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Aspects of mathematics and music in Ancient Greece

SOFIA KONTOSSI

Department of Music Studies

Athens University

K. Paxinou, 19B, 19009, AttikaGREECE

[email protected] http://www.sofiakontossi.com

RAZVAN RADUCANU

Department of Mathematics

Al. I. Cuza University

Bulevardul Carol I, Nr.11, 700506, Iasi

ROMANIA

[email protected] http://www.rraducanu.ro

Abstract: - In this paper, through the study of the different theories concerning the calculation of intervals sizes which

were developed in Ancient Greece, we are going to undertake a concise historical overview of the relationshipestablished between music and mathematics through the Pythagorean, Euclidean and Aristoxenian tradition.

Key-Words: -Ancient Greeks, Aristoxenus, Euclid, Pythagora, mathematics, musical intervals, music scales, musictheory

1 IntroductionAncient Greeks did not have todays knowledge

of sound wavelengths and frequencies, so they could notunderstand the musical phenomenon as the physicalexplanation of the harmonic series and pitches. Theirunderstanding of music science came initially throughmathematics. They noticed that the sound produced by a

string depends upon its length, tension and density. Inorder to be able to reproduce the same relationship

between two sounds (a concept they defined as musicalinterval) they were studying ratios of string lengths. Dueto the need to operate with intervals for musicalpurposes -like tuning, creating scales etc.-, theorists in

their effort to divide the tone (i.e. the distance betweennotes A and B) indirectly shaped the conception of ratio.In this paper, through the study of the different

theories concerning the calculation of intervals sizeswhich were developed in Ancient Greece, we are goingto undertake a concise historical overview of the

relationship established between music and mathematicsthrough the Pythagorean, Euclidean and Aristoxeniantradition.

2 The PythagoreansAccording to the Pythagoreans conception about

cosmos, numbers are the ultimate reality. Therefore,musical science was not to be explained on the basis of

the real musical phenomena temporal manifestation of

which, according to their view, conferred them traces of

imperfection but on the harmonious reflection ofnumbers.

The first to conceive the relationship underlyingbetween music and mathematics, establishing thus theidea of the numerical base of acoustics, was Pythagoras,a philosopher, mathematician and musician from Samos(580b.c. - c.500b.c.) who believed that every value including pitches of notes since they are related to thenumber of motions of a string could be expressed as aratio. Among his greatest discoveries (or those of thePythagorean school, the distinction seems to be hard), bymeans of the monochord,

1 a string fastened across a

movable bridge to facilitate changes in pitch, is that thechief musical intervals are expressible in simplemathematical ratios between the first four integers [6].Thus, the octave, the fifth and the fourth the mostimportant consonances in ancient Greek music were

1Questions concerning the existence of the monochord or the

credibility of Nicomachus of Gerasa famous story according

to which Pythagoras discovered the simple ratios underlying

musical consonance by noticing the intervals produced by

workmen pounding out a piece of metal upon an anvil with

hammers of different weight are not of importance in thepresent article so they will not be considered.

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produced from the ratios 2:1, 3:2 and 4:3 respectively.From a musical point of view, adding a fifth to a fourth,

which requires multiplying their ratios, results in theoctave, also true when expressed in mathematical terms(3/2 x 4/3 = 12/6 = 2/1). Considering the importance thatthe tetractys of the decade (represented by the numbers1, 2, 3, 4, the sum of which equals sacred number 10)

had to the Pythagoreans as the key to the understandingof the universe, these ratios were the reflection of both

mathematical and musical harmony. The underlyingconcept of such connections was projected also to thecosmos harmony and the planetary spheres were seen as

parts of a vast musical instrument attuned following thesame ratios governing musical intervals. As M. L. West

mentions, Plato's harmony of the spheres is not someunimaginable, transcendental passacaglia or fugue, butthe naked glory of the diatonic octave [9].

