Soundscapes & Other Worlds

8/12/2019 Soundscapes & Other Worlds

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Contents

Diagrams

page 2 Soundscapes and Other Worlds2 On Narrative and the Book3 Rhythmic Tools3 Open Music: Losing Control4 Texture Objects4 Heterophonies5 Alternative Notations5 Graphical Notation6 A Brief Guide to PC Set Theory6 Dramatis Personae7 Kh Sub-Complexes

8 Similarity Relations9-10 Mapping the PC Set Universe

11 New Tonalities12 An Analysis13 Dead or Live?13 Weaving the Web

5-6 Hauer-Steffens and derived notations7 Forte PC Sets and Kh Sub-Complexes8 Similarity Relations for same-sized sets9 PC Sets size 4 with shared triads9 PC Set 7-35= as a gauge of tonicality

10 End-sets

10 Interval-Class Vectors compared to End-sets12 PC Sets fromThe Bridge of Follies

Introduction

This collection of essays examines an of nearly 300compositions, grouped under 40+ cycles and written over a period ofmore than 20 years. Together they explore a variety of 'non-narrative' structures in which the linear development of idea orargument has been abandoned; instead the focus of attention shiftsalmost casually as if viewing an object or landscape from a newperspective or in a different light.

The influence of the spatial arts is evident everywhere but despite(or perhaps because of) this, the music is concerned above all withour perception of time - questioning the nature of change, chanceand coincidence - and with ideas of precognition and conflictingmemories. Thus all the pieces are open-form (allowing performerschoice in the ordering and shaping of events) and are of flexibleduration and instrumentation; this freedom is reflected in the highlyvisual layout of the scores, whose terse notation is designed to firethe imagination of players and lay bare the methods of composition.

The music employs pitch-class set theory to examine the universe of12-tone harmonies and to link these together to suggest newtonalities. Whilst some pieces apply a variety of textural ideas to asingle harmony, others relate new harmonies to an unchanging

texture; the latter will often juxtapose harmonies with differentdegrees of tonal 'loyalty', creating a sense of distance or movementthrough space. Yet other pieces transpose and recombine sets,kaleidoscope-like, to yield new background harmonies or landscapes.

Textures are based not so much on repetition as on reconstruction,mimicry and paraphrase. Recent pieces especially are impelled bythe so-called 'chaotic' patterns associated with natural processes andemploy huge leaps in register to suggest a myriad of unfurlingmelodies or 'journeys'; rhythm here tends to be non-metrical, withall beats in theory carrying an equal stress. Canonic and otheralgorithmic devices abound - retrograde, inversion, transposition,'key' signature change - as a means of generating a hoard of newbut kindred ideas.

Richard Cooke had the good fortune to studycomposition with David Lumsdaine, whose music and teaching havebeen an abiding source of inspiration. Later, with Helen Roe andPhilip Blackburn, he was a founder member of , anOxford-based composers' cooperative which pioneered new forms ofconcert programming and, during its first year of existence,premi red more than 50 new works. He has taught music in schoolsin Yorkshire and London, published numerous articles on new musicand has worked as a teacher of English in Italy and other countries.

uvre

The composer:

Soundpool


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The introduction on page 1 incorporates an openingparagraph which appears to make short shrift of'narrative' structures in music, whilst failing to explainwhat they are or to offer reasons for their rejection.This text is designed to remedy that matter.

Program-music, in the manner of Liszt or Strauss, is a

clear example of narrative: it aspires to depict eventsand arrange them according to some extra-musicalargument or theme. More generally, however, narrativemusic can be seen to be music which follows somekind of agenda - whether literary, ideological,documentary or driven by so-called self-expression orrhetoric - that is, it attempts to corral the listener orchannel him into one line of thinking.

Narrative, in its compression of time, simplifies andthereby distorts events in the real world; almost bydefinition it embodies a lie. It promotes the idea of theArtist as omniscient demi-god and, after a century

scarred by totalitarian excess, such projects andprograms should be viewed with suspicion. (Inliterature, it must be admitted, a plot need not be sosimple - it may embrace sub-texts and discontinuities,employ different narrators or invite imaginative leapson the part of the reader. It is also possible to claimthat certain individuals at certain times - Schoenberg,for example, who in Vienna witnessed some of themost tumultuous decades in history - may be forgivenfor feeling they have a message for mankind.)

For a composer, narrative is no more than a structuraltrestle and, in abandoning it, we are left with a music

which refuses to badger the emotions. Instead itattempts to create a small ironic space, one fromwithin which its audience can reflect on the affairs ofthe world; it may at times be likened to a paralleluniverse which the listener is invited to enter andexplore in order to make his or her own vitalconnections. From other points of view, it may be seenas a probe, a test, a free enquiry or a celebration ofthe complex adventure of life.

It is no exaggeration to say that music sinceRenaissance times has nearly always been narrative incharacter and no surprise to learn that composers have

chosen to publish it in the form of a book; by tradition,the musical score is laid out as a manual or series ofinstructions. That this has not always been the case isconfirmed by Baude Cordier's Circular Canon

or his equally intriguing, which is notated in the shape of a heart.

Both rondeaux appear in the early 15th-centuryChantilly Manuscript; they are perpetual canons andappear to share the medieval view of polyphony as amirror of the universe.

Dispensing with narrative liberates the score. It canreinvent itself or assume new guises: a list of

suggestions, a batch of anecdotes, an aide-mmoirefor improvisers, a game, a riddle, a toolkit or set ofbuilding blocks or an attestation to past performances.At its best - as is the case with Cordier - it addssomething to our understanding of the music itself orcan even stand as a work of art in its own right.

(Tout parcompas suy composs) Belle,bonne, sage

Soundscapes and Other Worlds

Soundscapes, soundwebs, sound tapestries, sound sculptures,music in four dimensions: there is no agreed term to span theintersection of music and the visual arts, either in English or inother languages (Klangarchitektur, Tonkunst, Tongemlde,Tonmalerei and so forth). Soundscape itself, following RaymondMurray Schafer, is commonly used to denote acousticenvironments in the real world, whether produced by naturalelements, animals, humans or machines; hence in music it tends

to mean composition using "found" sounds or sonic objects,something akin to .

For the musician, there are obvious ways in which this can beachieved, and some of these enjoy a surprisingly long history:

. Graphical scores are usually associated with John Cage

but date back at least to the late Middle Ages. Well-knownexamples include the heart-shaped rondeauand the circular canon , both byBaude Cordier, in the Chantilly Codex. Created around 1400,Cordiers designs are particularly satisfying as they give usinsights into the music itself.

. Melodic retrogrades and inversions can be seenas mirror images of the original line. The terminology isrevealingly visual: crab canon, sloth canon, mirror canon, spiralcanon. Puzzle canons hark back to Guillaume de Machaut, andBach wrote table canons (a retrograde inversion which can beread by two musicians seated on opposite sides of a table).

. This is non-narrative music where beginnings

and ends are unimportant and often arbitrary. The infinite canonor round (e.g. ) provides an early illustration,while Cubist-like cut-and-paste structures (as in Saties orsome of Stravinskys works , offer another solution. Normallythere is little sense of harmonic progression, consonance anddissonance, or tension and release.

By this we mean a clear separation in space ofplayers or sound sources. Again, this is hardly new and can betraced back to the Gabrielis, Tallis ( for eight 5-voice choirs) and, indeed, the first dawn chorus. In recordedmusic, stereo pan can be enhanced through a careful choice ofcontrasting timbres. Timbre, or tone-colour, was of course aparticular concern of Debussy, the Impressionists and others.

musique concrte

Belle, Bonne, SageTout par compas suy composs

Sumer is icumen inParade

)

Spem in alium

n the West seems to lean heavily towards the literaryrather than the pictorial arts, but this may simply reflect the longshadow of the Romantic movement, which still bulks large inconcert programming. Reinforcing this is the use of music inritual, often accompanying religious texts. Nevertheless there arefigures from more recent times - Kandinsky, Klee, Scriabin, Satie,Debussy, Cage and others - who have endeavoured to blur thedistinctions between sound and vision.

Listening may be lateral (between voices)or linear (in time). This music belongs not in a concert-hall ofcaptive audiences but in a gallery or series of spaces wherelisteners wander freely. This may be how Charles Ives envisagedthe performance of the and other works.The concert format looks authoritarian and ; it is thefreedom of the gallery that musicians covet.

Notation

Canonic devices

Static structures

Spatialisation.

Public presentation.

Unanswered Question

Music i

divisive

Amongst composers with close links to the visual arts we findSch nberg and Gershwin (both of whom painted), Mussorgsky,Hindemith, Varse and Feldman. Links to architecture are striking:

, in words commonly attributed toGoethe. Besides the Venetians, we have Nono, Stockhausen (notethe use of layers in or ) and Xenakis(who worked as an assistant to Le Corbusier), as well asspectralists such as Grisey and Murail. Musical painters, such asVermeer, Duchamp, Kandinsky and Klee, are fewer in number buttheir influence has been huge. While Picasso started from natureand gradually "removed the traces of reality" (quoted in HerschelB. Chipps , 1968), Kandinsky took music"unfettered by nature" ( 1911) ashis creative starting-point. It is Kandinsky whom we see as thefather of abstract art, and perhaps music is the mother.

Architektur ist gefrorene Musik

Kontakte, Gruppen Stimmung

Theories of Modern Artber das Geistige in der Kunst,

On Narrative and the Book


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Open Music: Losing Control

Indeterminate or open-form art - Calder's wind-drivenmobiles (1932), Earle Brown's(1953), Michel Butor's antinovels - has been around forsome time now but it remains a minority interestamongst creative artists. Nevertheless, Umberto Eco( , 1962) was surely right to argue that artwhich limits itself to a single unequivocal reading is less

likely to reward than art which is open, ambivalent orpolymorphic.

Determinate or "closed" art may betray a somewhattotalitarian mindset on the part of its creator or, moreprobably, a lingering Romantic view of the artist asdemi-god with a burning message for humankind. Inmusic and theatre it brings with it power structures andpecking orders - writer, conductor/director, performerand last (and in all likelihood least), audience. Whereart is commodity, artists pander to the predictablewhilst bureaucrats and middle-men flourish. Freedomcan be frightening and it often seems that scientists

alone have the courage to dream, pondering an infinityof universes, parallel worlds or new types of infinity.

The internet might be expected to change all this, sincechoice (interactivity) and variation (using randomprocedures) are things that the computer does best. Todate, however, the results have not been impressive,and this is doubtless because web technologies havebeen slow to develop. In music, Thomas Dolby's

and Sseyo's have quietly vanishedand in the visual arts, the world's most-used browserhas only recently lent support to Scalable VectorGraphics (SVG). It remains to be seen whether HTML 5,

with its native tag, will provide the stableplatform that artists require.

Freedom itself is open-ended and, since its inception,open-form music has ranged from the totally random -often based on word scores or graphic designs - to thetoken cadenza-type improvisatory interlude. (Ingeneral, composers have preferred to transfer authorityto - sometimes unwilling - performers rather thanlisteners; music which invites audience participation ismore properly defined as "interactive".)

