalcircle,itssidesbeingequaltothelengthofthecircle’sradius, and is constructed by walking the compass around the circlecircumference. The octagon is associatedwith the cubewith its eight verticesandistheancientsymboloftheelementearthwithitsqualityofstability.Thesetwo earthen principles are joined to form the pottery vessel. Musically thenumber 8 refers to the eight tones in an octave fromC toC'.From these areformedthesevenintervalsandsixwholetonesofthebasicscale.Thus6and8arejoinedintheoctaveandembodiedinourpotteryform.

*

Thus, the vesica constructions seem to be natural structures for the bringingtogether of the three tools of ancient harmonic science: tone, geometry, andnumber.Ibelievethatworkingwiththevesicaconstructionscanbringusclosertothespiritofancientharmonics.Moreimportantly,however,istheopportunitythat this geometry presents to the creative imagination in the realm ofharmoniousdesignand spacedivision.Themusical intervalsbeing inherent inbothnatureandthehumanpsychepresentafoundationuponwhichasystemofdesignbasedonLawcanbeconstructed—asystem thatwill resonatebetweentheinnerandouterworlds.

9

THEMUSICALUNIVERSE

THE REVIVAL OF THE ANCIENT PRACTICE of arranging phenomena in systems ofcorrespondencecancontributemuchtoourunderstandingoftheworld.Theideathatseeminglyseparatethingsindifferentscalesandmodesofbeingarerelatedto one another resonates deeply with the human soul that hungers for ameaningfulvisionof theworld inwhich thingsareconnected inways thataresimpleandunderstandable.Anexampleofacorrespondencestructurethatgivesa sense of this idea is shown in drawing 9.1. Here the relation of diversephenomenadrawsforthadeeperunderstandingof theunity that joinsdifferentcycles of nature and of human life. Its purpose is simplification andunderstanding. The words are combined within a geometric matrix and arejoinedbyimpliedproportionalexpressions,forexample,

spring:childhood::winter:oldage.

The elements are connected through the mechanism of correspondence thatarisesfromarelationalharmonythatpermeatestheUniverse.

ASABOVE,SOBELOW

Theancienttheoryofcorrespondenceisnowbeingrevivedasanalternativetoareductive science that finds itself increasingly unable to explain the world interms understandable to the majority of people. Physicist Richard Feynmannwrote:

Drawing 9.1. An example of a correspondence diagram demonstrating therelationofplantandhumangrowthandthelunarandsolarcycles.

It always bothersme that, according to the laws as we understandthem today, it takes a computing machine an infinite number oflogicaloperationstofigureoutwhatgoesoninnomatterhowtinyaregionofspace,andnomatterhowtinyaregionoftime.Howcanallthat be going on in that tiny space?Why should it take an infiniteamount of logic to figure out what one tiny piece of space/time isgoingtodo?1

Thisisanexampleofhowwecangetsoinvolvedinexplainingthedetailsthatwemissthewhole—howwecan’tseetheforestforthetrees.Correspondenceinthecontextofharmonicscienceoffersanalternativewayofunderstanding theworld as a whole, just as holistic medicine offers an alternative, not areplacement,forallopathicmedicine.

Systems of correspondences were an important part of ancient sacred sciencethatemergednaturallyfromtheperceptionsofsimilarpatternsatdifferentlevelsofbeing.Suchself-similarityisacharacteristicof,forexample,theever-smallerbranching patterns in blood vessels, in plant branching, and in river systems.Thusarteriesandcapillariesofanimalscorrespondinformandinfunctiontothepatterns created bymain channels and streams of river systems.The principlethatiselucidatedisliquidflow,whichalsocorrespondstononmaterialprocessessuchas themany streamsof thought,memory,dreams, and so forth, that feedinto the river of creative work.We begin to see that correspondence is oftenexpressed throughmetaphor, the language of poetry and myth. In the mythicvision our life becomes a meaningful journey filled with synchronicities andmagicalinitiationsratherthanaseriesofrandomevents.

*

Therewas a desire among the ancients to validate theirmetaphorical insightswith exact knowledge—that is, to find numerical correspondences that couldsupporttheirfindings.Inthisbookwehaveexploredsomeofthesenumerical-physical correspondencesusing, for themost part, the numbers of themusicalscaleandthenumbersderivedfromthegoldensection.

The inversecorrespondencesbetween themovementsof thesunand themoonareinteresting.InmidwinterwhentheSunislowintheskyanditslightisatitsweakest, theMoon is at itshighest in the skyand shines thebrightest.This isreversed in midsummer. At the summer solstice the Sun and Moon set inopposite locations, theSunnorthofwest and theMoon southofwest.This isreversed in midwinter. At the equinoxes the Sun and Moon set at the samepoints.

Thereisaharmonyandadancebetweenthetwo.Onelightsupthenightmostbrightly at the darkest time of year.One lights the days the longestwhile theotherdimsherlight.Justaswiththetidesinwhichtheworkisshared,thetaskoflightingtheworldisharmoniouslydividedbetweenthetwoluminaries.Thesemysteriousgoings-onintheskyledthemegalithbuilderstoerectobservatoriesso that they might better understand the principles at work. The numericalknowledge that resulted became the foundation of ancient sacred science, socalledbecausetheevents in theskywereconsideredtobe theworkingsof thegods.

*

Theuseofmusicalnumberstodescribecelestialeventsseemsappropriate inauniversewhereharmonyanddissonanceareever-presentrealities.The intervalofunisonisconsideredthehighestharmony,andtheexactunisonofthesizesoftheSunandMoonintheskyisasignificantmessagetousabouttherelationshipof the two that, for themajority of ancient civilizations, have represented themasculine and feminine principles. They share their duties according to theirpowers.TheSundispensesthegreaterlightwhiletheMoonprovidesthegreaterpull on the waters. While they appear to be the same size in the sky, theirdiameters are in the ratio of exactly 400 to 1. The sizes of their diametersexpressedintheharmonicnumerationoftheimperialsystemare864,000milesand 2,160miles. Their essential numbers are thus 864 and 216, which are indouble-octaverelationusingthedoublingsystemofoctavecalculation.Thustheoctavefrom128to256contains the216toneat thePythagorean-sixth interval(16/27). Similarly, 864 is at thePythagorean sixth in the octave 512 to 1,024.Thecentral256–512octaveliesbetweenthem.

Furthermore, we have seen in chapter 2 how the yearly cycles of Earth andMoon are related by 20/ϕ and how the diameter ratio of Earth/Moon is 3.66,whichcorresponds to the temporalmeasureof366 revolutionsofEarthduringits 365-day journey around the Sun. We mention these again in order toemphasize that the beautifully choreographed dance of Sun,Moon, and Earthhasafoundationinnumericalcorrespondences,whichareuniversalinscope,afactthatwillbecomeapparentinthefinalchaptersofthisbook.

Figure9A.Thetaiji,oryin-yang,symbolisanexpressionoftheprincipleofcorrespondencethatmanifestsasreciprocation,inversion,

complementarity,resonance,andreflection.

THEWATEROFLIFE

Leonora Leet in the Secret Doctrine of the Kabbalah writes about thecorrespondencebetweenthebiblicaldaysofCreationandmusicalintervals.Forexample, the third day ofCreation inwhich dry land, or “earth,” is created isassociatedwith the interval of the fifth.Leet then shows how the fifth day ofCreation,inwhich“creaturesinhabitingtheflowingelementsofwaterandair”arecreated,correspondstothemajor-thirdinterval.

Thefifthinterval,aswehaveseen,isassociatedwiththethirdharmonicandthethirdwiththefifthharmonic.Theintervalofthefifthalsogeneratesthevesicapiscis,whoseinternalanglebetweenpolesandequatorendsis120degrees,anangle fromwhichahexagoncanbeconstructed.Thehexagon in thisCreationscenario is the first form tobedifferentiated from thecircle, theoriginal formsymbolizing the fundamental tone. The hexagon emanates from the circle asspontaneouslyasahexagonalsnowflaketakesshapefromitsdustmoteseed,or

astheintervalofthefiftharisesfromthesingingvoice.Theradiusofthecirclebecomes the sides of the hexagon whose vertices are located with six equalgestureswiththecompass.Thefifth,thefirsttoneafterthefundamental,canbesaid to correspond to the element of earth, the biblical “land,” the symbol fornonorganicmatter.The“waters”arecreatedfromthemajor-thirdintervalwhosegenerating vesica has an internal angle of 104.5 degrees (drawing 4.18), thebondangleofthewatermolecule.

Drawing 9.2. In the top drawing the vesica piscis generated from themusicalintervalofthefifth(seedrawing1.4)containsthefirsttwosidesofthehexagon.Inthemiddledrawingthelengthofahexagonsideissteppedaroundthecircleto complete the hexagon. In the bottom drawing the outline of the hexagon isshaded, resulting in a three-dimensional representation of a cube, the Greeksymbolfortheelementofearth.

Drawing9.3.Theicosahedron, thewatersymbolof theancientGreeks,hasaninternalstructureconsistingofthreeintersectingϕrectangles.

According toLeet the fundamental is associatedwith the first day.Shewritesthat“thefirstdaycanbeassociatedwithheaven,”whichcombinedwiththeland(thefifth)ofthethirddayandwater(thethird)ofthefifthday,producesamajorchord containing the essential elements of a living creation: spirit, water, andland.FromthisharmoniousbeginningthesongofCreationcanproceed.

With water there are other correspondences—to both 5 andϕ.We recall thediagonal construction that locates the major third on the musical string(chapter 1) inwhich the crossing of two half-square diagonals locates the 4/5thirdnodeon thestring.Thehalf-squarediagonal is thegeneratorofϕ,whichitselfhasanaffinitywiththenumber5boththroughthepentagonandthe√5—thatis,

(√5+1)÷2=ϕ.

TheGreeks assigned the symbol forwater to the icosahedronwith its twentyequilateral triangular sides. Twenty and its first fewmultiples and divisors allreduce, throughoctaves, to4/5 (.80), themajor third—1.25,2.5,5,10,20,40,80,160:

1÷2.5=.4and.4×2=.80,1/20=.05and.05×2×2×2×2=.80,1/80=.0125and.0125×2×2×2×2×2×2×2=.80,andsoon.

The twenty triangularsidesof the icosahedron join to formtwelve intersectingpentagons,whichagainemphasizesthethemeof5,thenumberofthethird,andofϕ.Andtheinnerstructureoftheicosahedronconsistsofthreeintersectingϕrectangles(drawing9.3).

“Earth”fortheGreekscorrespondedtothecubewithitssixfacesrecallingthesix sides of the hexagon—from which an isometric cube can be drawn(drawing 9.2). The association of water with the number 5 and the pentagonshows its affinitywithϕ and its relation to life-forms such as flowers,whichfrequentlyhaveapentagonalpattern,andtothoseplantssuchassunflowersthatincorporatetheϕ-basedFibonacciseriesintheirseedheadpatterns.Themajor-thirdintervalgives“life”totheprimalscaleofthefundamentalandthefifthaswaterjoinedwiththemineralmatterofEarthgiveslifetotheburiedseed.

THEGREATCENTRALSUN

InesotericlorethereexistsanothersunaroundwhichourSunorbits.ThisistheGreat Central Sun (GCS), an entity of transcendent power. It has beenconsideredvariouslytobeAlcyone,astarinthePleiadescluster,aswellasthestarSirius.Othersholdthatitisaspiritualenergythatresideswithinus.SomebelievethattheGCSisatthecenterofourMilkyWaygalaxy,aregionwhosemassisfourmilliontimesthatofourSunandisthesiteofasupermassiveblackhole. We can affirm that our Sun and all the other stars in our galaxy orbitaroundthisregionandthus,ifthereissuchanentityastheGCS,thiswouldbeagoodplaceforit.

WemayfindmorecluestotheideaoftheGreatCentralSuninancientnumberlore.Letushypothesize that thisGCS is indeed locatedat theGalacticCenterandthatitisthesourceofcosmicenergiesthatradiateintoourgalaxy.ThereisanumberinnumerologythatseemstodescribetheeffectsoftheGCS,andthatisthe number 3,168, one of themost sacred numbers of gematria. JohnMichelldescribesitthisway:

The general character of the number 3,168, as conveyed by itspositioninancientcosmologicaldiagramsandthephrasesassociatedwithit throughgematria, is that itrepresentsthespiritwhichpassesthroughandencirclestheuniverse,Plato’sworldsoul.TheChristiantermforthisspirit,developedfromthenumber3,168,wasLordJesusChrist....Thepre-Christianreferenceofthisnumberseemstohavebeentothetwelvegodsofthezodiac.2

Thesymbolicnumber3,168,whichwewillnowidentifywiththeGreatCentralSun,happenstobeinrelationtotheSun-diameternumberastheEarth-diameternumberistotheMoon-diameternumber.UsingtheSun(864),Earth(792),andMoon(216)numbersaspreviouslyestablishedwehave

3,168/864=792/216=11/3or3.666(366beingthenumberofyearlyrevolutionsoftheEarth).

TheimportantsynchronicityhereistheEarth/Moondiameterratiohasthesameessentialnumbersas thatof theyearly revolutionsofEarth.Thisverynumber

becomesasymbolrelatingtheGCStotheSunandthusconnectingusbymeansofnumbertotheGalacticCenter.

Italsohappensthat3,168isinmajor-thirdrelation(thewatertone)totheEarthnumber:

7,920×2/5(octavethird)=3,168.

ThiscouldtellusthatEarthbaptizedbycosmicwater(carriedbytheprimordialrain of water-bearing meteorites and comets onto Earth) attains life andconsciousness.

Drawing 9.4. On the left is a drawing of Earth with its outer core, whosediameter is4,320miles,which isequal to2×2,160miles,or twice theMoondiameter. In decimal terms the Moon diameter is .272 x the Earth diameter,whichhereisequalto1.IfweexpandtheEarthcircletothesizeofthegalaxy(picturedinanartist’sconceptionontheright),estimatedas100,000light-yearsindiameter, then theMooncirclewouldequal27,200 light-years indiameter,whichhappens tobe theestimateddistanceof the solar system to thegalacticcenter.Thescaling-upoftheouter-corecircleofEarth,consistingofadoubledMoon, then represents the ideal circular orbit of the solar system around theGalacticCenter.Wehaveplacedthisoutercore/orbitcircleinthecenterofthegalaxy. The position of the Sun and its solar system is shown by the arrow

pointingtotheorbitalcircle.Inthiswehavecontinuedwithourcorrespondencepractice as a means of bringing the heavenly down to earth, in this case,literally. The intention is not tomake scientific claims but rather to open ourmindstothepossibilitythatuniversalcorrespondencemayexistamongdifferentlevelsofrealityastheancientsbelieved.

“Behold,reader,theinventionandsubstanceofthislittlebook!InmemoryoftheeventIamwritingdownforyouthesentenceinthewordsfromthatmomentofconception:‘TheEarth’sorbitisthemeasureofallthings.’”

JOHANNESKEPLERFROMMYSTERIUMCOSMOGRAPHICUM3

Drawing9.5.Preliminaryconstructionsfor theGalacticTemple(seetheEarthdrawings inchapter3).The threeprimarycirclesareestablished: thoseof theouter (1) and inner (2) cores within the containing circle representing bothEarth and the galaxy. In (3) they are intersected, and in (4) lines are drawninscribingcirclesandvesicas.Deletionsandrefinementsresultindrawing9.6.

In chapter 3 we constructed the inner layers of the Earth. The inner coreconstruction,generatedfromthejust-sixthintervalof3/5stringlength,resultsinaninnercoreof1,584milesindiameter,or1/5thediameterofEarth:

1/5×7,920=1,584.

The inner core is a sphere of solid iron and nickel of tremendous density andwith a temperature equal to that at the surface of the Sun. Its diameter is inoctaverelationtothenumberoftheGreatCentralSunsince

1,584×2=3,168.

Thephysicalpositionof the innercoreand itsconcentrationofmassandhightemperature at the center ofEarth corresponds to theGreatCentralSun at thecenterofthegalaxy.SurroundingtheinnercoreofEarthistheseethingmoltenringoftheoutercorewithitsdiameterof4,320miles,equaltoadoubleMoondiameter.Therelationofdiametersofoutercoretoinnercoreis

Drawing 9.6.TheGalactic Temple. The ring shape of the innerEarth and itscorrespondence to the position of the solar system within the galaxy (top)

producethistempleplan(bottom).

4,320/1,584=2.72=1/.3666,

anumberwearefamiliarwith—366isthenumberofrevolutionsofEarthinoneyear and the number that relates the diameters of Earth (7,920) and Moon(2,160):

7,920/2,160=3.66.

It is interesting that theGreatCentral Sunnumber 3,168 is locatedwithin theEarth (in octave form) as 1,584, the inner core diameter. In the ancientgeocentricviewtheSundoes indeedorbitaround thecoreofEarth,around itsinternal Great Central Sun. (We can add that the inner core revolves in aneasterly direction, and the outer core turns around it in a westerly direction.)Thus

3,168/864=792/216=GreatCentralSun/Sun=Earth/Moon

=1,584(innercore)(×10)/4,320(outercore)=3.66.

THEGALACTICCORE

Wehavenoted that the outer core ofEarth has a diameter equal to a doubledMoon.IfweconsiderEarth’sdiameterequalto1,thentheoutercoreequals2×3/11(.272),or6/11ofEarth’sdiameter.IfweexpandEarth’sdiametersothatitequals the diameter of the galaxy of 100,000 light-years (LY), then the outercorediameterwouldequal

100,000LY×6/11=54,545LY,withradius=27,272LY,

whichisabouttheestimateddistancefromtheSuntotheGalacticCenter.SothecircumferenceofEarth’s outer core corresponds to anorbit of theSun aroundtheGalacticCenter—that is, theouter core is slowly rotatingaround the innercore,andthiscorrespondstothemovementofaringofstars,whichincludesour

solar system, around the Galactic Center. The outer stars of the galaxy inincreasingly less dense concentrations correspond to Earth’smantle and crust,whichalsobecomeslessdenseasitsdistancefromthecenterincreases.

10

THETREEOFTIME

INTHISCHAPTERWEWILLUSEOURGEOMETRYTOINVESTIGATEthesubjectoftime.Wehavementionedthissubjectinpastchaptersinwhich,forexample,wehavefoundthatthenumberofyearlyrevolutionsoftheEarth—366—resonateswiththe 3.66 diameter ratio between the Earth and the Moon. We have alsodiscoveredthatthediscrepancybetweenthelunarandsolarcyclesproducesaϕ-familyrelationship:20/ϕ,or12.36,thenumberoflunationsinayear.Wehavealso used tonal numbers, an aspect of time, in our vesica constructions toinvestigatepropertiesofspace.

*

We will now undertake a project that involves the Mayan calendar and itselucidation by Carl Calleman in his book The Purposeful Universe. The titlecomes from his theory that the purpose of the universe is to generate and toevolvelife.Inthistheoryhedevelopsaconceptionofcosmictimeoriginatedbythe Maya that consists of nine “underworlds,” each of which has a durationtwentytimeslessthantheprevious(older)underworldandtwentytimesgreaterthanthefollowingunderworld(seefigure10A,below).

Each underworld consists of thirteen “heavens,” each of which is ruled by aspiritual energy or god. For example, the tenth heaven in each underworld isruledbyTezcatlipoca,thedarklordwhopresidesoverdestruction.Theeleventhheaven signifies a flowering time and is ruled byYohualticitl, the goddess ofbirth.ThedurationsofthecalendarwerespecifiedbytheMayainmultiplesofthetun—aperiodof360days—ahighlydivisiblenumberthatenablesthelarge

numbers of the calendar to be simply expressed and calculations to be simplymade.HerewehaveusedCalleman’stranslationofthetun-baseddurationsintoyears.Forthoseinterested,themechanismofthetunsystemisreadilyavailableinbooks such asCalleman’sor on the Internet. (Wemight add that there is atonalrelationbetweenthetunandtheyear:360/365=72/73,virtuallythesame(within99percent)asthePythagoreancommaof73/74thatbringsthecircleoffifthsintocongruencewiththeoctave(seedrawing6.6).

Figure 10A. The Periodic System of Evolution of the Mayan calendar asproposed by Carl Calleman. The diagram is from his book The PurposefulUniverse.Herethefirstfour(outofatotalofnine)underworldsofthecalendarare shown. Each underworld consists of thirteen periods—seven days and sixnights—called heavens. The heavens have themes that correspond to thegrowingcyclesofplants.Thedaysareperiodsofmanifestation,whilethenightsareperiodsof preparation that often involvedestruction.Thedurations in theCellular Underworld are in billions of years (BY). The durations of theMammalian, Anthropoid, and Human Underworlds are in millions of years(MY).IntheCellularUnderworldeachofthethirteenheavenshasadurationof1.26 BY. In the next underworld, the Mammalian, each heaven is 1/20 thedurationoftheCellularUnderworldheaven:1/20×1.26BY=63MY.Soeachsuccessive underworld and its heavens are reduced by a factor of 20. (Tableusedbypermission)

“Sincethesacredandstrongtimeisthetimeoforigins,thestupendousinstantinwhicharealitywascreated,wasforthefirsttimefullymanifested,manwillseekperiodicallytoreturntothatoriginaltime.Thisritualreactualizingoftheilludtempusinwhichthefirstepiphanyofarealityoccurredisthebasisforallsacredcalendars.”

MIRCEAELIADE1

Drawing10.1.(1)Weconstructthe21/40=.525tonalpoint(thereciprocalofthe harmonicmean between 1 and 20) with crossed diagonals. (2) From thispointthevesicaconstructionisgenerated.Theratioofitsvesicais4.472=2×√5(3)Thearearatiosbetweenadjacentcirclesintheringequal20.

Drawing 10.2.Drawing 10.1 is applied to the first three underworlds of theMayancalendar,whoseareasarein20-timesrelation.Wecanimaginesimilarpatternsrepeatingwithin thecircleof theAnthropoidUnderworldassixmoreunderworldswhose circle areas are 20 times smaller at each iteration. Thesecircleareasrepresentboth thedurationsof theunderworlds,whichare in20-timesratioand,accordingtoCalleman,the20timesfasterrhythmsoftimethatare experienced by the inhabitants of the Universe during each successiveperiod.

The tun-basednumbersof theMayancalendarproceed ina rhythmofoctavestimes 10 (2 × 10), a process that has the benefits of both the doubling anddecimalsystems.Thedecimalcomponentallowsforanexpotential increaseordecrease in the duration of underworldswhile the doubling component allows

for the durations to retainmusicalmeaning in the sequence of their numbers.Viewedinthiswaytheyaresimplyinoctaverelation,afamiliarwayoflookingatnumbers.Thenumber16.38BY(whichweroundoffto16.4)isthedurationoftheCellularUnderworldandofthecalendarasawhole.Thenumber1,638istheeighthoctaveof64—that is,64,128,256,512,1,024,2,048,4,096,8,192,16,380.So16.38BY÷20=819MY,thedurationofthenextunderworld,theMammalian,andsoon.

Myownpurposeinthischapteristodrawforthhiddenpropertiesofthecalendarusingtheharmonictoolsofnumber,geometry,andmusicalinterval.Oneresultof this analysis is adate for thebigbang that corresponds to the scientificallydetermined date. A number of new possibilities results from this approachincludingapre-big-bangeraandatimestructurethatIcalltheTreeofTime.

Drawing 10.3. Beginning with the same geometry as in drawing 10.1, weconstructastraight-linespiralrepresentingthesuccessionofunderworlds.Theadjacent segments of the spiral are in 4.472 (√20) relation. The verticesrepresent the critical points of the beginning of an underworld and itsflowering/fruitionendingwhenthepurposeoftheunderworldisattained(sixthday)and refined (seventhday).Precedingeach sixthday flowering is the fifthnight, a time of destruction, which prepares the way, for the flowering of thesixthday.

ThereareseveralgeometricwaystoimaginetheMayantimeconception.Usingourvesica-ringconstructionwedrawaninnercircleC4containedwithinalargercircle C3 in which the area ratio of C3 to C4 equals 20. Then a circle C1circumscribesbothofthem,resultinginarearatiosof

C1/C3=20,andC1/C4=400.

Drawings 10.1 and 10.2 show the three oldest underworlds imagined as circleareas.Thisringstructureofthesedrawingsisfamiliartousastheproportionalstructureoftheringsinourvesicaconstructions,althoughwehavebeenusedtocomparingthediametersratherthantheareasaswearedoinghere.Tohavetheadjacentcircleareasin20/1ratiowerequirearatiobetweenadjacentdiametersof4.472(=2√5=√20).This4.472ratiowillthenequalthelength-to-widthratioof thevesica (drawing10.1).From this factwe can find a tonewithwhich togenerateavesicaconstructionusing thevesica-circleformulasdescribedat thebeginningofchapter8.

