Aristoxenos and the Geometrization of Musical Space

8/2/2019 Aristoxenos and the Geometrization of Musical Space

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3 Aristoxenos and the geometrization ofmusical space

ARISTOXENOS WAS FROM the Greek colony of Tarentum in Italy, the homeof the famous musician and mathematician Archytas. In the early part ofhis life, he was associated with the Pythagoreans, but in his later years hemoved to Athens where he studied under Aristotle and absorbed the newlogic and geometry then being developed (Barbera 1980; Crocker 1966;Litchfield 1988). He was the son of the noted musician Spintharos, whotaught him the conservative musical tradition still practiced in the Greekcolonies, if not in Athens itself (Barbera 1978).The geometry of musicThe new musical theory that Aristoxenos created about 320 BeE differedradically from that of the Pythagorean arithmeticians. Instead of measuringintervals with discrete ratios, Aristoxenos used continuously variablequantities. Musical notes had ranges and tolerances and were modeled aslociin a continuous linear space. Rather than ascribing the consonance of theoctave, fifth, and fourth to the superparticular nature of their ratios, he tooktheir magnitude and consonance as given. Since these intervals could beslightly mistuned and still perceived as categorically invariant, he decidedthat even the principal consonances of the scale had a narrow, but stillacceptable range of variation. Thus, the ancient and bitter controversy overthe allegedly unscientific and erroneous nature of his demonstration that theperfect fourth consists of two and one half tones is really inconsequential.Aristoxenos defined the whole tone as the difference between the twofundamental intervals of the fourth and the fifth, the only consonances

smaller than the octave. The octave was found to consist of a fourth and a

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3-1T he g en era o f A risto xen os. T he d esc rip tio ns o fA ri st ox en o s (M a& r lJ n 1902) in term s o f tw elfth s o fto ne s h av e b ee n c on ve rte d to c en ts , a ss um in g 500cents to th e e qu ally te mp ere d fo urth . T he in ter-p re ta ti on o f A ri st ox en o s's f ra ct io n al t on es a s t hi rt yp ar ts to th e fo ur th is a fte r th e s ec on d c en tu ry th eo -rist CkonUks .

fifth, two fourths plus a tone, or six tones. The intervals smaller than thefourth could have any magnitude in principle since they were dissonancesand not precisely definable by the unaided ear, but certain sizes weretraditional and distinguished the genera known to every musician. Theseconventional intervals could be measured in terms of fractional tones bytheear alone because musical function, not numerical precision, was thecriterion. The tetrachords that Aristoxenos claimed were well-known areshown in 3-I.Aristoxenos described his genera in units of twelfths of a tone (Macran

1902), but later theorists, notably Cleonides, translated these units into acipher consisting of 30 parts (moria) to the fourth (Barbera 1978). Theenharmonic genus consisted of a pyknon divided into two 3-part micro-tones or dieses and a ditone of 24 parts to complete the perfect fourth. Nextcome three shades of the chromatic with dieses of 4,4.5, and 6 parts andupper intervals of 22,21, and 18 parts respectively. The set was finished withtwo diatonic tunings, a soft diatonic (6 + 9 + 15 parts), and the intensediatonic (6+ 12+ 12parts). The former resembles a chromatic genus, but thelatter is similar to our modern conception of the diatonic and probably

ENHARMONIC INTENSE CHROMATIC----0 50 100 500 0 100 100 500

3 + 3 + 24 PARTS 6 + 6 + 18 PARTS1/4 + 1/4 + 2 TONES III + III + I III TONES50 + 50 + 400 CENTS 100 + 100 + 300 CENTS

SOFT CHROMATlC SOFT DIATONIC----0 67 133 500 0 100 250 500

4 + 4 + IIPARTS 6 + 9 + 15 PARTS1/3 + 1/3 + I 5 /6 TONES III + 3/4 + I 1/4 TONES67 + 67 + 333 CENTS 100 + 150 + 250 CENTS

HEMIOLIC CHROMATIC [NTENSE DIATONIC----0 75 150 500 0 100 300 5004.5 + 4.5 + 2I PARTS 6 + rz + IIPARTS

3/8 + 3/8 + I 3/4 TONES III + I + I TONES75 + 75 + 350 CENTS 100 + 200 + 200 CENTS

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3 - 2. . O t he r genel 'o men t ioned b y Aris toxenos .