The music here takes an intermediate approach, aiming

to give performers maximum licence consistent withpreserving the identity of the piece. Hence pitchstructure (harmony) is the parameter most likely to becircumscribed whilst timbre, register, tempo, dynamicsand event-order tend to remain free. The resultingscores represent a rich mix of text, graphics andtraditional notation.

Twenty-five Pages

Opera aperta

Beatnik Koan/Noatikl

Rhythmic Tools

Music distinguishes itself from the other arts in itsspecial relationship to Time; almost uniquely, it has thecapacity to release its audience from the tyranny of thechronometer or perhaps, in Thoreau's words, 'kill timewithout injuring eternity'.

How it does this is a matter for psychologists but it

seems that rhythm plays little part. Rather moreimportant is the rate of harmonic change or eventextural modulation or what is sometimes called the'super-rhythm' of a work but which in reality is its form.Rhythm, at least in this music, is concerned with thehere-and-now and not with expectations of things tocome; it is just another object - a pattern of durations -significant only as an engine of texture. Often it canhave a life of its own: the durational series of Messiaenor Boulez - derived from pitch series, but not heard assuch - are quite arbitrary.

Amongst the pieces here, four types of rhythmic

'construct' or device can be identified and these can besummarized as follows:

1 : Here performers play at differenttempi, usually linked by simple ratios such as(8):(4):2:1 or 3:2. These are canons at the 'unison' inboth senses of the word, that is to say, there is notranposition between parts and voices commencesimultaneously. combinesduple and hemiola relationships (6:3:2) but only

involves values which mighttax performers (4:3 or, more precisely, 16:12:9) andhere strict accuracy is not demanded.

2 : Three pieces fromare defined as canons by 'insertion', which

means that players may interpolate rests at variouspoints in the line; certainly where one tempo reigns, asin , the effect is similar to hocket.Hocket does not necessarily involve truncation: in

or , notes aresustained in one voice while the other part moves.3 : These are discussed in the program-notes for and .Players read a rhythmic 'ground' (a sequence of note-events and rests) in 'shifts' of varying lengths.4 : In , all pieces, on the

surface, share a consistent tempo characterized byequal quaver articulations. However, huge leaps inregister create the illusion of compound melodies builtof different durations (and intensities); perhaps theseare best described as 'virtual' rhythms.

The first two result from voice-exchange or theinterplay of parts, whilst the others relate to a singleline. Both (2) and (4) are purely local in effect whilst(1) and (3) could conceivably shape whole sections. Ineither case, changes to durational values are unlikely tohave major impact except where the total length of thepattern is redefined.

Lastly, we may speak of 'fuzzy' or 'casual' rhythms,which arise throughout . Playersmay be widely separated in space; whilst sharingsimilar material, they are not obliged to follow asynchronized beat or fixed tempo.

Time Canons

Hocketing

Shift Patterns

Streaming

An Embrace of Summer

Carnival of Poetry and Lies

The Cauldron ofPlenty

Chorus of Hesitations

Dream Odyssey Dominion of Light

Rain Talisman Shores of Contention

Music by Omission

The Book of Encounters

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Texture Objects

Texture is generally held to refer to the vertical aspects of amusical structure: it is conditioned both by the number ofsounding lines or voices (density) and by the nature of theinteractions between those lines. Such inter-relationshipsmay be characterized by agreement or altercation, imitationor independence and they are heard largely in terms ofrhythm. Timbre, however, is also a crucial factor - especiallyif we extend that term to include register and spacing - as it

governs the relative projection of a line within a texture.This again can be influenced by intensity (dynamics) andarticulation.

Texture may of course comprise just one line (monophony)or voices may be aligned in distinct groupings or 'layers'.'Pointillism' (and its converse the sound continuum) is alsoa question of texture, so that texture can be seen to have ahorizontal dimension which again can be measured in termsof density.

Thus far we have made no mention of pitch. In fact textureis concerned chiefly with those parameters of music otherthan pitch and especially with rhythm and timbre, in that

order. From this standpoint, it requires just one small stepin imagination to see texture as

. (Percussion sounds haveindefinite, rather than no, pitch.)

If we can construct textural entities or objects which aretotally autonomous (unruled by pitch-content), it followsthat a musical composition can use these objects time andtime again by simply applying different (albeit related)harmonies or pitch-class data. There are certainantecedents in twentieth-century music, notablyStravinsky's sound-blocks or the sound 'masses' - volumes,shapes, planes, galaxies - of Varse or Xenakis or, earlierstill, amongst the Venetians.

There are analogies with the visual arts, especiallysculpture. Applying new harmony to a texture is likeviewing an object from a different perspective. Theintroduction of new pitch-classes (whether throughtransposition or transformation) resembles a change indistance and, new interval-class content a shift of angle ora change of light.

Perhaps the closest comparisons are with the objects ofOOP (object-oriented programming) languages such asSmalltalk or C++. Objects in OOP combine data andbehavior (methods) in a single package - a concept knownas encapsulation. The methods and procedures of a musicalobject are the rhythmic devices it employs and to ensure

maximum object re-usability these devices need to be fairlyelastic. This is most easily achieved by allowing performersfreedom of interpret-ation: ambiguity in rhythmic groupingor phrasing is an obvious formula; the opportunity tochange register (even instrument) might be another.

Major changes (modulation) of rhythm or texture are notout of the question but it makes more sense to explore suchchanges through the creation of fresh objects. Then thereremains the possibility of building new objects on thefoundation of old - inheritance, as it is called in OOP -perhaps through changes in density, vertical and/orhorizontal. This could involve the introduction, replacementor removal of a whole line or layer of music.

Composition with objects can result in such a high degree ofintegration that overall structure ceases, in certainsituations, to be of paramount importance; the order ofobject-appearance can often be left in the hands ofperformers.

the projection of (mainly)pitch material into space and time

Heterophonies

Musicologists differentiate four types of musical texture:- a single melodic voice without harmonic

accompaniment (eg Gregorian chant);- the simultaneous variation in two or more

voices of one melodic line (as in Gamelan music);- two or more largely independent melodic voices

(as in a Bach fugue);- one dominant melodic line with supporting

accompaniment (as in a hymn or chorale)., which consists of internally-changing

cluster chords and is usually identified with Gyrgy Ligeti,may be considered a fifth type.

These are, of course, very broad definitions, and muchinteresting music inhabits the areas of overlap. The first twotypes are common to all musical cultures whilst the otherstend to be confined to Western music. It is worth noting thattwo major revolutions in 20th Century music -dodecaphony/serialism and minimalism - are linked to arenewed concern with, respectively, polyphony andheterophony. Terry Riley's (1964), which allowsperformers to move through the same material in their own

time (though using the same pulse), is essentiallyheterophonic. The same is true of Steve Reich's morerigorous phasing techniques.

In other cultures, as well as jazz and folk-music, heterophonymay be associated with improvisation, rubato andspontaneous ornamentation, but these are not definingcharacteristics. Similarly, staggered entries, rhythmicimitation, inversion, retrograde and transposition betweenparts are linked to polyphony but are not always vital.Minimalists such as Riley, Reich and Rzewski show howdifferences can be blurred: much of the interest of liesin the polyphonic interplay between patterns in differentvoices. takes this further, applying a

variety of techniques - including mirror, crab, table andmensuration canon, as well as reordering, augmentation anddiminution - to mostly heterophonic textures.

Heterophony and polyphony are both 'democratic' in thesense that they give players a degree of independence. Forthe composer, they offer economy of means and an easy wayto combine unity with diversity. The interplay between partscan help create a 4D effect, especially where timbres aremixed or instruments are spatially separated. When furtherfreedoms - a degree of improvisation or indeterminacy - areadded to the mix, we start to hear new ideas perhaps notimagined by the composer or apparent from the score.

monophony

heterophony

polyphony

homophony

Micropolyphony

In C

In C

Music from Objects

Fig 15

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D e G C c

b c D G f e f G C cG b c f ee g b D c

C D e b GD f G C bG c C f DG b C f e

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Musical notation springs from the needs to preserve andcommunicate: historically, it has often served as a tool ofinstruction or analysis, a compositional test-bed, a testimonyto a past performance or, for sight-readers, the impetus to animaginary one. Nevertheless its paramount task is to act as achannel of communication between composer and performer.

Since these are unlikely to meet, notation needs to convey notjust individual notes but the crux or essence of the whole piece

and it has to do this in a manner which is fast, simple andexplicit. Regrettably, notation has emerged as a source offriction in recent years, not only in music which is highlyprecise but also in scores which offer the player a degree offreedom. In either case, the composer who desires asympathetic realization of his work has little choice but toaccept that the performer, like the proverbial customer, isalways right, since idiosyncracies in notation can wastevaluable rehearsal time or distract a player's attention.

A code rather than a language, notation can support newdiacritics but not new dialects. This means that composersneed to employ traditional notation wherever possible andintroduce extensions only where their meaning is absolutelyclear. Western notation - an eclectic mixture of graphical,

numerical, verbal and alphabetical symbols - is amazinglyefficient in representing important features such as pitch(including microtones) and time but less transparent when itcomes to matters such as articulation, dynamics andperformance style. From this we may infer that innovationsbased on graphics can provide examples of some of the mosteffective and also some of the most ambiguous new notations.

No-one would dispute the fact that notation can influence thequality of performance and this can lead us to suppose that acomposer can improve the chances of a faithful interpretationby providing different notations for different players - anonerous task in earlier times, but feasible now with the help ofa computer.

It has to be acknowledged that certain aspects of traditionalnotation appear increasingly anachronistic and at timesirritating. The use of G, F and C clefs (as well as parts fortransposing instruments) is one example. Restricting ourselvesto a single clef, with register indicated by figures (c3, c4 anda4 meaning 'cello c middle c and violin a, respectively), it soonbecomes clear that a four-line staff is more than sufficient toaccommodate an octave.

On the other hand, there are pieces on this site (for examplein ) which exploit different clefs (usuallycombined with changes of 'key' signature) in order to generateharmonic transformations which would otherwise have to benotated separately.

Accidentals can often present problems of notation since, inmuch twelve-tone music, g sharp and a flat are equivalent but,for earlier composers and even now for string players, they arequite distinct notes. There is a solution, and it has beenavailable for eighty years, namely the Hauer-Steffens staff,which is laid out like a (vertical) piano keyboard, with linesrepresenting the black keys, and spaces the white. Figure 16(where middle c is c4) shows the final section of

notated in such a manner, in an attempt to avoid, or atleast delay, suggestions of a tonal center.

There is one case here where traditional notation causes muchconfusion and that is in the melodic transformations of

. These five-tone patterns require just

a two-line stave and could be better represented as shown infigure 15 (where upper-case letters represent sharpenednotes). However, the same notation could not extend to otherparts of the piece and so, as is often the case, a ratherunsatisfactory compromise has had to be struck.