Sowhenthevesicaratio=4.472,theC2generatingcirclediameter=21/40.

Thefrequencyratio40/21happenstobetheharmonicmeanbetween1and20:

2(1×20)÷21=40/21.Andthevesicaratioisthegeometricmeanbetween1and20:

√(1×20)=4.472.

These important mean properties demonstrate the harmonious position of the21/40ratiointhevigesimalsystem.

*

SowehavegeneratedavesicaconstructioninwhichthearearatiosoftheC1toC3circlesandtheC3toC4circlesareboth20to1,whichrepresentstherelationofoneMayanunderworldtothefollowingunderworld.

Thisvesica-ringstructurealsoreflectstheMayantimestructureinthattheolderperiodscontain themore recent. It is their beginnings thatprecede and followoneanother.

TheMayan timeconception results froman intention that evolutionaryhistoryshouldbedividedintoconsecutiveperiodswhoserelationwillbeinmultiplesof20.Why20?Wehaveseentheusefulnessofthenumber20,whichcombinesthecomputing power of the decimal system with the descriptive power of theoctave.Thereisanothermoresubtlefactaboutthenumber20.If,asCallemanhaswritten,theuniverseispurposefulandthatitspurposeisthegenerationandevolution of life, then at its very inception there must have been some seed,some intention, some indication of the need for the element of water, thesubstanceof life.Wehavenotedinchapter9 that there isevidencefora tonalconnection between the interval of the major third and water. Intuitively weknowthatthereisnolifewithoutwater,andthereisnotrueharmonywithouttheintervalofthethird.And,intheMayancalendar,whosestructureisbasedonthethird, we again find intimations of the connection between this interval andwater.Specifically, aswehave seenand shall furtherdiscover, thenumber20anditsmultiplesrelatetothethirdandthustotheelementofwater.

Drawing10.4.Theangleofthebondbetweentheoxygenandhydrogenatomsinthe water molecule of 104.5° is formed within the vesica construction at themajorthird.

Drawing10.5.Theinnerstructureofthevesicawhose4.472(2√5=√20)ratiowe have used as a factor to locate the beginnings of the underworlds of theMayan calendar. The geometry of the rectangle that surrounds the vesica isfilledwithϕ,√5,and√20harmonies.(1)The4.472ratiovesicaisshown.Itisthegeometric enginedriving theMayancalendar.The rectangle that enclosesthevesicahasthefollowinginternalproperties:(2)Thecircle-in-the-square,theprimalgeneratingformofgeometry.(3)Itsfirstgenerationofthesquare-rootof5rectangleconsistingof2intersectingϕrectangles.In(4)isanotherversionofthe√5 rectangle consistingof aϕ anda1/ϕ rectangle. (5)This√5 rectangleleavesa√5/2rectangle.Theratioofthisrectangleis1.118=1/.8944..8944isthe string-length of the meantone whose function in the Mayan calendar isdescribedinthetext.(6)showsthe√5rectangleframedby√5/2rectangles.(7)The4.472(2√5)rectangle.

ASPIRALOFTIME

Thesuccessionofunderworldsfollowsaprogressionbasedonaparticulartonalinterval called themeantone.This is oneof anumberofwhole-tone intervals,themostcommonbeingthejustmajorwholetoneofstring-lengthratioequalto8/9.Anotherwhole tone is theminorwhole toneof 9/10 ratio.Thegeometricmeanbetweenthetwoisthemeantone,anditisderivedthus:

√(8/9×9/10)=.8944...

The meantone is an irrational number closely approximated by the fraction25/28. We will use its .8944 equivalent in our equations. The major-thirdintervaliscomposedoftwomeantones:

.8944×.8944=4/5,

whichisthestring-lengthratioofthemajorthird.

Themeantone is thus intimately involvedwith the third, and this involvementrelates it to the factorof20,which is thegeneratingnumberbetweenadjacentMayan underworlds. The ratio 1/20, the ratio between the duration ofunderworlds,isthefourthoctaveofthethird:4/5,2/5,1/5,1/10,1/20.Thus

1/20×16.4BYA,thebeginningofCellularUnderworld=820MYA,thebeginningofMammalianUnderworld

and1/20×820MYA=41MYAbeginningofAnthropoidUnderworld.

Divisionby20,thefourthoctaveofthethird,thustakesustothebeginningofeachunderworld.Wecanusethecomponentofthethird,themeantone,totravelwithintheunderworlds.Inparticularweusethesecondoctaveofthemeantone,or.2236,forthisfunction:

.8944÷2=.4472(firstoctave)and

.4472÷2=.2236(secondoctave).

(Thereciprocalofthisstring-lengthratioof.2236is4.472,thefrequencyratioofthisdouble-octavemeantone.)

Whenwemultiplyanybeginningunderworldnumberby .2236ordivide itby4.472wefindourselvesinthesixthdayofthecycle,thefloweringperiodwhenthe intention of the particular underworld achieves realization. This process isshown in drawing 10.3 inwhich a straight-line spiral, whose segments are in4.472 ratio, demonstrates the time progression from the beginning of anunderworld to its sixth day and then to the next underworld. (We have, inchapter6,usedthistechniqueinwhichtheratioofprogressionequalsthevesica

ratio,togeometricallydemonstratethecircleoffifths.)

Drawing 10.6. The Ship of Time Temple formed from the 4.472 (√20) ratiovesica that embodies the time ratios derived from the rhythms of the Mayancalendarasdescribedintheprevioustextanddrawings.Thecentercirclecouldbeapoolofwaterbeneathacleardomeoracircularspaceopentothesky.

Thenumberthatemergesatthisdouble-octavemeantonepositionintheCellularUnderworldisasurprise:

16.4BYA÷4.472=3.66BYA.

Wearefamiliarwith3.66asbeingthediameterratioofEarthtoMoonaswellas366beingthenumberofrevolutionsofEarthinoneyear.Therethusappearstobea reference toEarth in this floweringof lifeday,orperhaps it isEarth thatreferences,throughnumber,thisdaysixwhenitspurposewasfirstmanifested.At this time the intentionof theCellularUnderworldwasbrought forth in theform of the first cells, prokaryotes or one-celled organisms. Just before this,duringthe1.26billionyearsofnightfive,Earthitselfwasformed,andabouta100 million years later theMoon is believed to have been formed. This wasfollowed by a long period of meteorite bombardment of Earth that causedwidespreadchaosbutalsobroughtmeteorite-bornewatertoEarth,theabsoluteprerequisite for life to emerge. It is believed that the presence of the MoongraduallystabilizedtheerraticrotationofEarthinpreparationforthefloweringoflife.

Another iteration of our 4.472 factor brings us to the beginning of theMammalianUnderworld:

3.667BYA÷4.472=820MYA.

THEVESICAOFTIME

Atthecoreofourconstructionisthevesicawithitsratioof4.472.Letuslookmorecloselyatthisfigure(seedrawing10.5).Atthecenterofthevesicaisaninscribedcircleof1/20diameterinrelationtotheenclosingC1circle.Wenowdrawasquarearoundthisinnercircleandwithhalf-squareradiiwedroparcsonboth sides to the extended baseline of the square (drawing 10.5[3]).We have

generated two points from which two 1/ϕ rectangles can be drawn that arejoinedwithacentralsquare.Werecognizethiscentralelementasa√5rectangle.The remaining rectangles attached to the ends of the √5 rectangle each has aratioof1.118=√5/2.This1.118ratioisthelengthofthediagonalofthehalf-squarefromwhichϕisgenerated.Itisalsothefrequencyofourmeantonewithstringlengthof.8944:

1/1.118=.8944.

Whenour“canonical”3.667BYAdatefortheemergenceoflifeismultipliedby1.272,thesquarerootofϕ,weget4.667BYA,whichcloselyapproximatesthedatingoftheformationofthesolarsystemandEarth,whichhasbeenestimatedtobeabout4.6BYold.ItisbelievedthattheMoonwasformedabouta100millionyearsaftertheEarth.AllofthisactivityiswithinthedramaticnightfiveoftheCellularUnderworld,whichlastedbetween5.1and3.9BYA.RecentanalysesoftracesofbacteriainancientrocksinAustraliaandGreenlandindicatethatlifewaspresentonEarthbetween3.5and3.7BYA.

TheAboriginesofAustraliapossessthemostancientcosmologydescribingtheoriginoftheCreation.“Intheirview,theancestorfirstdreamshisobjectificationswhilesleepinginthecamp.Ineffect,hevisualizeshistravels—thecountry,thesongs,andeverythinghemakes—insidehisheadbeforetheyareexternalized.Objectificationsareconceivedasexternalprojectionsofaninteriorvision:theycomefromtheinnerselfoftheancestryintotheouterworld.”

NANCYD.MUNN2

This inner vesica structure is full of geometrical and numeric power and thusservesas a fittingengine togenerate thevital changes required for theMayancalendar.The structurecontains the√5 function that is thegeneratorofϕ, thegoldenproportion,whichisencodedintolife-forms.Thestructurealsocontainsthe meantone, the root of the major-third interval, which is associated withwater,theelementoflife.Theratioofthe4.472rectangleequals√20,whichsetstherhythmfortheunfoldmentoftheunderworlds.Thisstructureisheldwithinthevesica,asymbolicinterferencepattern,therepositoryofinformation,whose

generatingcircles/waveshavediametersof21/40,which is theharmonicmeanbetween1and20.

THEERABEFORETIME

Taking recent pre-big-bang cosmological hypotheses into account, the 16.4-BYlongMayancalendarallowsforanerabeforethePrimordialExpansion(bigbang), which we have called the Dreamtime, a period of preparation for theCreation that recalls the Aboriginal sequence of events as described in theaccompanying quote: “the ancestor first dreams his objectifications . . .” Thisdreamingor imagining is the firststageofCreationwhetherof theworkofanindividualortheworkofthecreatorgods.Theuniversallypervasiveprincipleofcorrespondencethatisatthecoreofharmonicsciencefreesustoimaginesuchconclusions particularly, as in the present case, when we are working withnumericalevidence.SincetheWilkinsonMicrowaveAnistropyProbe(generallycalledtheWMAPprobe),whichstudiedthemicrowavebackgroundtoascertainadateforthebigbang,gaveadatefortheExpansionof13.73BYA±0.12BY,our13.66BYAharmoniccalculationiswithinthescientificallyderiveddateforthebeginningofcreation.

Thegreatquestionsarewhenandhowuniversalcreationbegan.Itmakessenseto me that there must have been a time of preparation before the PrimordialExpansion.Perhapsduringthefirstday,thesowingtimeoftheMayancalendar—theDreamtime—the ideaofauniversewasplantedwithin theCosmicSoul.Moderncosmologistshaveconsidered thepossibilityofsucha timebefore theExpansion.One scenario is that therewas a time of contraction of a previousUniversebacktoasingularitythatthen“bounced”outintoanewexpansion—thepresentUniverse.

Drawing 10.7. A diagram showing the primary time periods in universalcreationandtheiraffinitywiththeEarth/Moondiameterratio:

EraofFormation10BYA÷Dreamtime2.73BYA=Earth/Moonratio;EraofLife3.66BY÷1BYsegmentofEraofFormation=Earth/Moon

ratioof3.66.

The central “ladder” divides the cosmic formation time, which occurredbetween thePrimordialExpansionand the emergenceof theprokaryotes, intoten1-billion-yearsections—possiblythecosmicsourceoftheubiquitousdecimalsystem.TheMoon,whichseemstobeamajorfactorinmeasurement(therootofthewordmoon comes frommens,whichmeans “tomeasure”) happens to fitperfectly into one of the sections, which reinforces this possibility. OthermeasuresrelatingtotheMoon:the2.73BYDreamtimerelatestothe27.33-daysidereal lunarcycle;halfof thisequals13.66days,whichrelates to the13.66BY period fromExpansion to the end of the calendar. TheMoon, the ancienttimekeeper ofEarth, thus appears to be in resonancewith universal time.Wewillfurtherexplorethispossibilityindrawing10.8.

OtherharmoniesincludetheDreamtimemultipliedbytheEraofLifeequalstheEraofFormation:2.73×3.66=10.TheDreamtimeisexactly1/6ofthewholecalendarand1/5ofthetimefromtheExpansiontotheendofthecalendar.

Drawing10.8.Inthisdrawingeventsincosmichistoryaresetinastructurewithlunar correspondences. The lunation circles demonstrate the correspondencebetweentheExpansion-to-Presentdurationof13.66BYandthe1/2lunarorbitalperiod(27.33days)thatequals13.66days.Whenweaddanewmoondayfromwithin theDreamtimewegeneratehalf a lunation cycle (29.53days)of 14.75days.

Thesedarknewmooncircleswithinthe2.73BYDreamtime,whichcorrespondtotheapproximatethreedaysthatthenewmoonisinvisible,couldsymbolizethefinalstageofablackhole,aremnantofapreviousuniversethathascontractedtoasingularityandisnowabouttoexpandoutintoanewuniverse.

Thelunarcyclethusservesasahighlyvisiblereminderofuniversalcreation,acorrespondencethatisbothmetaphoricalandrestsonverifiablenumbers.Itisinterestingthattheendofthecalendaris2/3ofthewaythroughthefullmoon,perhaps indicating that though we have come to the end of this evolutionaryphaseoftheMayancalendarwewillstill,forquitesometimetocome,bewithinthebrightinfluenceofitsclimax.

Atthispoint,intheabsenceofanyproof,wewillallowourselvestoimagineatime before the Expansion that generated our Universe. The numbers of theMayancalendar,asmentionedabove,suggestseveralclueswithwhichwemaywork.Whatwasthecharacteroftheperiodbeforetime?Perhapsitwasanerainwhichnumberwas theonly reality apart from theCosmicSoul itself.PerhapstheCosmicSoulimaginedauniversefilledwithtrillionsoflife-formsandthencontemplatedwhatmustbeinplaceforthistohappen?Weretherecreatorgodssuch as the Aboriginal ancestral spirits whoworked in collaborationwith theCosmicSoul?Atthispointwemustturntotheimaginationtotakeustoregionsthatarebeyondourrationalunderstanding.

Yet, there arenumerical clueswithin theMayancalendar thatmightguideus.Wehave,inchapter7(anddrawing10.9),describedsomeaspectsoftheminor-thirdratioof5/6andhowitcanbeusedtotransformπintoϕ:

ϕ=√5/6π.

The value π is the most ancient relation. It arose with the circle, the primalPlatonicform.Inorder,attheverybeginning,toinfusetheintentionoflifeinto

the cosmos itwas necessary that π be transformed toϕ, the ratio that shapeslivingforms.Aratio,atone,withintheeternalscalewasemployedtobringthisabout. When the minor third was sounded on the 16.4 BY musical string ofcosmic time in the presence of the seed, the intention for a livingUniverse—whichhadbeensownduringdayoneandwhichlaydormantduringnightone—this seed germinated, and the living Universe radiated into being during themorningofdaytwoat13.66BYA:

5/6×16.4BYA=13.66BYA.

Thedatealsohappenstoresonatewiththedateofthebeginningoforganiclifeat 3.66BYA as described above, the two dates being exactly 10 billion yearsapart. This 10-billion-year period poised between material creation and thearising of life firmly establishes an imperishable decimal standardwithwhichwecanmakesenseofthenumericalharmoniesintimeandspace.TheleftoverperiodintheMayancalendarbeforetheExpansionis2.73BY,anumberthatweexplore here.We have also hypothesized a musical aspect of the calendar asdemonstratedindrawing10.9.

Drawing10.9.Ifweconsiderthedurationofthe16.4BYMayancalendarasamusicalstring,thenwediscoverthattherearecertaintonesassociatedwiththemainsectionsofthecalendar.Wefirstplucktheunstoppedstringoflength16.4andsoundthefundamentaltone.Thenexttonetosoundisthewholetonecalledthemeantonesituatedat .8944ofthestringlength.Thenumber.8944isatthegeometricmeanbetween themajorwhole tone(8/9)and theminorwhole tone(9/10).Themeantonehasimportantproperties.Itisrelatedto√5,since

1÷.8944=1.118and1.118=√5/2.

The .8944 meantone is also the square root of 4/5, the major-third intervalwhosefourthoctave,1/20,setstherhythmfortheunfoldingoftheunderworlds.

The 1/6 interval, second octave of 2/3, the fifth, defines the duration of theDreamtime.

Thenextintervaltoconsideristheminorthirdwithstringlengthof.8333(5/6).The minor third is the tone through which π is transformed to ϕ. Thus(5/6×π)=ϕ2.

In our diagram it is located on the string at 13.66 BYA, the time of thePrimordialExpansion.

The next interval has a string length of .2236, the second octave of themeantone,which locates thebeginningofbiological life.The frequencyof thisintervalis4.472=2×√5.Asdemonstratedindrawing10.3,thisintervalcanbeused to locate the beginnings of each underworld as well as the floweringperiodswithinunderworlds.

Thefundamental(1)establishestheFieldofCreationwhosedurationis16.4BY.

The meantone, .8944, establishes the potential for √5 and ϕ and thethird.

Thefifth,indouble-octaveformof1/6,establishestheseedingtime,theDreamtime.

Theminor-thirdtone5/6,whichtransformsπintoϕ,whenappliedtothe

calendar as awhole,would signal the expansion, the beginning ofuniversalcreation:5/6×16.4BY=13.66BYA.

The next tone, 4/5, the third, would, in octave form, determine therhythmof theevolutionaryperiods, theunderworlds.And4/5also,in its vesica construction, generates the blueprint of water, theelement of life (see drawing 10.4). The duration of each heaven,1.26BY,isalsoafrequencynumberofthethird.

There may have been others who dwelled with the Cosmic Soul and whoassistedintheworkofdreamingauniverseintobeing,dreamingeachatomandcellandeachconstantofnature—whotunedtheseedsothatthestrongandweakforcewouldseparateattherightinfinitesimalmomentoftime,andthatprotonswouldformintheten-thousandthofasecondaftertheExpansion.Andthegreatcycles must also be tuned so that the trillions of beings necessary for thefulfillmentoftheDreamoftheCosmicMindwouldevolveandacttheirpartsinthe play whose climax would be when the created ones themselves becomecreators.

THETREEOFTIME

Callemannotestherecentdiscoverybycosmologistsofavaststructurethathecalls theCentralAxis of theUniverse.He then demonstrates that thisCentralAxismaywellbewhatmanycultureshavecalled theTreeofLife, thecentralregulatingphysicalandspiritualpresencefromwhichCreationemanates.IntheMayan calendarwe can also perceive the form of a tree completewith roots,trunk, and branches. It is a Tree of Time composed of numerical harmonies,thosePlatonicelementsfromwhichspringtheeternalsacredgeometrythatisthefoundation of space and time. Many of the ratios that we have studiedthroughout this book can be applied to the structure of the Tree of Time, ofwhichwewillhereattemptaroughsketch.

Drawing10.10.Aconstructionshowing,atthetop,thebranchesoftheTreeofTimethat,alongwiththeall-inclusiveCellularUnderworld,liststhenextthreeoftheremainingeightunderworlds.Insteadofthefactorof20thatistheactualmultiplier from one underworld to the next, we have here used a factor of 2betweenthebranchings.(WenotetheimpossibilityofrepresentinganaccurateTree of Time with its successive branchings in 1:20 ratio. If a branch in thefinal, or universal, underworld were one foot in length then the height of thewhole tree would be several million miles!) The present drawing is asimplification togetasenseof theevolutionarydynamicsof thecalendar.Thedual branching after the establishment of root and trunk recalls JosephCampbell’scommentonthestructureofcreationmyths:“ThefirsteffectofthecosmogonicemanationsistheframingoftheworldstageofSpace;thesecondisthe production of Life within the frame: life polarized for self-reproductionunderthedualformofthemaleandfemale.”3

Themiddlestructureisthetrunkofthetree,aperiodof10billionyearsduringwhich matter was organized into galaxies, solar systems, and planets inpreparationfortheevolutionoforganiclife.Atbottomistherootofthetreeoftime,anerathatwehavecalledtheDreamtime,referencingthelegendarytimebefore creationof theAustralianAborigines. Stayingwith our plantmetaphorwecan imagine thisasa timeof seeding.Whatwere the seeds?Perhaps theywereharmonicnumbersorheavenlymusicoradreamofalivingUniverse.

ItreflectswellontheimaginationoftheMayancalendarmakerswhoperceivedaCreationwhoseevolutionaryprocessescorrespond to the lifecycles its chiefliving inhabitants, theplants.Theplantmetaphors thatwesee in the“PeriodicSystem of Evolution” table (figure 10A) give credence to the larger plantmetaphorofthetree.DuringdayoneoftheMayancalendartheseedoftheTreeof Timewas planted. This period is the foundation time characterized by theintervalofthefifth(drawing10.9),whosestring-lengthratioequals1/6:

1/6×16.4BY=2.73BY.

On day two it germinated and on day three it sprouted creating the infantuniverse.Thedateis13.66BYA.TenbillionyearspassduringwhichtimestarsarebornanddieinnovasthatspreadessentialelementsthroughouttheUniverse.Newstars,thengalaxies,andfinallysolarsystemsandplanetsareformed.The

firstorganiclife-formsonEartharise3.66BYA.TheDreamtimewithdurationof2.73billionyearsistherootofthe16.4BYhighTreeofTime.The10billionyears that follow is the trunk. The remaining 3.66-billion-year period ofbiologicalevolutionformsthebranchingofthetree(seedrawing10.10).

WITHINTHEROOTOFTIME

TherootoftheTreeofTimemustbe,ifthemetaphorhasvalidity,asourceofnumericalknowledgefromwhichwecanpredictthemaintemporalpatternsofcreation. Sincemany of the numbers under consideration are based on Earth-Moonsystemmeasuresthereisanassumptionthatisrequired.Wemustassumethat the Earth-Moon system is a repository of numerical proportions andmeasures that describe not just our own system but the Universe as a whole.There is no scientific proof of this, yet this idea resonateswith some esotericsources thathold thatEarth is a libraryof cosmicknowledge.Perhaps it is anidealtypeofplanet,amodel,fortheemergenceandflourishingoflife.

We have determined that the duration of the root is 2.73BY,whichwe havesubdivided into three sections, two of 1 BY and one of .73 BY (seedrawing10.8).First,workingwiththe2.73number:

“Hereisoneofthemostimportanttasksofharmonicstudy:toprovideasensiblegraspoftemporalforms,especiallyperiodicities.Asafoundationforthis,harmonic‘chronometry’or‘formmathematics’—analogoustoharmonicgeometry—mustbeestablishedorre-created;thetime-formsmusthavetheirowndiscipline,analogoustothatofthespace-forms.Theestablishmentofpsychicvaluesinthenumbersofthetime-formswillnotonlyallowthesenumberstobeunderstoodinadeepersense,butwillalsoplacetheminadomainofourcognitionthatiscloselyconnectedwithtime:temporaltoneperception.”

HANSKAYSER4

2.73=1÷.366,and366days=numberofEarthrevolutionsinoneyear3.66=Earth/Moondiameterratio

3.66BYALifebeginsonEarth

13.66BYAPrimordialExpansion(bigbang)2.73 − 1 =within 99 percent of 1.732, the √3, the ratio of the vesica

piscis. The vesica piscis generates the regular polygons aswell astheFlowerofLife,symboloftheFieldofCreation.

73 × 2 × 2 × 2 × 2 = 11.68, the time, in billions of years, from thebeginning to the formation of Earth at 4.72BYA: 11.68 + 4.72 =16.4.

.73 ÷ 16.4 = .0445, the ratio of the fractional part of the root to thecalendar period. This is in octave relation to the “Mayan tritone,”discussedinchapter2:

.0445×2×2×2×2= .7122=52/73,whichdetermines theMayanritualperiod,thetzolkin:52/73×365=260days;and

52/73happenstoequaltheArabictritoneof729/1024:729/1,024÷52/73=.999.

When thenumerator anddenominator of the ratio729/1,024 are factoredwiththe numbers 2 and 3, the result produces all the numbers necessary forgeneratingthePythagoreanscale.

