Pythagorean Mathematics and Music

CROCKER RL. \a Ca RICHARD L. CROCKER M nqucmen: CROCKER R.L., Pythagorean mathematics andmusic: JAAC 22 1963 189-198, 325-335. L'auteur cherche à décrire quelques-unes des opérations fondamentales de l’arithmétique pythagoricienne, particulièrement celles sur lesquelles s’appuie la théorie pythagoricienne de la musique. La seconde partie de l'étude analyse, selon le témoignage d’Archytas et de Platon, les gammes — autres que celle des Pythagoriciens anciens — fondées sur les notions de moyennes géométrique et « harmonique ». I THROUGHOUT WESTERN HISTORY Number has from time to time been taken as the rule of art. Aithough never accepted by all artists of a given period and soon replaced by alternate aesthetic theories, mathematical explanations persistently and recurrently crop up in forms both old and new—sometimes based upon a recently developed mathematical operation, sometimes simply making a new application of an old, familiar one. Vitruvian proportions of architecture, Leonardo's canon of human proportions, Augustine’s analysis of poetic rhythm, the lure of the golden rectangle, all illustrate the use of mathematics to clarify the bases of art. Such attempts go back to the Greeks—at least, they tried it first and in some cases most thoroughly—and among the Greeks the earliest and most thorough of all were the pythagoreans. Pythagoras—so the story goes--invented the theory of music. He, or his disciples, drew attention to the fact that musical intervals could be expressed as numerical ratios and that the more consonant intervals had ratios with very small numbers, like 1:2. Pythagoreans are usually credited with dividing the octave by a fourth and a filth, RicHaro L. Crocker teaches in the department of music at the University of California, Berkeley. He is general editor of the Music Theory Translation Series and has had several articles published in music journals. constructing a scale using the whole-tone 8:9, and discovering the “pythagorean comma” or difference between 12 fifths and 7 octaves. There is, however, no easily available account of how these theorems are related to cach other and to the mathematics of which they are a part. In this article I will try to describe a few of the basic operations of pythagorean arithmetic and such special operations as have to do with the pythagorean theory of music. It is not easy to make out the history of pythagorean thought before 400 B.C., nor is it clear, even now, how much Pythagoras himself (who flourished around 532 B.C.) contributed to the arithmetic and musical theorems handed down under his name. There were a group of thinkers called “pythagoreans” active around the end of the fifth century B.C., shortly before Plato, Our information about pythagorean thought comes from or through this group—and not directly but through sources like Plato and Aristotle. Some scholars, Erich Frank in particular,’ attribute most if not all pythagorean science to these latter-day pythagoreans. Others, however, are willing to grant that Pythagoras himself taught the theorems traditionally associated with his name, while some argue that certain theorems did not originate with Pythagoras but came from the Egyptians, Babylonians, or still more ancient layers of Near-Eastern civilization.? Even if these problems cannot be solved at the present time, we can try to

RICHARD L. CROCKER and Music constructing a scale using the whole-tone 8:9, and discovering the "pythagorean WESTERN HISTORY num- comma" or difference between 12 fifths and THROUGHOUT ber has from time to time been taken as 7 octaves. There is, however, no easily availthe rule of art. Although never accepted by able account of how these theorems are reall artists of a given period and soon re- lated to each other and to the mathematics placed by alternate aesthetic theories, math- of which they are a part. In this article I ematical explanations persistently and re- will try to describe a few of the basic operacurrently crop up in forms both old and tions of pythagorean arithmetic and such new-sometimes based upon a recently de- special operations as have to do with the veloped mathematical operation, sometimes pythagorean theory of music. It is not easy to make out the history of simply making a new application of an old, familiar one. Vitruvian proportions of pythagorean thought before 400 B.C., nor architecture, Leonardo's canon of human is it clear, even now, how much Pythagoras proportions, Augustine's analysis of poetic himself (who flourished around 532 B.C.) rhythm, the lure of the golden rectangle, all contributed to the arithmetic and musical illustrate the use of mathematics to clarify theorems handed down under his name. the bases of art. Such attempts go back to There were a group of thinkers called "pythe Greeks-at least, they tried it first and