The Cauldron of Plenty

Habitations ofFire

Roomfor Rhetorical Discourse

Alternative Notations

Graphical and word scores are usually associated withcomposers such as Cage, Cardew and Bussotti, working in the1960s; they are designed principally as a stimulus to impro-visation and (intentionally) the resulting performances canvary considerably. The most rewarding tend to mix graphicswith musical notation and text: see

The scores here, in contrast, are impelled not so much bychance as by performer choice, with players given a certain

freedom in the shaping and ordering of events; independentperformances, though different, share a clearly audiblecommon source. The link is not to Cage but to the Renaissance- Baude Cordier ( , 1400), Josquin des Prez, JohnBull - or Baroque - the 'figured' bass and the 'enigma' canonsof Bach and others.

Contemporary music, at least in the West, is much concernedwith 'development', 'argument' and 'endgames' but manywould claim that these by nature are literary, not musical,ideas. Modern scores routinely appear in book-form, which hasa tendency to disguise structure, especially recapitulation,variation and transformation. Nevertheless music, whilst time-based, has the singular power to take us out of time and toembrace the visual arts, giving us Debussy (Impressionism),

Satie (Cubism, Dada) and Varse (Futurism). Developmenttakes place not in the music itself, but in the listener'sunderstanding of musical ideas and their contexts.

A pithy and compact notationproducing apparently limitless musical material: generative,algorithmic, repetitive and minimalist music spring to mind.Note, however that there is no true repetition here - ratherparaphrase, mimicry and reconstruction - and the closestanalogy would be with the reusable objects (hence

) of object-oriented programming languages. Theterseness of the notation helps make the composition-processtransparent and its flexibility means the music is nottechnically difficult to perform.

: Landscapes - real and imaginary,literary, mythical, magical - dominate these sound-worlds andbiospheres. The influence of the spatial arts is evident every-where but despite (or perhaps because of) this, the music isconcerned above all with our perception of time - questioningthe nature of change, chance and coincidence - and ideas ofprecognition and conflicting memories. Duration is undefined;there are no beginnings and no ends - just atmospheres,expectations and mysteriously wandering objects.

Mainstream art music, still bewitchedby Schoenberg, has largely become the preserve of a pseudo-intellectual elite and, whilst post-serialist language hasproduced some undoubted masterpieces, it has also alienated

millions of ordinary music-lovers. Using some fairly crudemathematics, Schoenberg and his disciples abandoned major-minor tonality when it might have been more interesting to tryto extend it.

pushes the boundaries of traditionaltonality and explores neglected harmonies brought to light bymeans of set theory. Lacking microtones, the result is by nomeans a global musical language, but one which is capable byturn of sounding jazzy, medieval, African, Arab or Oriental. Atthe same time, the music strives to appeal to the whole person- physically, emotionally and intellectually - something earliercomposers would have considered axiomatic and somethingindispensable, if we are to begin to rebuild lost audiences.

www.notations21.net/.

Circular Canon

Music fromObjects

Music from Objects

Paraphrase and Generativity:

Experiments with time

Is anyone listening?

Graphical Notation

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A Brief Guide to PC Set-Theory

Most music-lovers will have heard of the Diminished 7th orScriabin's 'mystic' chord (not to mention the whole-tone and

scales or Messiaen's impossibly charming) and many will have wondered what other

combinations of pitches lie out there.

An oft-quoted sum suggests that (12 factorial) 479,001,600'tunes' can be gleaned from the twelve-tone scale, but this

figure hides an immense number of reorderings andtranspositions. By focusing on unordered note-groups (sets)and treating transpositions as equivalent, we can pare thisfigure down to a more modest 336; further, by linkingcomplements (each -note set has a note complement)and inversions (the minor and major triads being inversions ofone another, for example), we arrive at the highly manageabletotal of 129 pitch-class sets.

Once they have been identified and recorded as prime forms,we can start to seek similarities between pc sets; the mostobvious method is to look at their interval-content. Again, the11 intervals contained in the chromatic scale can be reduced to6 interval-classes by pairing off inversions (for example, Minor3rd and Major 6th) and the icv or interval-class vector (an array

of ic occurrences) for each set can be established. Bycomparing these vectors cell by cell, we can calculate similarityrelations between sets of both equal and unequal size.

Standard set nomenclature follows Allan Forte's(Yale, 1973), and here the icv is central: thus

pc set 6-01 is the 6-note set containing the highest total ofs (Minor 2nd or Major 7th). The index is slightly flawed in its

treatment of Z-related pairs (apparently unrelated sets sharingthe same icv): thus set 6z36 might be better designated 6z03band 4z29, 4z15b. Forte excludes sets of size 2/10 and 1/11,with a total membership of 14; since there is but 1 setcontaining 11 pitch-classes, it is interesting to speculate whySchoenberg opted for the 12-tone row.

Concerned as they are with analysis (and particularly of workswhich hitherto may have defied analysis), Forte and hisfollowers have tended to obscure distinctions between a set andits inversion, even where the sound may be quite different.(The same is true of intervals and their inversions, hence theicv may be less useful than the 11-entry interval-vector.)Nevertheless, Forte's outstanding achievement has been theidentification of large-scale associations of sets, in particularthe highly-coherent Kh sub-complexes which embrace setslinked by reciprocal complement inclusion, that is, sets whichappear in both the reference or nexus set and its complement.(See figure 1 below.)

Other writers, notably Karel Janecek in(Foundations of Modern Harmony, Prague, 1965,

which includes a summary in German), have employed pitch-class set theory to construct universal theories of harmony,embracing tonal as well as post-tonal music. Janecek details350 harmonies of 1 to 11 pitches, using a notation based ondirected interval content: thus Forte's 4z15 and 4z29 become132(6) and 231(6), 124(5) and 421(5). He spotlights theinclusion of major/minor triads and proposes four categories ofharmony based on compounds of consonant and dissonantintervals.

In many ways, pitch-class set theory can be seen as theSchoenberg system extended from pitches to pitch relation-ships, a kind of serialism in 3 dimensions. For composers, it can

serve as a taxonomy, a gazetteer, a catalogue of possibilities oran extended palette of harmonic resources, since compositionwith sets is nothing if not flexible. Sets can be used to organizeboth global and local harmony in works which may be built ondramatic juxtaposition, seamless transformation or organicdevelopment; they can be shaped into chords, melodies ormotives, modes, scales or ragas or indeed anything in-between.

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The Structureof Atonal Music

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Dramatis Personae

Six essays here require a knowledge of pitch-class set theory.Allan Forte's (Yale, 1973) is

he authoritative text but for those who have not had theopportunity to read it, we offer a brief glossary.

Middle c and violin a are examples of pitches. Pitch class c ishe set of all possible cs, in all octaves. We use lowercase c tondicate c natural whilst C means (equal-tempered) c# or d flat.

A pc set is an unordered collection of pitch-classes. A set of twoelements is often called a dyad, for example [0,3], whichndicates any pitch plus the pitch three semitones above (annterval of a Minor 3). Sets of higher cardinality or size areometimes known as trichords, tetrachords, pentads, hexads,

heptads, octads, etc..

Since the musical octave contains 12 semitones, pitch classesemploy mod 12 arithmetic. Thus any number above 11 isepresented as a figure from 0 to 11. It is usual to notate 10nd 11 in hexadecimal form, as A and B.

Pitch-class (or pc) sets are often related by transposition and/ornversion. Thus set [1,4,8] and [3,6,A] are transpositions of0,3,7] and [0,4,7] is an inversion. Groups of pc sets like these

belong to the same set class. Surprisingly, there are only 208different set classes of cardinality 3 - 9. The simplest notation -here 0,3,7 - is known as the prime form. Forte gives each prime

orm a set name, where set 3-01 is (0,1,2), 3-11 is (0,3,7) and5-02 is (0,1,2,3,5).

Any pc set of size n has a complement, consisting of the 12-nmissing pitch-classes. Thus the complement of set 3-11 is 9-11which has the prime form (0,1,2,3,5,6,7,9,A).

An involution is a set which duplicates itself on inversion, and isndicated in this paper with an '=' sign, e.g. 3-01=. Pc set0,1,2,3,5) inverts to (0,2,3,4,5), and these may be designated

5-02o (original) and 5-02i (inversion). These extensions aremportant to composers and listeners but not to analysts.

An interval is the distance in semitones between two pitches.Thus a Major 3rd comprises two pitches which are 4 semitonespart.

ntervals have inversions: the minor 6th (b up to g) is thenversion of the Major 3rd (g-b). Together they form intervallass 4. There are only 6 interval classes from minor 2/Major 7o tritone.

The icv is an array which counts all the interval classes 1-6ound in a pc set. Thus (0,1,2) includes two semitones (ic 1)

nd one tone (ic 2). The icv for set 3-01 is therefore 110000;or 3-11 (0,3,7) it is 001110 and for 5-02 (0,1,2,3,5) 332110.

Comparing icvs is a common way of looking for similaritiesbetween pc sets. Z-related pairs share the same icv, whilstsomorphs are linked by switched icv entries.

The Structure of Atonal Music

Pitch and pitch-class

Pitch-class (pc) set

Modulo 12 or "clock" arithmetic

Pitch-class set class

Complements

Extended set names

nterval

nterval class

nterval class vector (icv)

Similarity Relations

N.B. In a departure from Forte, the suffixes =, o and i are hereused to distinguish pc set involutions, primes and inversions.Similarly, Z-related pairs are explicitly denoted 4z15, 5z18 etc.Elsewhere, uppercase letters CDFGA indicate sharpened notes.


7/14

4-

123456789012345678901234567890102030405060708091011121314151617

181920212223242526272829303132333435

12345678901234567890123456789010203

04050607080910111213141516171819202122232425

26272829303132333435363738

12345678901234567890123456789

0 1 2 5-

123456789012345678901234567890123456780102030405060708091011121314151617

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12345678901234567890123456789012345678

0 1 2 3

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5-n

Khsub-complexes:indicat

espcsetslinkedbyreciprocalcomplementinclusion

4-n Table of Pitch-Class Sets0102030405060708

091011121314151617181920212223242526272829

3-n

Name Pc Set Vector *Iso

5-01=01234 432100 355-02 01235 332110 235-03 01245 322210 27

5-04 01236 322111 295-05 01237 321121 145-06 01256 311221 205-07 01267 3101325-08=02346 232201 345-09 01246 231211 245-10 01346 223111 255-11 02347 2222205z12=01356 222121 365-13 01248 221311 305-14 01257 221131 055-15=01268 2202225-16 01347 213211 325z17=01348 212320 375z18 01457 212221 385-19 01367 2121225-20 01378 211231 065-21 01458 2024205-22=01478 2023215-23 02357 132130 025-24 01357 131221 095-25 02358 123121 10

5-26 02458 1223115-27 01358 122230 035-28 02368 1222125-29 01368 122131 045-30 01468 121321 135-31 01369 1141125-32 01469 113221 165-33=02468 0404025-34=02469 032221 085-35=02479 032140 015z36 01247 222121 125z37=03458 212320 175z38 01258 212221 18