Thus729/3=243;243/3=81;81/3=27;27/3=9;9/3=3;and1,024/2=512;512/2=256;256/2=128;128/2=64;64/2=32;32/2=16;16/2=8;8/2=4;

4/2=2;2/2=1givingus

thenumbersrequiredforbuildingthePythagoreanscale:

unison:1/1minorsecond243/256majorsecond:8/9minorthird:27/32majorthird:64/81fourth:3/4tritone:512/729

fourth:3/4tritone:512/729fifth:2/3minorsixth:81/128majorsixth:16/27minorseventh:9/16majorseventh:128/243octave:1/2

For the Pythagoreans, number, and particularly musical numbers, form thefoundation of the cosmos. Here we see an intimation of this idea in thenumericalcontentsofourmetaphoricalRootof theTreeofTime.Wementionthat the spiral of the Pythagorean tones,whichwe constructed in drawing 6.6usingthegeneratingratioof13/18(atritone),relatestotheMayantimesystemsince13/18equals260/360,thetzolkindividedbythetun.13/18,aspreviouslymentioned,istheinverseofourϕ-tritoneratioof9/13.

Thenumber73itself,aswehaveseeninchapter2,hasinterestingproperties.

73days×5=365days73÷29.53(lunation)=2.472(4/ϕ)73 × 29.53 = 2,156, within 99 percent of lunar diameter of 2,160miles72.6=7,920(Earthdiameter)÷109(Sun/Earthratio)73.3=7,920(Earthdiameter)÷1,080(Moonradius)

Andthenumber73,aswehavenoted,isthedenominatorofthetzolkin-toyeartritone ratio of 52/73 (in musical terms the most dissonant interval). Whenapplied to ayear’sduration the resulting260daysmarks thebeginningof thetime of travail for the pregnant mother, being the minimum period fromconceptiontobirth.Letusnowapplythetzolkin52/73tritoneratiototheMayanLongCountcalendar,aperiodof5,125yearsthatbeganin3114BCEandendedin2011or2012:

52/73×5,125years=3,650years,thetimefromthebeginningoftheLongCounttoitstonalcenter.

SubtractingtheLongCountstartingdatefrom3,650givesthetonalcenter:

3,650–3,114=536CE.

TheLongCountthusdividesintotwodurationswithtemporalresonances:

5,125years=3,650years+1,475years,and3,650yearsreferencesthe365-dayyear,and

1,475yearsreferencesthehalf-lunationcycleof14.75days.

Theirratioequalsthe4/ϕratiooftheApollotemple:

3,650÷1,475=2.474,within99percentof2.472(4/ϕ).

Anadditionalinterestingsynchronicity:

3,114−1,475=1,639,whichreferencesthe16.38BYlengthoftheMayancalendar.

Theyear536CE,thetonalcenteroftheMayanLongCountcalendar,embodiedthe dissonance of both the tritone interval and the influence of this period’srulingdeity,Tezcatlipoca(theAztecnamefor the lordofdarkness, theMayanname having been lost). It was a time of worldwide catastrophe generated, itseems,byasuper-volcanothatdispersedadensecloudofasharoundtheglobe,darkening the Sun and causingmany seasons of frigidweather, drought, cropfailures, and famine and contributing to political and economic collapsethroughout theworld. In theMayanworld the catastrophemay have been thecauseofthedeclineandeventualabandonmentofTeotihuacan,thepoliticalandculturalcenteroftheempire.Theresultsoftheeruption,ifthatwasindeedthecauseofthechaos,wasfeltacutelyforanumberofyears.FortheMayaitledtoapoliticalrealignmentoftheircity-statesandagradualculturalflowering.Thebook Catastrophe by David Keys describes in detail the global effectsexperienced as a result of this event that is so precisely pinpointed when thetzolkin ratio is applied to the Long Count calendar. In the Mayan world,accordingtothescholarofMayansculptureTatianaProskouirakoff, therewas,after534andcontinuingto593“adarkperiodinthehistoryofMayasculpture.”

Shecontinues“. . . thechangedartisticmodeseemstobeareflectionofsomemomentoushistorical event thatdisturbed thenormal artisticdevelopment . . .andwasfollowedbyarestorationoforderandanewpulseofcreativeactivity... . It may be tentatively assumed that the historical events which acted asincentivetochangeaffectedallaspectsofculturesimultaneously....”5

When the 52/73Mayan tritone ratio is applied toCalleman’sMayan PeriodicSystemofEvolutionthefollowingdataemerges:

52/73×16.4BY=11.682BY.

Takenfromthebeginningthisgivesadate:

16.4BY−11.682BY=4.718BYA,

whichisthemodernestimatefortheformationofthesolarsystemandofEarth.Like the yearly and Long Count tzolkins, the time ratios are in 2.472 (4/ϕ)relation:

260days÷105days=4/ϕ,3650years÷1,475years=4/ϕ,and11.682BY÷4.718BY=(4/ϕ).

And,alongwiththetzolkin-indicatedmaternallaborcrisisfollowedbythebirthof a child, they allmark a periodof travail followedbynew life: the536CEeruptionfollowedbyaMayanculturalrenewal.

The4.718BYAperiodwhenEarthwasinastateofcrisis,sufferingbirththroesin the form of constant meteorite bombardment, a toxic atmosphere, and achaoticorbit. InMayantermsthisperiodwasalsoruledbyTezcatlipoca,who,through destruction, prepares the way for a flowering. With the help of thenewly bornMoon,who helped stabilize her orbit andwith the preciouswaterbrought by the bombarding meteorites and comets, Earth gradually became asuitable vessel for life as itmoved into the flowering time, the time ruled byYohualticitl, goddess of birth, and around 3.66 BYA brought forth the firstsingle-celledlife-forms.

11

THECOSMICFLOWER

THE SUN GENERATES ENERGY AT ITS HIDDEN CORE andmanifests it as light thatstreams forth from its blazing surface.This is theuniversal radiant gesture. Inlike manner each being has its own expression of the energy at its core: theslitheringofthesnake,thedelicatewalkofthedeer,theswoopingdanceoftheswallow,theripplingofwheatinthewind,thesparklinggeometryofthecrystal.Theuniquesignatureofeachtypeofbeinghasitssourceinthespeciessoul,thedevic energy that animates it. Their bodily expressions are their ways ofradiating,orunfolding,theirlifeenergiesasabudunfoldsintoaflower.Itmaybesaid thatallEarth life-formsknowinglyorunknowinglypayhomage to theSunintheirradiantunfoldings; it is intheflower,however,wherethis ismostclear.Andthepolygonalshapesofmanyflowersthatunfoldfromvesicalbudscanserveasgeometricmetaphorsfortheunfoldingoftimeandlife.

Figure11A.TheradiantgestureoftheSunismirroredintheformsoflifethatexpandoutfromacentertoabodilyperipheryandevenbeyondtospheresofsoundandfragranceandawareness.Beesonaglobe

thistle,Echinopsritro.

THESYMBOLINTHESKY

Forourunderstandingoftheprinciplesofcreationtotranscendmentalconceptsandlanguagewerequiresymbolicimagerythatopenspathwaystodeepermodesofknowing,withfeelingandwith intuition.This iswhat thegeometricartcanprovide tous.Agreat symbol forms in the skywhen thenewmoonpasses infront of the Sun—the solar eclipse. The equality of their apparent sizes thenbecomesobvious—asynchronicitybeyondallrandomcoincidencethatnaturallydemands our attention. Like all symbols there are several or manyinterpretations.

Figure11B.ThenewmoonintheprocessofeclipsingtheSun.

Thetwo—thebrightmasculineSunandthedarkfeminineMoon—becomeone.Then the separation begins as the Moon continues to move across the Sun’ssurface. The image is one of scission similar to the movement between thedaughter somatic cells during mitosis. This is a gesture of creation, of Unitydividingintwo.Asgeometersweexaminethesymbolinitsdetails.WereduceSunandMoontosimplecirclesandwitnessindrawing11.1theseriesofvesicasformedbythetonalnodesthattheedgeoftheMoonsilentlysoundsasitmovesalongthemusicalstringoftheSun’sdiameter.Theyinandtheyangjoinandareparted again, and during this process much has happened. The “ten-thousandthings” have arisen in the form of the innumerable vesica shapes with theirassociated ratios and their tones that have symbolically sounded. Before thepartingofthecircles,however,wenoteanimportantpositionintheprogression:when the circumference of the Moon encounters the center of the Sun, and,conversely, thecircumferenceof theSun touches thecenterof theMoon, theyformthevesicaknownasthevesicapiscis.

Drawing11.1.AsymbolicmeaningofthegeometryoftheeclipseoftheSunbytheMoon.

At totality darkness completely dominates, a surprising and shockingdevelopmenttotheancientswhoperceivedheavenlymessagesandwarningsinthesky.Darknesscouldbeinterpretedastheeradicationofconsciousnessoroflife.SucheventsarepossibleandhavenearlyhappenedduringthecataclysmichistoryoftheEarthfilledwithmeteoriteandcometstrikes,massextinctionsandpost-glacial floods.Yet themessage isnotnecessarilysodire.Thedarkness isnotevil.Itisapartofwhoweare.Theeclipseremindsustoconsiderthedarkhiddenpartsofourselvesandtointegratethemintoourconsciousness,ourlight.

Drawing11.2.Theprimalunfolding.Part(1)isagesturaldrawingshowingtheunfolding,orflowering,thatisabstractedfromtheclassicalconstructionin(2)inwhich thisdualunfoldingsymbol isplacedwithin thematrixof intersectingcircles representing theAkashicField, a section of theFlower of Life symbol(seedrawing4.7).Inthiscontextwecanconstructthepolygons(here:triangle,square,pentagon,hexagon)usingtheinherentpointsandlinesofthefigure.Thesidesofthecentraltriangleconnectingequatorendsandpolesofthevesicacanbe seen to be folding out and around the vesica-forming circles like a flowerblooming from the vesica bud. Next the sides of the square unfold. They are

verticalsdrawnperpendicular to thebaseline that intersect thevesica-formingcircle and are then connected with a horizontal line. Next, the crossed linesintersecting points on the field as shown are extended to the vesica-formingcirclestoestablishpentagonverticesfromwhichtwosidesofthepentagoncanbedrawn.Withoursidelengthsknownwecan,withcompass,crossarcsontheextensionsoftheverticalaxistoestablishthepointwithwhichwecancompletethepentagon.Forthehexagontherightandleftfield-vesicapolesprovidepointstowhich the lowersidescanbedrawn.Thepointswhere theextensionsof thesidesoftriangleandsquareintersectprovidepointstowhichtheuppersidesofthe hexagon can be drawn. The ends are connected, and the hexagon iscomplete.

The vesica piscis is the figure from which polygonal geometryunfoldsoutwardfromthetriangularinnerstructureofthehalfvesica,like a flower unfolding from a bud. Drawing 11.2 illustrates thegeometry that gives rise to this opening gesture. In placing thisgeometrywithinamatrixofvesicapiscesweareable to locate theconstruction points from which many regular polygons can begenerated.

SACREDCOSMOLOGY

“As above, so below.” This is the ancient esoteric guide to those seeking tounderstandtheworkingsoftheworld.Theseekerwhoacceptsthisdictumlooksfor“theworldinagrainofsand”—forcluestothewholecontainedwithinthepart, for eternity within the moment. The stillness that we can connect withinsideourselves, like thequantumvacuumofdeepspace,contains tremendouspower.Thelightwithinisourinnersunthatwearedrawntoasasunflowerisdrawn to theSun in the sky.The soundof thewind in the trees is continuouswith the inner beehive hum of our neurons. The outer reflects the inner. Themacrocosmisreciprocaltothemicrocosm.

Let us now create a symbolic geometric map of cosmic evolution with theassistance of the powerful tools of geometry, number, and the musical scale.Theseancienttoolshavenotbeendisplacedbymodernscienceandtechnology.Infact,theyhavebeenenhancedbythem.Onapracticallevelcomputerdrawingprogramshaveremovedmuchofthedrudgeryofcalculationandtheattainment

ofaccuracywhilestillallowingaclassicalapproachtoconstruction.Onadeeperlevel the resurgence of the harmonic vision of the world owes much to thediscoveries ofmodern science that havemade available accuratemeasures oftime and space, which enhance the accuracy of harmonic symbology.Harmonics,inturn,hasbroughtabreathoffreshairtoasciencethatisinneedof simplification so that the ordinary person can partake of our commonbirthright:someunderstandingofourlivingUniverse.

Musical geometry is able to accomplish this by creating symbols that awakenpartsofus thatpurelymentalconstructscannot reach.Theseare the realmsofthe sacred thatwere, untilmodern times, theprovincenotonlyofmysticsbutalsoofthosewhostudiedthenaturalworld.Thisstudywascalledsacredsciencebecauseitdidnotlimititselftothematerialworldbutenteredtherealmsofsoulandspiritthatwerealsoconsideredpartsofthenaturalworldandthusworthyofstudy.

In thepreviouschapterwepresentedapossibleharmonic time linebehind theevolutionof theUniverse as portrayed in theMayan calendar. It is interestingthat the period before Creation, which we have called the Dreamtime, had adurationof2.73BY,whichis1/6thedurationofthewholecalendar(16.4BY)and1/5 the duration of the period (13.66BY) since thePrimordialExpansion(bigbang).These ratios suggest that theremightbehexagonal andpentagonalproperties in universal creation. Both polygonal structures (see drawing 11.2)unfoldfromthevesicapiscis,whichitselfistheresultofthescissionofcircularunity into two circles. Drawing 11.2 also reveals these circles to be part of amatrix of intersecting circles whose finished symbol is the Flower of Lifeillustrated in drawing 4.7. As we have noted, this symbol can represent theuniversalenergyfieldknownasthequantumvacuum,or,asLaszlocallsit,theAkashicField,aseaofcolossalenergyandinformationfromwhichCreationisbelieved to emanate. In our present context we can imagine the polygonalstructuresoftimeandlifearisingfromthisfield.

Drawing11.3.Fromtheuniversalfield.shownontheleft,thetriangleemerges.ThisisPlato'sdemiurge,theCreativeSpiritwhotakesonthetaskofcreation.Asthe sides of the triangle unfold outward in an opening gesture (center), thesquareisgenerated.Thisisthesymbolofconsciousness,theawarenessofwhatneedstobeaccomplishedbeforethemainpurposeathand,thecreationoflife,can unfold.When the ground is prepared the pentagon (right), symbol of life,unfolds from the square. The construction lines that form the symbols alsoproducetheoutlineofawaterdrop,below,thesubstancethatisessentialtotheevolutionoflife.

Ourconstructionofthepolygonsthatblossomforthfromthebudofthevesicapiscishavethefollowingpropertiesandinterpretations.

(1)Thegenerating form is the triangle, theunderlying structureof thevesica piscis, which unfolds from the heart of the vesica bud andsweeps out arcs in both directions locating the construction pointsfor square, pentagon, and hexagon. Musically, the number 3 isrepresentedbytheintervalofthefifth.

Figure11C.Naturalhexagons frommicrocosm tomacrocosm: (a)micrographofcrosssectionoffibersinafly’sflightmuscles;(b)cellsinabeehive;(c)thegreathexagonalcloudstructureatthenorthpoleoftheplanetSaturn

(2) The second regular polygon to be constructed is the square, thesymbol of reason and consciousness, the organizing principlesbehindCreation.Thesquare isalso the traditionalsymbolofEarthandprovidesaunifyingsymbolofourfourfoldperceptionsofspaceandtime:thefourdirections,thefourseasons,thefourphasesoftheMoon, and the four daily positions of the Sun—dawn, noon,sundownandmidnight.Thefourfoldsquarealsorepresentsthefourorganizing forces of nature: gravity, electromagnetism, the strongandweaknuclearforces; thefourlettersoftheDNAcode;andthefourmainsectorsof thehumanbrain—reptilian, limbic,neocortex,and prefrontal lobes. The inner structure of the square is amatrixthat can generate innumerable geometric constructions. We haveseenthroughoutthebookafewexamplesofthisintheintersectionsoftwoofthemanykindsofdiagonalsthatcrisscrossthesquareandwhose intersectionpoints,whenprojected to the string/diameteroftheenclosedcircle,giveus the tonalnodes fromwhich toproducevesicaconstructions.Musically,4referstotheoctave.

(3)Thenextpolygonisthepentagon,theembodimentofthenumber5,the number of themajor-third interval,which, as noted in chapter10,playsanimportantroleintheMayancalendarandisassociatedwith thesubstanceofwater.The inner structureof thepentagon isdominatedbyϕ-familyratiosthatareinvolvedwithbiologicallife-

forms. (See appendix drawing A2.16 for a demonstration of thisproperty.)

(4)Thenextpolygongeneratedbythetrianglewithinthevesicabudisthehexagon,symbolofprimalmatter,oftheatomicandcrystallinerealms thatmake upmaterial creation. In our time analysis of theMayancalendarthehexagondescribestheentirecalendar,includingthepre-CreationDreamtime.The inner structure of the hexagon isfilled with relations based on √3 (1.732), which is the height-to-width ratio of the vesica piscis (see appendix drawing A2.17).Musically,thenumber6,likethenumber3,relatestotheintervalofthefifth.

“Thesymbolawakenspresentiment;languagecanonlyexplain.Thesymbolplucksallthestringsofthehumanspiritsimultaneously;languageisalwaysobligedtodevoteitselftoasinglethought.Thesymbol’srootsreachintothemostsecretdepthsofthesoul;languageonlytouchesthesurfaceofunderstandinglikeaslightbreathofwind.Oneisdirectedinward,theotheroutward.Onlythesymbolcanconnectthemostvariedthingsintoaunifiedcollectiveexpression.”

J.J.BACHOFEN1

Drawing 11.4. A hexagonal time structure representing the evolution of theUniverse from thepre-Creationperiodwehavecalled theDreamtime throughthePrimordialExpansion(bigbang)untilthepresent,adurationof16.4billionyears,theproposedtimespanoftheMayancalendar.Thesidesofthehexagonare2.73billionyears induration,which isalso the lengthof thepre-CreationDreamtime (bottom) period that establishes the rhythm of the calendar. Theprogressionproceedsclockwisein2.73BYincrementsandgivesaroughideaofthetimingofevents,likestarformation,whichtookplaceoverbillionsofyears.Thediagramismeanttogiveageneralpictureofevolutionarytime.Theouterheptagon showing the stages of plant growth and their correspondences with

Creationandevolution ismore in the spiritof theMayancalendarand in thespiritofthisbook,whichseekscorrespondencesbetweenaboveandbelow,andhere,betweenthegrowthoftheUniverseandthegrowthofaplant.

Figure11D.Thenumber5manifestsofteninlife-forms:(a)starfish,(b)flower,(c)applecrosssection,(d)hand,(e)blackberryleaves.

We now return to the generating equilateral triangle at the heart of theconstruction. It is intimately related to the hexagon, being a sixth part of itsstructure.Inthedrawingitoccupiesthelowerpartofthehexagon,theareathatwehavereferredtoastheDreamtime,the2.73BYperiodbeforeCreation.Theposition and function of this triangle recall a feature of theGreat Pyramid inwhichthebedrockofthebaseareawasmostlyleveledexceptforasectionthatwas allowed to protrude into part of the lower section of the pyramid. Thisprovided a stabilizing anchor for the structure as well as a connection to theEarthanditsenergies.AsimilarfeatureisfoundintheDomeoftheRock,thetemple in the Old City of Jerusalem, in which the rough, exposed bedrockoccupies much of the floor of the temple. In a similar way the triangle indrawing11.3representstheCreativeSpiritemanatingfromtheDreamtime,theeternalmatrixofexistencefromwhichitgeneratesconsciousness,life,andtime

continuestoinformevolutionasauniversalcollectiveunconscious.

Figure11E.Sixpolygonaldiatomsembodyingthenumbers3to8.Fromleft,(a)Triceratium, (b) Perissonoe, (c) Actinoptychus, (d) Aulacodiscus, (e)Asterolampra,(f)Asterionella.

“NoneofthehundredsofAboriginallanguagescontainawordfortime,nordotheAborigineshaveaconceptoftime.Aswithcreation,theAboriginesconceivethepassageoftimeandhistorynotasamovementfrompasttofuturebutasapassagefromasubjectivestatetoanobjectiveexpression.ThefirststepinenteringintotheAboriginalworldistoabandontheconventionalabstractionoftimeandreplaceitwiththemovementofconsciousnessfromdreamtorealityasamodelthatdescribestheuniversalactivityofcreation.”

ROBERTLAWLOR2

ThepropertyofthreenessinherentinthisperiodrelatestomanyoftheCreationmythsofhumanity thathavea triunecreatororacreator,preserver,destroyer,representingthemostuniversaltrinityofexistence—birth,life,anddeath—that

are intertwinedwith the three dimensions of space and the three properties oftime:past,present,andfuture.

THESYMBOLSPEAKS

Ournestedpolygonsmight be interpreted as follows:The field of intersectingwaves, theAkashicField,produces thevesicapisciswhentwowaves intersecteach other’s center. The inner structure of the half vesica is the equilateraltriangle,theCreativeSpirit,whoassumesthetaskofcreation.Itsfirstcreationisconsciousnessorpsyche,representedbythesquarethatmediatesbetweenspiritandlife.

Lifeisthenextemanationandisrepresentedbythepentagon,theembodimentof thegoldenratioandof thefivefoldpatternsfoundthroughout thebiologicalworld, themore advanced realmof life.The next emanation is time,which isrepresentedbythehexagon.Perhapstimeservesasacontainerandregulatorofevolutionoratoolthatlifeusestoperfectitself.Ortimemightbecomparedtoashell that contains a sea creature, life, which precedes time as the pentagonprecedesthehexagoninthegeometricalsymbol.

It seems that while life appears contained in time it is, in some ways,independentoftime.Thus,“timeless”momentscanoccurspontaneouslyinourlives.Orwecanchoose to enter into timeless statesofmind.Thedreamstateand deep meditation are portals into these timeless regions. The trianglerepresentssuchaplace.Itistheplaceofdreamingwhereexistencewasdreamedinto being. According to the symbol, it is embedded within the ring of time,consciousness, and life, as night dreams are embeddedwithin our daily lives.This place is a part of us and thus is forever accessible to the seeker, to thedreamer.

CEREMONY

We have seen thatwith a foundation of sacred number and geometrywe canenter confidently into regions that are inaccessible to ordinary reason. Thetraditionalpurposeofthesetoolsisthecreationofsymbolsthatinvokespiritualenergiesforpersonalandsocialbenefit.Apracticalapplicationofthispurposeis

the building of temples proportioned by cosmic principles. George MichellwritesofthephilosophybehindthebuildingofHindutemples:

Drawing11.5.Awheel of tones andpolygons,a construction, generated fromdrawing 11.2, in which the polygonal angles are framed in terms of tonalintervals.Thesemicirclewithangleof180°isconsideredtobeamusicalstringequaltoanopenstringthatsoundsthefundamentaltone.Thedifferenceisthatthismusicalstringiscurvedinsteadofstraight.Thelefthalfsoundstherealmofharmonicsfrom1/∞to1/2.Therighthalfsoundsthetonesofthemainoctave.At the 1/3 point of the semicircle is our first regular polygon, the equilateraltriangle,whosebaseangleof60°is1/3ofthefundamentalangleof180°,whichdesignates itasanoctavefifth.Thesemicircularstring(a logical impossibilityin thephysicalworldbutaconceptual tool in the realmofharmonics)canbecomparedtoacurvedguitarstring.Whenyoustopitat1/3itslengthandpluckit,asyouwouldwithaguitar,youwillsoundanoctavefifth.The1/3lengthinthepresentconceptionnotonlysoundsanoctavefifthbut,beingpartofacircle,also creates an angle. Next, at 90°, the octave, the next regular polygon, thesquare,isgenerated.Thepentagonwithits108°angle(108°/180°=3/5)isthemajor-sixth ratio. At the fifth ratio, at 2/3 of the perimeter, the hexagon is

formedfromthe120°angle.Wehavealsoincludedseveralothertonesthatarisefrom the angles of regular polygons: octagon, the fourth; decagon, theminorthird;andan18-gon,whichrepresentsthemajorwholetonewithratioof8/9.Betweenthe8/9toneat160°andthefundamentalat180°areinnumerabletonesandtheirassociatedpolygonswhoseperimetersgraduallyapproachtheperfecthalf-circle of the fundamental. (The number of sides of each superparticular-ratio-derived [superparticular ratios are those with the numerator anddenominatorseparatedby1]polygonistwicethenumberofthedenominatorofthetonalratio.)

At the other end of the fundamental semicircle we see the realm of theharmonics.Theseincreasinglysmallerintervalsrepresentthehigherandhigherfrequenciesoftheharmonicseries.Nearthelimit,whichisataninfinitelyhighvibration,arethevibrationsofthequantumworld.