UNNAMED CHROMATIC

a lOa4 + 8 + 18 PARTS

1/3 + 2/3 + 1 112 TONES67 + 133 + 300 CENTS

DIATONIC WITH SOFT CHROMATIC DIESIS

o 3004 + 14 + 12 PARTS1/3 + I 1/6 + 1 TONES67 +233 + 200 CENTS

500

DIATONIC WITH HEMIOLIC CHROMATIC DIESIS

a 75 5000045 + 135 +].I PARTS3/8 + I 1/8 + I TONES75 + 225 + 200 CENTS

REJECTED CHROMATIC

a 100 ISO6 + 3 + 2I PARTS

Ih + 1/4+ 13/4 TONES100 + 50 + 350 CENTS

500

UNMELODIC CHROMATIC---------------o 75 133

45 + 35 + 22 PARTS3/8 + 7/24 + I 5/6 TONES75 +5 8 + 367 CENTS

500

500

represents the Pythagorean form. Two such jo-part tetrachords and awhole tone of twelve parts completed an octave of 72 parts.Several properties of the Aristoxenian tetrachords are immediately

apparent. The enharmonic and three chromatic genera have small intervalswith similar sizes, as if the boundary between the enharmonic and chro-matic genus was not yet fixed. The two chromatics between the syntonicchromatic and the enharmonic may represent developments of neu-tral-third pentatonics mentioned in chapter 2.The pyknon is always divided equally except in the two diatonic genera

whose first intervals (half tones) are the same as that of the syntonicchromatic. Thus Aristoxenos is saying that the first interval must be lessthan or equal to the second, in agreement with Ptolemy's views nearly :fivehundred years later.The tetra chords of 3 -2 are even more interesting. The first, an approved

but unnamed chromatic genus, not only has the 1:2 division of the pyknon,but more importantly, is extremely close to Archytas's chromatic tuning(Winnington-Ingram 1932). The diatonic with soft chromatic diesis is avery good approximation to Archytas's diatonic as well (ibid.). OnlyArchytas's enharmonic is missing, though Aristoxenos seems to allude to itin his polemics against raising the second string and thus narrowing thelargest interval (ibid.). These facts clearly show that Aristoxenos understoodthe music of his time.The last two tetrachords in 3-2 were considered unmusical because the

second interval is larger than the first. Winnington-Ingram (1932) hassuggested that Aristoxenos could have denoted Archytas's enharmonictuning as 4+ 3 + 23 parts (6 7 + 5 0+ 3 83 ), a tuning which suffers from the samedefect as the two rejected ones. A general prejudice against intervalscontaining an odd number of parts may have caused Aristoxenos to disallowtetrachords such as 5 + II + 14, 5 + 9 + 16 (ibid.), and 5 + 6 + 19 (M acran1902).The alleged discovery of equal temperamentBecause a literal interpretation of Aristoxenos's parts implies equal tem-peraments of either 72 or 144 tones per octave to accommodate thehemiolic chromatic and related genera, many writers have credited himwith the discovery of the traditional western European ra-tone intonation.This conclusion would appear to be an exaggeration, at the least. There is



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no evidence whatsoever in any of Aristoxenos's surviving writings or fromany of the later authors inhis tradition that equal temperament was in-tended (Litchfield 1988).Greek mathematici answould have had no difficulty computing the stringlengths for tempered scales, especially since only two computations for each

tetrachord would be necessary, and only a few more for the complete octavescale. Methods for the extraction of the square and cube roots of two werelong known, and Archytas, the subject of a biography by Aristoxenos, wasrenowned for having discovered a three-dimensional construction for thecube root of two, anecessary step for dividing the octave into the 12, 24, 3672, or 144 geometric means as required by Aristoxenos's tetrachords (Heath[1921] 1981, 1:246-249). Although irrationals were a source of great worryto Pythagorean mathematicians, by Ptolemy'S time various mechanicalinstruments such as the mesolabium had been invented for extracting rootsand constructing geometric means (ibid., 2:104). Yet neither Ptolemy norany other writer mentions equal temperament.Ptolemy, infact, utterly missed Aristoxenos's point and misinterpreted

these abstract, logarithmic parts as aliquot segments of a real string of 120units with 60 units at the octave, 80 at the fifth, and 90 at the fourth. Hisupper tetrachord had only twenty parts, necessitating the use of com-plicated fractional string lengths to express the actually simple relations inthe upper tetrachords of the octave scales.There are two obvious explanations for this situation. First, Aristoxenos

was opposed to numeration, holding that the trained ear of the musicianwas sufficiently accurate. Second, Greek music was mostly monophonic,with heterophonic rather than harmonic textures. Although modulationsand chromaticism did exist, they would not have demanded the para tacticalpitches of a tempered gamut (polansky 1987a). There was no pressing needfor equal temperament, and if it was discovered, the fact was not recorded(for a contrary view, see McClain 1978).Later writers and Greek notationAlthough most of the later theorists continued the geometric approachtaken by Aristoxenos, they added little to our knowledge of Greek musictheory with few exceptions. Cleonides introduced the cipher of thirty partsto the fourth. Bacchios gave the names of some intervals of three and fivedieses which were alleged to be features of the ancient style, and Aristides