6-01=012345 543210 326-02 012346 443211 336z03 012356 433221 256z04=012456 432321 376-05 012367 422232 186z06=012567 421242 066-07=012678 4202436-08=023457 3432306-09 012357 3422316z10 013457 333321 466z11 012457 333231 116z12 012467 332232 126z13=013467 324222 506-14 013458 3234306-15 012458 323421 316-16 014568 3224316z17 012478 322332 17

6-18 012578 3222426z19 013478 313431 196-20=014589 3036306-21 023468 242412 346-22 012468 2414226z23=023568 234222 236z24 013468 233331 396z25 013568 233241 366z26=013578 232341 376-27 013469 2252226z28=013569 224322 286z29=013689 224232 426-30 013679 2242236-31 013589 2234316-32=024579 1432506-33 023579 1432416-34 013579 1424226-35=02468A 060603

4-01=0123 321000 234-02 0124 221100 224-03=0134 212100 264-04 0125 211110 144-05 0126 210111 164-06=0127 2100214-07=0145 201210 204-08=0156 200121

4-09=0167 2000224-10=0235 1220104-11 0135 1211104-12 0236 112101 274-13 0136 1120114-14 0237 111120 044z15 0146 111111 294-16 0157 110121 054-17=0347 1022104-18 0147 1021114-19 0148 1013104-20=0158 101220 074-21=0246 0302014-22 0247 021120 024-23=0257 021030 014-24=0248 0203014-25=0268 0202024-26=0358 012120 034-27 0258 012111 124-28=0369 0040024z29 0137 111111 15

3-01=012 210000 093-02 013 111000 073-03 014 101100 113-04 015 1001103-05 016 1000113-06=024 0201003-07 025 011010 023-08 026 0101013-09=027 010020 013-10=036 0020013-11 037 001110 033-12=048 000300

Name Pc Set Vector

6z36 012347 4332216z37=012348 432321

6z38=012378 421242

6z39 023458 3333216z40 012358 3332316z41 012368 3322326z42=012369 324222

6z43 012568 322332

6z44 012569 313431

6z45=023469 2342226z46 012469 2333316z47 012479 2332416z48=012579 232341

6z49=013479 2243226z50=014679 224232

7-01=0123456 6543217-02 0123457 5543317-03 0123458 544431

7-04 0123467 5443327-05 0123567 5433427-06 0123478 5334427-07 0123678 5323537-08=0234568 4544227-09 0123468 4534327-10 0123469 4453327-11 0134568 4444417z12=0123479 4443427-13 0124568 4435327-14 0123578 4433527-15=0124678 4424437-16 0123569 4354327z17=0124569 4345417z18 0123589 4344427-19 0123679 4343437-20 0124789 4334527-21 0124589 4246417-22=0125689 4245427-23 0234579 3543517-24 0123579 3534427-25 0234679 345342

7-26 0134579 3445327-27 0124579 3444517-28 0135679 3444337-29 0124679 3443527-30 0124689 3435427-31 0134679 3363337-32 0134689 3354427-33=012468A 2626237-34=013468A 2544427-35=013568A 2543617z36 0123568 4443427z37=0134578 4345417z38 0124578 434442

8-01=01234567 7654428-02 01234568 6655428-03=01234569 6565428-04 01234578 6555528-05 01234678 6545538-06=01235678 6544638-07=01234589 6456528-08=01234789 644563

8-09=01236789 6444648-10=02345679 5664528-11 01234579 5655528-12 01345679 5565438-13 01234679 5564538-14 01245679 5555628z15 01234689 5555538-16 01235789 5545638-17=01345689 5466528-18 01235689 5465538-19 01245689 5457528-20=01245789 5456628-21=0123468A 4746438-22 0123568A 4655628-23=0123578A 4654728-24=0124568A 4647438-25=0124678A 4646448-26=0124579A 4565628-27 0124578A 4565538-28=0134679A 4484448z29 01235679 555553

9-01=012345678 8766639-02 012345679 7776639-03 012345689 7677639-04 012345789 7667739-05 012346789 7666749-06=01234568A 6867639-07 01234578A 6776739-08 01234678A 6767649-09=01235678A 6766839-10=01234679A 6686649-11 01235679A 6677739-12=01245689A 666963

010203

04050607080910111213141516171819202122232425

26272829303132333435363738

010203040506070809101112

*Iso= Isomorphs:Like theseg roupsshare the sameKh-complex size and BIP (basic interval pattern) count.They canbe identified byswitchingicv entries 1 and5.

Z-related pairs,

3-

0102030405060708091011121314151617

181920212223242526272829303132333435

123456789012010203

04050607080910111213141516171819202122232425

26272829303132333435363738

1234567890120102030405060708

091011121314151617181920212223242526272829

123456789012

0 1

123456789012

Sub-Complexes KhFigure 1 above shows pitch-class sets linked by reciprocal complementinclusion. It includes complexes surrounding sets of size 3/9, thoughthese are generally too large to be of much practical interest.


8/14

Name Pc Set Vector *Iso

5-01=01234 432100 355-02 01235 332110 235-03 01245 322210 275-04 01236 322111 29

5-05 01237 321121 145-06 01256 311221 205-07 01267 3101325-08=02346 232201 345-09 01246 231211 245-10 01346 223111 255-11 02347 2222205z12=01356 222121 365-13 01248 221311 305-14 01257 221131 055-15=01268 2202225-16 01347 213211 325z17=01348 212320 375z18 01457 212221 385-19 01367 2121225-20 01378 211231 065-21 01458 2024205-22=01478 2023215-23 02357 132130 025-24 01357 131221 095-25 02358 123121 105-26 02458 122311

5-27 01358 122230 035-28 02368 1222125-29 01368 122131 045-30 01468 121321 135-31 01369 1141125-32 01469 113221 165-33=02468 0404025-34=02469 032221 085-35=02479 032140 015z36 01247 222121 125z37=03458 212320 175z38 01258 212221 18

6-01=012345 543210 326-02 012346 443211 336z03 012356 433221 256z04=012456 432321 376-05 012367 422232 186z06=012567 4212426-07=012678 4202436-08=023457 3432306-09 012357 3422316z10 013457 333321 466z11 012457 3332316z12 012467 3322326z13=013467 324222 506-14 013458 3234306-15 012458 323421 316-16 014568 3224316z17 012478 3223326-18 012578 3222426z19 013478 3134316-20=014589 3036306-21 023468 242412 346-22 012468 2414226z23=023568 2342226z24 013468 233331 396z25 013568 233241 366z26=013578 232341 376-27 013469 2252226z28=013569 2243226z29=013689 224232 426-30 013679 2242236-31 013589 2234316-32=024579 1432506-33 023579 1432416-34 013579 1424226-35=02468A 060603

4-01=0123 321000 234-02 0124 221100 224-03=0134 212100 264-04 0125 211110 144-05 0126 210111 164-06=0127 2100214-07=0145 201210 204-08=0156 2001214-09=0167 2000224-10=0235 1220104-11 0135 1211104-12 0236 112101 274-13 0136 1120114-14 0237 111120 044z15 0146 111111 294-16 0157 110121 054-17=0347 1022104-18 0147 1021114-19 0148 1013104-20=0158 101220 074-21=0246 0302014-22 0247 021120 024-23=0257 021030 014-24=0248 0203014-25=0268 0202024-26=0358 012120 034-27 0258 012111 124-28=0369 0040024z29 0137 111111 15

3-01=012 210000 093-02 013 111000 073-03 014 101100 113-04 015 1001103-05 016 1000113-06=024 0201003-07 025 011010 023-08 026 0101013-09=027 010020 013-10=036 0020013-11 037 001110 033-12=048 000300

89

8989

89

89

89

89

Name Pc Set Vector

6z36 012347 4332216z37=012348 432321

6z38=012378 421242

6z39 023458 3333216z40 012358 3332316z41 012368 3322326z42=012369 324222

6z43 012568 322332

6z44 012569 313431

6z45=023469 2342226z46 012469 2333316z47 012479 2332416z48=012579 232341

6z49=013479 2243226z50=014679 224232

7-01=0123456 6543217-02 0123457 5543317-03 0123458 5444317-04 0123467 544332

7-05 0123567 5433427-06 0123478 5334427-07 0123678 5323537-08=0234568 4544227-09 0123468 4534327-10 0123469 4453327-11 0134568 4444417z12=0123479 4443427-13 0124568 4435327-14 0123578 4433527-15=0124678 4424437-16 0123569 4354327z17=0124569 4345417z18 0123589 4344427-19 0123679 4343437-20 0124789 4334527-21 0124589 4246417-22=0125689 4245427-23 0234579 3543517-24 0123579 3534427-25 0234679 3453427-26 0134579 344532

7-27 0124579 3444517-28 0135679 3444337-29 0124679 3443527-30 0124689 3435427-31 0134679 3363337-32 0134689 3354427-33=012468A 2626237-34=013468A 2544427-35=013568A 2543617z36 0123568 4443427z37=0134578 4345417z38 0124578 434442

8-01=01234567 7654428-02 01234568 6655428-03=01234569 6565428-04 01234578 6555528-05 01234678 6545538-06=01235678 6544638-07=01234589 6456528-08=01234789 6445638-09=01236789 6444648-10=02345679 5664528-11 01234579 5655528-12 01345679 5565438-13 01234679 5564538-14 01245679 5555628z15 01234689 5555538-16 01235789 5545638-17=01345689 5466528-18 01235689 5465538-19 01245689 5457528-20=01245789 5456628-21=0123468A 4746438-22 0123568A 4655628-23=0123578A 4654728-24=0124568A 4647438-25=0124678A 4646448-26=0124579A 4565628-27 0124578A 4565538-28=0134679A 4484448z29 01235679 555553

9-01=012345678 8766639-02 012345679 7776639-03 012345689 7677639-04 012345789 7667739-05 012346789 7666749-06=01234568A 6867639-07 01234578A 6776739-08 01234678A 6767649-09=01235678A 6766839-10=01234679A 6686649-11 01235679A 6677739-12=01245689A 666963

Similarity RelationsSets may be compared with respect to pitch-class and interval-class content,as well as sub-/supersets held in common. Unlike the sub-complexes Kh, suchrelations can reveal similarities between sets of the same or complementarycardinality. In figures2-5above, the top-right-hand tables useGalton's

to compare theinterval-class vectors of sets of equal size; thefinal tallies have been halved, with a low total indicating a close relationship.Bottom-left tables measure the sum of shared pitches in all 12 transpositions,andhereit ishighcounts whichsuggestan affinitybetween sets.