Only if the temple is constructed correctly according to amathematicalsystemcanitbeexpectedtofunctioninharmonywiththemathematicalbasisof theuniverse.The inverseof thisbelief isalso held; an architectural text, the Mayamata, adds “if themeasurementofthetempleisineverywayperfect,thattherewillbeperfection in the universe as well.” Thus the welfare of thecommunity and the happiness of its members depend upon thecorrectlyproportioned temple, and the architectural texts stress thatonly work that is completed “according to the rules” will gain thedesiredmeritforitsbuilder.3

Figure 11F. Itzhak Bentov wrote, “The seed carries not only the informationabout the shapeof the treebutalso its unfoldment in time. . . . The seed is aunique structure because in it space-time has been condensed and stored,awaitingtheproperobjectivetimeforitsunfoldment....Itistherepresentationofthetreeinanalteredandhigherstateofconsciousness.”4

The seed contains the plant in a highly compressed form, demonstrating thereciprocalrelation:thelargerthequantity,thesmallerthereciprocal.Thepointis the seed of geometry. The geometric point contains the whole of geometrywithinitselfasaseedcontainstheplantwithinitself.Thepointradiatesoutinall directions creating a sphere in which all geometric figures can beconstructed.Thisprinciplegaverisetothemoderncreationmythofthebigbanginwhichaninfinitesimalpointof infinitedensityexpandsoutandbecomestheUniverse.

And, if we follow Bentov’s insight, the singularity/seed represents the higherconsciousnessfromwhichtheUniverseemanates.Thisseedgestateswithintheseedingperiod,theDreamtime.Inourpreviousdrawingsitisrepresentedbythetriangle,thesymbolofSpirit.Above,pumpkinseeds.

The beliefs that generated temple building had their origins in prehistoricshamanic cultures inwhich ceremonial practiceswere developed for the samepurpose:toinvokehealingenergies.Likethetemplebuilders,theshamans’firstprioritywastoestablishsacredspacewithinwhichceremoniescouldunfold.Acommon practice was (and is) the calling in of the energies of the fourdirections, thereby creating a symbolic square. Within this square a circularritualareamightbescribedontheground,asinsomevision-questceremonies,or built, as with the round sweat lodge, or suggested, as with a central firearoundwhich ceremonialists gather or dance. In such sanctified ringswemayperceive the primordial symbol of the circle within the square (mentioned inchapter1)representingSpiritheldwithinthevesselofmatter.Insuchspacestheprayersof theceremonialistsare infusedwithpower.The templearchitectandtheshamanthusshareacommonpurposeintheirwork—thereestablishmentofharmonyintheworld.

Figure11G.

ThusopenouttheTaoanditenvelopsallofspace

anditenvelopsallofspaceandyethowsmallitisnotenoughtofillthehand!

HUAINANTZU

APPENDIX1

THEVESSELOFCEREMONY

ORDINARYWAKING EXISTENCE IS, FORMANYOF US, a fairly unconscious journeythroughthedaysofourunexaminedlives.Moreandmorepeople,itseems,havedecided that thisway of life is unacceptable and that theremust be away todeepenourexperienceofthisamazingopportunitywehaveasparticipantsinthelivinguniverse.Onewaythathumanshavetraditionallyaddressedthis issueistoparticipateinsacredceremonies—toselectaprecisepieceoftimeandcraftitintoameaningfulandbeautifulworkofart—thatwillrestoremeaningtolifebyinfusingitwiththesacred.

Wehave, in thisbook,givena fewexamplesofhowmusical ratios andothermeaningfulnumbersandgeometryareembedded incosmic structure.There isan ordered harmony that pervades the cosmos, a cosmos that seems to beconsciousandwhosepurposeseemstobetheflourishingoflife.Whenwelookatnaturalformsitishardtoimaginetheswalloworthebearorthesharkortheredwood wanting anything other than a long and flourishing life. They arecompletely pro-life. This alignmentwith life’s purpose so natural for animatebeings can be a challenge for humans, particularly for those living incivilizations.

*

Wemustrememberthat inourUniverse, inwhichtheflourishingof life is thepriority, help is available for the asking. At some unknown period humansrealizedthevalueofrequestinghelpfromSpiritinaformalway.Thepowerofasincere prayer, which often arises spontaneously in times of travail, had longbeennotedbyhumans for its efficacy in transformingwhatever difficult issue

has presented itself by injecting a mysterious energy, which humankind hascalled“grace,”intothesituation.Aftersuchexperiencesitwasnaturaltostudythematterforfuturereferencewheninevitablyasimilarsituationwouldarise—asituation requiring grace-full help from the unseen world. But perhapsspontaneousprayerwouldnotalwaysbe forthcoming,and regularceremonies,formal prayers, were arranged so that there was less need to depend on anunpredictable soul-wrenching traumatic situation to draw forth our prayer. Ofcourse, therewillalwaysbesuchsituations,but in themidstof themweoftenforgetabouttheavailabilityofhelp,orwehavenotformedthehabitofasking.Andwhenwe just think aprayer itoftendoesnot reachourheart,our feelingcenter. Ceremony can help us form the habit of asking by tuning our wholeselves—our bodily senses, our minds, our feelings, our imaginations—toresonatewiththeworldofSpirit.Ourprayerisnotsimplythoughtorspokenbutis integrated within a ceremonial context designed to elicit the essentialcomponents of true prayer such as sincere intention and gratitude from theparticipants.Ceremoniesare symbolicprayers thathave thepower toopen theheart and give us access to the hidden depthswithin ourselves. Therewe canfind the resources to approach the spiritual realities being invoked in theceremony.

“Symbolismintheancientworldwasalwaysrelatedtopracticalfunction,andthesymbolicfeaturesofthetemplewereintendedtoassistthepurposeforwhichthebuildingwasdesigned,asaninstrumentofinvocation.”

JOHNMICHELL1

CEREMONYWITHINGEOMETRICSPACE

Ancienttemplearchitectureoftenembodiedanabstractsymbolismcomposedofgeometrically designed spaces that represented the ratios derived from cosmicstructures of space and time. Such geometric spaces as the ones describedthroughout this bookwill be our vessels of ceremony. In creating a vessel ofinvocationfromgeometryweareattuningourceremonytothatwhichis—likethemusicianwhotuneshisstringstotheeternalintervalsthatareforevernew.Carefully listening, he turns the tuning pegs just as Zalzal or Pythagoras haddone,fortheyknewthattomakemusicthestringsmustbesettotheuniversal

order, must resonate with the ratios that are unchanging. Spaces can also betuned to resonate with the cosmic order through the careful setting of theirproportions to resonate with the eternal ratios. Space is thus transformed tosymbol.Whenwe enter symbolic spacewe aremore likely to experience thefeelingofthesacredandthusbeabletoplayapartinthetheaterofceremony,sincethefeelingofthesacrediswhatdrawsthesacredtous.

The theater of the ancient Greek mystery plays was staged within that mostpowerfulgeometricform,thecircle.Humanshadsatordanced,drummed,andsung in fire circles since the earliest times, as today they still do.This primalsymbolofhumansgatheredaroundafirecorrespondstoandresonateswiththeorbit of the Earth around the central fire of the Sun or the orbit of the solarsystemaroundtheGalacticCenter.AmoreintimateinterpretationofthissymbolistheideaofthecentralfireoftheSelfaroundwhicharegatheredthecontentsoftheunconsciouspsycheandtheconsciousmind-body.

There are other shapes besides the circle that can hold sacred space, can bevessels of ceremony. These are the temple spaces created by ancientcivilizations, such as Indians, Egyptians, Mayans, medieval Europeans, andmanyothers,whosebuildersknewthesacredratiosandencodedthemintotheirtemples so that their ceremonial spaces would resonate with the geometry ofCreation.Wepreviouslyconsideredanexampleofthisinchapter2:thetempleof Apollo Epikourios at Bassai, Greece. These ceremonial spaces weredescendants of older spaces, the megalithic observatory/temples, such asStonehenge, which embodied the dynamics of the Sun and Moon in theirstructures. Still older were the ceremonial spaces within Paleolithic caves,sanctifiedby thewallpaintingsofspiritanimals,messengersof thedivine.Ashumans we are inheritors of these ceremonial traditions. They are encodedwithinuswaitingtobereactivated—likeafinepowderedtea—bythewaterofceremony.

Ceremonyisthusawaytoexperiencethesacredthroughtheuseofappropriatesymbols. The principal symbols that we will use in our ceremonies are largegeometric drawings that represent the subjects of the ceremonies. By literallyplacingoneselfwithintheseconstructionstheparticipantcanmoreeasilymaketheshifttoanimaginalmodeinwhichrevelationbecomespossible.

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When we carefully make representations of cosmic structures such as Earth,Sun,Moon, thewatermolecule, and the structure of evolutionary timewe arehonoring and appreciating the Creation and are setting the stage for theinvocation of the energies of nature to be present in our ceremonies. Theirpresenceassuresus that thequalityof the sacredwillpermeateour ceremony.With this assurance we may feel confident that our prayers of honoring andgratitudewillbeheard,anddivinefeelingsofwonderandjoywillhaveagoodchanceofmanifestingfortheparticipants.

We can use these formal prayers, these ceremonies, for anythingwewant—ifnone thereby shall be harmed. Thus financial success, health, lovingrelationships, blessing for a home, and so forth, are all subjects for sacredceremony.Andyetthereareceremoniesinwhichtheprayerissimplytowalkinbeauty or to be of service in helping to transform the world. Wheneverceremonies have this kindof intention, a certain vibration seems to take over.The ceremony canmagically draw forth the trust that is required towalk thebeautyway—thepathofloveandservice.

Wehave, in thepreviouspagesof this book, learned touse somegeometricaltoolsthatwewillnowputintopracticebycreatingceremoniesofinvocationandhonoring directed toward the sacred entities that are foundational to ourexistence: Earth, Moon,Water, Sun, and evolutionary time.We will learn togenerate large-scale geometries that are based on the dimensions of theseentities.Andwewillsuggestsomewaysofinvocationandjourneyingthatwilldrawusintothefeelingofthesacrednessofthesepowersthatsustainourlives.

CLASSICALGEOMETRY

Torepeatwhatwehavementionedearlier,inourexplorationsofsacredspaceswe will be using what is called classical geometry—that is, geometry drawnusing only compass and straightedge. In addition, we will not be laboriouslymeasuringouttheelementsofthedrawingbutratherlettingthepotenciesofthelines and circles themselves take us from one stage of the construction to thenext.Whatismarvelousaboutthisapproach,asweshallsee,isthatbyusingafewsimpletechniques,drawingsnaturallyarisethataccuratelyrepresentactualcosmicstructures.

Practically speaking, theconditionsof thegroundarenot alwaysconducive toreceivingdrawings.Whiletheancienttemplebuildersmayhaveleveledbedrockto accurately scribe giant geometric building layouts, we oftenmust work onunevenground.Inthiscasewecanusemeasurementswiththeconditionthatalldrawingscouldhavebeenconstructedusingonlycompassandstraightedge.

CREATINGSACREDSPACE

In the Earth-Honoring Ceremony, as in all of our ceremonies, the cardinaldirections are first established and then the invocation can be made. For theindigenousceremonialistthecallinginoftheenergiesofthefourdirectionsandabove and below has always been an essential element in the establishing ofsacred space, the field within which the ceremony can unfold. The principlebehindthisisthatcertainuniquecreativeenergiesresideinthenorth,south,east,andwestandinheavenandearthandthatthesemaybecalledupontobepresentwithin the ceremonial space. Their presence ensures that the ceremony willunfoldinasacredway.

When the invocation is made, the geometric construction can begin withconversationlimitedtotheworkathand.Theparticipantsintheceremonywhoarenot creating thegeometrycan silentlywatch theunfoldingof thedrawing.Or,ifsodesired,theycanjointheirvoicesinasimpletoningaccompanimenttothework.Thefacilitatorcouldbeginbyintoninganoteandparticipants,iftheyso choose, could join inby imitating thenoteor improvisingvariationsof thesound.Thispracticehasacalmingeffectaswellasunitingtheparticipantsintheceremonialwork.Also,whenappropriate, themeaningof thegeometrycanbeexplainedtothemasitproceeds.

ANEARTH-HONORINGCEREMONY

TherestoringofhonortotheEarthisanessentialstepinbringingaboutbalancewithintheecosystemofourworld.TohonortheEarthmeanstolivewithinherlaws,tonotpolluteorwaste,andtoalignourselveswithherwisdom.Howcanwebringsuchconsciousnessintoourlivesinarealway?Thereisadisconnectbetweenwhatweknowtobetrueandourabilitytoact.Wehaveanabundanceofinformationaboutenvironmentaldestruction,butweseemtobeunabletoact.

Thereasonforthisisthatwehavelosttheabilitytofeelwhatishappeningandsoremainpowerless.Knowledgeandfeelingmustbejoinedforrealchangetomanifest. The awakening of the critical feeling power is the function of thisceremony.

IntheEarth-HonoringCeremonywhenwebeginbycallinginthedirectionsweareinvokingthepowersoftheecospheretoworkwithusinsuchawaythatwemay be more in tune with the laws of nature. This kind of invocation is anessential element inallourceremonies, forwithoutadirect collaborationwithNatureandherlimitlesswisdomwewillbeworkinginavacuum.

Therearemanywaystocallinthedirections.Findonethatyoulikeandworkwithit.Hereisacompositedrawnfromdifferenttraditions(ahshayistheDagarawordfor“Sobeit!”).

TothewindsoftheSouth,greatSerpent,spiritofhealing,youwhoknowthesecretplacesoftheEarth.Hearus!HelpustomovesoftlyontheEarthasyoudo,leavingnotraceofourpassing.Ahshay!TothewindsoftheWest,motherJaguar,hearus!Protectourceremonialspace.TeachusthewayofthepeacefulwarriorthatwemayreleaseallthingswithinusthatdonotservetheCreation.Ahshay!

TothewindsoftheNorth,ancientancestorsandhelpersofhumanity,hearus!Showusthewayofwisdom,thejoiningofheadandheart.Ahshay!

TothewindsoftheEast,greatEagle,youofeternalvision,hearus!Teachustoseeourtruepath.Giveusavisionofbeauty.Ahshay!

MotherEarth,greatGaia,sweetcradleoflife,hearus!Wehonoryouandyourchildren,theplantpeoplewhonourishandhealus,thewingedoneswhoteachusthewayoffreedom,thefour-leggedswhoteachusthewayofgrace,themany-leggedswhocleantheearthandspreadtheseeds,themysteriousstonepeoplewithmagicencodedintheirgeometriesandthetwo-leggedswhohaveheardyourcalland

havecometorestoreyourhonor.Ahshay!

GrandfatherSky,FatherSun,GrandmotherMoon,StarNations,hearus!Wereceiveyourblessingsingratitude.GreatSpirit,creatorandpreserverofLife,hearus!ThankyouforgatheringustogetherandallowingustosinganddancetothemusicofLifeforonemoreday.Ahshay!

An importantpartof ceremony is the individualoffering.Eachparticipant canbringaphysicalobjecttooffertotheenergiesbeinginvoked—perhapsanaturalobject, a stick or a flower or awritten poemor a talisman—something one iscomfortable with giving away. Or, conversely, the participant could bring atreasuredobjectthatismoredifficulttopartwith.Itmightbeaburnableobjectiftheintentionistoburntheofferings.Otherwiseitcouldbeburied.

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The actual conduct of the ceremony can takemany forms. First the geometry(seeEarthCeremonydrawings1–14onthefollowingpages).Iftheceremonyisto take place on the beach or on a cleared and leveled piece of ground, thegeometry can be marked out with lines inscribed in the sand or dirt using astake-and-cord compass for circles and a chalk line for straight lines. If theceremonyisheldindoorsthecirclescanbemadewithstake-and-cordcompassbut with the difference that the lines are generally not scribed into the floor.Instead,onepersoncanhold thecenterof thecompassandanotherperson thescribingend.Asthescribeslowlyandlightlypassesthescriberaroundthefloorathirdpersoncarefullydepositspebblesorcornkernelsclosebehindthescriberto form the arcs and circles. While a chalk line could be used indoors, analternative method may also be used: a line is held taut from both ends andbeneathitthecornkernelsarelaidout.Ifsodesired,whenthefinaldrawingisproduced theconstruction linescanbeerasedor removed leaving theessentialgeometrytostandalone.

When the geometry is completed, the drawing can be decorated by all theparticipants. For example, in this Earth-HonoringCeremony the circles of thefinisheddrawingcanbeoutlinedwithflowersornaturalobjectsgatheredfromthesurroundinglandorbroughtbyparticipants.InEarthCeremonydrawing14)weseethatportalsleadingfromonelayertothenexthavebeenmaderesulting

inaspiralpathwayintothecenter.Theportalscanalsobedecoratedwithnaturalobjects. If the ceremony takes place at night candles could outline the ringcircles. In general, the decorations can be left to the discretion of theparticipants.

ThepurposeofthisceremonyistojourneytothecenteroftheEarthandtheretoconnectwiththeSpiritoftheEarthtogivehonorandgratitudeforallthegiftsthatwehavereceived.Thejourneycanalsobea timeofquestionsthatcanbedirected to the Spirit of the Earth. Other intentions may arise, and theparticipantsarefreetofollowtheirowninclinationsinthisregard.Inthesameveinwe should note that the details—such as thewords of the journey or themusical accompaniments—are left to the discretion of the facilitator and/orparticipants.Thefollowingareonlysuggestions.

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A possible scenario: After the drawing is complete and an explanation of theconductoftheceremonyismade,theparticipantsgatheraroundtheoutercircleoftheEarthdrawingandbeginmeditation.Atacertainpointtheybegintoning—perhapsthevowelsoundaaahh.Adrumbeginsandthetoningfadesoutandthefacilitatorbeginstheguidedjourney.

Closeyoureyesand feel yourself relaxing.Beginbreathing in to thecountof seven:one, two,three,four,five,six,seven...holdingforacountofseven:one,two(etc.)...breathingoutforacountofseven...holdingforacountofseven...

(facilitatorcontinuesforseveralrounds).

Feelyourspiritbodyrisingupfromyourphysicalbody.Youarewalkinginameadow.Thesunisshininginabluesky.Lookatyourfeetastheywalkthroughthegrass.Arethereflowers?Youlookaroundyourselfasyouwalkalong.Whatdoyousee?Youapproachagreattreerisingoutof themeadow. Itsbranchesaresohigh that theydisappear into theclouds.Gigantic rootsentertheearth.Youwalkuptothetreeandplaceyouhandonitstrunkandgreetit.Tellthetreethatyouwouldliketoentertheearth.Youseeacavelikeopeningatthebaseofthetree,andyouenter it and feel yourself goingdown, down into theEarth, passing the roots of thegreat tree,passing huge boulders. You pass through the geological layers, through the crust and uppermantleanddownintothelowermantletowardthecore.Youpassthroughthemoltenriveroftheoutercore.Asyoupassthroughtheliquidfireyoufeelittakingawayanyimpuritiesthatyouhavebroughtfromabove....AndnowyouenterthesolidironoftheinnercoreandcometorestatthecenteroftheEarth.HereyougreettheGreatMotherandmakeyourprayerofgratitude....YouaskMotherEarthifthereisanythingyouneedtoknowtoserveHer.Youlistenforheranswer....YoureceiveananswerandthankMotherEarth.

Nowyoubeginyourjourneybackup.Youascendoutofthesolidinnercoreandthroughthemoltenoutercore, through the loweranduppermantleand thecrustandbackout theopening

beneath the tree.Youwalkoutonto themeadowandacrossthegrassandreturn toyourbodyseatedaroundtheEarthsymbol.

(Thedrumstops.)

Takeadeepbreathandmoveyourfingersandtoesandbegintoningasyou integratethemessagefromtheEarthMother.

Thetoninggraduallyfadesout,andtheparticipantssitinmeditationforawhile.The toningbeginsagain, andonebyone theparticipants enter the firstportal,wheretheyarepurifiedwithsagesmoke.Theytravelthespiralpathilluminatedbycandles,throughthesecond,thethirdportals.TheyenterthefinalportalandplacetheirofferingtoEarthMotheronthealtar.Thentheyreturnthewaythattheycamebacktotheoutercircle.Whenallhavemadethejourneythetoningfades and the ceremony closes. The offerings can be burned or buried in theearth.

EarthCeremonydrawings1and2.(1)Drawaneast-west lineThiswillbethediameterofthecirclerepresentingtheEarth.(2)Drawtwointersectingcircleseachwithacenteratanendoftheeast-westlineandaradiusextendingtotheotherendof the line.Join the intersectionpointsof theresultingvesicawithaline that locates the center of the original line and also gives the north-southline.

Earth Ceremony drawings 3 and 4. (3) We now have a center for our Earthcirclewithitsdiameterthelengthoftheoriginalline.Drawthecircle.(4)Withthe fouraxis endsas centersdraw four circleswith the samediameterasourEarthcircle.ThenconnecttheintersectionsofthesecirclestogenerateasquarethatenclosestheEarthcircle.

EarthCeremonydrawings5and6.(5)Findthemidpointsoftwoquarter-squaresidesanddrawtwodiagonalsasshown.DroptheintersectionofthediagonalstothehorizontaldiametergivingapointthatdefinestheradiusoftheinnercoreoftheEarth.(6)Drawtheinnercore.

EarthCeremonydrawings7and8.(7)Drawacirclearcwithhalf-squareradiusto intersect the extended baseline of the Earth square. Mirror this on the

opposite side and draw a rectangle adjoining the Earth square. (8) DrawanotherarcwithradiusthelengthoftheEarthsquareplusrectangle.Mirrorasin(7)anddrawanewrectanglewithinit.Withinthissmallerrectangledrawacircle.This circle is the sizeof theMoon in relation to the large circlewhichrepresentstheEarth.

Earth Ceremony drawings 9 and 10. The diameter of the Moon circle fromdrawing (9) becomes the radius of a circle drawn in the center of the Earthcircle. Draw this new circle. (10) This circle represents the boundary of theoutercoreoftheEarth.

EarthCeremonydrawings11and12.(11)Usingthe1/philengthfromdrawing(7) section off the Earth square three times as shown. (12) Connect the phisectionpointswithdiagonals.Droptheirintersectiontothehorizontaldiameterand draw a circle, whose center has been determined by the bisection of thehorizontaldiameter,betweentheresultingpointandthefarendofthediameter.

EarthCeremonydrawings13and14.CenterthenewcircleontheEarthcircle.(13) This represents the boundary between the inner mantle and the outer

mantle/crust.(14)Cutoutportalsintoeachlayerasshown.

ABIRTHOFTHEMOONCEREMONY

This ceremony will educate us, using sacred geometry and the tools ofceremony,abouttheoriginoftheMoonanditsimportanceinthehistoryoftheEarthandthehistoryofhumanity.

ThestudyoftheMoonisamostancienthumanoccupation.TheMoonwithitsregularphaseswasprobablytheearliesttimekeeperforhumans.Archaeologistshave found lunarnotations in the formofcounting lines incisedonbones thathavebeendatedtothePaleolithicera.Later,thestudyoftheMoonledhumansto erect stone-circle observatories to predict lunar positions during annual andmultiyearcyclesandtoobservethechangingrelationsoftheSunandMoon.

WeknowthattheMoonworkingwiththeSunraisesandreleasesthewaterlevelinharbors andonbeaches twice a day and, in a slower rhythm, increases anddecreasesthevolumeof thiswater twiceamonthatfullanddarkmoontimes.AndweknowthattheMoon,breakingtheEarth’shold,drawsthemoisturefromthe ground up through the soil to dampen and help awaken the dormant plantseeds.As it becomes clear that theMoonhas an intimate connectionwith lifeanditscycles,wecanbegintounderstandwhyourancestorsstudiedtheMoonforthousandsofyears.

The modern scientific story of the birth of the Moon tells us that relativelyshortlyafterEarthwas formedaplanetabout thesizeofMars (whichwewillcall the Wandering Planet) crashed into Earth. Some of the debris from theexplosionremainedonEarth,somewentintospace,andtheremainderwentintoorbit around Earth. Through gravitational attraction this debris graduallycoalescedintotheMoon.

Asshown in thedrawings,wewillconstructageometric space thatwillbringtogethertheelementsofthestory.Withinthisspacetherewillbethefourphasesof theMoon,aswellasEarthandtheWanderingPlanet.Andthere isafigurethatwewillusetoconsolidatetheseelementsintoaunifiedwhole.Thisfigureisthewell-known chacana, the symbol of the nativeAndean peoples,whichwewillmodifytocontaintheelementsofourmodernmythicMoonstory.