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3 -3 . T wo m ed ie va l I sla mic f o r m s . T he se tw o m ed -ieval Islamic t et ra cb o rd s a re A ri sto xe ni an a p -p r ox im l lt io n s t o P t ol em y 1quable d ia to n ic . T h eA I 'a b s a ls o l is te d A r is to x eno s' s o th e r t et ra c bo r ds i nthe ir treatises .

NEUTRAL DI,l.TONIC

a 200 350 50012 + 9 + 9P,l.RTS1+ 3/4 + 3/4 TONES200 + 150 + ISO CENTS

EQU,l.LDIATONIC

a 167 334 50010 + 10 + 10 PARTS

5/6 + 5/6 + 5/6 TONES167 + 167 + 166 CENTS

Quintilianus offered a purported list of the ancient harmoniai mentionedby Plato in the Timaeus .One exception was Alypius, a late author who provided invaluable

information on Greek musical notation. His tables of keys or tonoi weredeciphered independently in the middle of the nineteenth century byBellermann (1847) and Fortlage (1847), and made it possible for the fewextant fragments of Greek music to be transcribed into modern notationand understood. Unfortunately, Greek notation lacked both the numericalprecision of the tuning theories, and the clarity of the system of genera andmodes (chapter 6). Additionally, there are unresolved questions concerningthe choice of alternative, but theoretically equivalent, spellings of certainpassages. Contemplation of these problems led Kathleen Schlesinger to theheterodox theories propounded in The G reek Autos .Others have simply noted that the notation and its nomenclature seem

to have evolved away from the music they served until it became anacademic subject far removed from musical needs (Henderson 1957)' Forthese reasons, little will be said about notation; knowledge of it is notnecessary to understand Greek music theory nor to apply Greek theory topresent-day composition.Medieval Islamic theoristsAs the Roman empire decayed, the locus of musical science moved fromAlexandria to Byzantium and to the new civilization of Islam. Aristoxenos'sgeometric tradition was appropriated by both the Greek Orthodox church todescribe its liturgical modes. Aristoxenian doctrines were also included intheIslamic treatises, although arithmetic techniques were generally employed.The tetrachords of 3-3 were used by Al-Farabi to express 3/4-tone scales

similar to Ptolemy's equable diatonic in Aristoxenian terms. Ifone subtracts10 + 10 + 10 parts from Ptolemy's string of 120 units, one obtains the series120 110 100 90, which are precisely the string lengths for the equablediatonic (1211 I . 11110' 10/9). It would appear that the nearly equaltetrachord I r/ro- IIlIo. 400/363 was not intended.The tetrachord 12 + 9 + 9 yields the permutation 120 108 99 90, or

10/9' I Z/II 1I1I0. This latter tuning is similar to others of Al-Farabi andAvicenna consisting of a tone followed by two 3/4-tone intervals. Othertetrachords of this type are listed in the Catalog.

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3 -4 , B yw ntim and G reek O rthodox tetrllChords.A th an ll .fO p ou kJ s's e nh ar m on ic a nd d ia to nic g en er ac on sis t o f v ar io us p er mu ta tio ns o f 6+ I2+12, i.e, 12+6+12. X en akir p erm its p erm uta tio ns o f th e 12+I I +7 l ind 6+12+I 2 g ene1 'l l. A c lo se r , bu t non-s up er pa rt ic ulo r, a pp ro xim a tio n t o X e na kir 's in te ns etb romat ic ' Wo ul d b e 22h1' 6/5' 35/33.

