The Infrastructure of Hellenistic Harmonika

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    INFORMATION TO USERS

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    THE INFRASTRUCTURE OF HELLENISTIC HARMONIKA

    by

    Jonathan Brady Spiegel

    copyright Jonathan Brady Spiegel 1998

    A Thesis Submitted to the Faculty of the

    DEPARTMENT OF CLASSICS

    In Partial Fulfillment of the Requirements

    For the Degree of

    MASTER OF ARTS

    In the Graduate College

    THE UNIVERSITY OF ARIZONA

    19 8

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    STATEMENT BY AUTHOR

    This thesis has been submitted in partial fulfillmentof requirements for an advanced degree at The University ofArizona and is deposited in the University Library to bemade available to borrowers under rules of the Library.

    Brief quotations from this thesis are allowable withoutspecial permission, provided that accurate acknowledgement

    of source is made. Requests for permission for extendedquotation from or reproduction of this manuscript in wholeor in part may be granted by the copyright holder.

    This thesis has been approved on the date shown below:

    APPROVAL BY THESIS DIRECTOR

    ftJon Solomon

    PrOTessor of Classics

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    TABLE OF CONTENTS

    I. LIST OF TABLES 4

    II. ABSTRACT 5

    III. INTRODUCTI N 6

    IV. BODY

    . The fifteen (pdoyyoL of the disjunctive GPS 9

    / ? . Miar] = Octave Species Configuration A 14

    y. The 5 L C t r e a o o i p c i i v Interval 19

    5. The Three Genera 21

    e. The K O I VQ V , or 'Stringed Instrument Ruler' 27

    e l b t ] and T O V O L 37

    1 7 . dvva^ei ' ; , 0eaet9 50

    . AojpLOv and 'Standard Attunement' 6

    V. WORKS CITED 76

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    4

    LIST OF TABLES

    TABLE I 13

    TABLE II 22

    TABLE III.a,b,c,d 44-46

    TABLE IV 52-53

    TABLE V 59

    TABLE VI 60

    TABLE VII 62

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    7

    treatise predominantly in the Pythagorean 'ratio

    tradition. Aristides Quintilianus (3rd century C.E.)

    examined music theory in the Aristoxenian tradition.

    Alypius provided a tablature which standardized the T O V O L of

    Aristoxenus, and he lived 3rd-4th century C.E. Bryennius

    flourished circa 13 00 C.E. and left us with a very detailed,

    though somewhat flawed, summary of Ptolemaic music theoiry.

    Many others of this era wrote on the infrastructure of

    harmonics: the reader may refer to the works cited.

    There exists in common among all the Hellenistic

    theoreticians certain common structural configurations.

    These configurations, the ancient equivalents (or, more

    properly, approximations) of modern keys and modes,

    represent the method of order of the notes within the

    avaTfi^cxTOi, often translated "scales", but actually meaning

    one of two "systems," one of fifteen notes (two octaves,

    ametabolic disjunctive Greater Perfect System) and one of

    eleven (octave and a fourth, conjunctive Lesser Perfect

    System) The principal auaTT]/xa with which I shall be

    c o n c e r n e d i s t h e a m e t a b o l i c d i s j u n c t i v e { d le^evypievov)

    ^ By 'ratio' I mean to refer to th e employment of fractions todescribe intervals. Bounding any interval there were two notes, a lowerand a higher, corresponding to two string lengths which sounded those notes(on the stringed instrument) a longer and a shorter, respectively. Thedenominator represented the length of the shorter string, and th e numeratorrepresented the length of th e longer string.

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    Greater Perfect System { a v a T T j u a TeXetov) , since the

    conjunctive {avvT\inievov) system seems to be a later and

    lesser important offspring of the former. Henceforth the

    " a m e t a b o l i c G r e a t e r P e r f e c t S y s t e m " , o r avaT-qiia reXecou,

    shall be referred to as the "GPS."

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    9

    a. The fifteen

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    10

    except for the unison (frequency ratio H/F = l/l).

    II. Seven of the fifteen (functional) notes are

    " f i x e d " [ E OT UNE ) ; eight are "moveable" {K LV ov IIEV o L) . The

    fixed notes are the following: irpoaXaiiPavofxevo'^, uiraTT]

    u - n o L T O i v, u i r a T T j/ z l c r c j i / , i i e < y q , i r o i p a i i i a T j ,v f \ T T ]5 l e ^ e u y f j - i v u v , and

    vfiTTj uireplSoXoituv. The fixed notes represent the octave

    species configurations A,B, and E {v. infra ). The moveable

    notes are those eight which remain. The fixed notes remain

    in the same position within the range of the three genera,

    whereas the moveable notes change position per genus.

    III. All the Hellenistic theorists distinctly-

    postulated that each of the fifteen notes in the disjunctive

    GPS corresponded to specific respective positions within

    each of the two octaves, that is, they represented the

    notes, in their dynamic or functional form {icaTa bvvoniLv, a

    Ptolemaic term^) , as octave species, or modal,

    configurations. The modal configuration of the octave from

    dynamic i x e a r } to dynamic i / f j T T jUTTep^oXaluv, and likewise from

    dynamic rcpoa\a.ii av6fxevo to dynamic nearj, was equivalent in

    configuration to the modern Aeolian mode, that is, the

    For example th e notes F/C would create a fourth interval; C represents th eoctave tonic and F represents the note a fourth interval above it. H/F =ratio of pitch = frequency ratio; the inverse is F/H, = ratio of stringlengths = inverse frequency ratio. Of. Handschin (15).

    This will be explained and addressed extensively below (section 1 7 ) .

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    11

    modern letter designation A-A' , commonly called by modern

    musicians the "natural minor scale."' ' The reader must

    understand, however, that the designation of this octave

    species configuration as "mode" is inaccurate, since by the

    word "mode" we mean a particular species defined by a tonic

    note. The tonic note, however, as we know it, was unknown

    to the ancient theorists, and therefore it is misleading to

    assume automatically that they used the idea.' Clearly,

    however, the ancient theorists utilized functional notes

    By "A-A" I mean to make reference to the "C scale", which containsno sharps and no flats. The notes A-A thus refer to th e modem notes inth e diatonic scale A,B,C,D,E,F,G,A', which sounds out the modem Aeolianmode. In this treatise for simplicity of reference I use this C-scalesystem to refer to th e octave species of th e ancient theore ticians. Thefollowing letters will be recognized as (but do not necessarily indicate

    the sound of) th e octave species configuration of the following modemmodal scales, respectively: A-A' = Aeolian, B-B' = Locrian, C-C = Ionian(major scale), D-D' = Dorian, E-E' = Phrygian, F-F' = Lydian, G-G' =Mixolydian.

    ^ Perhaps the most important identifying factor of ancient Greekmusic. Cf. Shirl aw (17): "If such a conclusion as that of the identity ofthe Greek Perfect System with our minor scale can be established beyonddoubt, musical research has brought to light no fact of greater importanceconcerning Greek music an d tone-systems. If, however, it be unfounded,nothing could be more calculated to lead to their completemisapprehension." In section 3 I