Methodof Least Squares

02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 3401 1 2 3 7 11 16 4 5 4 5 7 7 8 7 9 9 11 11 20 9 11 8 8 10 11 11 10 11 12 11 16 13 13 32

02 1 2 5 9 13 3 3 2 3 4 4 7 5 7 6 8 9 19 5 7 4 5 7 8 7 6 7 7 8 13 9 8 2503 1 2 5 9 2 2 1 1 2 2 4 3 4 3 4 5 14 6 7 3 3 4 5 5 4 4 5 5 10 7 8 29

04 2 4 7 3 2 1 2 2 4 3 2 2 2 4 4 11 4 4 5 3 5 4 8 5 6 7 4 11 8 6 2305 1 3 5 3 3 2 1 3 5 4 3 1 1 4 13 8 7 5 4 4 4 7 5 4 5 5 11 8 9 31

06 1 7 4 6 4 2 7 7 7 4 2 1 6 15 11 8 9 6 5 4 12 9 7 9 7 11 9 11 3307 12 7 10 8 4 11 12 11 7 4 3 10 20 13 9 13 10 9 7 17 13 11 12 11 16 13 13 32

08 1 2 1 3 5 4 5 5 5 5 7 16 7 7 4 2 2 3 7 6 5 8 5 4 3 7 2809 2 1 1 5 5 5 4 3 3 7 17 5 4 4 2 2 2 8 6 5 7 5 5 3 5 23

10 1 2 2 2 1 2 2 4 3 10 3 4 2 1 3 3 4 2 3 4 2 8 5 4 2211 1 2 3 3 3 2 2 4 13 6 6 2 1 1 2 4 3 2 4 3 5 3 6 28

12 3 5 4 3 1 1 5 15 5 4 3 2 2 2 6 4 3 4 4 7 4 5 2413 5 3 5 3 4 4 13 7 9 1 3 4 6 1 1 1 1 4 11 7 8 31

14 1 1 3 5 1 4 7 7 6 2 4 3 7 4 5 8 1 8 7 7 2815 1 2 5 1 5 4 5 4 2 5 4 5 2 4 5 1 11 8 5 23

16 1 3 1 5 5 4 6 2 4 2 8 4 5 7 1 9 7 5 2317 1 2 9 5 4 4 2 3 2 6 3 3 4 2 9 6 5 24

18 4 13 9 7 5 3 2 2 7 5 3 5 4 7 5 8 3119 3 8 8 6 3 5 4 6 3 4 6 1 11 9 8 30

20 15 15 16 10 14 11 15 10 13 16 5 20 19 15 3621 1 5 4 8 6 9 5 8 7 5 13 8 1 9

22 7 4 7 4 12 7 9 9 5 11 7 1 923 2 3 5 1 1 1 1 4 8 4 5 25

24 1 1 4 2 2 4 1 4 2 3 2225 1 5 4 2 5 3 2 1 6 29

26 8 5 4 7 2 3 2 4 2327 1 1 1 5 11 7 9 33

28 1 1 2 10 6 5 2529 1 3 7 4 7 31

30 5 12 7 7 2831 7 5 4 23

32 1 9 3233 5 25

34 9

02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 3401 1 2 2 4 6 11 3 4 4 5 5 6 8 9 6 8 7 8 10 13 11 9 8 8 8 10 8 10 10 11 10 17 11 16

02 1 1 2 4 8 2 2 2 2 2 4 4 6 4 5 4 5 6 10 8 4 4 4 5 5 5 5 6 8 6 14 6 903 1 2 2 7 2 2 2 1 2 2 4 5 2 2 2 4 4 5 4 5 4 4 3 4 4 5 4 7 4 13 6 10

04 1 2 5 2 2 1 2 1 3 3 4 2 4 2 2 4 8 5 5 4 3 4 5 3 4 5 5 4 14 6 10

05 1 2 4 2 3 2 1 3 1 2 4 4 2 2 2 8 5 4 3 4 5 4 4 3 4 8 5 14 6 806 2 5 3 4 2 2 2 2 2 3 2 1 2 1 4 2 6 4 5 4 4 4 4 3 8 4 14 7 10

07 10 6 8 6 4 6 2 2 8 7 4 3 2 10 6 8 6 8 9 7 7 5 6 12 8 18 10 1108 1 2 3 3 2 6 5 3 5 4 5 7 9 7 6 3 4 2 6 2 6 4 6 5 7 4 11

09 3 2 2 1 3 2 4 4 3 4 4 8 6 4 1 4 2 4 2 4 2 8 5 6 3 810 2 1 4 4 6 1 4 2 2 5 8 5 4 4 1 3 4 2 3 5 2 2 14 4 8

11 1 2 2 4 2 1 1 3 2 4 3 2 2 2 2 1 3 2 2 6 2 12 3 512 3 1 3 2 3 1 1 2 7 4 2 2 1 3 2 2 1 3 4 2 13 3 5

13 4 2 3 2 2 4 3 4 3 6 2 5 1 4 2 5 1 8 4 6 4 1014 2 5 4 2 2 1 8 5 2 2 3 5 2 4 1 3 8 4 14 4 4

15 6 5 3 3 2 8 5 6 2 6 4 5 3 4 2 10 6 8 5 916 2 1 2 4 4 2 6 5 2 2 4 2 4 4 2 1 14 5 10

17 1 4 2 1 1 5 4 4 2 2 4 4 2 7 2 13 5 818 1 1 3 1 4 3 2 2 2 2 2 2 4 1 13 4 7

19 2 7 3 5 4 2 4 4 2 2 4 3 2 15 5 820 4 2 4 3 4 4 2 4 2 2 8 3 14 5 6

21 1 10 8 8 4 5 7 8 4 10 4 16 9 1322 8 6 5 3 4 4 5 3 6 2 15 7 11

23 2 2 5 1 5 1 4 8 4 14 2 124 3 2 2 2 2 1 8 4 6 1 4

25 3 2 2 1 4 2 1 14 2 426 3 1 4 1 5 2 6 2 8

27 4 1 2 7 2 13 2 228 3 2 3 2 7 2 8

29 3 5 2 14 2 230 8 3 6 2 6

31 2 18 6 1132 14 3 6

33 7 1734 3

35

02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 2801 1 2 2 3 4 5 6 7 3 3 4 4 5 4 6 7 6 9 8 8 7 9 10 9 8 7 13

02 1 1 2 4 3 5 7 2 1 2 3 3 2 4 4 4 5 5 4 4 7 5 5 5 4 1103 1 3 5 2 5 7 2 2 1 2 3 2 5 2 2 4 4 7 5 8 7 7 4 3 7

04 1 2 1 2 4 2 1 2 2 1 1 2 2 2 3 2 6 3 5 6 6 3 3 1005 1 2 1 2 4 2 3 3 2 1 1 4 3 4 3 5 4 6 5 4 5 4 12

06 4 1 1 4 3 5 3 2 2 1 6 4 7 4 8 4 4 9 7 5 5 1307 2 5 5 3 3 4 2 2 3 1 2 1 1 8 5 8 6 7 4 4 11

08 1 6 4 5 4 2 2 1 4 3 4 2 9 5 6 8 7 5 5 1309 7 6 6 4 4 3 2 7 4 8 5 11 7 7 11 8 7 6 12

10 1 2 1 2 2 4 4 3 7 5 6 2 3 8 7 2 2 711 2 2 1 1 2 3 3 4 3 3 1 3 4 4 2 2 10

12 1 3 1 4 2 1 4 4 5 4 7 5 4 3 1 413 2 1 3 3 1 6 4 7 3 4 8 6 2 1 4

14 1 1 2 2 3 1 6 1 2 6 6 1 2 1015 1 2 1 3 2 4 2 4 4 3 2 1 7

16 4 3 4 2 5 2 3 5 4 3 3 1217 1 1 1 8 4 7 6 7 2 2 7

18 3 2 8 4 6 7 6 2 1 419 1 7 5 9 4 6 4 4 12

20 8 3 5 6 7 2 3 1121 4 8 1 1 7 5 15

22 1 5 5 1 2 1123 10 9 2 4 13

24 1 7 5 1525 7 4 12

26 1 727 4

02 03 04 05 06 07 08 09 10 11 12

01 1 2 2 2 3 3 3 4 5 4 702 1 2 2 2 1 2 3 2 2 603 1 2 3 2 2 4 2 1 3

04 1 3 2 2 2 4 1 305 4 2 2 2 3 2 6

06 2 1 3 5 3 407 2 1 2 1 6

08 3 3 2 309 5 2 7

10 2 711 3

02 3803 36 3404 36 34 32

05 34 32 32 3206 32 30 32 32 3207 28 28 28 32 34 3408 36 34 32 32 30 30 2809 34 32 32 32 32 30 30 3410 34 32 32 32 30 30 28 32 3011 32 32 32 30 30 30 28 30 30 3012 32 32 30 32 32 30 32 30 30 32 3013 32 30 32 30 30 32 30 32 32 30 30 3014 30 30 30 30 32 32 34 28 30 30 30 32 3015 28 28 28 30 32 32 36 30 32 28 28 30 32 3216 32 30 32 32 30 30 28 32 30 32 30 30 30 28 2817 30 30 32 30 30 32 28 30 30 30 32 30 32 30 28 3218 30 30 30 30 30 32 32 30 30 30 30 30 30 30 30 32 3219 28 28 28 32 32 32 34 30 30 32 28 32 30 32 32 32 28 3220 28 28 30 30 32 32 34 28 30 28 30 30 30 32 32 30 32 32 3221 28 28 32 28 28 32 28 28 28 28 32 28 32 28 28 32 36 32 28 3222 28 28 30 30 30 32 32 28 28 30 30 30 32 30 30 32 34 32 32 32 3623 30 32 30 30 30 28 28 30 30 30 32 32 28 32 28 28 30 30 28 30 28 2824 30 30 30 30 30 30 30 32 32 30 30 30 32 32 32 28 30 30 30 30 28 28 3225 30 30 30 30 30 28 28 30 30 32 30 32 28 30 28 32 30 30 32 30 28 30 32 3026 30 30 30 30 28 30 28 32 32 30 30 30 32 28 30 32 32 30 30 30 32 32 30 32 30

27 28 30 30 28 30 30 28 28 30 30 32 30 30 32 28 30 32 30 28 32 32 30 34 32 32 3028 28 28 28 30 30 30 30 32 32 32 28 30 32 30 32 32 28 30 32 30 28 30 28 32 32 32 2829 28 30 28 30 30 30 32 28 30 30 30 32 28 32 30 30 30 30 32 32 28 30 34 32 32 30 32 3030 28 28 30 28 30 30 30 30 32 28 30 30 32 30 32 30 32 30 30 32 32 32 30 32 30 32 32 32 3031 28 28 28 32 28 28 28 32 28 34 28 32 28 28 28 34 28 32 34 28 28 32 28 28 34 32 28 34 32 2832 28 28 30 30 28 30 28 30 28 32 30 30 30 30 28 32 32 32 32 30 32 32 30 30 32 32 32 32 32 30 3433 28 28 28 28 28 28 28 36 36 28 28 28 36 28 36 28 28 28 28 28 28 28 28 36 28 36 28 36 28 36 28 2834 28 30 28 28 28 28 28 32 32 30 30 30 30 30 30 30 30 30 30 30 28 28 34 34 32 32 32 32 32 32 32 32 3635 26 30 28 28 30 28 28 28 30 30 32 32 28 34 28 28 30 30 28 32 28 28 38 34 34 30 36 28 36 32 28 32 28 36