Eastern Orthodox liturgical musicThe intonation of the liturgical music of the Byzantine and SlavonicOrthodox churches is a complex problem and different contemporaryauthorities offer quite different tunings for the various scales and modes(echoi) . One of the complications is that until recently a system of 28 partsto the fourth, implying a 68-note octave (28 + 12 + 28 = 68 parts), was inusealong with the Aristoxenian 30 + 12 + 30 parts (Tiby 1938).Another problem is that the nomenclature underwent a change; the term

enharmonic was applied to both a neo-chromatic and a diatonic genus, andchromatic was associated with the neo-chromatic forms. Finally, many ofthe modes are composed of two types of tetrachord, and both chromaticismand modulation are commonly employed in melodies.Given these complexities, only the component tetra chords extracted

from the scales are listed in 3-4. The format of this table differs from thatof 3-1 through 3-3 inthat the diagrams have been omitted and partiallyreplaced by the ratios of plausible arithmetic forms. The four tetrachordsfrom Tiby which utilize a system of 28 parts to the fourth are removed tothe Tempered section of the Catalog.

PARTS RATIOS GENUSENTSATHANASOPOULOS (1950)

9 + IS +6 150+ 2S 0 + 100 CHROMATIC6+ 18+6 100+ 300 + 100 CHROMATIC6+Il+Il 100+ 200+ 200 DIATONIC12+12+6 200+200+ 100 ENHARMONIC

SAVAS(196S)8 +14+ 8 133+ 133+ 133 CHROMATIC10+ 8+ 12 167+ 133+ 100 DIATONIC8 + 11. + 10 133+ 1.00+167 BARYS DIATONICI1.+I1.+6 200+ 100 + 100 ENHARMONIC8 +16+6 133+ 1.67+ 100 BARYS ENHARMONIC6+ 20+4 100+ 333+67 PALACE MODE (NENAN

XENAKIS (1971)7 +16+7 1I7 + 266+ 1I7 16lrS . 7/6. ISlr4 SOFT CHROMATIC5 +19+ 6 83+ 317 + 100 25 611 .43' 6 /S . 135/128 INTENSE CHROMATICI1.+U+7 200 +183+ II7 9/8. 10/9' 16/IS DIATONIC6+11+12 100+ 100 + 200 1 s 6 1243 . 9/8. 9/8 ENHARMONIC

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The tetrachords of Athanasopoulos (1950) are clearly Aristoxenian inorigin and inspiration, despite being reordered. One of his chromatics isAristoxenos's soft diatonic and the other isAristoxenos's intense chromatic.The rest of his tetra chords are permutations of Aristoxenos's intensediatonic.Savas's genera (Savas 1965) may reflect an Arabic or Persian influence,

as diatonics with intervals between 133 and 167 cents are reminiscent ofAl-Farabi's and Avicenna's tunings (chapter 2 and the Catalog). They mayplausibly represent I2/II and IIIIa so that his diatonic tunings are in-tended to approximate a reordered Ptolemy's equable diatonic. Hischromatic resembles 14113 .8/7' 13hz and his Barys enharmonic, 15h4'716. 16h5' Savas's ordinary enharmonic may stand for either Ptolemy'sintense diatonic (1019' 918 . 16115) or the Pythagorean version (2561243 .918. 9/8). The palace mode could be 15114.615 . 28127 (Ptolemy'S intensechromatic). The above discussion assumes that some form of just in-tonation is intended.The tunings of the experimental composer Iannis Xenakis (1971) are

clearly designed to show the continuity of the Greek Orthodox liturgicaltradition with that of Ptolemy and the other ancient arithmeticians, thoughthey are expressed in Aristoxenian terms. This continuity is debatable;internal evidence suggests that the plainchant of the Roman Catholicchurch is derived from Jewish cantillation rather than Graeco-Romansecular music (Idelsohn 192 I). It ishard tosee how the music of the Easternchurch could have had an entirely different origin, given its location andcommon early history. A case for evolution from a common substratum ofNear Eastern music informed by classical Greek theory and influenced bythe Hellenized Persians and Arabs could be made and this might give theappearance of direct descent.The robustness of the geometric approach of Aristoxenos is still evident

today after Z300 years. The musicologist James Murray Barbour, a strongadvocate of equal temperament, proposed 2 + 14 + 14 and 8 + 8 + 14 asAristoxenian representations of49/48. 8/7.817 and 141r3 . 13hz. 8/7inhis1953book on the history ofmusical scales, T u nin g a nd T em p er am e nt. 'WithXenakis's endorsement, Aristoxenian principles have become part of theworld of international, or transnational, contemporary experimental music.In the next chapter the power ofthe Aristoxenian approach to generate newmusical materials will be demonstrated.


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Aristoxenos and the Geometrization of Musical Space