InBirthofMoondrawing15wehaveinsertedtheSunintotheblankspaceleftwhen the Moon was thrust from the Earth and into space. It fits perfectlybecause the sizes of the Sun and the Moon in the sky are the same, bothoccupying a half a degree—a congruence that defies any sort of standardexplanation. Perhaps, as the ancients believed, the sky speaks to us about thenatureofthings.TheSunwassymbolofthemasculineprinciple,theMoon,thefeminineprinciple.Thoughquitedifferent in theirwaysofexpression, the skytells us they are ultimately equal. There are many such correspondences thatrelate to the feminine Moon and the masculine Sun. In a feminine way,moonlightmysteriouslytransmutesthelightoftheSunandbathesthelandscapeinasoftbeautyinstarkcontrasttotheharshradianceofnoon.Andthepowerofthe feminine Moon is not to be belittled. Regularly, during eclipses, shecompletelyblockstheSun.Yet,duringa lunareclipse,evensheiseclipsedbytheshadowpowerofherMotherEarth.Thusspeaksthesky.

These things did not go unnoticed by early humans living within a mythicconsciousnessinwhichthehappeningsintheskyhadmeaning.Theywerealsoaware that theMoon influencedhumanbehavior, aphenomenonconfirmedbymodern cosmo-biologists as well as many, including myself, who haveexperiencedhereffects.Forexample,around the timeof the fullmoonhumanenergy is amplified, and this energy often is expressed in both creative anddestructiveways.Forfragileearlysocieties,thiskindofknowledgewasofgreatimportance and led them to erect the standing-stone sky observatories so thattheycouldbetterunderstandthecosmicforcesthatinfluencedtheirlives.

While constructing the geometry for our ceremony we will follow theproceduresexplainedpreviouslyregardingthecallinginofthedirectionsandthemaintaining of silence except for necessary communications. When thegeometry is complete the participants can decorate the Moon phases and thechacanawithflowers,candles,andnaturalfoundobjects.Thentheycangatheraroundthechacanaandmakethemselvescomfortable.Thefacilitatorcanbegintoninganoteandotherscanjoininifsodesiredinordertohelpclearthemindand relax the body for the journey to come. The drum begins and the toningfadesout,andthefacilitatorbeginstheguidedjourney.

“Withtheuseofceremonyandritual,weareengagingwith,playingwith,thesymbolsandpoetryofaprimitiveconsciousness,bypassingthequestion-and-answerdiscourseofourrationalminds.Thereare

nohard-and-fastrules.Ouronlyobligationistobefullypresent...tofreeourselvestowhateverweexperience.”

ALBERTOVILLOLDO2

Relaxandbreathedeeply...longbreathin...longbreathout....Feelyourspiritbodymovingoutfromthisplacetoagrassymeadow.Youwalkacrossthemeadownotingwhatisaroundyou.Are there flowers?Are thereanimals?You lookupat thesky. . . .Are thereclouds?Are therebirds?Inthedistanceyouseeagreattreerisingintothesky.Itissohighthatyoucannotseeitstop.Youwalkup to the treeandplaceyourhandon its trunkandgreet the tree: “Greetings,OgreatTreeofLife.IwouldliketojourneybacktothebirthoftheMoon.”

Youwalkaroundthetreeandtakeflight.Yousailovermountainsand lakesandback intotime. You see the Moon rising over ancient Rome where citizens stand in front of a marblecolumn.Then you fly over a forest and come to a cliffwherepeoplewearinganimal skinsarehuddledintheentrancetoacave.TheystareupattheMoonthatyouseereflectedintheireyes.You travelback further in time towhendinosaurswalked theEarthand furtherback toancientswamplandswheregiantdragonfliesflitamongthefernswith thepaledaytimemoonoverhead.Backfurtherintime,youflyoveranoceanthatcoversthewholeEarth.TheSunissettingandthefullmoonisrising.YouflybillionsofyearsfurtherbackintimewhentheairlessandlifelessEarthwasachaoticplacefilledwithgreatearthquakesandthefireofvolcanoes.Hermotioniserraticonheraxisandsheseemsonthevergeofdestruction.AgreatsphereapproachestheEarththenfillsthesky.YouflyfaraboveandlookdownonthecolossalexplosionasthesphereimpactstheEarth.Theexplosionejectsbillionsoftonsofrockintospace,muchofitheldinorbitbytheEarth.Now,asyoumoveforwardintime,youseetherockfragmentscoalescing,formingasphere.TheMoonisborn.TheimpacthastiltedtheaxisoftheEarthtoitsorbitandsotheseasonsbeginandtheicecapsbegintoform.AsyouflybacktowardthepresentyounotehowtheMoonhashelpedstabilizethemotionoftheEarth,whichisnowcyclingthroughregulardaysandnights.Thegreatrainsbeginandcontinue for thousandsofyears.Youcomecloser to theEarthandflyover theworldoceanandseebeneaththewaveshowlifeisformingundertheinfluenceoftheMoon.Firsttinyalgae,thenplantformswavinginthecurrents,thentinyanimalforms,thenfish.Andnowthecontinents arise from theworld oceanand theplayof life continueson the land.Closer to thepresent you seehumansplanting fields according to theMoon rhythms.And theyare erectinggreat stone circles to study and to honor the power of theMoon so intimately connectedwithwaterandthuswithalloflife.YoucomebacktotheWorldTreeaftersundownasthefullmoonisrisingilluminatingthemeadowwithasoftglow.YoulookupattheMoonandaskitaquestion....Youwait and receive the answer, thenwalk back across themeadowand enter your physicalbodyhereinthisplaceinthepresenttime.

Theparticipantsbegintoningtointegratetheexperience.Maintainingthetoningtheparticipantsriseandwalkaroundthechacana,leavingofferingsattheMooncircles.Then,onebyone,theyentertheportalthathadbeenconstructedearlier.Theyseat themselveswithinthechacanaaroundtheSuncircle, inwhichafirehasbeenbuilt,foraperiodofmeditation.Theofferingsof theparticipantscanbeplacedinthefire,afterwhichtheceremonyends.

BirthofMoondrawings1 and2. (1)Drawaneast-west line.Thiswill be thediameterofthecirclerepresentingtheEarth.(2)Drawtwointersectingcircleseachwithacenteratanendoftheeast-westlineandaradiusextendingtotheotherendof the line.Join the intersectionpointsof theresultingvesicawithaline that locates the center of the original line and also gives the north-southline.

BirthofMoondrawings3and4.(3)WenowhaveacenterforourEarthcirclewith its diameter the length of the original line.Draw the circle. (4)With thefouraxisendsascentersdrawfourcircleswiththesamediameterasourEarthcircle.ThenconnecttheintersectionsofthesecirclestogenerateasquarewhichenclosestheEarthcircle.

BirthofMoondrawings5and6.(5)Locate1/5ofthesideofthesquaretobeused in the drawing (6) by constructing the diagonal crossing as shown. (6)Usingthe1/5measurestepoffthe1/5and3/5distances.Drawthediagonalsasshown.Drop their crossingpoint to thehorizontaldiameter to locate the7/11sectionpoint.Mirrorontheothersideoftheverticalaxisofthecircle.

BirthofMoondrawings7and8.(7)Afterfindingthecirclecentersbetweenthesectionpointsanddiameterendsdraw the two intersectingcircles to form thevesica.(8)Drawacircleinsidethevesica.ThisisexactlythesizeoftheMooninrelationtotheEarthrepresentedbytheenclosingcircle.

BirthofMoondrawings9and10.(9)Drawacirclearoundthevesica.Thisisthe relative estimated sizeof theWanderingPlanet,which impacted theEarthand resulted in the formation of the Moon. (10) Draw a square around theWanderingPlanet.

Birth ofMoon drawings 11 and 12. (11) Draw theMoon above the Earth to

representtheresultoftheviolentimpactoftheWanderingPlanetontheEarth.(12) Draw a squaring circle through the Moon. The squaring circle’scircumferenceequalstheperimeterofthesquarearoundtheEarthandgivesuscenterstodrawthefourphasesoftheMoon.(Thesquaringcircleisanancienttoolof invocation that symbolicallyexpressesunionbetweenSpirit (thecircle)andmatter(thesquare).

BirthofMoondrawings13and14.(13)DrawlinesconnectingthephasesoftheMoon.(14)UsingtheelementsoftheEarth,MoonandWanderingPlanetshownindrawing(13),constructachacana.

BirthofMoondrawing15.(15)Cutoutaportalintothechacana.ThecentralcircleisleftandwillrepresenttheSun,whichisthesameapparentsizeastheMooninthesky.

AWATER-HONORINGCEREMONY

Wenowenter themicrocosm to illuminatea sacred form: thewatermolecule.To construct a scalemodel of thismoleculewewill proceed according to therulesofclassicgeometrythat,limitedtothetoolsofcompassandstraightedge,isthesimplestapproachtotheheartofgeometry.

Sacred geometry, alongwithmusic, mythic stories, poetry, and sacred dance,partakes of a simple and profound way of knowing. In all of these we usemimesis, or representations of reality, to alter our consciousness such that wemayenterintothepresenceofthesacred.Humanshaveusedpicturesorstatuesofgodsforthispurpose.Indigenoushuntershavewornfurorfeatherstobecometheirpreyintribaldancesbeforethehunt.Shamansmayputonbirdwingstoaidin shape-shifting to access hidden knowledge. In our present example wegeometricallydrawanaccurate representationof thewatermolecule to invokethespiritofwater.

In our water molecule construction we once again use the Earth/Moon sizerelation, andwhile themoon’s size propertieswill be necessary factors in thegenerationofthemoleculeofwaterthepresenceofthemoonisalsoanecessaryceremonial element since themoon andwater are energetically connected.Asusualweinvokethefourdirectionsandaboveandbelowaftertheeast-westandnorth-southlinesareestablished(drawings1and2).Wethenconstructthecircleinthesquareandgeneratethemoonfromthehalfsquarethroughtheagencyofthe golden proportion (drawings 4–7).A squaring circle is drawn through thecenterof themoon—bothaceremonialgestureandapracticalwayof locatingthecentersofthefourMoonphasesthatarethenconstructed(drawing8).

Wenowdrawavesica-toneconstruction(drawings9and10)(seechapter1foradescriptionofthisfigure)derivedfromthemajor-thirdintervalthatproducesaninnerangleofthevesicaof104.5degrees—thebondangleofthewatermolecule(drawing11).Wethenconsiderthelengthofthedistancebetweenthehorizontalandtheverticalvesicaendsasthebondlengthofwater.Wenowhavetheangleand bond length of water. The relative sizes of the atoms of oxygen andhydrogen in relation to the bond lengths are derived from the Earth/ Moondrawingasshowninwaterdrawings12and13.Amazingly,theproportionsturnouttoquiteaccuratelycorrespondtotheatomicsizes.

AchacanarepresentingtheEarthcontainingthemodelofthewatermoleculeisnowdrawn(drawing14).ThisisthegroundplanfortheTempleofWaterthatissurroundedbytheMoonphases.(Thisconstructionwasdemonstratedinchapter4.)At the center of the chacana abowlofwater is placed anddecoratedwithflowers.Theparticipantsgatheraroundtheinnerwallsofthechacana.Theysitand make themselves comfortable, breathe deeply, and close their eyes.Followingtheinstructionsofthefacilitatortheybegintoningtheintervalofthe

major third, which, as we have noted, is geometrically associated with thestructureofthewatermolecule.Halfthegrouptonesafundamentaltone,andtheotherhalftonesthethirdabove.Thefacilitatorspeaks.

As you tone imagine that the tones are generating walls of falling water in the shape of thechacana,creatingatranslucentshimmeringcathedral.Youareco-creatingthisarchitecturewiththeSpiritofWaterandwithsacredharmony.

(Asthetoningfadesthejourneydrummingslowlybegins.)

Now,withyour intention, leavethewallsofwater inplaceandcomeintosilence.Letyourspirit-bodywalkaroundtheTempleofWater....Explorethesecretplacesthatcalltoyou....Andnowletyourspirit-bodymovetothecenterofthetemplewheretheSpiritofWaterresides.GreettheSpiritofWaterandthankitforitsblessings.Ifyouhaveaquestion,askit.Ifnot,simplyaskforguidance....Whenyoureceiveyouranswerletyourspirit-bodycomebacktowhereyouaresittingandjoinwithyourphysicalbody.

(Thedrummingendsasthetoningbegins.)

Begintoningtheintervalofwater,andwithyourintentionslowlybringthewaterwallsofthetempleback into theearth.Nowcome into silenceand integratewhat youhave learned in theTempleofWater.

Theparticipantscannowblowtheirthankstowaterintoaflowerandplaceitinthebowlofwater.Thebowlcanthenbeemptiedintoanearbyriverorocean.Ifnobodyofwaterisavailabletheflowerscanbeputintothefireandthewaterpouredontotheearth.

WaterCeremonydrawings1and2.(1)Drawaneast-westlineThiswillbethediameterofthecirclerepresentingtheEarth.(2)Drawtwointersectingcircleseachwithacenteratanendoftheeast-westlineandaradiusextendingtotheotherendof the line.Join the intersectionpointsof theresultingvesicawithaline that locates the center of the original line and also gives the north-southline.

Water Ceremony drawings 3 and 4. (3)We now have a center for our Earthcirclewithitsdiameterthelengthoftheoriginalline.Drawthecircle.(4)Withthe fouraxis endsas centersdraw four circleswith the samediameterasourEarthcircle.ThenconnecttheintersectionsofthesecirclestogenerateasquarethatenclosestheEarthcircle.

WaterCeremonydrawings5and6.(5)Usingahalf-squarediagonalasradiusdropanarctoanextensionofthesideofthesquare.Mirrorontheoppositeside

togenerateaϕrectangle.(6)Withradiithelengthoftheϕrectangleconstructacirclearc to intersect therectangle.Mirroron theoppositesideanddrawasmallerrectangleinsidethelargerfromthepointsofintersection.

WaterCeremonydrawings7and8.(7)Acircleis inscribedwithinthesmallerrectangle.ThiscirclerepresentstheMooninproperproportiontothecirclethatrepresents the Earth. (8) A larger circle centered on the center of the Earthcircle is drawn through the center of theMoon.Draw the four phases of theMoonfromthepointsgeneratedfromthislargercircle.

Water Ceremony drawings 9 and 10. (9) Draw two half-square diagonals asshown.Dropalinefromtheirintersectiontothehorizontaldiameteratx.Bisectthe line xy at z. Thiswill be the center of the first vesica-forming circle. (10)Withpointzascenterdrawacirclethatintersectspointxandthediameterend.Mirror the construction on the other side of the vertical axis to generate avesica.

WaterCeremonydrawings11 and12. (11)The internalangleof the vesica is104.5°,whichisthebondangleofwater.(12)Inthisdrawing,tofindthesizesofthe oxygen and hydrogen atoms that comprise the water molecule we have

repeatedtheconstructionshowninwaterceremonydrawings5–8,butnowweconsidertheMoon,whichwehavealreadygeneratedinthepreviousdrawings,as the Earth (a). (b) Now draw a circle around both these figures—that is,aroundthenew,smaller,Earth/Moonfigure.Thiscircumscribingcirclewillbeouroxygenatom.(c)Dividetheoxygencircleintwoanddrawacircle.Thiswillbeourhydrogenatom.

WaterCeremonydrawings 13 and14. (13)Center the hydrogenatomson theendsofthelinesthatrepresentthebondlinesofthewatermolecule.Centertheoxygenatomatthevertexofthebondlines.(Thismodelquiteaccuratelyreflectstheproportionsoftheactualwatermolecule.)Placeanotheroxygencircleintheleftsideofthecomposition.Thiswillserveasaguidefortheconstructionofthechacanaandwilllaterbedeleted.(14)Drawthechacana:drawalinefromtheEarthsquaretotheintersectionofthehydrogenandvesicacircles, thentotheEarthcircle,thentotheoxygenperimeterandfinallybacktotheEarthsquare.Mirrorhorizontallyandvertically.

Water Ceremony drawing 15. The completed chacana. Cut out entry and exitportals.TheMooncircles canbe filled inas shownwith thenewmoon in theeastwhereitinvisiblyriseswiththeSun.

ABIRTHOFTHECOSMOSCEREMONY

For thisceremonywewillcreateaceremonialvesselbasedonthestructureof

cosmic time as revealed by the Mayan calendar. This is our Tree of Timedescribedinchapter10.Thepurposeofthisceremonyistheritualcleansingofthe accumulated dark energy of the past from the energy fields of theparticipantsbyjourneyingbacktothe“FirstTime”beforematerialcreationandthencomingforwardthroughevolutionarytimebacktothepresent.

Thebeginningoftheceremonywilltakeplaceintheregion,describedinchapter10, thatwe have called theDreamtime, after theAboriginal term for the timewhen the ancestor gods were dreaming the world into being. We havehypothesizedthroughharmonicanalysisoftheMayancalendarthatsuchatimebeforetheCreationmayhaveexisted(seechapter10inthetext).Thisisnotatoddswithmodern scientific theory that admits thepossibilityof a timebeforethebigbang, although its character remainsamystery, amystery thatwewillattempttoenterwiththetoolsofimaginationandsacredceremony.

*

The structure of our ceremonial vessel will be a Tree of Time whose rootsextend into thedarksoilof theDreamtime.After thegeometry is laidout, thearea of theDreamtime can be enclosed by a sweat-lodge-type of circular tentwithaportalopeningtothebeginningofthePathofFormation(thetrunkoftheTree of Time),which leads to theEra ofBiological Life (the branches of theTree).TheEraofBiologicalLifehasafirecircleenclosedbystonesinitscenterinthemiddleofabranchingpatternthatcanbedecoratedwithleaves,flowers,shells,andsoforth.

Theparticipantswill sit or lie down in thedarkness and silenceof theDreamLodge.Aslowdrumbeatbegins.Thefacilitatorspeaks.

Breathealongbreathin.Breathealongbreathout.(Repeattentimes.)Youareinthedarknessofthevoidinanerabeforetime,beforethecreationofmatter.ThisisatimeofdreamingwhenonlySpiritexists,thatSpiritwhichisyou.Youaredreamingofaseedlyingwithinthedarkvoid.Itisfilledwithinfiniteenergyyetisdormant,waiting,atpeace.Youexaminetheseed,admiringitsbeautiful form,feeling itspotency. . . .Nowadropofcrystal-clearwaterfallsontotheseed,andtheseedcracksopenandlightradiatesoutfromitsdepths.Thelightstreamingoutfromtheseedis filledwith starsandgalaxiesof stars that fill thedark voidwith shiningbrilliance.Within thisbrilliant light you see our Sun with a cloud circling around it. You look more closely and seeplanetsformingwithinthecloud.YouseetheearlyEarth,amoltensphereoflavawithasteroidsrainingontoit.Agiganticplanet-sizeobjectcrashesintotheEarthcausingahugeexplosionandsendingupmoltenrocksthatsurroundtheEarth.NowtheMoonformsfromthecirclingrocks,andtheEarthbecomescalmerasitcirclestheSunwithitscompanionMoon.Youcomeneartothe

Earth.Anoceanhas formed.Youwalk to theedgeof theoceanand lookcloselyat the foamyedgeofawave.Youseeatinywrigglingcellandthenothercellsaroundit.Timepasses.Afishleapsintheair.Alizardsunsitselfonarock.Youturnandwalkbackacrossthebeachandintotheforest.Miceplayontheforestfloor.Monkeysswingthroughthetrees.Nowyouwalkoutontoaplain.Humansaregatheredarounda fire.Theywave toyouandgesture thatyou join them.Yousitdownatthefireasyouwakefromyourdream.

Theparticipantsrise,andthedoortotheDreamLodgeisopened,andtheywalkthe Path of Formation and enter the Era ofBiological Lifewhere they gatheraroundthefireinwhichtheyplacetheirofferingstotheUniverse.Theceremonyends.

Tree of Time drawings 1 and 2. (1) Draw an east-west line. This will be thediameterofthecirclerepresentingtheEarth.(2)Drawtwointersectingcircleseachwithacenteratanendoftheeast-westlineandaradiusextendingtotheotherendof the line.Join the intersectionpointsof theresultingvesicawithaline that locates the center of the original line and also gives the north-southline.

TreeofTimedrawings3and4.(3)WenowhaveacenterforourEarthcirclewith its diameter the length of the original line.Draw the circle. (4)With thefouraxisendsascentersdrawfourcircleswiththesamediameterasourEarthcircle.ThenconnecttheintersectionsofthesecirclestogenerateasquarethatenclosestheEarthcircle.

Tree of Time drawings 5 and 6. (5)Our foundational form, the circle-in-the-squarebeginsthispartofourconstruction.(6)Withhalf-squarediagonalradiusswing an arc to the extension of the square. Mirror on the other side andcompletethegoldenrectangle.

TreeofTimedrawings7and8.(7)Withthelongsideofthegoldenrectangleasradius swing an arc to the opposite side.Mirror, as in (6), to forma smallerrectangle.(8)Drawacirclewithin thenewrectangle.Thissmallercircle is insizerelationtothelargercircleastheMoonistotheEarth.

TreeofTimedrawings9and10.(9)Enclose theMooninasquare. (10)Withhalf-square radii draw 2 arcs to the baseline as shown. Complete the newrectanglewhoseratioequals√5.

TreeofTimedrawings11and12.(11)Duplicatethe√5rectangle9moretimesforatotalof10inaverticalcolumnasshown.(12)Drawacirclearoundthecolumnandcircumscribeasquarearoundit.(Seedrawing4)

TreeofTimedrawings13and14.(13)Withradiusofthehalf-squarediagonalofthelargecircleswinganarctotheextensionofthesquareasshown.Mirrorontheoppositesideandcompletethegoldenrectangle.(14)Withradiusofthelongsideoftherectangleswinganarctotheoppositesidetogiveapointfromwhichtoconstructasmallerrectangle,as in(7).Within thissmallerrectangleinscribeacirclerepresentingtheMooninrelationtothelargerEarthcircle.

TreeofTimedrawings15and16.(15)Drawinthebranchingpatternasshownon the completed Tree of Time. (16) Make a fire circle in the center of thebranching.

ASUN-HONORINGCEREMONY

ItisnotdifficulttounderstandwhysomanycultureshavehonoredtheSun.FortheQuechua peoples of theAndes it is Inti. For the Shinto Japanese it is thefeminine goddess Ameratasu. For the ancient Egyptians it was Ra. It’s notdifficulttounderstand,becausetheSun’senergybringsforthfromtheearthallthatweneedtosurvive.Forusitistheever-givingsourceofallthatwehave.TohonortheSuninceremony—thatis,toconsciouslygiveourthanks—seemsthemostnaturalofactions,andithasbeensoformostofhumanitysincetheearliesttimes.

Aswehaveforotherceremonieswewillagain,usingsacredgeometry,createanimage of the object of our ceremony. Beginning with an accurate geometricimagewearealreadyhonoringthewaythingsare:theharmonyinherentinsizerelationsthatarisefromsacrednumber.Incarefullymakingtheconstructionwearealreadyestablishingtheheightenedstateofawarenessthatwewillmaintainthroughouttheceremony.

Webeginasusualbylayingoutthefourdirectionsandmakingourinvocation.A largecircle isdrawn—itcouldbealmost any size—let’s say it is20 feet indiameter. This is our Sun circle. Next we want to draw a circle to representEarth,acirclethatwillbeinproperproportiontotheSuncircle.Weknowthatitwould take 109 Earths laid end to end to stretch across the Sun—that is, theSun/Earthdiameter ratio is109.Becausewewant to find thesizeof theEarthgeometrically rather than through measure we can use a method of octavereduction. Thuswewill continually halve the diameter of a circle constructedwithintheSuncircleuntilwearriveatacirclewiththerelativesizeoftheEarthdiameter.Thisisdemonstratedinourdrawings.

WhenwefindthesizeoftheEarth(forexample,iftheSundiscwere20feetindiametertheEarthdiscwouldbearound2inchesindiameter)arepresentationoftheEarthcanbemade—perhapscutoutfromapieceofcardthatcanthenbe

paintedwith theseasandcontinents. Itcouldalsobeasphereofpaintedclay.TheSuncircleitselfcanbeoutlinedinstonesanddecoratedwithfoundobjectsand flowers. Candles can be placed around the periphery. A fire circle,representingthesolarcorewherethefusionreactionstakeplace,isconstructedinthecenter.Thediameterofthecoreisgenerallythoughttobeabout1/5oftheSundiameter.ThefireislitandtheparticipantscanbeseatedaroundtheouterperipheryofthemainSuncircle.