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

02 6803 65 6304 64 62 6105 59 59 60 6006 57 57 59 60 6407 54 56 58 60 66 7008 65 62 61 60 57 57 5409 62 61 60 60 59 60 60 6210 62 61 60 60 58 57 56 60 5911 61 60 60 59 59 59 58 61 60 5912 58 59 59 59 61 62 64 58 60 58 5913 59 60 60 58 60 58 58 57 57 59 59 5914 61 58 59 60 57 57 54 61 58 60 59 56 5715 60 59 59 60 58 57 56 58 57 60 58 57 59 6216 58 57 58 60 59 60 60 58 58 59 58 58 57 62 6117 56 57 58 59 61 62 64 56 58 58 58 60 59 58 59 6018 55 56 58 58 62 64 66 57 59 57 59 61 59 57 57 59 61

19 57 56 58 59 59 59 58 57 56 59 58 57 59 63 62 62 60 5920 57 54 57 60 57 57 54 57 54 60 57 54 57 69 66 66 60 57 6921 59 61 58 60 57 56 58 57 59 60 57 59 58 57 60 59 59 56 57 5722 57 59 57 60 58 59 62 57 60 59 57 60 56 57 59 60 60 58 57 57 6623 58 60 59 57 58 56 56 58 58 59 59 59 62 56 58 56 58 58 57 54 60 5824 58 58 58 58 57 57 56 60 59 59 59 58 58 60 59 59 58 58 59 60 59 59 5925 57 57 58 57 58 59 58 61 60 58 60 59 58 59 57 58 58 60 58 57 56 57 59 6026 56 56 57 58 58 60 60 60 60 58 59 59 56 60 58 60 59 60 59 60 58 60 57 60 6127 57 59 59 56 58 55 54 57 56 59 59 58 64 57 59 56 58 58 59 57 58 55 64 59 59 5628 56 58 58 57 58 56 56 56 56 59 58 58 62 58 60 58 59 58 60 60 60 58 62 59 58 57 6429 55 57 58 56 59 58 58 57 57 58 59 59 62 57 58 57 59 60 59 57 57 56 62 59 60 58 64 6230 54 58 58 56 60 58 60 54 56 58 58 60 64 54 58 56 60 60 58 54 60 58 64 58 58 56 66 64 6431 56 56 57 58 57 57 56 58 57 59 58 57 58 62 61 61 59 58 62 66 59 59 58 60 59 60 59 60 59 5832 57 56 57 56 55 57 54 65 62 58 61 58 55 61 56 58 56 59 57 57 55 57 58 62 65 64 57 56 59 54 6033 56 57 57 56 56 57 56 62 61 58 60 59 57 58 56 57 57 59 56 54 58 59 60 61 63 62 59 58 60 58 59 6834 55 58 56 58 56 56 58 57 59 59 57 59 57 57 59 59 59 57 57 57 66 66 60 60 58 60 58 60 58 60 60 59 6135 54 60 54 60 54 54 60 54 60 60 54 60 54 54 60 60 60 54 54 54 78 78 60 60 54 60 54 60 54 60 60 54 60 78

02 1903 18 1704 17 16 1605 16 15 14 1506 16 14 13 15 1707 15 15 16 16 15 1408 14 13 13 15 17 18 1609 14 12 12 14 18 20 14 2010 17 16 16 15 13 14 13 12 1211 16 16 15 15 14 14 14 13 12 1612 15 15 16 14 14 13 14 13 14 15 1413 15 14 15 14 14 15 13 14 16 16 14 1614 14 14 14 15 14 15 15 15 14 15 15 13 1415 14 14 14 14 15 15 14 15 16 14 14 15 15 1416 13 13 12 14 16 17 14 17 18 13 14 13 14 15 1517 13 14 16 15 13 12 17 14 12 14 14 15 14 15 14 1318 13 13 15 14 14 14 15 15 16 14 13 16 16 14 15 14 1619 12 14 15 15 14 12 18 15 12 12 14 14 12 15 14 14 18 1520 12 13 14 15 14 14 17 16 14 13 14 13 13 16 14 15 17 15 1821 14 16 13 13 15 13 12 12 12 14 16 15 13 13 15 15 12 12 14 1222 13 14 13 14 13 14 13 13 12 16 16 13 14 16 14 15 14 13 14 15 1623 13 13 12 14 13 16 12 14 14 17 16 12 15 17 14 16 13 13 12 15 14 1924 12 15 13 13 15 12 14 13 12 12 15 15 12 13 15 15 14 13 17 14 22 15 1225 12 14 12 12 16 14 12 14 16 12 14 16 14 12 16 16 12 14 14 12 22 14 12 2226 12 13 14 14 12 13 14 13 12 16 15 14 15 16 14 14 16 15 15 16 13 17 18 13 1227 12 13 14 13 13 13 13 13 14 15 14 16 16 14 15 14 15 16 14 14 15 15 15 15 16 1628 12 12 16 12 12 12 12 12 16 16 12 20 20 12 16 12 16 20 12 12 12 12 12 12 16 16 20 28

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

02 803 6 604 6 6 605 6 6 6 606 6 6 6 6 407 6 6 6 6 6 608 6 6 6 6 6 8 609 6 6 4 6 6 6 8 610 4 6 6 4 6 4 6 6 411 4 6 6 6 6 6 6 6 6 612 4 4 8 8 4 8 4 8 4 4 8

Fig 2

Fig 3

Fig 4

Fig 5


9/14

Fig 7

6- B C D

17 0 737 1 964 4 16

07 64 2 1308 72 5 2021 34 75 8 2822 75 8 28

14 78 15 2679 6 1080 5 1082 5 10

02 33 93 11 12 2695 5 1098 5 11

z19 44 101 15 1530 101 4 1915 31 103 15 30

104 5 1016 105 15 30z17 107 11 16z36 107 12 17z47 107 12 17

z10 46 112 12 15

113 5 10113 5 10113 5 10

27 113 7 23

z11 40 116 13 14 21z12 117 11 16z03 25 118 10 11 17z43 125 12 16

352001 32

z37 48z28z45

z29 42z06 38

z04 26

z13 50z23z49

z24 39 101 14 15

09 107 13 29

z41 116 13 16

05 18 125 13 29

5- B C D

11 3 1018 3 9

22 4 5 11

21 22 6 1426 4 10

13 30 27 8 1826 27 8 18

02 23 28 8 1507 29 7 1411 31 9 16z36 32 8 15

03 27 33 8 1409 24 34 8 1704 29 36 8 1619 38 6 1528 38 6 15z1838 39 8 15

06 20 40 8 1316 32 40 8 13

22 4 10

22 5 10

28 5 731 28 5 14

05 14 33 7 14

10 25 40 7 8 13

3301 35

34 08

15

z17 37

22

z12

4- B C D

3 0 44 3 14

6 2 126 3 14

19 6 3 20

6 3 148 2 148 2 148 2 148 2 14

04 14 10 4 5 2405 16 10 5 2611 10 5 2612 27 10 5 6 2513 10 5 2518 10 6 24z15 29 12 6 26

2824

01 2306

2103 2607 201017

0925

08

6 1 86 1 10

8 3 1302 22 8 5 25

3- B C D

1 2 02 4 02 4 32 4 4

03 11 3 6 204 3 6 205 3 6 202 07 3 6 308 3 6 4

1201 090610

Legend

n- Pc sets of cardinal and their Isomorphsn(found by swapping icv entries 1 and 5)

B Number of Basic Interval PatternsC Number of Invariant SubsetsD Kh Set-complex SizeE ICV compared to 6-32, 5-35, 4-23, 3-09

0

9

3

111118

11

73

4

7

5

2

7

E

322016164

1311

811108

137

114

1112111199

10258

118

1011757

109

11

E

171683

11139

1085

119

11558

108

108878

1010

08

11

6

1

4242

7466 E

743542233

0

2

1

E

131079949

88837675637464

0

25

123

4

6 20 26 18

8 7 3 13

19 109 17

5

4

2 1

11

2322

14

27

12

16

15

29

28

21 2425

5

5 4

4

4

3 11 311

3

11

7

711 11

3

59

4 9

411

411

911

7 11

711

34

2

3

1 4

45

58

8

58

45

5

7

711 10

11

2103

10

810

710

3

23

23

2

7

2

2

272

12

26

510

34

Fig 6

The pcs universe can be mapped in numerous ways.Figure 6 shows sets of size 4 which hold two of theirfour triads (indicated by small numbers) in common. Infigures 6-7, involutions are shown in red; note also thesymmetrical placement of isomorphs.

In figure 7, columns B-D indicate morphologicalaffinities between pc sets of the same size. The fifthcolumn shows how (measured ratherarbitrarily by icv similarity to cognates of pcs 7-35=)often cuts across these boundaries.

Figure 8 compares the icvs of selected 4-note sets withthose of -sets of cardinal 7. End-sets possesshighly distinctive icvs; they resemble Forte's setgenera and, more especially, Russom's referential scalecollections.

End-sets are related by simple algorithms which willtransform one into another, as shown in fig 9.

tonicality

end

Figure10 compares the icvs of sets of size 3-6 with those of 5end-sets. The totals have been converted to rankorders, so that 1 indicates the closest bonds.

Mapping the pcs Universe


10/14

Circle of Fifths

Expanding Intervals

(Capital letters indicate sharpened notes.)

Chromatic b c C d e f F 7-01=Diatonic f c g d a e b 7-35=Diatonic g c f A D G C 7-35=

Chromatic c C d D e f F 7-01=Whole-Tone c d e F G A c 6-35=Diminished c D F a c D F 4-28=

Augmented c e G c e G c 3-12=Diatonic c f A D G C F 7-35=

Diatonic c d e f g a b 7-35=Diminished c D F G b d f 7-31oAugmented c e G b D g b 6-20=Diatonic c f A d g c f 5-35=

Fig 9

5-24 5 4 9 2 95-25 5 7 2 7 95-26 9 1 5 2 9

5-27 3 6 8 6 115-28 9 2 3 3 95-29 3 7 5 7 115-30 7 1 9 2 115-31 12 8 1 11 125-32 7 4 2 7 115-33= 15 5 13 1 155-34= 4 5 6 3 125-35= 1 10 11 10 145z36 6 6 4 6 65z37= 9 3 8 6 95z38 8 3 4 6 8

6-01= 14 11 8 11 1

6-02 13 9 5 6 26z03 10 7 4 8 36z04= 11 4 6 4 46-05 11 6 4 10 76z06= 11 8 7 12 116-07= 14 10 9 11 146-08= 5 9 6 7 56-09 6 7 6 4 66z10 8 3 3 3 56z11 6 6 3 7 66z12 7 5 4 5 76z13= 11 6 2 10 76-14 8 3 6 7 86-15 11 1 4 4 76-16 9 1 6 4 9

6z17 9 2 4 5 96-18 7 6 4 10 116z19 11 2 4 9 116-20= 15 5 10 13 156-21 13 2 5 2 96-22 11 2 7 2 116z23= 8 6 2 6 86z24 5 3 3 3 86z25 3 7 4 8 106z26= 4 4 6 4 116-27 11 8 1 12 116z28= 10 3 2 6 106z29= 7 6 2 10 116-30 12 6 1 7 12