Measuring smaller intervals than the fourth wasof capital importance in ancient Greek music considering

both the facts that the tetrachord was the unit ofconstruction of musical scales and that the different

modalities of tetrachords subdivision defined thegenera. For that purpose the Pythagoreans usedmathematical processes. By extracting the 4th from the

fifth, they defined the ratio of the tone (9:8). Thenvarious semitones sizes occurred: the difference

between the fourth and two tones (or the differencebetween three octaves and five perfect fifths),represented by the ratio 256:243, was called limma

(remainder, the diatonic semitone of the intemperatesystem) and the difference between the tone and the

limma (or between seven 5ths and four octaves) wasnamed apotom (segment, ratio 2187:2048, thechromatic semitone). As a result, the Pythagorean scale,consisting of two disjointed consecutive fourths, wasexceeding the octave by a small interval known as thePythagorean comma (531,441:524,288), which couldalso be thought of as the discrepancy between twelvejustly tuned perfect fifths and seven octaves.

The size of the semitone and the addition oftones and semitones to create the consonant intervals

became a subject of heated controversy between thePythagoreans, with their fundamentally arithmeticapproach, and the Aristoxenians, who adopted ageometric approach to the measurement of musicalspace [7]. An equal division of the tone, meaning froma mathematical point of view to find x so that x

2=9/8,

leads to irrational numbers using todays terminology,an unacceptable disturbance in the Pythagorean musicalsystem and conception of the world.

3 Euclid (4th-3th century BC)

3.1 The description of the Pythagorean scale by

Euclid

Euclid, the famous mathematician and geometer, in atreatise attributed to him entitled The Division of theCanon, describes the steps of the construction of the

Pythagorean scale. Known as intense diatonic, this scalehas been the subject of great theoretical discussion fromantiquity to our days. The scale can be produced on amonochord with the exclusive use of two consonantintervals, the octave and the fifth. A short description ofthe procedure follows:

a) Halving the string, we take the upper octave of theinitial sound.

b) Descending from the octave a fifth we take thefourth.

c) Ascending an octave and descending a fifth arises

the seventh.d) Descending from the seventh a fifth appears the

third.e) Ascending an octave and descending a fifth the sixth

is obtained.f) Descending from the sixth a fifth we take the

second.g) Ascending an octave and descending a fifth appears

the fifth.

Expressed in ratios, the aforementioned scale would berepresented as following:

[1] 256/243 [2] 9/8 [3] 9/8 [4] 256/243 [5] 9/8 [6] 9/8[7] 9/8 [8]

Two observations can be made upon this scale. Firstly,that, when started from the second degree, the scalecoincides with the usual major type scale of westernmusic and that two similar tetrachords (tone-tone-limma) are formed. Secondly, that the scale constructionis based only on tone (~204 cents) and limma (~90cents), which actually means that the difference between

the small step (~ semitone) and the big step (tone) isthe wider possible (thus justifying the sound impressionas being intense). From antiquity though, musicalpractice suggested a finer subdivision of the diatonicscale, a more gradual transition from one step to another.

This could be obtained by replacing in a tetrachord thelimma with the apotome (~114 cents), thus, the second

tone resulting to be quite smaller than the first one (~180cents). This interval can be considered as the elassontone(smaller tone) of Byzantine music. The new scale,

called mild diatonic, had the disadvantage that, themathematical representation of the degrees resulted into

very big numbers (i. e the ratio of the elasson tone was216

/ 310

). The problem was solved by Dydimus (1st


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century B.C) who replaced the Pythagorian ditone by themild diatonic third (5/4). This ratio for the meizon

(major) third, generally accepted since Hellenistictimes, was also adopted from the Arabs theoreticians.Both the intense and the mild diatonic scale were used inByzantine music, the last one especially beingwidespread met in the musical practice of all people of

the Eastern Mediterranean basin and the Middle East [8].

3.2 Extending Euclidean thought beyond

mathematics. The application of Euclidean algorithm

on music theory

Among other significant mathematical achievements,

Euclid remained known in history for his famousalgorithm. In the XXth century, researchers connected theEuclidean algorithm to music theory. Norwegian

mathematician Viggo Brun used Euclidean algorithms toexplore tuning matters [4]. On the other hand, the

Euclidean algorithm was related to rhythms and scales intraditional music. Its structure can be used to

automatically generate, very efficiently, a large family ofrhythms used as timelines (rhythmic ostinatos), intraditional world music. 2

Here we will present an application of thealgorithm to highlight, a significant problem that had a

direct impact on musical practice, namely theidentification of the mathematical relationship(proportionality) between two musical intervals ratios

(i.e. how many times a fifth goes into an octave, meaninghow many times do we have to multiply 3:2 to get 2:1).