TheEarthdiscisnowpassedaroundtheSuncirclefrompersontoperson.ThiscangiveasenseoftheorbitoftheEarth; theperipheryoftheSun,duringthistime,becomestheorbitoftheEarthandthecorecirclebecomestheSunitself.Asparticipants take theEarthdisc theycanexpress to thegroupor silently tothemselveswhytheyarethankfulfortheSun.

AnaltarcanbemadeontheperipheryoftheSuncirclewheretheEarthsymbolcanbeplaced.Burnableobjectsbrought fromhomeor found innature canbeplacedonthealtarasofferingstotheSun.NexttothealtarisasmallbreakintheperipheryoftheSuncircle,aportalintotheinterioroftheSun.

Fromthisanarrowpathleadstothecentralfire.Thepathisdecoratedwith,inorder, red, blue, and white candles representing the higher and highertemperatures as one approaches the white-hot interior of the Sun. Theparticipantssitaroundtheoutersuncircleandthefacilitatorbeginsthejourney.

Breathe in deeply and relax with your attention on your breath. Imagine a golden ball of lightwithinyourbodyatyoursolarplexus.Breatheagainexpanding firstyourbelly thenyourchest.Holdandrelease.Andagainbreatheindeeply,holdandrelease.

(Thetoningoftheaaahhsoundbeginsandcontinuesforseveralminutes.Asthetoningfadesthejourneydrumbegins.)

Feel your spirit body lift out of yourphysical body.Youare inagrassymeadow.YouarewalkingalonginthebrightsunshinefeelingthebeautifulradianceoftheSunonyourfaceasyoulookupintotheclearbluesky.FeeltheSunwarmyourskinandyourbones.Youstopandcloseyoureyes.Youopenyourarmsand,withyour righthandyoudraw thesunlight intoyourheartand whisper “love.” With your left hand you draw the sunlight into your belly and whisper“courage.” Now you feel yourself rising toward the Sun, leaving the Earth behind. You movethroughtheetherfeelingthesolarwindonyourfaceasyouapproachtheSun.YouseegiganticflaresrisefromtheSunasyoucomecloserandcloser.NowyoucomedownonthesurfaceoftheSunandaluminousangelapproachesyou.YoubowtotheangelandasktobetakentotheSpiritoftheSun.Theangelturnsandbeckonsyoutofollow.ThereisagapwheretheflamespartandyoufollowtheangeldownadazzlingcrystaltunnelleadingintotheheartoftheSun.Theflamesof the tunnelchange fromred toblue,andyoustop foramomentandyou feelenergymoving

throughyou,cleansingyourbodyandsoulofanyimpuritiesthatyouhavebroughtfromtheEarth.Theangelmovesondownthetunnel,andyoufollow.Thebluecolorturnstowhiteasyoucometoacircularspaceat thecenterof theSun.Theangel leavesandyoumakeyourprayertotheSpiritoftheSun.IfyoulikeyoucanasktheSpiritifthereisanymessageorperhapsagiftthatithasforyou....Youwaitforananswer.WhenyoureceiveyouransweryouthanktheSpiritoftheSun,and theangel reappearsandbeckonsyou to followbackup the tunnel.Asyou reach thesurfaceyouthanktheangelandflyintospaceandseetheballoftheEarthgrowbeforeyou.YoulandinthegrassymeadowandlookbackupattheSun,feelingitwarmyourface.YounowcomebackintothistimeandplaceattheSun-HonoringCeremony.

(Thejourneydrumstops.)

Breathedeeplyandwiggleyourfingersandtoesandopenyoureyes.Againwewilltonetheaaahhsound.

Asthetoningfadesparticipantscansitinmeditationforafewmomentsand,onebyone, riseand take theiroffering from thealtarandenter theSuncircleandplaceitinthefireattheSun’score.Afterthisisdonetheparticipantscanleave,orthosewhowouldliketostayandsharetheirexperiencecandoso.

SunCeremonydrawings1and2. (1)Drawaneast-west line.Thiswillbe thediameter of the circle representing the Sun. (2)Draw two intersecting circleseachwithacenteratanendoftheeast-westlineandaradiusextendingtotheotherendof the line.Join the intersectionpointsof theresultingvesicawithaline that locates the center of the original line and also gives the north-south

line.

SunCeremonydrawings3and4.(3)WenowhaveacenterforourSuncirclewith its diameter the length of the original line.Draw the circle. (4)With thefouraxisendsascentersdrawfourcircleswith thesamediameterasourSuncircle.ThenconnecttheintersectionsofthesecirclestogenerateasquarethatenclosestheSuncircle.

Sun Ceremony drawings 5 and 6. (5) Draw crossed diagonals of the half-squaresasshowntoestablisha1/5segmentofthediameter.Thissegmentwillbeusedtostepoff2/5and1/5segmentsofthesquarein(6).Drawanewsetofdiagonalsasshown.Nowprojectthisintersectionpointofthesediagonalstothemainhorizontaldiametertogiveusa64/109(.5872)sectionpointof themaindiametertowhichacirclewillbedrawnin(7).

Sun Ceremony drawings 7 and 8. (7) Draw the circle between A and thediameterend.Thisisthe64/109diametercirclethatmustbereducedinsizeby

6timestoequalthe1/109sizeoftheEarthinrelationtotheSun.BisectthenewcircleatB.(8)Drawthenextcirclebetweenthecenterofthe64/109circleatBandthediameterendtogiveacirclewithdiameterof32/109.

Sun Ceremony drawings 9 and 10. (9) Continue the process through 16/109,8/109,4/109,2/109,andfinally1/109,whichistheEarthcirclewhosesizeisinproperrelationtothemainSuncircle.(10)AnenlargementofadetailoftheSunandtheEarthintheircorrectsizerelationshipof109to1.(Thiscanbebetterappreciated ina large-scaleEarthdrawing inwhich the full-sizeSundrawingcan be made.)When the Earth circle is drawn a representation of it may bemadeasnotedinthetext.

SunCeremonydrawings11and12. (11)Bisect thehalf-sidesof thesquareasshown and draw a diagonal. Draw a half-square diagonal as shown thatintersects the previous diagonal. Their crossingpoint is projected to themaindiameteratapointtowhichacircleisdrawn.ThiscirclerepresentstheinnercoreoftheSun.(12)ConstructportalsintotheoutersurfaceoftheSunanditsinnercore.

APPENDIX2

AMISCELLANYOFNUMBERSANDFORMS

THIS SECTION PRESENTS MORE EXAMPLES OF MATERIAL from the text includingsculptural andarchitectural analyses, lunarnotes, vesica constructions, and thegolden proportion, all subjects that have been explored throughout the book.Thereisalsoadditionalinformationonthetechniqueofdiagonalgeometryandideas on the creation of harmonious forms from the vesica including potteryfigures,bookpagelayouts,andirregularshapessimilartoseedsandeggs.Thisappendixalsogivesseveralexamplesof thesculpturalandarchitecturaluseofthe crossed-vesica constructions in several ancient cultures that we brieflymentioned in the text. The history of the gothic arch, a half-vesica shape, isconsidered along with some geometric arch constructions. In general, thisappendixismeanttoaddafewpracticalexamplestothetextofthebookfortheuse of the designer of spaces and for the philosophers and students of art andnaturewhoaredrawn tosymbolicexpressionand to themanypossibilities forcreationofferedbysacredgeometry.

DrawingA2.1.Adrawingshowingthecirclesofthevesicaconstructionwithatable containing the property values of the vesica circles for twelve standardtonesofajust-intonationscaleaswellassomeothertonesthatarerelevanttothegeometryofvesicaconstruction.Thisisasummationofthenumericalaspectofthevesicaconstruction.

Drawing A2.2. The square possesses an unlimited wealth of rational ratiosgenerated from the crossing of itsmostly irrational diagonals. Themostwell-knowndiagonalsare thoseof the full square,half-square,andquarter-square.Thehalf-squarediagonalisfrequentlyusedinsacredgeometryasageneratorofthe golden rectangle and the √5 rectangle. Crossings of these principaldiagonalscangenerate,attheircrossingpoints,tonalratioswhenthepointsareprojectedtothemainstring/diameter—thatis,tothehorizontaldiameterofthecircle-in-the-squareor,here, to thehorizontalaxisof the square.Examplesof

this phenomenon are demonstrated in the present drawing—a division-by-2“diagonal weaving” where the main consonances of the scale are generatedfromthecrossingsofthesebasicdiagonals.Furthermore,the1/3,1/4,1/5,and1/6 segments, left over from the 2/3, 3/4, 4/5 and 5/6 tonal ratios on thestring/diameter,providethelengthsthatwemaystepoffonthesideofasquaretoproducedivision-by-three,four,five,andsixdiagonalweavingsfromwhichagreatmanymoretonesmaybegenerated.Severalofthesearedemonstratedinthe drawingsA2.3,A2.4, andA2.5. Similarly, the geometrically produced 1/ϕlengthwhen applied to the sides of a square, as in drawing A2.6, provides ageneratingmatrixforphi-familynumbers.

In this drawing the square has been divided in half both horizontally andvertically by the axes of the circle-in-the-square, and the following tones areproducedfromthegenerateddiagonals3/5(.60),thesixth;2/3(.667),thefifth;3/4(.75),thefourth;4/5(.80),thethird;and5/6(.833),theminorthird.

Whilethediagonalsareconstructedwithoutmeasurementwemustmeasuretheresulting string-length todetermine their ratiosandhence their intervals. Idothiswithacomputerdrawingprogram,whichfacilitatestheexactmeasurementofthegeneratedstring-lengths.

DrawingA2.3.Adivision-by-threediagonalweaving.Thenumberedpointsarelocatedbythecrossingsandwhenprojected(bydroppingperpendiculars)ontothe horizontal diameter (considered to be a musical string) produce thefollowingtones.(1)8/15(.5333),justmajorseventh;(2)7/13(.5385),septimalseventh;(3)6/11(.5455),neutralseventh;(4)5/9(.5556),acuteminorseventh;(5) 4/7 (.5714), just minor seventh; (6) 7/12 (.5833), septimal sixth; (7) 5/8(.6250), just minor sixth; (8) 5/7 (.7143), septimal tritone; (9) 8/11 (.7273),undecimal flattened tritone; (10) 11/15 (.7333), undecimal augmented fourth;(11)7/9 (.7778), septimal third; (12)4/5 (.8000), just third; (13)9/11 (.8182),

undecimal neutral third; (14) 9/10 (.9000), minor whole tone; (15) 11/12(.9167),undecimalneutralsecond.

Drawing A2.4.A division-by-four diagonal weaving. (1) 11/20 (.5500), acuteminorseventh;(2)9/14(.6429),septimalminorsixth;(3)7/10(.7000),Euler’stritone;(4)11/14(.7857),undecimaldiminishedfourth;(5)6/7(.8571),septimalminor third; (6) 7/8 (.8750), septimal whole tone; (7) 8/9 (.8889), just wholetone; (8) 13/14 (.9286), septimal semitone; (9) 19/20 (.9500), an ancientsemitone.

DrawingA2.5.Adivision-by-fivediagonalweaving.(1)3/5(.6000), justmajorsixth; (2) 7/11 (.6364), undecimal augmented fifth; (3) 7/10 (.7000), Euler’stritone; (4) 11/15 (.7333), undecimal augmented fourth; (5) 4/5 (.8000), justmajor third; (6) 13/15 (.8667), tridecimal 5/4 tone; (7) 9/10 (.9000), minorwholetone;(8)14/15(.9333),semitone.

DrawingA2.6.Aϕweavingcontainingseveralϕ-familynumbers.Thesereciprocalsandtheirratiosexpressedasϕnumbersareas

follows:

Reciprocal Ratio (1).5528 2/√5xϕ (2).559 4/√5 (3).618 ϕ

(4).691 1+1/√5 (5).7236 1+1/ϕ2

(6).809 2/ϕ (7).8542 ϕ2/√5 (8).8944 2/√5

Drawing A2.7. This drawing demonstrates another use of the diagonalweavings. Here we have used the full phi weaving as our design template inorder to layoutadoorway.Usingcrossingpointswehavegeneratedagoldenrectanglesurroundinga√5rectangle.Thesecloselyrelatedfigures—(√5+1)/2 = ϕ )—form a harmonious doorway with the √5 rectangle as the doorsurroundedby trimand step composedof theϕ rectangle.Further harmoniesarenotedinthedrawingontheright.

DrawingA2.8.On the upper left theϕ tritone construction ismadewith twonestedtriangles.Ontheupperrighttherelationsamongthepartsdemonstratetheϕharmoniesthatpervadethestructure.Onthelowerleftweconsiderhowthe nested triangles also createϕ harmonies in their matrix, the upper half-square.Onthelowerrightisthecompletedwallopening,perhapsanentrancetoaharmonicworkshop.

Drawing A2.9. A √ϕ vesica construction of the cross section of the GreatPyramid.Picturedin(1)istheconstructionoftheϕvesicafromthehalf-squarediagonals;(2)showstheconstructionofthe√ϕvesicawithintheϕvesica:thebisectionpointofABanditsmirrorpointarethecentersofthevesica-formingcircles thatgenerate the√ϕ vesica. In (3) thepyramidcross section isplacedwithinthe√ϕvesica,whichissurroundedbytheϕvesica.

FigureA2.A–C. (A2.A) a page from the psalter ofKingAthelstan of England

(924–940), reputed to have established the first guild of stonemasons inEngland; (A2.B) a carving from the tomb of Dona Sancha (ca. 1110, JacaMonasteriodelasBenedictinas)showingangelsliftingthevesica-shapedsoulofthedeceased;(A2.C)VirgininthevesicafromthePortadellaMandorlaatthecathedralofFlorence.

FigureA2.D

DrawingA2.10.AnanalysisoftheconstructionoftheAthelstanpsalter(figureA2.D, above) in which, an initial ϕ section of the half square (1) is made(drawing 2.8 gives a simple geometric method of finding the .691 point) andtransformedintoavesicaconstructionwithavesicawithϕratio.(2)Oneofthevesica-forming circles is copied and centered on the construction. (3)With aradiuscenteredonxandextendingtoyanothercircle isdrawnandmirrored,givinganenclosingvesicawitharatioof1.4472=1/.691.

Drawing A2.11. A variation on the vesica-within-a-vesica ring concept asillustrated in the Athelstan psalter in drawing A2.10. In this example wegenerate a vesica within another vesica whose axis ratio is the square of theoriginal. (1)Avesicaconstruction ismade. (2)AC4 circle is inscribedwithinthevesica.AlineisdrawnfromthehorizontalaxisendBtotheintersectionoftheC4circleandtheverticalaxis.Another line isdrawnperpendicular to thislinetointersectthehorizontalaxis.(3)LineABisbisectedgivingaC2center.(4)Thiscenterismirroredontheothersideoftheaxis,andtheC2circlesaredrawngeneratingthenewvesica.(5)Thetwovesicashaveaxisratiosinsquarerelation.Inthiscasethevesicaareashappentobein√5relation.

Drawing A2.12. Continuing our vesica-within-a-vesica theme leads us to astructure in which both the inner and outer vesicas have the same ratio. Theenclosing ringwould then be the gnomon of the original vesica, a shape thatHero of Alexandria defined as “that formwhichwhen added to another formresultsinanewformsimilartotheoriginal.”

(1)Avesicaconstructionisdrawn.AC3circleisdrawn,andlinesABandCBaredrawnconnectingtheC3diameterendswiththeendoftheverticalaxis.(2)Perpendicular bisectors of AB and CB intersect the horizontal diameter. (3)These intersectionpointsare thecentersof thenewC2 circles thatgenerateanew vesica with the same shape as the original vesica. The gnomon is thecoloredareabetweentheconcentricvesicas.

“Pointedarchesassuchwerenotnew.Eveninprehistoricornamenttheyappearautomaticallyastheproductofintersectingcircles.TheTreasuryofAtreushasadomewhosesectionisapointedarch.GreekmathematiciansandRomanarchitectsmusthaveknowntheform.Theimportantfactor,however,wasnotknowledgeoftheformbutthedecisiontouseitinarchitecture.TheEgyptiansmadeuseofitinsectionsofcanals,butthey,andafterthemtheRomansandtheByzantines,wouldnothavethoughtofusingininanexposedposition.Islamicarchitectswerethefirsttorecognizeitsaestheticandstylisticvalue.”

PAULFRANKL,GOTHICARCHITECTURE.1

FigureA2.E.TheCairoMosqueofal-Hakimca.690CE

DrawingA2.13.Adesign foranarchwhosedistancebetweencenters is1/5ththewidthofthearch.(1)The6/7pointislocated,and(2)thecenterpointofthe6/7 length is foundandmirrored,and the twoC2circlesareconstructed.Thewidthoftheresultingvesica=5/7(.7143)andthecentersareatadistanceof1/7 from each other.We note that 6/7 (a septimal minor third) is an ancienttonalratiocitedbyPtolemythathasbeenusedinArabicscales.

“Anewelementwasbrought—thepointedarch,ageometricalrevolutionoriginatinginIslamicsacredarchitecture.TheoriginofthepointedarchinEuropehasbeendeterminedasbeingattheItalianBenedictinemonasteryofMonteCassini,built1066–71....ItwasonlyamatteroftimeuntilallthegeometricsecretsoftheArabmasonswereincorporatedintoWesternsacredarchitecturetoformanewtranscendentstyle—nowknownuniversallybyitseighteenth-centuryderogatoryname,Gothic.”

NIGELPENNICK,SACREDGEOMETRY:SYMBOLISMANDPURPOSEINRELIGIOUSSTRUCTURES.2

FigureA2.F.PortalofAmiensCathedral,thirteenthcentury

Drawing A2.14.The construction of a “drop arch” a type of Gothic arch inwhichthedistancebetweenarchcentersis1/2thewidthofthearch.Thisarchisgenerated from the vesica construction at the minor-sixth tone resulting in avesicawithratio=2.

DrawingA2.15.Pottery constructions for nine tones generated from half-andquarter-squarediagonalswithin vesicaconstructions.On the left, thepotsarescaledsothattheirheightsareequal.

DrawingA2.16.Anillustrationofsomeofthemanyϕ-familypropertiesthatarecontainedwithinthepentagon,thevesselofthegoldenratio.Ontheupperleft:thediagonalscrossotherdiagonalsatϕpoints.Thebaseofthesmalltrianglesare .382=1/ϕ2.Upperright: the internalpentagonwithin themainpentagonhas sides of .809=ϕ/2.Perimeters andarea ratios all haveϕ-family values.Lowerleft:theperimeterofthestar=6.180=10/ϕ.Lowerright:theperimetersofalltheelementsarerelatedtoϕasarethearearatios.

Drawing A2.17.An analysis of the hexagon in which some of its √3 (1.732)propertiesareshown.Ontheleftweseehow√3,or1.732,isderivedfromthehexagon.Thehexagon,whilecontainingmany√3harmonies,also,asshowninbothdrawings,includestheratiosof2,5,and6,whichgivesitamoreuniversalquality than the pentagon whose inner dimensions seem to be completelydominatedbyϕ.Thisuniversalqualitybecomesevidentintheunfoldmentofthevesica piscis (see chapter 11) from which all the polygons, including thepentagon,takeform.

DrawingA2.18.Apracticaluse towhich rectangular spacedivisionusing thevesica constructionmaybeapplied is the layoutof thebookpage.Here is anexampleinwhichweusethe7/11intervalasthegeneratingtone.Thetwo-pagesheet is thebasicmoduleof thebook. In thecontextof thevesicaconstruction

thesheetbecomestherectanglethatcircumscribesthetwoC2circles.In(2)halfandfulldiagonalsaredrawnwithinthesheetrectangle,andapointaislocatedwherethehalf-pagediagonalintersectsaC2circle.(3)Averticallineisdrawnfromatointersectafulldiagonalatb.Ahorizontalisextendedfrombtoahalf-diagonal togivepointc.Avertical fromc intersectsahorizontal fromaatd,completing the text block that is thenmirrored on the other page (4). In thisexampletheratioofthetextblockandthepageareequalwithratiosof14/11.Margin ratios are shown in (5). While this construction uses the Americanstandardpapersizes,thesameconstructioncould,ofcourse,bemadeonmetric-systempaper.

DrawingA2.19.Morebook-pagelayoutsbasedonmusicaltonesandusingthemethodshownindrawingA2.18.

DrawingA2.20.The typeface designer or calligraphermay employ the vesicaconstruction in its many forms for designing letter-width modules. Afterconstructing (ormeasuring out) a ratio—in this case the 8/11 ratio—a vesicaconstruction isgenerated(1aandb)andthevariousvesicacirclesaredrawn.Throughexperiment, letterwidthsare foundbetweenvariousnodes formedbytheelementsoftheconstruction.

DrawingA2.21.Lettersandtextgenerated fromthemodulewidthsof the8/11tonedemonstratedindrawingA2.20.

DrawingA2.22.Thecrossed-vesicasconstructionthatwelookedatinchapter8mayhavehadawideuseinancientartandarchitecture.Formedfromarcsofthegeometrictritone(drawing8.6)acirclesquaringisestablishedgeneratingaquadrature geometry that sculptors used to structure the proportions of thehuman body so that the vital centers were subtly emphasized. This harmonicapproachwhenemployedbyconsummateartistsassisted them in theirgoalof

manifesting ideal types of humanity. Our first example is an analysis of asculptureofPharaohMenkaureofFourthDynastyEgyptusingacrossed-vesicaconstructionandquadraturethatlocatesthesecondchakra,throatchakra,andbrowchakra.The topof thecrownmaycorrespondto theseatof thesoul, the“skypoint”of theEgyptians.Theintersectionof fabricarcsonhiskilt locatestherootchakra.Thewholeiscenteredonthesecondchakraatthetopofhiskilt,thecenterofpowerinsometraditions.Thetwogeometricmeans,onefromeachendofthebody-as-monochord,areplacedattheendsofhisritualbeardandthetongueofhiskilt.

DrawingA2.23.TheCasselApollo,aRomanmarblecopyoftheGreekbronzeoriginalofca.470–460BCE. In thisand theotheranalyses inwhichwehaveoverlaid the crossed-vesicas construction we notice the importance of the

placement of the knees, whichmirror the placement of the vital center of theheart-lungsat thefourthchakra.Thehumankneesareessential toouruprightstanceandbipedalityandhavemuchtodowithwhatitmeanstobehumanasevidencedbythesenseofcelebrationthathappenswhenachildbeginswalking.Thehandsarealsofreedupatthistime.MirceaEliadewrites:“Itissufficienttorecall that the vertical posture alreadymarks a transcending of the conditiontypical of theprimates.Uprightness cannot bemaintained except in a state ofwakefulness.Itisbecauseofman’sverticalposturethatspaceisorganizedinastructure inaccessible to the prehominians: in four horizontal directionsradiating from an ‘up’‘down’ central axis. In other words, space can beorganizedaroundthehumanbodyasextendingforward,backward,toright,toleft,upward,anddownward.Itisfromthisoriginalandoriginatingexperience—feelingoneself ‘thrown’ into themiddleofanapparently limitless,unknown,and threatening extension—that the different methods of orientatio aredeveloped; for it is impossible to survive for any length of time in the vertigobroughtonbydisorientation.Theexperienceofspaceorientedarounda‘center’explains the importance of the paradigmatic divisions and distributions ofterritories,agglomerations,andhabitationsandtheircosmologicalsymbolism.”3

DrawingA2.24.TheMediciVenusbyafollowerofPraxiteles,marble,ca.300BCE.Thesamecrossed-vesicasgeometryaswehavepreviouslyseenisappliedtothefemalebody.Here,again,theessentialbodilyelementsareclearlysetout—theheadwith its regulating functions: therootchakraat thecenter;and,atthegeometricmeans,theheart-lungnode(centersofbreathandfeeling)andtheknees,whosefunctionproducestheuniqueuprightstanceofthehuman.

Drawing A2.25. Aphrodite from Frejus. Roman marble copy after Greekoriginal of ca. 430–400 BCE. Is it coincidence or angular illusion that theproposedguidelinesare tangent to theupraisedhandandtheball in theotherhand?Inanyeventsuchcueswouldmaketheaccuratecopyingofsculptureabiteasier.

Drawing A2.26.Michaelangelo’s David, sculpted about 2,000 years after theclassicalGreeksculpturesandofquitedifferentcompositionyetwithasimilararrangementofgroin,chest,knees,andheadasordainedbythecrossed-vesicasconstruction. This is an example of the rediscovery of classical proportionalstandardsbyRenaissanceartists.