6-31 7 1 4 4 116-32= 1 11 8 11 146-33 2 9 5 6 136-34 9 2 5 2 136-35= 16 12 11 1 166z36 10 7 4 8 36z37= 11 4 6 4 46z38= 11 8 7 12 116z39 8 3 3 3 56z40 6 6 3 7 66z41 7 5 4 5 76z42= 11 6 2 10 76z43 9 2 4 5 96z44 11 2 4 9 116z45= 8 6 2 6 8

6z46 5 3 3 3 86z47 3 7 4 8 106z48= 4 4 6 4 116z49= 10 3 2 6 106z50= 7 6 2 10 11

7-35Diatonic

6-35Whole-Tone

4-28Diminished

3-12Augmented

7-01Chromatic

Fig 10

Note the similarities between isomorphs, such as 6-01/6-32

7-35Diatonic

6-35Whole-Tone

4-28Diminished

3-12Augmented

7-01Chromatic

Fig 8

4-21=

7-33=W

hole -Tone

7-01=Chromatic

4-11

4-23=

7-35=

Diatonic

4-14

4z15/29

7-21Augmen

ted

7-31Dim

inished

3-01= 7 5 4 4 13-02 5 4 2 3 23-03 6 2 2 3 3

3-04 5 2 4 3 53-05 6 4 3 5 63-06= 4 3 4 1 43-07 2 4 2 3 53-08 6 2 3 2 63-09= 1 5 4 4 73-10= 7 5 1 6 73-11 3 2 2 3 63-12= 8 1 5 2 8

4-01= 7 6 7 8 14-02 6 3 5 4 24-03= 6 3 3 7 24-04 5 3 4 6 3

4-05 6 3 6 5 44-06= 5 5 7 8 54-07= 6 2 5 7 44-08= 6 3 7 8 64-09= 8 5 6 9 84-10= 3 5 3 7 34-11 3 3 4 3 34-12 6 3 2 5 44-13 5 4 2 8 54-14 3 3 4 6 54z15 5 2 3 4 54-16 4 3 6 5 64-17= 5 2 3 7 54-18 6 3 2 8 64-19 6 1 6 4 6

4-20= 4 2 5 7 64-21= 6 3 8 1 64-22 2 3 5 4 64-23= 1 6 7 8 74-24= 7 1 8 1 74-25= 8 2 6 2 84-26= 2 3 3 7 64-27 4 3 2 5 64-28= 9 7 1 10 94z29 5 2 3 4 5

5-01= 14 10 11 10 15-02 10 9 9 7 25-03 11 6 8 6 3

5-04 11 7 5 7 35-05 9 7 9 7 55-06 11 4 9 7 75-07 12 8 12 11 125-08= 12 5 6 3 45-09 9 4 9 2 55-10 9 7 2 7 55-11 6 5 7 5 65z12= 6 6 4 6 65-13 11 1 9 2 75-14 5 7 9 7 95-15= 10 3 10 4 105-16 11 4 2 7 75z17= 9 3 8 6 95z18 8 3 4 6 8

5-19 9 5 3 8 95-20 7 4 9 7 115-21 13 3 10 9 135-22= 12 2 6 8 125-23 2 9 9 7 10


11/14

In popular usage, the words 'tonal'and 'tonality' cover a plethora ofmeanings. In the strictest sense,'tonal' may refer to music written in aspecific key whilst, in its broadestimport, 'tonality' may suggest loyaltyor allegiance to a tonic or tonal center.Other definitions favor music of the

Classical period, or of the major-minor system, which may or maynot include 'modal' music and can even be stretched to embracecertain non-Western musics. Much confusion arises (as Dahlhauspoints out in his New Grove article) from the paucity of adjectivalforms corresponding to words such as 'note' and 'key', so that"'tonal' has to serve a wider area of meaning than 'tonality'".

Schoenberg's rejection of the term 'atonal' suggests that he leanedtowards the wider view and implies that he recognized the presenceof tonal references throughout his music. Whilst truly 'atonal' musiccan certainly exist - music in which pitch-content is of noimportance to structure or of less importance than other parameters- much music commonly called atonal might be better described asbeing of 'extended' or 'diffuse' tonality. In this respect, tonality maybe seen as relative, having analogies with perspective in painting or,more especially, physical gravity; hence it is may be possible, at

least in theory, to measure or . The mainproblem is that this pre-supposes a model, a paragon tonality towhich all others can be compared.

To pursue this argument, it will be necessary to refer by name tovarious pitch-groupings and the reader will need access to tables ofpitch-class sets and set-names, to be found in Figure 1 or in AllenForte's (Yale, 1973). In addition, itwill sometimes be useful to differentiate between sets and theirinversions and for this reason we shall introduce the suffixes 'o' (forprime forms or originals), 'i' (inversions) and '=' (for inversionallyequivalent sets or involutions). It should be remembered that pitch-class sets are always unordered so that for example the major scale(like any of the church modes) is not the same as pc set 7-35= butjust one arrangement of it.

Pc set 7-35=, indeed, is the most obvious candidate for a modeltonality: it is built from six superimposed Perfect 5ths, its five-notecomplement contains the ubiquitous pentatonic scale and it is one ofonly a small group of sets possessing unique interval-class vectorentries (254361). Furthermore, it is linked to subsets and supersetswhich are similarly built upon superimposed 5ths (6-32=, 4-23=, 3-09= and complements) and these together provide a frameworkwithin which sets of any size can be equated. There are variousmethods of comparing sets, involving both interval-class and pitch-class content, and they produce broadly similar, if not identical,results. All of them indicate, however, that the sets furthestremoved from (showing least similarity with) pcs 7-35= are sets 4-28= (known in certain contexts as the Diminished 7th) and 6-35=(the whole-tone scale). These two sets, using the same criteria,register maximum dissimilarity one with the other, underlining the

fact that pcs relationships are essentially 3-dimensional.

Major-minor tonality - still the world's best-loved tonal system - is ofcourse much more than pc set 7-35= and its associates and furtherexamination suggests that it is something of a dual or hybridsystem. The minor scales are forms of pc sets 7-32o (harmonicminor) and 7-34= (melodic ascending) but - more significantly - themain harmonic building blocks of the system derive not from pcs 3-09= but from sets 3-11i and 3-11o (the major and minor triads).The pre-eminence of these two triads is clearly related to theories ofconsonance and dissonance, since the Major and Minor 3rds (orinversions) appear early on in the harmonic ('overtone') series. Bycontrast, the superimposed 5ths model lends greater weight to theMajor 2nd or 9th, the first interval-class to assert itself after thePerfect 5th. Thus it is possible to view major-minor tonality as a

kind of compromise between two rather conflicting hierarchies - onederived from the overtone series and one from the Circle of Fifths.The tempered scale itself represents a similar type of compact.

It is salutary to bear in mind that whilst pc set 7-35= contains agrand total of six major and minor triads, this is also true of threeother sets of the same cardinality (7z17=, 7-22= and 7z37=). Pcsets 7-21o and 7-21i both contain seven, whereas the minor scales

(7-32o and 7-34=) have five and four respectively. It would not befanciful to suggest that any of these other sets could providestarting-points for building a new scale or mode (for presentpurposes, a scale without fixed final note) - indeed the same couldbe said of most sets of size 7 as well as numerous others of greateror lesser cardinality. So-called 'synthetic' scales abound throughoutthe Twentieth-Century repertoire: Bartk's 'acoustic' scale is anotherform of pc set 7-34= whilst all the sets yielding restricted numbers

of forms under transposition are well-represented, for example, inDebussy (6-35=), Bartk (4-09=, 6-20=, 8-28=) and Messiaen (8-28=, 9-12= and, more in theory than in practice, 8-09=, 6-07=, 8-25= and 10-06=).

A scale or mode is not the same thing as a tonal system, though itis certainly an essential ingredient of one. A tonal system impliessome kind of hierarchical ordering, together with a means offocusing in and out of different regions and perhaps a freedom todigress in a consistent manner beyond the narrow pitch boundariesdefined by the scale. Whilst a scale or mode (or tone-row) isessentially 2-dimensional, a tonal system can be seen to have atleast three dimensions.

Major-minor tonality is a human (collaborative) construct and it isdifficult to escape the conclusion that it represents one of mankind's

greatest cultural achievements. The question arises as to whether itis possible to carve other tonal systems out of the twelve-toneuniverse. ('Tonal system' is perhaps an over-ambitious concept andit may be better to speak of 'quasi-' or 'para-' tonalities.)

We can start to look for an answer in Forte's associations of sets orset-complexes and especially the Kh sub-complexes linking setswhich have reciprocal complement relations. Forte himself (in thework cited, p.48) talks of 'whole-tone' and 'diminished' formationsfrom which we can infer groupings centered round sets 6-35=/5-33=/3-06= and 4-28=/5-31/3-10= respectively. In a similar vein,we can point to 'chromatic' and 'two-tone' formations based on sets7/6/5/4/3-01= and 5-21/4-19. All of these sets prioritize intervalclasses other than the Perfect 4 or 5th and, like 7-35=, they arecharacterized by highly-distinctive interval-class vectors. A quite

different formation, perhaps best described as 'neutral', might belinked to one or more of the all-interval tetrachords: 4z15/4z29/5-19/5-28/3-05/3-08. Perhaps, instead of troubling ourselves withquestions of tonality and atonality, it would be better to think interms of different types of tonality, such as diatonic and chromatic,all-interval, whole-tone, diminished and augmented, and hybrids ofthese.

Mention of a 'chromatic' formation brings to mind Varse and apiece like suggests how an alternative tonal systemmight work in practice. The chromatic pcs 7-01=, of course, has aparticular relationship to diatonic 7-35=, whilst inhabiting a quitedifferent sound world: the two sets share the same Kh sub-complexsize and the same number of bips (basic interval patterns). In factsome 70% of Forte's Kh sub-complexes are paired in a similar way -partners ('isomorphs') can be identified by simply interchanging

(nexus set) ic vector entries 1 and 5.

In conclusion, it may be said that the conscious use in compositionof set complexes or associations gives great coherence to a piece ofmusic and provides a deep level of organization analogous totonality. Such groupings may be defined by inclusion or reciprocalcomplement inclusion or by similarities in pitch or interval content;they vary greatly in terms of membership size and thus supportboth small- and large-scale forms. Above all, this type of approachto pitch-structure can open up the complete harmonic palette of thetwelve-tone system, allowing the composer access to a wealth ofbeautiful harmonies.

tonicality tonal loyalty

The set reproduces itself under inversion followed by transpositionIntervals and their inversions are treated as equivalent, thus

producing 6 interval-classes. The ic vector is an ordered array of 6digits defining the total interval content of a set, starting on the leftwith the smallest (ic1 = Minor 2/Major 7).For example, by comparing icv entries using Galton's Method of

Least Squares.The complement of a set consists of all pitch-classes not found in

the set. The Kh subcomplex comprises only sets related by inclusionto both the nexus (reference) set and its complement.