thagoreans" active around the end of the in some cases most thoroughly-and among fifth century B.C., shortly before Plato. Our the Greeks the earliest and most thorough information about pythagorean thought of all were the pythagoreans. comes from or through this group-and not Pythagoras-so the story goes-invented directly but through sources like Plato and the theory of music. He, or his disciples, Aristotle. Some scholars, Erich Frank in drew attention to the fact that musical in- particular,' attribute most if not all pytervals could be expressed as numerical thagorean science to these latter-day pyratios and that the more consonant intervals thagoreans. Others, however, are willing to had ratios with very small numbers, like grant that Pythagoras himself taught the 1:2. Pythagoreans are usually credited with theorems traditionally associated with his dividing the octave by a fourth and a fifth, name, while some argue that certain theorems did not originate with Pythagoras but teaches in the department of came from the RICHARD L. CROCKER Egyptians, Babylonians, or music at the University of California, Berkeley. He still more ancient layers of Near-Eastern is general editor of the Music Theory Translation civilization.2 Even if these problems cannot Series and has had several articles published in music journals. be solved at the present time, we can try to

RICHARD CROCKER put together a summary account of early the pythagorean theory of music is dependpythagorean musical theory. Our account ent upon these truths, we must, for the will reflect the state of pythagorean mathe- moment, learn to count with points. matics in 400 B.C.; it cannot penetrate The simplest arrangement of points is in earlier than that except by speculative re- a straight line. construction. In 400 B.C. the creative phase Example 2 of pythagorean mathematics was drawing to a close, for thereafter it tended more and more toward academic numerology or mysticism. The same date, 400 B.C., also marks All integers, of course, can be represented the end of the great period of Greek music, in this way. But the linear picture, being a which flourished during the fifth century;3 trivial one, is for most purposes replaced by hence this summary of early pythagorean some other. Beginning with 3, points can mathematics gives a picture of the kind of be arranged in a triangle, that is, a plane instead of a line. Beginning with 4, theory that was associated with classic figure as we this plane figure can be a square; saw, Greek music about which we know so little. with an 6, oblong; with 8 a cube-a solid The pythagoreans are reported to have In general, integers that can be exfigure. with each numbers points, represented as the product of two factors (2 X point being a unit.4 These points were ar- pressed = 3 are 6) represented as rectangular numranged in some appropriate pattern to give those that are the product of three bers, a picture of each number. Here, for exfactors (2 x 2 x 2 = 8) as solid numbers. 4. the number is ample, But for the purposes of music theory we Example 1 need consider only plane numbers. * a Triangular numbers occur as the sums of series: 3 is the sum of 1 plus 2; 6, the next triangular number, is the sum of 1 plus 2 * a plus 3. It is important (Michel observes) to realize Example 3 that these points give a picture of the very essence of number as conceived by the pythagoreans. These points are not the points of geometry, for they occupy a certain amount of space in the picture: any given number contains only a finite number of points, the number 4 having four points. Clearly this series can be extended indefiYet these points are themselves not geo- nitely. The generation of numbers in series metric areas, but simply units. They are was one of the most important operations not divisible, but instead are "atoms," in- of pythagorean arithmetic. Square numbers, divisible particles. In this respect they re- too, were regarded as sums of series; even veal a concept of mathematics concerned though squares appear to us to be the prodonly with integers-that is, whole numbers ucts of their sides, the pythagoreans gener-the most important feature of the rela- ated square numbers from 1 in the followtionship between pythagorean mathematics ing way.5 and pythagorean theory of music. FurtherExample 4 more, the essence of a number is revealed by the way in which the point-units are ar? * a a ranged. Without thinking, we call the number 4 a "square," but the pythagorean picture above shows why it should be called *000* a square number. For us, the point-numbers * * a the for needless seem effort; pythagmay oreans, they made possible an intuitive grasp of many mathematical truths. Since