DrawingA2.27.Mannikkavachakar,a seventeenth-century Indian sculpture inwhichweseeavariationoftheclassicalcrossed-vesicasconstruction.Whileinmostofthepreviousanalysesthecenterwasattherootchakra,inthiscasethecenter is at the second chakra slightly below the navel (as in the PharaohMenkauresculpture).Theheart-lungarea,thefourthchakra,isatthegeometricmeanposition.Thenext circle locates the throat chakra,which isat the samedistancefromthecenterastheknees.Thenextcircleintersectsthebrowchakra.

Drawing A2.28. Four of the previous sculptural analyses with their goldensectionpointsintheareaofthesolarplexus,thethirdchakra.

Drawing A2.29. An example of doubling quadrature generated from crossedvesicasused togeneratea templeplan.This is thegroundplanof theancientOsiriontempleatAbydos,Egypt.

(1)Acontainingcircleisdrawntotheverticalendsofthestructure.(2,3,4,5)Thecrossedvesicasaredrawn.(6)Asquareisdrawntotheintersectingpointsofthecrossedvesicas.(7)Acircle,alsodrawntothesepoints,circumscribesthesquare.(8)Asquarecircumscribesthecircleof(7).(9)Acirclecircumscribesthesquareof(8).(10)Afinalsquarecircumscribesthecircleof(9).

DrawingA2.30.Asetting-outofsomeelementsoftheVaikunthaperumalHindutemple atKanchipuram, India, eighth centuryCE, using a crossed vesica andquadrature construction. (1) The building is composed of nested squares.Beginningwithacentralcirclethecrossedgeometrictritonevesicasaredrawn.Their components immediately give all the essential layout directions. (2) Theouter crossings of the vesica-generating circles provide points to draw thecontaining square.The inner squareandmiddle squarearealsodrawn to thepointsprovidedbytheoriginalgeometry.(3)Anothersquarebetweentheinnerandmiddlesquaresisdrawnwiththehelpofdiagonalsthatcrossthecontainingsquare.

Drawing A2.31. The shape of the wind when, as it streams out of the high-pressureareas,issuckedintothelow-pressureareas.ThewinddirectioninthedrawingoccursintheSouthernHemisphere,flowingcounterclockwiseoutofthehighandclockwiseintothelow.

Figure A2.G. A similar pattern to the shape of the wind (drawing A2.31) isdemonstrated when two hurricanes approach each other exhibiting the“Fujiwharaeffect.”ShownhereareIonaatlowerleftandKirstenoffthePacificcoastofMexicointhemonthofAugust1974.

Figure A2.H. The taiji symbol and liquids mixing in a cup have a similarinterlockingspiralpatternasthatofthewinds.

DrawingA2.32.Thetaijisymbol

Drawing A2.33. The taiji, when revolved, becomes a circle-dividing tool. We

needonlyhaveanangleofrotationtoconstructasecondpairofvanes.Avesicaconstructioncanbeusedtosetthisout.Givenanangle,inthiscasetheGreatPyramid base-to-apothem angle of about 51.827°,we first divide the angle intwoandthentakethe1/tanoftheresult,whichgivesourvesicaratiowithwhichwemaymakeaconstruction.Halfofthisangleis25.913°,andthereciprocalofits tangentequals2.058(ϕ×√ϕ).Afterdrawing thevesicaconstruction,withitsC2 circle= .618 (1/phi), the next step is to circumscribe the vesicawith arectangleandthentocircumscribetherectanglewithacircle(C6).This is thetaiji circle. The revolved taiji vanes are constructed to intersect the rectanglecorners and divide the C6 circle as shown. On the right are the resultingϕ-basedarearatios.

DrawingA2.34.Fibonaccinumbersinnature.Onthetopisapineconewithan8/13 system of opposing spirals. Thirteen rows spiral clockwise, each rowhavingeitherallredorallblackdots.Eightrowsspiralcounterclockwise,eachhavingalternatingredandblackdots.Onthebottomisanothersimilarpattern—a5/8arrayofopposingspirals.BothoftheseareFibonaccipatternsthatcanbefoundthroughoutnatureintheseedheadsofflowerssuchasthesunflower.

TheFibonacciseriesisasummationseriesinwhicheachnumberoftheseriesisthesumofthetwopreviousnumbers:

0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1,597,andsoon.

Forexample,onalargeseedheadtheremaybe89spiralsgoingonewayand55theotherway,or55onewayand34 theotherway.Thequotientsofany twoadjacentratiosgraduallyapproachyetneverquitereachtheirrationalratioofϕ=1.6180339....Thus:

1/0=∞;1/1=1;2/1=2;3/2=1.5;5/3=1.666;8/5=1.6;13/8=1.625;21/13=1.61538;34/21=1.61904;55/34=1.61764;89/55=1.61818;

144/89=1.61797;233/144=1.6180555;377/233=1.6180257;610/377=1.618037135;987/610=1.6180345;1,597/987=1.6180345;

and1.6180345/1.6180339...(ϕ)=1.000000371,thedegreeofaccuracyinrelationtoϕatthisstageoftheFibonacciseries.

WerecognizetheratiosoftheFibonacciseriestobemusicalintervals.Intermsofstringlength:1/1,theunison;1/2,theoctave;2/3,thefifth;3/5,thesixth;5/8,theminorsixth;followedbymoreunusualintervals.Thefirstinterval,1/0whosequotientequals∞,mightsuggestthattheserieshasitsoriginintheinfinite.Thenext interval, 1/1, the unison, is the relation of two identical tones. Levy andLevariesayofthisinterval,

Thereisonlyoneperfectconsonancebetweentwotones:theunison.

. . . All other intervals . . . are more or less dissonant. . . . Theunderlyingassumptionissomemusicalunity,aonenessofsound,inwhichthepartakingtoneslosetheiridentity.Acousticsoffersustwoprominentmanifestations of such a unity: the oneness of the stringwhich vibrates simultaneously as a whole and in parts, and theonenessoftheharmonicserieswhichformsonetone....4

AnexampleofunisoninnatureistheapparentequalityofsizesbetweenSunandMoon,whichismostobviousduringasolareclipse.Ingeometrywehaveseenastriving towardunison in thecircle-squaringpracticesdiscussedearlier in thebook.

Drawing A2.35. The vesica and the ellipse. The ellipse can be seen as the“fulfillment”ofthevesica,roundingoffitssharppolesandtransformingitfromageometricsymbolintoamorenaturalobjectwithinnumerablemanifestationsfrom the shape of certain galaxies to the orbits of planets to the forms ofmicrobes.Geometricallytheareaofthevesicaitselfseemsdiscordantwithinthevesicaconstruction,thatis,thereisnosimpleproportionthatjoinsittoitsinner(C4) and circumscribing (C3) circles. However, when we transform it into an

ellipse a geometric proportion emerges: C3/ellipse = ellipse/C4. The abovedrawing of our familiar ϕ-tritone vesica construction includes two ellipsesforming an ellipse ring, that is, an ellipse around the vesica and an ellipseconnectingtheC1containingcircleandtheC3circle.Wehavealsoaddedtwoothercirclestotheconstruction,theellipsefocuscircles,Cf,whichjointhefociofeachellipse.Thefollowingaretheareasandarearatiosoftheelementsoftheconstruction.

C4=.1146 (a)C1/Cf’=ϕ(1.618) Cf=.1854(focuscircle) (b)Cf’/C3=ϕ E=.1854 (c)C3/E=ϕ C3=.3 (d)E/C4=ϕ Cf’=.4854(focuscircle) E’=.4854 C1=.7854

Also, E’ is the arithmetic mean between E and C1; E is the geometric meanbetweenC3andC4.Theprogressionofnumbers.1146,.1854,.3,.4854,.7854isan additive progression inwhich each number is equal to the sum of the twopreviousnumbers.Theseriescanalsobetransposedintotheseries1,ϕ,ϕ2,ϕ3,ϕ4.Theeccentricityoftheellipsesis.7861=1/√ϕ.

DrawingA2.36.Threedrawingsoftheellipsethathelptodescribethisshape.In(1) we demonstrate how the square is to the circle as the rectangle is to theellipse—they form a proportion that equals 4/π, or 1.273. (2) The principalpropertiesoftheellipseareshown.Giventhelengthsoftheaxes,thefocicanbecalculatedwiththeformula:f2=a2−b2whereaisthesemimajoraxisandbisthe semiminor axis and f is the distance from the center to the focus. Theequationfortheareaoftheellipseisarea=abπ.Theeccentricityoftheellipserangesfrom0to1andisgivenbytheequatione=f/a.(3)Thefocuscircleisconstructed.

DrawingA2.37.DrawinganEarthellipse.Asimpleway todrawa large-scaleapproximateellipseonthegroundisbybeginningwithavesicaconstruction(1).In(2)radiiaredrawnfromthecentertotheC1circumference.(3)Verticalsaredrawn from the radii/inner circle (C4) intersections and horizontals are drawnfromradii/outercircle (C3) intersections. (4)The intersectionsofverticalsandhorizontalsformapathonwhichanellipsemaybedrawn.

DrawingA2.38.Amongtheinnumerableformsthatseedstake,acommonshapecan be drawn from a vesica combinedwith a circle or an ellipse. The vesicaconstruction enables us to create seed forms from the numbers of musicalintervals or, in the present case, fromϕ-based numbers. In (1) the seed is acompositeofthevesicaandtheC4circle.Itsareaisabout1/7thoftheenclosingsquare. Itsheight-to-width ratioequalsϕ2/2.Theareasof seedandC4circleare in1.236=2/ϕ ratio. (2)Acompositeofa vesicaandavesicaellipse. Itsheight-towidthratioequalsϕ. (3)The twoseedsareconjoinedwitharearatioa/b=4.0.

DrawingA2.39.Theshapeofeggs.Harmoniouseggshapescanalsobedrawnfromthevesica/ellipseconstruction.Herethetoppartoftheeggisahalfellipsederivedfromtheϕ-tritonevesicaconstruction.Thebottompartisahalfellipsewithalongaxisthatextendstothevesicaellipsefocuscircle.Theaxisratiooftheegg=1/.691=1/(1+1/√5).

FiguresA2.I.Many(ormost)shapesinnaturesaremoreorlessasymmetrical,a quality that gives them a one-of-a-kind beauty: (a) pumpkin seed, (b)Mitracardinalisshell,and(c)acornsquash.Thisfactofnatureinspiresustocreateoffsetshapeswithharmoniousproportions.

Drawing A2.40. Asymmetrical harmonies. Two asymmetrical figures thatseparatelyandconjoinedproduceavarietyofϕfamilyratios.Theareavaluesareshownnexttothefigures.(1)HalfofthevesicaellipseiscombinedwiththeC3semicircleproducinganoffsetshapewithareaof.2427=1/(2ϕ×√ϕ).

Arearatios: a/b=1.4471=1/√5 a/c=3.236=2ϕ b/c=2.236=√5

(2)A composite of the ellipse at the .691 tonal position and the ellipse at the.618tonalposition.Theareaoftheoffsetshape(c)=.15=1/2ϕ2×√ϕ.

Arearatios: a/b=1.236=2/ϕ a/c=5.236=2ϕ2

b/c=4.236=ϕ3

(3)Combining the figures in (1) and (2)generates a figurewith the followingratios.

a/b=1.4472 b/d=15.330=ϕ4×√5 a/c=13.706=2ϕ4 b/e=3.618=ϕ×√5 a/d=22.186=2ϕ5 c/d=1.618=ϕ a/e=5.236=2ϕ2 e/c=2.618=ϕ2

DrawingA2.41.Moon octaves. TheMoon in its phases has amusical qualitymanifestedinthesteadyadvanceofthedark/lightline,theterminator,acrossitsdiameterimaginedasamusicalstring.Inthisdiagramweseethediatonictonalpositions of theMoon as its string/diameter is stopped at specific points andplucked.(Ofcourse,allpossibletonesaresilentlysoundedduringthelunation,tonesthatcorrespondtoeachsecondofthemonthlycycle.)

Fouroctavesaresoundedduringthecourseofamonth.Withthisphenomenonwehaveamanifestationofbothtimeandspacemeasuresexpressedasaunity.For example, at the fifth (G) position between new and first quarter, theterminator touches theMoon diameter of 2,160miles at the 1,440mile point(2/3×2,160=1,440).Thenumber1,440isakeynumberintimemeasurementbeingthenumberofminutesinaday.Itisalsofourtimesthenumberofdegreesinacircle;and144is1/6of864,theSunnumber(i.e.,Sundiameter=864,000miles).

The 2,160-mile standard chosen for theMoon has, as previously discussed, atemporalsignificancesince2,160 is thenumberofyears inaGreatMonth,orAge, of the precessional cycle, the Great Year of 25,920 years. Thus ourintervalsseemtobesteppingoffcosmictimeduringthelunationcycle.

Besides these indicationsof the time-spacequality of the lunationwenote thewell-knownphysicalandemotionalinfluencesthattheMoonhasonlife-forms,including its effects on the human psyche. The creative process may also beenhanced when the creator becomes aware of the lunar cycles. From theauthor’s experience the time from new to first quarter is a time of seeding ofideasandofinspiration;fromfirstquartertofull,atimeofactionandcreation;fromfulltolastquarter,atimeoffinishingup;fromlastquartertonew,atimeofrest,contemplation,andintegration.

Drawing A2.42. TheMoon at theϕ tritone. Four tonal regions make up thelunation as described in drawing A2.41. Here we have concentrated on theshapeoftheMoonattheϕ-tritoneposition(betweenFandG)inregions2and3during thewaxingandwaninggibbousperiods of the lunation inwhich thelight and dark lunar areas are in a 2:1 ratio. This configuration results inseveralϕ-familynumbersasshown.

Wehavediscussedtheharmoniousgeometricpropertiesof thisϕ-tritonetonalpositionthroughoutthebook.Hereweseeitappliedtotherelationofthelight

todarkareasoftheMoonduringthelunation.Theϕtritoneisatransitionstagebetween, on the one hand, the complete darkness of the new moon and thebrillianceof the fullmoon;and,on theotherhand,betweenthe fullmoonandthereturntothedarknessofthenew.Inbothstagesthelightpredominatesoverthedarknessinaratioof2to1.

For the creatorworkingwithin lunar time the first of these two stages occursduringatimeofconcentratedworkthatisilluminatedbythegrowinglight.Thesecondstageoccursduring the timeofperfecting theworkbefore thegrowingdark prevails and the creator returns to rest within the profound peace ofstillness,ofdarkness.

FigureA2.J.

Theunseendarkplaysonhisfluteandtherhythmoflighteddiesintostarsandsunsintothoughtsanddreams.

RABINDRANATHTAGORE5

ENDNOTES

INTRODUCTION.AHARMONYTHATSTRUCTURESTHEUNIVERSE

1.VonFranz,NumberandTime,286.

CHAPTER1.THEFIELDOFHARMONY

1.Vitruvius,Dearchitectura, quoted inNeedham,Science andCivilisation inChina,4:202.

CHAPTER2.THEGEOMETRYOFTIME

1.Daniélou,MusicandthePowerofSound,156.

2.Leet,UniversalKabbalah,5.

3.Hambidge,TheElementsofDynamicSymmetry,84.

CHAPTER3.MOONANDEARTH

1.Eliade,TheSacredandtheProfane,156.

2.McClain,“TheShih-Chi,”142.

3.Needham,ScienceandCivilisationinChina,4:200.

4.LevarieandLevy,Tone,56.

CHAPTER4.THETEMPLEOFWATER

1.Thom,MegalithicSitesinBritain,2.

2.Ibid.,2–3.

3.Bentov,StalkingtheWildPendulum,110.

4.Laszlo,TheAkashicField,70.

5.Villoldo,TheDanceoftheFourWinds,116.

CHAPTER6.THESPIRALINGOCTAVE

1.JohnMichell,TheDimensionsofParadise,196.

2.Eddington,StellarMovementsandtheStructure,244.

CHAPTER7.HARMONICSTRUCTURES

1.Alberti,TheTenBooksofArchitecture.

2. The list of proportions of concert halls is taken from a chart in Tone, byLevarieandLevy,186,figure65.

3.Hale,SacredSpace,SacredSound,6.

4.Kayser,Akroasis,77.

CHAPTER8.THECIRCLEOFRESONANCE

1.Kayser,TextbookofHarmonics,1:29.

2.Heath,Richard,MatrixofCreation,42.

3.West,SerpentintheSky,40,42.

CHAPTER9.THEMUSICALUNIVERSE

1.Feynmann,TheCharacterofPhysicalLaw,57.

2.JohnMichell,TheDimensionsofParadise,197,199.

3.Kepler,MysteriumCosmographicum.

CHAPTER10.THETREEOFTIME

CHAPTER10.THETREEOFTIME

1.Eliade,TheSacredandtheProfane,81.

2.Munn,TheTransformationofSubjectsandObjects,59.

3.Campbell,TheHerowithaThousandFaces,234.

4.Kayser,TextbookofHarmonics,1:21.

5.Proskouriakoff,AStudyofClassicMayanSculpture,111.

CHAPTER11.THECOSMICFLOWER

1. Bachofen, Grabersymbolik der Alten, quoted in Kayser, Textbook ofHarmonics,lxxvii.

2.Lawlor,VoicesoftheFirstDay,37.

3.GeorgeMichell,TheHinduTemple,73.

4.Bentov,StalkingtheWildPendulum,151.

APPENDIX1.THEVESSELOFCEREMONY

1.JohnMichell,TheDimensionsofParadise,13–14.

2.Villoldo,TheJourney,41.

APPENDIX2.AMISCELLANYOFNUMBERSANDFORMS

1.Pennick,SacredGeometry:SymbolismandPurpose inReligiousStructures,84.

2.Frankl,GothicArchitecture,20.

3.Eliade,AHistoryofReligiousIdeas,1:3.

4.LevarieandLevy,Tone,199.

5.Tagore,TheEnglishWritingsofRabindranathTagore,1:475.

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Pennick, Nigel. Sacred Geometry: Symbolism and Purpose in ReligiousStructures.SanFrancisco,Calif.:Harper&RowPublishers,1980.

Proskouriakoff,Tatiana.AStudyofClassicMayanSculpture.Washington,D.C.:CarnegieInstituteofWashington.Pub.593.1950.

Thom,Alexander.MegalithicSitesinBritain.Oxford:OxfordUniversityPress,1976.

Tresidder, Jack. The Complete Dictionary of Symbols. San Francisco, Calif.:ChronicleBooks,2004.

Tagore, Rabindranath. The EnglishWritings of Rabindranath Tagore. Vol. 1,Poems. Edited by Sisir Kumar Das. New Delhi: Atlantic Publishers andDistributors,Ltd.,1994.

Villoldo,Alberto.TheDanceoftheFourWinds.Rochester,Vt.:DestinyBooks,1995.

———.TheJourney.CoconutGrove,Fla.:TheFourWindsSociety,2004.Von Franz, Marie-Louise. Number and Time: Reflections Leading toward a

UnificationofDepthPsychologyandPhysics.Evanston, Ill.:NorthwesternUniversityPress,1986.

Vicentino, Nicola. Ancient Music Adapted to Modern Practice. New Haven,Conn.:YaleUniversityPress,2011.

West, JohnAnthony.Serpent in the Sky: TheHighWisdomofAncient Egypt.Wheaton,Ill.:QuestBooks,1993.

ILLUSTRATIONCREDITS

Frontispiece.TheFirstToneofPlainsongfromWilliamFleming,ArtandIdeas(NewYork:HoughtonMifflinHarcourtSchool,1986).

CHAPTER1

Fig. 1B (a). Calendar Stone, from Nicholas Fry, Treasures of World Art(London:TigerBooksInternational,1986),218.

Fig.1B(b).Westrosewindow,NotreDamedeParis,fromMarcelAubert,ArtoftheHighGothic(NewYork:CrownPublishers,Inc.,1965),30.

Fig.1B(c).Pidisc,fromWenFongandJamesC.Y.Watt,PossessingthePast:Treasures from theNationalPalaceMuseumTaipei (NewYork:MetropolitanMuseumofArt;Taipei:NationalWestrosewindow,NotreDamedeParis,fromMarcel Aubert, Art of the High Gothic (New York, Crown Publishers, Inc.,1965),30.

Fig. 1C. Dividing cell, from Boyce Rensberger, Instant Biology (New York:BallantineBooks,1996),97.

CHAPTER2

Fig. 2A. Photo of Apollo temple, photos: Henri Stierlin; text: RolandMartin,LivingArchitecture:Greek(London:OldbourneBookCo.Ltd.,1967),101.

Drawing2.4.Apollotempleplan,fromIsabelHoopesGrinnell,GreekTemples(NewYork:MetropolitanMuseumofArt,1943),30.

CHAPTER4

Fig.4A,Fig.4B.Waterdrop,photosbyOscarKreisel,textbyJacobBronowski,TheAscentofMan(Boston/Toronto:LittleBrown1973),185.

Drawing 4.4 (1). Long Meg and Her Daughters, from Alexander Thom,MegalithicSitesinBritain(Oxford:OxfordUniversityPress,1976),151.

Drawing 4.4 (3). Flattened circles, fromAlexander Thom,Megalithic Sites inBritain(Oxford:OxfordUniversityPress,1976),28.

CHAPTER5

Drawing5.5(1).JarfromSusa,fromTrudiKawami,AncientIranianCeramics(NewYork:TheArthurM.SacklerCollection,1992),18.

Drawing 5.5 (2). The Sanctuary Rhyton, from Nikolaos Platon, Zakros (NewYork:CharlesScribnerandSons,1971),165.

CHAPTER6

Fig. 6A (b). Begonia ‘Duartei,’ from Kjell B. Sandred, photos, and GhilleanTolmePrance,text,Leaves(NewYork:CrownPublishers,1985),17.

CHAPTER8

Drawing8.7(1).Doryphoros, fromH.A.Groenewegen-FrankfortandBernardAshmole,ArtoftheAncientWorld(EnglewoodCliffs,N.J.:PrenticeHall,1972),318.

Drawing 8.7 (2). Amiens plan, from Paul Frankl, Gothic Architecture (NewHaven:YaleUniversityPress2001),91.

Fig.8A.Bruno’sFiguraMentis, fromFrancesYates,GiordanoBrunoandtheHermeticTradition(Chicago:UniversityofChicagoPress,1964),306.

CHAPTER10

Fig. 10A. Chart of the Periodic System of Evolution of theMayan Calendar,from Carl Calleman, The Purposeful Universe (Rochester Vt.: Bear andCompany,2009),113.

CHAPTER11

Fig. 11C (a). Fly flight muscle, from Jeremy Burgess, Microscosmos(Cambridge:CambridgeUniversityPress,1987),119.

Fig. 11C (b).Beehive, fromP.Passerin d’Entreves,TheSecret Life of Insects(London:OrbisPublishing,Ltd.,1976),142.

Fig.11C(c).Saturnhexagon,NASA.

Fig.11E(a–f).Diatoms,fromF.E.Round,R.M.Crawford,andD.G.Mann,TheDiatoms:BiologyandMorphologyof theGenera (Cambridge:CambridgeUniversityPress,1990),218,200,189,211,350.

APPENDIX2

Fig. A2.A. Athelstan Psalter fromMichelle Brown,Anglo-SaxonManuscripts(Toronto&Buffalo:UniversityofTorontoPress,1991),29.

Fig. A2.B. Tomb of Dona Sancha, from Roberto Salvini,Medieval Sculpture(Greenwich,Conn.:NewYorkGraphicSociety,1969),84.

Fig. A2.C. Porta della Mandorla, from Howard Hibbard, Masterpieces ofWesternSculpture(NewYork:HarperandRow,1966),figure33.

Fig.A2.E.CairoMosqueofal-Hakim,JohnHoag, IslamicArchitecture, (NewYork:HarryAbrams, 1975), 143, plate176.From the seriesHistoryofWorldArchitecture,PierLuigiNervi,generaleditor.

Fig.A2.F.AmiensCathedralportal, fromLouisGrodecki,GothicArchitecture(NewYork:RizzoliInternationalPublications,1978),66,plateVI.

DrawingA2.22.MenkaurefromPaulJohnson,Art:ANewHistory(NewYork:HarperCollins,2003),29.

DrawingA2.23.CasselApollo,fromH.A.Groenewegen-FrankfortandBernardAshmole, Art of the Ancient World (Englewood Cliffs, N.J.: Prentice-Hall,1972),280.

DrawingA2.24.MediciVenus,fromEleanorMunro,TheGoldenEncyclopediaofArt(NewYork:GoldenPress,1961),43.

DrawingA2.25.AphroditefromFrejus,fromH.A.Groenewegen-FrankfortandBernardAshmole,Artof theAncientWorld (EnglewoodCliffs,N.J.:Prentice-Hall,1972),335.