As we can see below, applying Euclids algorithm to findthe highest common factor for the fifth and the octavegives no integer solution.

Interval

a n b = c, where

b > c

Ratio

Corresponding equation

for ratios

octave - fifth = fourth 2:1 / 3:2 = 4:3 (1)

fifth - fourth = tone 3:2 / 4:3 = 9:8 (2)

2 Articles related to this subject: Toussaint, Godfried, The

Euclidean Algorithm Generates Traditional Musical Rhythms,extended version of Banff, Alberta,Proceedings of BRIDGES:

Mathematical Connections in Art, Music and Science, 2005, pp.

47-56, http://cgm.cs.mcgill.ca/~godfried/publications/banff.pdf

and Erik D. Demaine, Francisco Gomez-Martin, Henk Meijer,

David Rappaport, Perouz Taslakian, Godfried T. Toussaint,

Terry Winograd, David R. Wood, The distance geometry of

music, Computational Geometry 42 (2009), pp. 429454.

Fourth - 2 tones =

semitone4:3 / (9:8)

2= 256:243 (3)

tone 2 semitones =comma

9:8 / (256:243)2 =531441:524288 (4)

We could continue this indefinitely. Practically, theimpossibility to reach an integer solution means that,tuning based on successive fifths leads to octaves out of

tune. For instance, starting from C and performing thecircle of fifths [C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B#(=C)], which requires 12 fifths to be completed, wecover a musical distance of seven octaves. Inmathematical terms, considering the ratio for the fifth

being 3:2 and that of the octave 2:1, this should mean(3/2)

12equals 2

7, which fails to be true. The compound

ratio of (3/2)12

: 27

actually proves to be the renowned

Pythagorean comma, 531441:524288. In this point wemust notice that Greeks realized that, adding twomusical intervals required multiplying the correspondingratios and similarly, subtracting one interval from

another requires dividing their ratios. Some scholarsbelieve that Euclid and his predecessors probably

conceived the theory of ratio as a generalization ofmusical theory of intervals.

However, if we want to get some approximate

result that will be useful in musical reality, we could cutoff the algorithm at some point and pretend that one

interval went exactly into another with no remainder. Ifthe real remainder is small enough, that will give us auseful approximation. For example, suppose we pretendthat comma is equal to zero (tone - 2 semitones = 0).Then

tone = 2 semitones

fourth = 2 tones + semitone (by 3)= 5 semitones (by 5)

Fifth = fourth + tone (by 2)

= 7 semitones (by 5 and 6)

Octave = fifth + fourth (by 1)= 12 semitones (by 5 and 7).

So, using Euclid's algorithm we conclude that, areasonable approximation to the number of fifths in anoctave is 7/12, because a fifth is roughly 7 semitones andan octave is roughly 12 of them.

4 Aristoxenus (4th century B.C.)

A revolution in the history of musical thought in ancienttimes was made by Aristoxenus, a student of both


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Pythagoreans and Aristotle, who first considered music asun autonomous discipline and he is often referred to as

the father of Musicology. In the first part of theHarmonicElements, Aristoxenus criticizes the foundations of hispredecessors teachings, without naming them, probably atestament of respect. In opposition to the Pythagorean -Platonic ideal, according to which music is part of

mathematics and a musical interval is not perceived as amusical entity but as a ratio consisting exclusively from

whole numbers, Aristoxenus prime criterion for themusical phenomena was the ear. For him musicconsisted of sounds structurally organized within a sound-

space, and the function of the science of harmonics was todescribe and regulate their spatial and dynamic relations

[3]. He defined musical sound as distinct from noise orthe sounds of spoken language and conceived notes asmere points on a line of pitches and musical intervals

indirectly ratios also as one-dimensional and continuousmagnitudes that, following the rules of Euclidean

geometry, should be capable of being dividedcontinuously [1]. Thus, the octave was divided into six

tones and the tone into equal semitones or quarter-tones[10], as well as into 12 equal parts, notions which findno epistemological resonance with the mathematics of his

time but which betrays signs of the idea of the logarithmand also provide basic support for a mathematical

understanding of equal temperament [2]. Aristoxenusconsidered also as unmelodic and useless theHarmonicists katapyknsis (close packing) tradition,

according to the diagrams of which the octave could bedivided into 28 consecutive diesis.

