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RECENT ADVANCES in MATHEMATICS and COMPUTERS in BUSINESS, ECONOMICS, BIOLOGY & CHEMISTRY Aspects of mathematics and music in Ancient Greece SOFIA KONTOSSI Department of Music Studies Athens University K. Paxinou, 19B, 19009, Attika GREECE [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 relationship established between music and mathematics through the Pythagorean, Euclidean and Aristoxenian tradition. Key-Words: - Ancient Greeks, Aristoxenus, Euclid, Pythagora, mathematics, musical intervals, music scales, music theory the real musical phenomena –temporal manifestation of which, according to their view, conferred them traces of imperfection – but on the harmonious reflection of numbers. The first to conceive the relationship underlying between music and mathematics, establishing thus the idea 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 the number of motions of a string– could be expressed as a ratio. Among his greatest discoveries (or those of the Pythagorean school, the distinction seems to be hard), by means of the monochord,1 a string fastened across a movable bridge to facilitate changes in pitch, is that “the chief musical intervals are expressible in simple mathematical ratios between the first four integers” [6]. Thus, the octave, the fifth and the fourth –the most important consonances in ancient Greek music– were 1 Introduction Ancient Greeks did not have today’s knowledge of sound wavelengths and frequencies, so they could not understand the musical phenomenon as the physical explanation of the harmonic series and pitches. Their understanding of music science came initially through mathematics. They noticed that the sound produced by a string depends upon its length, tension and density. In order to be able to reproduce the same relationship between two sounds (a concept they defined as musical interval) they were studying ratios of string lengths. Due to the need to operate with intervals for musical purposes -like tuning, creating scales etc.-, theorists in their effort to divide the tone (i.e. the distance between notes A and B) indirectly shaped the conception of ratio. 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 relationship established between music and mathematics through the Pythagorean, Euclidean and Aristoxenian tradition. 1 Questions 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 the present article so they will not be considered. 2 The Pythagoreans According to the Pythagoreans’ conception about ‘cosmos’, numbers are the ultimate reality. Therefore, musical science was not to be explained on the basis of ISSN: 1790-2769 40 ISBN: 978-960-474-194-6 RECENT ADVANCES in MATHEMATICS and COMPUTERS in BUSINESS, ECONOMICS, BIOLOGY & CHEMISTRY 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 the octave, also true when expressed in mathematical terms (3/2 x 4/3 = 12/6 = 2/1). Considering the importance that the tetractys of the decade (represented by the numbers 1, 2, 3, 4, the sum of which equals sacred number 10) had to the Pythagoreans as the key to the understanding of the universe, these ratios were the reflection of both mathematical and musical harmony. The underlying concept of such connections was projected also to the cosmos’ harmony and the planetary spheres were seen as parts of a vast musical instrument attuned following the same ratios governing musical intervals. As M. L. West mentions, “Plato's harmony of the spheres is not some unimaginable, transcendental passacaglia or fugue, but the naked glory of the diatonic octave” [9]. Measuring smaller intervals than the fourth was of capital importance in ancient Greek music considering both the facts that the tetrachord was the unit of construction of musical scales and that the different modalities of tetrachord’s subdivision defined the genera. For that purpose the Pythagoreans used mathematical processes. By extracting the 4th from the fifth, they defined the ratio of the tone (9:8). Then various ‘semitones’ sizes occurred: the difference between the fourth and two tones (or the difference between three octaves and five perfect fifths), represented by the ratio 256:243, was called limma (‘remainder’, the diatonic semitone of the intemperate system) and the difference between the tone and the limma (or between seven 5ths and four octaves) was named apotomē (‘segment’, ratio 2187:2048, the chromatic semitone). As a result, the Pythagorean scale, consisting of two disjointed consecutive fourths, was exceeding the octave by a small interval known as the Pythagorean comma (531,441:524,288), which could also be thought of as the discrepancy between twelve justly tuned perfect fifths and seven octaves. “The size of the semitone and the addition of tones and semitones to create the consonant intervals became a subject of heated controversy between the Pythagoreans, with their fundamentally arithmetic approach, and the Aristoxenians, who adopted a geometric approach to the measurement of musical space” [7]. An equal division of the tone, meaning from a mathematical point of view to find x so that x2=9/8, leads to irrational numbers –using today’s terminology–, an unacceptable disturbance in the Pythagorean musical system and conception of the world. Euclid Euclid, the famous mathematician and geometer, in a treatise attributed to him entitled The Division of the Canon, describes the steps of the construction of the Pythagorean scale. Known as intense diatonic, this scale has been the subject of great theoretical discussion from antiquity to our days. The scale can be produced on a monochord with the exclusive use of two consonant intervals, the octave and the fifth. A short description of the procedure follows: a) Halving the string, we take the upper octave of the initial sound. b) Descending from the octave a fifth we take the fourth. 