The square number 4 was formed by adding onto 1 the three points enclosed by the right-angled lines. The next square number, 9, was formed by adding on the next 5 points similarly enclosed-and so for all subsequent square numbers. The pythagoreans gave a special name to this group of added points: they called it the gnomon, ("index"; compare gnosis). When a gnomon of a fixed length L is added to any square of length L - 1, it produces the next square. The gnomon for square numbers forms its own series, 3, 5, 7, 9, 11, etc. The series of square numbers has remained a basic tool of mathematics right up to the present day, while other series studied by the pythagoreans have been discarded. Of these other series perhaps the most important, and the next in order, was formed by adding a rectangular gnomon onto the initial term 2 instead of onto 1. Example 5 191 than 2; as long as a rectangular gnomon is added, the result will be a series of oblong numbers. Each series will have its own type of ratio that holds for every number in the series; the ratios of the series just described, for example, always show a difference of 1 between their terms-or stated more generally, the larger term of the ratio will exceed the smaller by an aliquot part ('/, /s, 14, etc.) of the smaller. But each new series of oblong numbers will be characterized by a different type of ratio, unless the new series is simply a previous series expressed in terms twice as great, three times as great, or some higher multiple. The pythagoreans determined that there were six types of ratios in all.8 equal multiple epimore multiple-epimore epimere multiple-epimere Example 6 1:1, 2:2, 3:3... 1:2, 1:3, 1:4... 2:3, 3:4, 4:5... 2:5, 3:7, 4:9..,2:7..,3:10 3:5; 4:7; 5:7, 5:8, 5:9;... 3:8; 4:11, 4:15; 5:14,... The types "equal" and "multiple" are selfexplanatory, being the ratios of the sides of square numbers and of the multiples of square numbers. Epimores have also been The resulting numbers are, of course, no discussed, being ratios whose lowest terms longer square but oblong, originally called differ by 1. In multiple-epimore ratios, this promeke or heteromeke.6 The series, as can difference between the terms is understood be determined from example 5, is (2), 6, 12, to be between the larger term and some 20, 30, 42, 56, 72, 90... which by itself does multiple of the smaller: 2:5 is understood not seem especially promising. In order to as 2:(2 x 2) + 1, whence the ratio is both grasp its significance, we must shift our multiple and epimore. In epimere ratios, attention to the pictures of the point-num- the difference is not an aliquot part of the bers themselves, in particular to the rela- smaller term, but some more complex part tionship of the sides of the rectangles to -not "a part" but "parts." Multiple-epieach other-their ratios. It is clear at once meres are analogous to multiple-epimores. that whereas the sides of any square numIf the reader has the patience and inclinaber are always equal, the sides of oblong tion, he can verify for himself that all numbers (of the series started in example integer ratios fall into one or the other of 5) always differ by 1, for the sides are suc- these six groups. This six-fold classification, cessively 2 and 3, 3 and 4, 4 and 5, and so no small accomplishment in itself, was a on. Thus the shape of successive square tool for handling ratios, for comparing numbers is always the same, but the shape their size and structure-an especially useof numbers in this oblong series is always ful tool when the ratios were large or comdifferent.7 The oblong shape, however, gets plex. It is interesting to observe that these more and more square as the oblong num- six kinds of ratios become more and more bers get larger diffuse as one goes from equal to multipleThere is only one series of square num- epimere, just as does the integer series itself bers, but many series of oblong numbers, as it departs from unity towards infinity. In for one can start from some number other fact, the series of integers seems to have

provided the pythagoreans with a model: not only did it furnish them with their subject-matter but in some sense with their method as well. Again and again one discovers that the truths of pythagorean science are the relationships inherent in the series of integers. The pythagoreans based their musical theorems directly on their arithmetic. As a result, the kind of problem that could be taken up, as well as the kind of solution that could be found for it, depended upon the nature of pythagorean arithmetic. Since the pythagoreans dealt exclusively with integers (fractions being ratios of integers), they dealt with that aspect of musical sound that could be numbered with integers. This meant that their attention was focused on musical intervals, for these lend themselves readily to numerical expression. But-and this is the crux of the matterthe pythagoreans could deal only with those musical intervals that could be expressed as the ratios of integers. Not