DrawingA2.26.David,fromWilliamFleming,ArtandIdeas(NewYork:HoltRinehart&Winston,1974),181.

Drawing A2.27.Manikkavachakar, from Balraj Khanna and George Michell,HumanandDivine:2000YearsofIndianSculpture(London:HaywardGalleryPublishing,2000),49.

DrawingA2.29.TheOsirion, fromRobertLawlor,SacredGeometry (London:ThamesandHudson,1982),61.

DrawingA2.30. Vaikunthaperumal Temple, fromGeorgeMichell,TheHinduTemple(NewYork:HarperandRow,1977),135.

Fig.A2.I (b).Mitracardinalis shell, fromA.GordonMelvin,Seashellsof theWorld(NorthClarendon,Vt.:TuttleBooks,1966).

AbouttheAuthor

JOHNOSCARLIEBENisageometerwhohasresearchedsacredgeometryandharmonics for over 25 years. A graduate of New York University, he alsostudied at Oregon School of Arts and Crafts and Roehampton College inLondon.Adesignerandbuilderofgardensandceremonialspacesproportionedbysacredgeometry,helivesinBellingham,Washington.

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INDEX

Allpagenumbersarerefertotheprinteditionofthistitle.

Aborigines,138Actinoptychus,157AkashicField,53,54,152,154,

158Alberti,LeonBatista,95–96AmiensCathedral,110,209Aphrodite,219Apollotemple

defined,26

dimensions,26

groundplan,26–27

ground-planratio,31,47

rectangle,drawing,28,29arches,209

arcs,16,31,55–56,154,168,216arithmeticmean

defined,90

formulafor,90

illustrated,91ArtemisatEphesustemple,30“asabove,sobelow,”59,116–19,152ashleafvesica,73Asterionella,157Asterolampra,157asymmetricalharmonies,231Athelstanpsalter,206,207Aulacodiscus,157

Bachofen,J.J.,155Bentov,Itzhak,54,160biologicallife,beginningof,142BirthoftheCosmosCeremony

beginningof,187

defined,187

drawings1and2,188

drawings3and4,189

drawings5and6,189

drawings7and8,190

drawings9and10,190

drawings11and12,191

drawings13and14,191

ending,188

guidedjourney,187–88

vesselstructure,187BirthoftheMoon,The,

defined,104

illustrated,107

modernscientificstory,174

momentofbirth,107

overview,104

parts1-6illustrated,105

parts7-9illustrated,106BirthoftheMoonCeremony,

The.Seealsoceremony

drawing15,175,180

drawings1and2,177

drawings3and4,177

drawings5and6,178

drawings7and8,178

drawings9and10,179

drawings11and12,179

drawings13and14,180

geometricspaceconstruction,174

geometryconstruction,175

guidedjourney,175–76

offerings,177

overview,174

Sunin,175

toning,177body-as-monochord,110book-pagelayouts,213Bruno,Giordano,112,113

CairoMosqueofal-Hakim,208Calleman,Carl

Mayancalendar,6,127,128

PeriodicSystemofEvolution,

128,149

tun-baseddurations,129

underworldsand,127–28CasselApollo,217,222CellularUnderworld,130,137,

144ceremonialspaces,165ceremony,158–61

BirthoftheCosmosCeremony,

187–92

BirthoftheMoonCeremony,

174–80

Earth-HonoringCeremony,

167–73

experienceofsacredthrough,

165

formalprayersin,166

asformalrequestforhelp,

163–64

withingeometricspace,164–66

overview,6–7

sacredparticipationin,163

spiritualrealitiesinvokedby,

164

Sun-HonoringCeremony,

192–97

assymbolicprayer,164

theaterof,164–65

trustand,166

vesselof,163

Water-HonoringCeremony,

181–86chacana,62,63,65,182chromaticscale,18–19circle-in-the-square

firststepsingenerating,9

ashumanbeing,2

asseedofvesicaconstruction,3

TempleofWater,61circleoffifths,135circles

withinacircle,10

drawing,9

flattened,51

ofhigheroctaves,16

inholdingsacredspace,165

innerEarth,43–45

multipleintersectionsof,54

nested,16

orvesicaconstruction,100–101,

198–99

primordialpoint,13–15

resonance,102–3,108

inseedform,230

TempleofWater,57,60–65

ofvesicaconstruction,17

water-area,47–48circlesquaring,8,108,216classicalgeometry,3,166Cohen,Leonard,79

“comma,”79,80,82contemplation,3contraharmonicproportion,97correspondence

exampleof,116,117

expressionthroughmetaphor,

118

inharmonicscience,117

inverse,118

revivalof,116–17

systemsof,117–18

water,121correspondenceprinciple,6cosmicformationtime,139CosmicSoul,143cosmictime

concept,127

cycles,2

fromMayancalendar,6Creation

honoringandappreciating,165

lunarcyclein,140

primaryperiodsin,139

temporalpatternof,145

vesicalconstructionasmodel

of,104CreativeSpirit,154,157,158crossed-thetavesicaconstructions,

110crossed-vesicaconstruction,111,

216,217,218,220,221crossedvesicas,109–13,223,224crossings,108–13crystallineelementarymatter,

114

David(Michaelangelo),220,222diabolusinmusica,24diagonals

division-by-fiveweaving,202

division-by-fourweaving,202

division-by-threeweaving,201

full-square,200

half-page,212

half-square,200,210

quarter-square,200,210directions,callingin,167–68division-by-fivediagonalweaving,

202division-by-fourdiagonalweaving,

202division-by-threediagonal

weaving,201

DNA,94DomeoftheRock,157donAntonio,55DonaSanchatombcarving,206doublesquare,timeproperty,31doublinggeometricprogression

series,84Dreamtime,143,153,156–57

defined,138

duration,145

inlunarcycle,139,140

seedgestationduring,160droparch,209

Earth

attaininglifeandconsciousness,

123

crustasfloating,47

diameter,75,77

ellipse,229

forGreeks,121

“ideal,”79

inner,43–45

innercorediameter,124,125

Marsconstructioninrelation

to,69

measuredversusideal,80

meteoritebombardmentof,137

orbitasmeasureofallthings,123

outercorediameter,126

relativesize,76

Spiritof,169

inSun-HonoringCeremony,193

tonalratio,67

Venusconstructioninrelation

to,68

Venusdiameterand,43

vessel,70EarthandMoon

diameterratio,41–43

eclipse,35

geometricpresenceinside,43

ratio,36,108,181

size,34Earth-HonoringCeremony.See

alsoceremony

callinthedirections,167–68

onclearedlevelground,168

drawingdecoration,168

drawings1and2,170

drawings3and4,171

drawings5and6,171

drawings7and8,172

drawings9and10,172

drawings11and12,173

drawings13and14,173

forms,168

guidedjourney,169–70

individualoffering,168

indoors,168

meditation,170

offerings,170

overview,167

purposeof,169Earth-Moon-Sunsystem,5,43Earth-Sunconstruction,75–76,

77–78

diameters,75,78

ratio,78,85

vesica,75,76eggs,2308/11ratio,19,21,26,106,214,215electromagneticfields,54Eliade,Mircea,32–33,129,217ellipse,7,99,231

drawingsdescribingshape,229

Earth,229

eggshapeand,230

inseedform,230

vesicaand,228emergenceoflife,137EraofBiologicalLife,188euthynteria,26,29

Feynmann,Richard,116–17Fibonaccinumbers,226Fibonacciseries,227fifth

doubleoctave,89

earthelementand,121

foundationtimeand,145

frequencyratio,83

generationof,13,68

grave,70

harmonicsandtonalratio,

17–18,19

number3and,154

perfect,71,82,92

Pythagoreanscaleand,83

inseventonescale,37–38

sounding,18

spiral,83

thirdharmonicand,119

undecimalaugmented,21

vesicaconstruction,17

vesicapiscisgenerationfrom,

113,119–20FiguraMentis,112,113firstoctave,15,16,19,78,1355/6ratio,97FlowerofLifesymbol,53,54,146,

154fourthoctave,19,135,142Frankl,Paul,208frequencyratios,17–18,135“Fujiwharaeffect,”225full-squarediagonals,200fundamentaltone,18,19,23,68,

82,89

GalacticCenter,122,126,165GalacticTemple,124,125geometricmean,91,92geometricspaces

ceremonywithin,164–66

defined,164

transformedtosymbols,164geometrictritone,14

geometry

artof,2

classical,3,166

derivedfrommusicalscale,2

asharmonicstool,88,89,90

reciprocalconcept,92

seedof,160

ofsolareclipse,151goldenratio,138,158,211goldenrectangle,87Gothicarch,209grace,164gratitude,165–66GreatCentralSun(GCS)

clues,122

defined,122

Earthinnercorediameterand,

124,125

massandtemperatureatEarth

centerand,124

number,125

symbolicnumber3168,122–23GreatPyramid,157,205,225

Hale,SusanElizabeth,97half-pagediagonals,212half-squarediagonals,200,210harmonicmean

defined,90

examplesof,90–92

formula,90

illustrated,91harmonics

defined,1,88

disciplineof,2–3

asintegralpartsoftone,89–90

studyof,2

tableof,19

toolsof,88–89harmonicscience,113–15

correspondenceincontextof,117

temporalformsand,145harmonicseries

arisingof,89

membersof,89

nearthelimitof,159

octave,fifth,andthirdand,18

overview,5

reciprocal,92

vesicaconstruction,96harmonicstructures,88–99harmonicsymbology,153harmoniousrectangles,26–31harmony

ofEarth-Moonsystem,4

fieldof,8–19

thatstructurestheuniverse,1Heath,Richard,106–8HeroofAlexandria,208hexagon

ingeometricalsymbol,158

inpotteryvesselconstruction,

114

√3properties,211hexagonaltimestructure,156higheroctaves,16

historicalpractices,7huangchung,35,41human,94hydrogenatom,63

icosahedron,120“idealEarth,”79innerEarth,43–45innerSun,153inversecorrespondences,118

Jenkins,JohnMajor,23Jupiter

Moonand,106

number25,107–8

synodicperiod,107

Kayser,Hans,92,97,145Kepler,Johannes,123

land,waterand,47–49Laszlo,Ervin,54–55,154Lawlor,Robert,158leaves

ash,73

primal“tones”of,72

purplebeech,73

scaleof,72

southerncatalpa,73

asvesicaform,72Leet,Leonora,24–25,121letter-widthmodules,214–15LongCount,147–48lunarcycle.

SeealsoMoon

creativeprocessand,232

defined,20

Dreamtimeand,139

inuniversalcreation,140lunardiameter,20

macrocosm,153majorsecond,19,146majorseventh,19,69,73,201majorthird

harmonicsandtonalratio,19

meantones,135,138

number5,155

string-lengthratio,135

vesicaconstructionfrom,

60,86

watermoleculeand,4,119–23,

133MammalianUnderworld,137Mannikkavachakar,221marginratios,212Mars

inrelationtoEarth,69

tonalratio,67

Venusratioand,68

vessel,71Mayancalendar

cosmictimefrom,6

durationsof,129

evolutionaryphase,140

hiddenpropertiesof,130

LongCount,147–48

majorthirdandwaterand,133

musicalaspectof,141–42

numbersof,141

PeriodicSystemofEvolution

of,128

ShipofTimeTempleformed

from,136

sowingtimeof,138

tun-basednumbers,130MayanCalendarRound,24Mayanculture

centerofthecosmosand,23

nine“underworlds,”127–28

Teotihuacan,148

Tezcatlipoca,148,149

tzolkin,23,147Mayantime,132Mayantritone,53,146mean,90–92meantone,134–35,137MediciVenus,218,222Mercury

relativesize,67

tonalratio,67

vessel,70mesocosm,93Michell,George,160Michell,John,34,35,164microcosm,153mimesis,181minorsecond,19,83,146minorseventh,19,70,71,201minorsixth,19,70,82,83,201,

202minorthird

characteroflonging,97

crossed-vesicaconstructionand,

111

5/6interval,97,143

groundplanlayoutand,5

harmonicsandtonalratio,19

TreeofTimeand,141,142,

143,146

vesica-harmonicconstruction,

97–99

inwidth-to-heightratio,96“mitoticspindle,”104monochord,15Moon

areaofcircle,42

birthof,104–8

characteristicsof,32–33

diameter,20,79

eclipseshadow,35

eclipsingSun,150–51

feminine,powerof,175

geometricpresenceinsideEarth,

43

asGrandmotherMoon,32–33

Jupiterand,106

asmeanterm,94–95

new,140

number,84

octaves,232

atφtritone,233

physicalandemotional

influencesof,232

inshapingEarth,33

Sunand,119

Sun/Earthreciprocaland,

80

tonalratio,67

vessel,71Moon-Earthsystem,32Munn,NancyD.,138musicalintervals.

Seealsospecific

intervalsandtones

concerthalldesignand,96

numbersof,13

ratiosofFibonacciseries,227musicalnumbers,118musical-numericalconception,2musicalrectangles,95–96musicalstring

firsteightharmonicdivisions

of,89

intervaloffifthon,13

rateofvibrationsof,18musicalvessels

overview,66–67

pottery,67–72

worldingrainofsandand,72

nature

asymmetricalshapesin,231

cyclesof,116

intersectionofwavesand,3

musical-numericalconception,2

musicalsystemsfrom,42

organizingforcesof,155

ringshapein,10

nautilusshell,86Needham,Joseph,34–35nestedsquares,224neutralseventh,21number2,43number3,43,154number5,38,155,156number11,21,85number20,23,133number73,21,24,147–48number108,77number216,84number366

areaoftheMooncircleand,42

frequencygeneration,42

importantroleof,4

asnumberofEarthrevolutions

inoneyear,21number.691,27,29number864,50,79,84number1,232,33number2,160,20–21,24,37,

39–42,79–80,119,232numbers

“5-limit,”84

derivedfrommusicalscale,2

asharmonicstool,88

ofMayancalendar,141

ofmusicalintervals,13

formusicalscale,43

octavedetermination,113

octavesystemof,42

astones,113

octaveratio,96octaves.Seealsospecificoctaves

asbeginningpointofvesicas,18

constructionof“idealEarth,”79

Moon,232

systemofnumbers,42

vesicaformswithin,74octavespirals

offifths,83

minorsixthtone,82

progression,82

inthreedimensions,86

tonesegments,82one-from-twomodel,104one-to-threeharmony,51organization,thisbook,3–7Osiriontemple,223oxygenatom,63

Palladio,Andrea,95PappusofAlexandria,91Parthenoneuthynteria

construction,92

constructionillustration,30

dimensions,29

groundplan,29

φtritone,29

tritoneconstruction,31PathofFormation,188pentagon

inTheBirthoftheMoon,107

liferepresentationby,158

φ-familyproperties,211

asvesselofthegoldenratio,211PeriodicSystemofEvolution,128,

145,149Perissonoe,157PharaohMenkaure,216,221phenomena

arranginginsystemsof

correspondence,116

correspondencesamong,88

diverse,relationof,116

FlowerofLifesymboland,53

ringsymbolconnectionof,10phi,20–21φtritone

construction,28,29,205

defined,29

intersection,30

Moonat,233

octaveconstructionat,81

sizesofEarthandMercuryand,

67

waterdropshapegeneration

from,52φweaving,203,204πvalue,141Plato,78polygonaldiatoms,157PortadellaMandorla,206potteryvessels

areasof,68

compositeconstructionof

hexagonandoctagon,114

Earth,70

illustrated,69

Mars,71

Mercury,70

Moon,71

forninetones,210

planetary,67–72

ratiosamongtheareas,71

Susa,thirdmilleniumBCE,69

tones,68

Venus,70prayers

inceremonies,166

formal,164,166

thinking,164PrimordialExpansion,138,139,

153,156prokaryotes,emergenceof,139Proskouirakoff,Tatiana,148purplebeechleaf,73Pythagoras,164Pythagoreancomma,82Pythagoreanscale,84,146–47Pythagoreansemitones,38Pythagoreanseven-tonescale,

37–39Pythagoreantritone,26,48

quadrature,109quantumvacuum,153,154quarter-squarediagonals,200,210

raindrops

fallingintowater,53–54

shapeof,49–56reciprocal

concept,92,93–94

constructionillustration,93

constructions,93–94

ratioformsand,92–93

relation,94rectangles

golden,87

harmonious,26–31

musical,95–96rectangularspacedivision,212redwoodtreeroots,54resonance

asbindingforce,102

circle,102–3,108

defined,101

Kayserviewof,101–2

overview,5

principle,5,101–2ringform

defined,10

illustrated,11

vesica-within-a-vesica,207rootchakra,216,218,221RootofTreeofTime,145–49

rootsystemintersections,54,56“rope-stretchers,”57

sacred

experiencethroughceremony,

165

feelingof,164sacredcosmology,6,152–58sacredspace

creating,167

shapesthathold,165scales

chromatic,18–19

existenceasmembersof,66

leaves,72

numbers,43

Pythagorean,84,146–47

seven-tone,18,37–39

twelve-tone,75secondoctave,19,135,142seeds,160,230self-similarity,117semicirclestring,159septimalfourth,1137/11tone,21,22seven-tonescale,37–39shamaniccultures,161shamanism,181sharpenedtritone,27siderealday,21siderealperiod,79“skypoint,”216solareclipse,150–51solarplexus(thirdchakra),222sound

generationof,103

asprimordialcreativeenergy,

102southerncatalpaleaf,73space

flattened-circle,51

geometric,164–66

measuresof,34–35

sacred,165,167

visual,12space-dividingtechnique,109spatialmeasure,20spiralingoctave,77–87spirals

golden,87

logarithmic,86

octave,82–83

pineconewith8/13system,226

straight-line,86,135

three-pointcirclearcs,86,87

time,134–37

“WostaofZalzal,”85SpiritoftheEarth,169√φvesicaconstruction,205squaringcircle,8,181standing-stoneflattened-circle

rings,50straightline,9straightlinespiral,135string-lengthratio,135Sun

diameter,75,77

distancetoGalacticCenter,126

energygeneration,150

GreatCentral(GCS),122–26

Moonand,119,150–51

number,50,79,84

radiusnumber,84

relativesize,76Sun/Earthreciprocal,80

Sun-HonoringCeremony.Seealso

ceremony

altar,193

beginningof,193

drawings1and2,195

drawings3and4,195

drawings5and6,196

drawings7and8,196

drawings9and10,197

drawings11and12,197

Earthrepresentation,193

guidedjourney,194

overview,192–93

passingofEarthdisc,193

toning,194–95symbols

ofconsciousness,154

FlowerofLife,53,54,146,154

geometricspacetransformation

to,164

hexagon,158

invokingspiritualenergies,

160

inmusicalgeometry,153

pentagon,158

practicalfunctionrelationship,

164

triangle,158

vesicasas,100

yin-yang,119“syntoniccomma,”39

Tagore,Rabindranath,234taijisymbol,225Tao,103templebuilding,161templegarden

cosmicrelationshipswithin,59

defined,57

geometricdesign,58

overallshapeof,58

φ-familydimensionswithin,57

vesicasymbol,57

TempleofWater

chacana,62,63,65

circle-in-the-square,61

constructionmeasurements,65

Earthsquare,61–62

4/5node,62

groundplan,57

hydrogenatom,63

innerangleofmajor-third

vesica,63

largercircleonaxis,62

linesbetweenpolesandequator,

60,61

molecularcomponents,64

Mooncircles,57

Moongeneration,60,61

orientation,65

oxygenatom,63

raisedgardenbed,65

simpledesign,64

watermoleculemodel,61,64templespace,165theaterofceremony,164–65Theia,104thirdharmonic,119thirdoctave,18,19,21,106Thom,Alexander,513/11tone,21,22threemasters

part1,36–37

part2,39–41threeness,158time

centerof,23–24

cosmic,2,6,127,139

erabefore,138–43

hexagonalstructure,156

measuresof,34–35

primaryperiodsinuniversal

creation,139

propertyofdoublesquare,31

rootof,145–49

spiralof,134–37

TreeofTime,130,143–45

vesicaof,137–38TimeTemple,136tonalnodes,13,15–16tonalratios.

Seealsospecificratios

generating,3–4

table,19tone,17–19

Chineseand,34–35

frequencyratiosand,17–18

asharmonicstool,88–89

octave,18

vesicashapeand,16tones.Seealsospecifictones

book-pagelayoutsbasedon,213

descendingglissandoof,15

nineteenharmonics,18,19

numbersas,113

ofPythagoreanscale,84

ofstandardscale,18

twenty-seventhharmonic,18,19

undecimal,85

whole,38toning,177,182,194–95TreeofTime

construction,144

dayoneoftheMayancalendar,

145

defined,130,143

harmonies,143

Rootof,145–49

structureof,143–44TreeofTimeCeremony.Seealso

ceremony

beginningof,187

defined,187

drawings1and2,188

drawings3and4,189

drawings5and6,189

drawings7and8,190

drawings9and10,190

drawings11and12,191

drawings13and14,191

ending,188

guidedjourney,187–88

vesselstructure,187Triceratium,157tritone

compositionof,24

defined,23

Mayan,53,146

inmodernmusic,26

φ,28–30,52,67,81,205

aspointertowardssacredcenter,

25

Pythagorean,26,48

regionof,25–26

sharpened,27two-from-onemodel,102–3tzolkin,23,147

undecimaltones,85underworlds

beginningof,131,135,142

defined,127

durationof,130

imaginedascircleareas,132

PeriodicSystemofEvolution,

128

travelwithin,135unfolding

defined,150–51

primal,152

vesicapiscis,152

VaikunthaperumalHindutemple,

224Venus

Marsratioand,68

inrelationtoEarth,68

tonalratio,67

vessel,70vesica-circleformulas,132vesicaconstruction,11–17

circlesof,17,100–101,198–99

creativeprocesssymbolizedby,

101

crossed,111,216,217,218,220,

221

crossed-theta,110

defined,3

Earth-Sun,75–76,77–78

ingeneratingsizesofinner

planetsandmoon,5

inletter-widthmoduledesign,

214–15

majorthird,60,86

minorthird,98

asmodelofcreation,104

propertyvaluesofcircles,198–99

Pythagoreantritone,26

resonanceprinciplein,5

seedformof,3

√φ,205

varietyof,17

waterdropshape,49–51

inWater-HonoringCeremony,

181

water-to-landratio,4vesica-constructionmatrix

establishmentof,3

generationbytonalratios,

100–101

octavegenerationof

fundamentaltone,82vesicaform

introductionto,11

symbolof,15vesica-harmonicconstruction,96,

97–99vesicapiscis

defined,11

generatedfromtonalfifth,56

illustrated,12

withinscribedrhombus,112

twowavesintersectingateach

other’scenters,158

unfolding,152vesicas

arcs,74

crossed,109–13,223,224

Earth-Sun,75,76

ellipsesasfulfillmentof,228

formswithinoctave,74

generation,13–15

height-to-widthratio,16,82

joiningtoanotherperson

within,11

leaves,72–76

nested,ofstandardtwelve-tone

scale,75

nestedgroupingof,75

octaveasbeginningpoint,18

assymbolforform,100

oftime,137–38

oftwelvetones,74vesicashapes

infinitevarietyof,3

toneand,16

typesof,16

visualspacewehaveincommon,

12vesica-toneconstruction.Seevesica

constructionvesica-within-a-vesica

innerandoutervesicaratio,208

ringconcept,207Villoldo,Alberto,175vonFranz,Marie-Louise,1

water

correspondences,121

tolandsurfaceproportion,47

majorthirdand,133

numericalsynchronicities,47

shapeofraindropsand,49–56

soulassociationof,46

TempleofWater,57–65waterandlandconstruction,

47–49water-areacircle,47–48waterdrops

Mayantritone,53

rationalforms,52

shapegeneratedbyφtritone,

52

shapeillustration,49

takingform,49

vesicaconstruction,49–51Water-HonoringCeremony.See

alsoceremony

chacana,182

drawing15,186

drawings1and2,183

drawings3and4,183

drawings5and6,184

drawings7and8,184

drawings9and10,184

drawings11and12,185

drawings13and14,186

overview,192

toning,182

vesica-toneconstruction,181watermolecule

construction,181

model,61,64

inTempleofWater,57waveinterference,55wave-interferencepatterns,56weavings

division-by-fivediagonal,202

division-by-fourdiagonal,202

division-by-threediagonal,201

φ,203,204wholetones,38WilkinsonMicrowaveAnistropy

Probe,138wind,shapeof,225“worldinagrainofsand,the,”153“WostaofZalzal,”85

yin-yangsymbol,119

Zalzal,52,85,164

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