by Plato but brought to light by the musicologist, Ernest McClain [3,4,5], will be shown to be the basis of the Just scale, another ancient musical scale. I shall refer to this table as the Harmonic Matrix, or simply as HM. In his books and papers, 19-32 McClain has made a strong case for music serving as the lingua franca of classical and sacred texts, ANCIENT providing plausible explanations of otherwise too HARMONIC LAW difficult to understand passages and providing metaphors to convey ideas and meaning. In this paper I will focus primarily on the mathematics and music at the basis of ancient musical scales, Jay KAPPRAFF and this paper will serve as an introduction to the work of McClain. An original contribution of the Abstract author is a collaboration with McClain to create an The matrix arithmetic for ancient harmonic algorithm for the construction of an approximation theory is presented here for two tuning systems to 2 good to six decimal places using the musical with opposite defects: ‘Spiral fifths’ as presented scale. This approximation was used in Vedic India by Nicomachus, a Syrian Neo-Pythagorean of the with a geometric derivation. The author will also second century A.D., and Plato’s ‘Just tuning’ as demonstrate that the musical scale functions as a reconstructed by the ethnomusicologist, Ernest vector system simplifying musical computations. McClain, from clues preserved by Nicomachus 1 2 4 8 16 32 64 ... th and Boethius (6 century AD). These tables lie 3 6 12 24 48 96 ... behind the system of architectural proportions 9 18 36 72 144 ... used during the Renaissance, and their basic ratios 27 54 108 216 ... now pervade modern science as the foundation of 81 162 324 ... a ‘string theory’ formally presented first in Euclid. 243 486 ... Calculation employs an early form of a log table 729 ... governed by vectors of 2-3-4 in the first, and by Table 1. Nicomachus’ table for expansions of the ratio 3:2 3-4-5 in the second. (as relative frequency). The square root of 2 plays a central role in integrating these systems governing 12-tone theory The construction of ancient musical scales will from the perspective of four primes--2, 3, and 5 be shown to be based strongly on primes: 2, 3, 5 generate all ratios under the overview of 7— as and 7. There were no easy notations for fractions disciplined ‘self-limitation’ within a ‘balance of in ancient civilizations so the problem was to perfect opposites’. represent ratio in terms of whole numbers. To this Plato states: Introduction ‘For surely you know the ways of men who Nicomachus, writing about 150 AD, forms are clever in these things. If in the argument someone attempts to cut the one itself [i.e., one of the best links to what survived from his use a fraction], they laugh and won’t permit it. day about Greek theory of numbers and music If you try to break it up into small coin, they [1,2]. I shall describe how the sequence of integers multiply…’ Republic 525 shown in Table 1, and attributed to Nicomachus, defines perfect octaves, fifths, and fourths, the Ancient musical scales must be assumed to only consonances recognized by the Greeks, be elegant solutions in smallest integers to this and lies at the basis of ancient tuning systems problem. preserved by Ptolemy. A second table inferred Two of Plato’s foremost concerns were 20 ANCIENT HARMONIC LAW limitation, preferably self-limitation, and balance chosen as the first or fundamental, and since any of opposites. McClain states [5] that tone is perceived by the ear to be identical when ‘In political theory as in musical theory, played in different octaves, each tone of the scale both creation and the limitation of creation pose refers to a pitch class of tones differing by some a central problem. Threatening infinity must be number of octaves. contained. What musicians have shown possible in The D mode, or ancient Phrygian mode, was tones, the philosopher should learn to make the preferred self-symmetric scale of ancient possible in the life political.’ times (notice the symmetric pattern of the keys on both sides of D in figure 1), and I will represent As for the balance of opposites, in the all scales in the D mode. The heptatonic scale Republic, Plato states: beginning on C: CDEFGABC is the famous do-re- ‘Some things are apt to summon thought, mi... scale, and common reference now in Western while others are not…Apt to summon it are those that strike the sense at the [same] time as civilization. The intervals of the piano are equal- their opposites.’ tempered so that beginning on any tone, after Republic 524 twelve perfect fifths one arrives again at the same One of the themes of this paper will be to tone seven octaves above or below in pitch. show how the musical scale expresses the concepts Because of the periodicity of the chromatic of lmitation and balance of opposites. scale, it can be comfortably represented on a tone Figure 1. The piano keyboard Our story begins with a look at the 88 keys of circle as shown in figure 2 with semitone intervals the piano shown in figure 1. Three tones, A, D, and between adjacent tones, and with pairs of G, have been highlighted. D is located at the center semitones equaling a wholetone. Each semitone of of symmetry of the black and white pattern of the equal-tempered scale is worth 100 cents now keys with G and A seven tones below and above D. assigned on a logarithmic scale with 1200 cents to The interval from D to G spans five tones DCBAG the entire octave. The relationship between cents called a falling fifth while from D to A is a rising and frequency is discussed in the appendix. The fifth (DEFGA). In the opposite direction these equal-tempered tritone located at 6 o’clock has a intervals are perfect fourths (DEFG and DCBA). relative frequency of 2. Proceeding clockwise The pattern of black and white keys repeats every around the circle leads to a rising scale while a counter-clockwise rotation leads to a falling scale. 12 tones, called the chromatic scale, with the white The numbers 0-12 on the clock positions number keys repeating every seven keys and assigned the the tones of the chromatic scale, and they can be tone names DEFGABCD’, the interval from D thought of as numbers in a mod 12 number system to D’ being called an octave, and with the black where any multiple of 12 can be added to one of # keys augmented or diminished by sharps ( ) and the integers to get another tone in the same pitch flats ( ) to insert semitones between wholetones. class. I will refer to the numbers on the face of the The white keys produce heptatonic scales in seven clock as the digital roots of the pitch class more different modes depending on the reference pitch. commonly called principal values in mathematics. The black keys naturally provide five pentatonic Notice that the digital roots of two tones modes, repeating on the sixth tone. Because of the symmetrically placed around the tone circle sum to periodicity of the musical scale, any tone can be 12. Such tonal pairs are called complementary. Jay KAPPRAFF 21 Arithmeticization of the ancient musical scale begins with the numbers 1, 2, 3, and 4. The Sliding bar reference length of a vibrating string is arbitrarily assigned the value of 1 unit which is taken to be the fundamental tone when bowed or plucked. If the length of the fundamental is multiplied or divided by any power of 2, the resulting tone sounds to the ear identical in implied harmonic function to the fundamental by some number of octaves. In other words, shortening the string by half gives Unison Fourth Fifth Octave D a tone one octave C# 0 E 11 1 higher in pitch while C E 10 2 doubling the length of Figure 3. A sliding bridge on a monochord divides the string the string results length representing the fundamental tone into segments 9 3 B F in a falling octave. corresponding to the perfect fifth (2:3) fourth (3:) and octave Therefore each tone is (1:2). 8 4 B F# a representative of a associated with the ratio 3:1. This ratio is equivalent, 7 5 A 6 G pitch class of tones all after suppressing octave powers of 2, to relative A=# differing by a multiple string lengths 3/2 and 3/4 of a falling perfect fifth Figure 2. The Tone Circle. of 2 in string length. or a rising fourth. In this case, taking D to be the That all tones in a pitch class sound alike to the ear fundamental, 3:1 is represented by the pitch class G in harmonic function is considered to be the miracle (a fifth below D). Alternatively, 3 can be associated of music. In figure 3, if you place your finger at the with the ratio 1:3 which in turn corresponds to 2/3 point on the string and sound the string it gives relative string lengths 2/3 and 4/3, a rising perfect rise to a perfect fifth; lengthening the string by one fifth or a falling fourth, i.e., inverse ratios. Now half results in the interval of a falling fifth. If you with D as the fundamental, 3 is represented by shorten the string by one fourth a perfect fourth pitch class A.