The Structure of Atonal Music

Hyperprism

1

2

3

4

1

2

3

4

New Tonalities

This essay looks at pitch-class set theory as a means

of bridging the gapbetween tonal and non-

tonal language.


12/14

07G d A c*1 4-21=

13G d b a*1 4-13o

05G d A g*1 4z29o

01d G a C*1 4-08=

09C g b d*1 4z 29 i

08 b g a d*3 4-22o

16a c d g*4 4-23=

12A d g c*3 4 -22 i

03 d G b C*1 4 -13 i

06 C d b a*2 4 -11 i

14 A a d g*2 4-14o

04b c d g*2 4 -14 i

11 G d c g*1 4-16o

15C g a d*1 4 -16 i

02 A a c d*2 4-11o

10d a b c*2 4-10=

Fig 12

da

C

G

c

da

b

cg

da

F

cg

e

b

DA

f

4-08=

g

e

FC

G

DA

f

4-10=

e

bF

C

G

DA

f

4-23=

Fig 14

Differencessquared

Squaressummed

Differencesre 254361

Ic Vector

Fig 13

4 - 2 14 - 0 84 z 2 94 - 1 34 - 1 64 - 1 04 - 1 14 - 1 44 - 2 24 - 2 3

0 3 0 2 0 12 0 0 1 2 11 1 1 1 1 11 1 2 0 1 11 1 0 1 2 11 2 2 0 1 01 2 1 1 1 01 1 1 1 2 00 2 1 1 2 00 2 1 0 3 0

2 2 4 1 6 00 5 4 2 4 01 4 3 2 5 01 4 2 3 5 01 4 4 2 4 01 3 2 3 5 11 3 3 2 5 11 4 3 2 4 12 3 3 2 4 12 3 3 3 3 1

4 4 16 1 36 0 610 25 16 4 16 0 611 16 9 4 25 0 551 16 4 9 25 0 551 16 16 4 16 0 531 9 4 9 25 1 491 9 9 4 25 1 491 16 9 4 16 1 474 9 9 4 16 1 434 9 9 9 9 1 41

1

2

3

45

Fig 11

Carnival of Poetry & Lies

A Corridor of Time

A Tower with Terrass Round

Ic VectorSet

9-10=5-31oi8-21=7-33=9-12=7z17=

6 6 6 6 41 1 1 1 24 4 6 4 32 2 6 2 36 6 6 6 34 3 4 4 1

84

76

95

Though billed as an analysis, this article aims also to fleshout some of the ideas presented in earlier essays and at thesame time discuss ways in which a typical piece here mightbe put together. is a representativepiece in terms of level of complexity though, with threedistinct melody and two descant lines, it calls for a largerensemble than most. Perhaps what makes it particularlyinteresting is the brazen way in which it flirts withtraditional tonality.

The sixteen short staves (figure 12) which make up thebridge each represents a new section or block of texture,formed by repeating a four-note figure at full, half and one-quarter speed. In each of the three melody lines, a playercan articulate this figure using any of six rhythmic patterns(built from 2 crotchets + 2 quavers) and he or she has acertain freedom to re-order notes. In addition there are twolines of descant, one anchored to the current harmony andthe other (made up of glissandi) free-standing. Sectionsmay be of any reasonable duration but in any givenperformance it is appropriate that all four pitches secureequal projection. Pitch-structures within a section do notform hierarchies - rather, a section represents a flat surfacebathed in strong color, much as in a Mondriaan painting.

Whilst free to debate the order of sections, performers areadvised not to juxtapose similar harmonies or neighboringstaves, that is, they should move across the score inirregular zigzags. Notwithstanding this, the arch-like designof the bridge powerfully suggests an overall movementfrom bottom-left to bottom-right.

At this point, it may be useful to take a brief look at theother pieces which make up and toexamine their global and other salient harmonies. In fig 11,the icv entries reveal that the three remaining piecescultivate harmonies with a high content of interval-class 3(minor 3/major 6), ic 2 (major 2/minor 7) and ic 4 (major3/minor 6). takes a slightly differentapproach insofar as it focuses on subsets of size 4 which

are shared by pitch-class set 8-01= (comprising 7superimposed semitones) and 7-35= (6 superimposedfifths). In other words, it attempts to bridge two extremesof chromatic and diatonic harmony.

The intersection of pc sets 8-01= and 7-35= reveals a totalof twenty-six shared subsets which, by removal oftranspositions, can be whittled down to sixteen unique sets.Figure 12 shows that all sixteen sets share the pitch d but itis also possible to isolate a similar collection which holds gin common - rejected for reasons which will becomeapparent later. D is the notional tonal center of the work,yet few of the sets here would find a natural home in atraditional d major or d minor piece, even one whichmodulates.

For sake of convenience, subsets of 7-35= may be called'diatonic' sets but it is clear that some are more diatonicthan others, since they may occur at a number oftranspositions (*1 - *4 in fig 12). It is this relative'diatonicism' which determines the position of each set onthe score. (Sets 4-08= and 4-21= have been separatedbecause they hold so little else in common.)

There are other ways of measuring the diatonicism of a set,most obviously by comparing its interval-class vector withthat of 7-35= (254361). Galton's(see figure 13) produces five bands which parallel theaforementioned set-occurrences 1 - 4 of figure 12.

Another method is to look at the manner in which a set'straddles' (occupies) the . From figure 14(where again upper-case letters denote sharpened notes),it is clear that highly diatonic sets can squeeze into arelatively small sector of the circle. Sets associated withnon-tonal language (such as 4-28= and 6-35=) spread farmore evenly.

The Bridge of Follies

Streets & Broad Spaces

The Bridge of Follies

Method of Least Squares

Circle of Fifths

An Analysis

There may be implications here for melodic construction, since ageneral anti-clockwise path along its 'tonal arc' gives a melody aclear feeling of closure. This is a contentious issue, since melodyis shaped by so many other factors, notably rhythm andrepetition, and there is seldom point in classifying, say, thetwenty-four paths a four-note figure might take. Nevertheless,there are sometimes occasions when it may be useful tocharacterize melodic motion as 'progressive' or 'recessive' or evendip into the language of cadences to pirate terms such as 'perfect'

or 'plagal', 'imperfect' or 'interrupted'.


13/14

In July 2012 Tate Modern opened the Tanks, the world's first'permanent' gallery dedicated to 'live' (aka performance, aka time-based, aka action) art. Two months later the Louvre unveiled its newIslamic arts wing. France, it seems, has yet again perfidiculed Albion -not just artistically this time but, thanks to Saudi funding, diplomaticallyand financially as well.

In Britain, Germany and elsewhere, performance art seems to be flavour

of the month, indeed the new century, but why? Is it really that good?Plainly, it chimes in with the big beasts of contemporary culture - realityTV, e-networking and the Me-Now credo of the post-Christian West. Butis it Art/art? Is it new? Is it best served by art galleries? Does it reallyneed to squeeze out other forms of self-expression?

The head of Tate Modern (The Independent, 16 July 2012), says: Weare a little bit fed up with people and organisations doing things at us...We want to give not only artists a voice but the audience a voice." Butwho, precisely, are we? Do Manet or Mozart or Molire or Mrquez dothings at us, and are we really fed up? Has Chris Dercon thrown out theBeatles and Beethoven, and filled his CD collection with Cardew andCage? Do contemporary audiences really gain a voice, or just a handfulof extroverts and artists' friends? Might some not prefer to give theirviews quietly through polls or surveys, if only they had the chance? Wecan all support Josef Beuys (jeder Mensch ist ein Knstler/everyone is

an artist), but not if it means low standards or if audiences are led tomake fools of themselves.

Performance art no doubt has important roles to play in education, inpsycho-therapy or in political protest. It can be stimulating and thought-provoking, but so can many things: that does not make it art. What isart, then? Despite the best efforts of Tolstoy and others, art appears todefy definition, and as soon as we think we have found a definition, newwork comes along to challenge it. It seems we must look at the creator'sintention; if he or she says it is art, it is art. (If we suspect he/she maybe a prankster or charlatan, we pretend not to notice.) In this sense,advertising copy, muzak, even birthday card poems can all aspire to art.

We may not be able to define art, but we can talk about some of itslikely (if not always vital) constituents:

Performance art can encompass all of these things, though it is difficultto find examples which articulate many. Too often, the work is one-dimensional, with a single reducible message. It is sometimes describedas 'time-based', meaning 'ephemeral', but shouldn't art somehow standoutside time, taking us to another world where we can quietly reflect onthe mystery of being? More seriously, performance art can appearcuriously dj vu, simply rehashing ideas which would not look out ofplace in the 1960s (Cage or Fluxus) or a century since (Futurism, Dada,Cabaret Voltaire).

Meanwhile, we live in what ought to be, thanks to education and newtechnology, an artistic Golden Age. Perhaps we indeed do, but artsadministrators and selection panels have long ceased to notice.Computers are everywhere, allowing artists to work with infinite loops,split screens, independent time-lines, previously unimaginable layers ofsound, and to mix media at will. Digital cameras and cell-phones enableus all to shoot images in endlessly different ways and compare them,even before editing, until we achieve exactly what we want. 3Dprinters/replicators, now costing 1400, will revolutionise... on ne saitquoi. Post-modernism is dead and change is in the air.

Where does that leave 'live' art? Not in museums or art galleries, onehopes, but out where it belongs, in hospitals, shopping malls, car-parksand care homes. Whether live art counts as art or not, it is clearly socialscience, since it is about people. We do not place buskers in concert-halls (yet); by putting live art in galleries, bureaucrats are making it

less, not more accessible. And if performance art is a backlash againstausterity and the art market (Nicholas Serota, op cit.), why do we needentrance charges?

From the end of October 2012, the Tanks will open "irregularly asbuilding work is carried out". What a performance!

- art embraces layers of meaning, provoking different interpretations and inviting return visits;- it reflects the age it inhabits and it attempts to say something unique or new;- it involves good management of time and/or space, briefly concentrating human experience;- in terms of information streams, it strives to balance entropy and familiarity;- it speaks to the whole person - body, mind, spirit - not just the intellect;- it is self-contained and should need no explanation, programme-note or apologia;- the creator displays some special talent, technique or vision or, at least, a great deal o f ingenuity.

Dead or Live?

For digital artists, the central question is: to w or not to w?, wherew means to welcome (weave or wonder at) the world-wide web.The web is indeed ww and it might seem perverse, even rude, toignore potential audiences of billions - unless the artist has verygood reason. As web languages begin to catch up with C++, Javaand Delphi, and as download times shrink, the technologicalreasons are fast disappearing.

Web-art brings certain advantages: compared to other art-forms,it is easy to revise, refine or update, perhaps in response to onlinecomment. Browsers include developer tools and error consolesand there is plenty of help available in programmer forums.Usually the entire piece can be downloaded and analyzed, and itoften runs better off-line than on-. Assets (in the form of externalgraphics or audio files) and indeed programming code/stylesheets,can be shared between pieces; reusing old code makes it easy toprototype new work. The backwards compatibility of web browsersmeans that completed

Soundscapes & Other Worlds

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