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 be represented 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 scale coincides with the usual major type scale of western music and that two similar tetrachords (tone-tonelimma) are formed. Secondly, that the scale construction is based only on tone (~204 cents) and limma (~90 cents), which actually means that the difference between the ‘small’ step (~ semitone) and the ‘big’ step (tone) is the wider possible (thus justifying the sound impression as being intense). From antiquity though, musical practice suggested a finer subdivision of the diatonic scale, a more gradual transition from one step to another. This could be obtained by replacing in a tetrachord the limma with the apotome (~114 cents), thus, the second tone resulting to be quite smaller than the first one (~180 cents). This interval can be considered as the elasson tone (‘smaller’ tone) of Byzantine music. The new scale, called mild diatonic, had the disadvantage that, the mathematical representation of the degrees resulted into very big numbers (i. e the ratio of the elasson tone was 216 / 310). The problem was solved by Dydimus (1st 3 Euclid (4th-3th century BC) 3.1 The description of the Pythagorean scale by ISSN: 1790-2769 41 ISBN: 978-960-474-194-6 RECENT ADVANCES in MATHEMATICS and COMPUTERS in BUSINESS, ECONOMICS, BIOLOGY & CHEMISTRY century B.C) who replaced the Pythagorian ditone by the mild diatonic third (5/4). This ratio for the meizon (‘major’) third, generally accepted since Hellenistic times, was also adopted from the Arabs theoreticians. Both the intense and the mild diatonic scale were used in Byzantine music, the last one especially being widespread met in the musical practice of all people of the Eastern Mediterranean basin and the Middle East [8]. Fourth - 2 tones = 4:3 / (9:8)2 = 256:243 (3) semitone tone – 2 semitones = 9:8 / (256:243)2 comma 531441:524288 (4) = We could continue this indefinitely. Practically, the impossibility 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 the circle of fifths [C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B# (=C)], which requires 12 fifths to be completed, we cover a musical distance of seven octaves. In mathematical terms, considering the ratio for the fifth being 3:2 and that of the octave 2:1, this should mean (3/2)12 equals 27, 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 we must notice that Greeks realized that, adding two musical intervals required multiplying the corresponding ratios and similarly, subtracting one interval from another requires dividing their ratios. Some scholars believe that Euclid and his predecessors probably conceived the theory of ratio as a generalization of musical theory of intervals. However, if we want to get some approximate result that will be useful in musical reality, we could cut off the algorithm at some point and pretend that one interval went exactly into another with no remainder. If the real remainder is small enough, that will give us a useful approximation. For example, suppose we pretend that comma is equal to zero (tone - 2 semitones = 0). Then 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 famous algorithm. In the XXth century, researchers connected the Euclidean algorithm to music theory. Norwegian mathematician Viggo Brun used Euclidean algorithms to explore tuning matters [4]. On the other hand, the Euclidean algorithm was related to rhythms and scales in traditional music. Its structure can be used to automatically generate, very efficiently, a large family of rhythms used as timelines (rhythmic ostinatos), in traditional world music. 2 Here we will present an application of the algorithm to highlight, a significant problem that had a direct impact on musical practice, namely the identification of the mathematical relationship (proportionality) between two musical intervals ratios (i.e. how many times a fifth goes into an octave, meaning how many times do we have to multiply 3:2 to get 2:1). As we can see below, applying Euclid’s algorithm to find the highest common factor for the fifth and the octave gives no integer solution. tone = 2 semitones Interval Ratio a – n × b = c, where Corresponding equation b>c for ratios fourth = 2 tones + semitone = 5 semitones (by 3) (by 5) octave - fifth = fourth 2:1 / 3:2 = 4:3 (1) Fifth = fourth + tone = 7 semitones (by 2) (by 5 and 6) fifth - fourth = tone 3:2 / 4:3 = 9:8 (2) Octave = fifth + fourth = 12 semitones (by 1) (by 5 and 7). 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. 