Studying intervals, according to Aristoxenus, isnot just a matter of measuring them as it has been for thePythagoreans and the Harmonicists. It involves mostlythe way intervals are combined in order to achieve theircoherent musical arrangement, thesynthesis. As it can beunderstood, in Aristoxenus approach to music not onlyintervals but the whole musical system was treated.According to him, the primary components of music arethe fourth and the fifth (not the octave). He defined thepositions of the movable interior notes (kinoumenoi) of

the tetrachord, invariable in size, because of its outernotes being immovable, (hesttes) [7]. He formulatedthe concept of genosand described the three genera, theenharmonic (a ditone followed by two quarter-tones,moving from top to bottom), the chromatic (a tone-and-a-half, and two consecutive semitones) and the diatonic (atone, a tone and a semitone), the last two of whichpresented various shades (chroai) [3]. According to theGreeks strong belief of moral and emotional impact ofmusic on people (ethos), the character of each genre andits effect are also considered. The enharmonic (meaningin tune), also called harmonia, the most beautiful andsophisticated according to Aristoxenus, was the standardtuning. Used in tragedy, it was praised as conducive to

manliness. The chromatic genre (chromameans colour),marked by its name as a kind of deviation, was associated

with professional chitarodes and believed that softenedmen. It seems that the diatonic genre has been typical ofcertain regions and that has enjoyed the primary status inmusical theory of the Pythagorean school [9]. Besidesgenera, Aristoxenus also described the tonoi, which

represented the scales transpositions.In theHarmonics Elements Aristoxenus discusses

music in a scientific way. He creates an independentscience for the field of harmonics and divides musicalknowledge into distinct subjects. He brings together all

the elements of earlier scholarship, which he organizesand judges. Applying the Aristotelian scientific doctrine

to the subject he defines the elements of its science, hegives a complete description of musical phenomena,setting out from the simplest of entities (musical sound)

and proceeding to increasingly complex combinations ofintervals and systems, thus justifying his significant role

as an innovator of the discipline of musicology [5].

References:[1] Oscar Joo Abdounur,A preliminary survey on theemergence of an arithmetical theory of ratios,Circumscribere, 7(2009), pp 1-8.[2] Oscar Joo Abdounur, Ratios and Music in the Late

Middle Ages in Music and Mathematics by PhillipeVendrix (edit), Brepols, 2008, p. 26[3] Annie Blis, Aristoxenus in Sadie, S. and J. Turell

(edit.), The New Grove of Music and Musicians,Macmillan, London, 2001, vol. 2, pp. 1-2.

[4] Viggo Brun, Euclidean algorithms and musicaltheory, Enseignement Mathmatique, 10:125137,1964.[5] Sophie Gibson, Aristoxenus of Tarentum and TheBirth of Musicology, Routledge, Oxford University / TheUniversity of California, Los Angeles, 2005, pp. 4-5.[6] G.S. Kirk & J.E. Raven , The PresocraticPhilosophers, Cambridge University Press, 1964, p.229[7] Thomas J. Mathiesen, Greece, 1. Ancient, 6.Music Theory in Sadie, S. and J. Turell (edit.), The New

Grove of Music and Musicians, Macmillan, London,2001, Vol. 10, pp. 337-341.[8] Marios D. Mavroidis, : , , [Musical modes in EasternMediterranean: The Byzantine Hhos, The ArabicMakam, The Turkish Makam], Fagotto, Athens, 1999,pp. 24-35.[9] M. L West,Ancient Greek Music, Oxford UniversityPress, 1992, pp. 234.[10] R.P. Winnington-Ingram,Aristoxenus in Sadie, S.and J. Turell (edit.), The New Grove of Music andMusicians, Macmillan, London, 1993, vol. 1, pp. 591-592.


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