429–454. ISSN: 1790-2769 So, using Euclid's algorithm we conclude that, a reasonable approximation to the number of fifths in an octave is 7/12, because a fifth is roughly 7 semitones and an octave is roughly 12 of them. 4 Aristoxenus (4th century B.C.) A revolution in the history of musical thought in ancient times was made by Aristoxenus, a student of both 42 ISBN: 978-960-474-194-6 RECENT ADVANCES in MATHEMATICS and COMPUTERS in BUSINESS, ECONOMICS, BIOLOGY & CHEMISTRY manliness. The chromatic genre (chroma means ‘colour’), marked by its name as a kind of deviation, was associated with professional chitarodes and believed that softened men. It seems that the diatonic genre has been typical of certain regions and that has enjoyed the primary status in musical theory of the Pythagorean school [9]. Besides genera, Aristoxenus also described the tonoi, which represented the scales’ transpositions. In the Harmonics Elements Aristoxenus discusses music in a scientific way. He creates an independent science for the field of harmonics and divides musical knowledge into distinct subjects. He brings together all the elements of earlier scholarship, which he organizes and judges. Applying the Aristotelian scientific doctrine to the subject he defines the elements of its science, he gives a complete description of musical phenomena, setting out from the simplest of entities (musical sound) and proceeding to increasingly complex combinations of intervals and ‘systems’, thus justifying his significant role as an innovator of the discipline of musicology [5]. Pythagoreans and Aristotle, who first considered music as un autonomous discipline and he is often referred to as the father of Musicology. In the first part of the Harmonic Elements, Aristoxenus criticizes the foundations of his predecessors’ teachings, without naming them, probably a testament 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 a musical entity but as a ratio consisting exclusively from whole numbers, Aristoxenus’ prime criterion for the musical phenomena was the ear. For him “music consisted of sounds structurally organized within a soundspace, and the function of the science of harmonics was to describe and regulate their spatial and dynamic relations” [3]. He defined musical sound as distinct from noise or the sounds of spoken language and conceived notes as mere points on a line of pitches and musical intervals – indirectly ratios also– as one-dimensional and continuous magnitudes that, “following the rules of Euclidean geometry, should be capable of being divided continuously” [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 find no epistemological resonance with the mathematics of his time but which betrays signs of the idea of the logarithm and also provide basic support for a mathematical understanding of ‘equal temperament’ ” [2]. Aristoxenus considered also as unmelodic and useless the Harmonicists’ katapyknōsis (‘close packing’) tradition, according to the diagrams of which the octave could be divided into 28 consecutive diesis. Studying intervals, according to Aristoxenus, is not just a matter of measuring them as it has been for the Pythagoreans and the Harmonicists. It involves mostly the way intervals are combined in order to achieve their coherent musical arrangement, the synthesis. As it can be understood, in Aristoxenus’ approach to music not only intervals but the whole musical system was treated. According to him, the primary components of music are the fourth and the fifth (not the octave). He defined the positions of the ‘movable’ interior notes (kinoumenoi) of the tetrachord, invariable in size, because of its outer notes being ‘immovable’, (hestōtes) [7]. He formulated the concept of genos and described the three genera, the enharmonic (a ditone followed by two quarter-tones, moving from top to bottom), the chromatic (a tone-and-ahalf, and two consecutive semitones) and the diatonic (a tone, a tone and a semitone), the last two of which presented various shades (chroai) [3]. According to the Greeks’ strong belief of moral and emotional impact of music on people (ethos), the character of each genre and its effect are also considered. The enharmonic (meaning ‘in tune’), also called harmonia, the most beautiful and sophisticated according to Aristoxenus, was the standard tuning. Used in tragedy, it was praised as conducive to ISSN: 1790-2769 References: [1] Oscar João Abdounur, “A preliminary survey on the emergence of an arithmetical theory of ratios”, Circumscribere, 7(2009), pp 1-8. [2] Oscar João Abdounur, “Ratios and Music in the Late Middle Ages” in Music and Mathematics by Phillipe Vendrix (edit), Brepols, 2008, p. 26 [3] Annie Bélis, “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 musical theory, Enseignement Mathématique, 10:125–137, 1964. [5] Sophie Gibson, Aristoxenus of Tarentum and The Birth of Musicology, Routledge, Oxford University / The University of California, Los Angeles, 2005, pp. 4-5. [6] G.S. Kirk & J.E. Raven , The Presocratic Philosophers, 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, Οι µουσικοί τρόποι στην Ανατολική Μεσόγειο: Ο Βυζαντινός ήχος, Το Αραβικό µακάµ, Το Τούρκικο µακάµ [Musical modes in Eastern Mediterranean: The Byzantine Hhos, The Arabic Makam, The Turkish Makam], Fagotto, Athens, 1999, pp. 24-35. [9] M. L West, Ancient Greek Music, Oxford University Press, 1992, pp. 234. [10] R.P. Winnington-Ingram, “Aristoxenus” in Sadie, S. and J. Turell (edit.), The New Grove of Music and Musicians, Macmillan, London, 1993, vol. 1, pp. 591592. 43 ISBN: 978-960-474-194-6