all intervals can be so expressed; many intervals, including all those drawn from our modern scale of 12 equal semitones to the octave, are "irrational" quantities having no exact expression in the realm of integers. Not only did integer arithmetic commit the pythagoreans to the study of particular kinds of musical intervals, but it also injected a certain sense of value into their dealings with these intervals. In working with the integer series, one automatically uses the numbers at the beginning of the series as tools with which to work on the larger numbers. One reduces ratios to lowest terms, one finds "lowest common denominators," one takes small-number ratios as ideal forms or prototypes of larger ones. Because they consist of fewer elements, the smaller numbers are obviously simpler; thence they easily acquire the virtue of being "better." The pythagoreans' reverence for unity is well-known, the resulting cult of number-magic often derided, but such mysticism seems merely to have been a crustation formed on a hard core of arithmetic truth. In any case, the pythagorean theorist of music was led, almost inevitably, to establish a hierarchy of value among musical intervals, favoring those with small- RICHARD L. CROCKER number ratios. This was not, however, completely a function of his arithmetic approach to music, for these same intervals actually sounded simpler than other intervals. In the case of music, the arithmetic method was solidly confirmed by commonsense, empirical evidence. No wonder that in music the pythagoreans saw a patch of the basic fabric of the universe. Pythagoreans classified intervals according to the types of ratios associated with them; this principle was applied differently, however, by different generations. Early pythagoreans, it seems, divided intervals into two groups, those whose ratios were multiple or epimore, and those whose ratios were more complex.9 Here again, there was empirical support for this grouping: the first group sounded simpler-or, as they said, the notes of these multiple or epimore intervals blended better with each other. But the early pythagoreans included among the consonances only the octave (1:2), twelfth (1:3), double octave (1:4), fifth (2:3), and fourth (3:4), the rest of the epimore series, from 4:5 on, not being considered consonant. For this the early pythagoreans have often been criticized, first by Ptolemy (130 A.D.),10 then by theorists and historians of later ages for whom the epimores 4:5 and 5:6 have become consonant. But the pythagoreans had good reasons for limiting the number of consonances, reasons no more arbitrary than those of the modern overtone theorist who stops short at the fifth overtone because he, in turn, cannot accept as consonant the interval 6:7. The pythagoreans limited the number of consonances because, as mathematicians, they were deeply involved with the properties of the small-number ratios; as long as these, the first and most obvious types of ratio, offered the mathematician an interesting field of study, there was no reason to move on to other types. One of the most striking facts of pythagorean arithmetic-let me say, of the integer series itself-is the extraordinary wealth of relationships at the beginning of this series. When the numbers are small and the differences between them relatively large, they are related to each other in manifold ways. The first number added to itself produces

the second, a relationship found nowhere else in the integer series. The second added to itself produces its own square, again a unique result. The first and second added together produce the third, while the second and third added together skip over the fourth to produce the fifth. The squares of the third and fourth add up to the square of the fifth-the first integer example of the "Pythagorean Theorem." The modern observer, exposed to the possibilities of non-Aristotelian logic and non-Euclidian space, may offer the objection that all these striking relationships among the small integers are true only by tautology, claiming that the integer series is by definition that series that produces such relationships. But perhaps integers still seem so "natural" to us that we can glimpse something of the power their relationships had for earlier minds. Having found the ratios of the most consonant intervals among the small numbers 1:2, 2:3, 3:4, the pythagoreans then saw that the ratios 2:3 and 3:4, when combined, circled back to form another octave, 2:3:4, which in turn formed a double octave with the first ratio, 1:2:4. This is the only case in the integer series where two consecutive ratios (2:3 and 3:4) produce the ratio imexmediately preceding (1:2)-another ample of a unique relationship among the small numbers. This relationship is no less striking in musical sounds than it is in numbers. But then the early pythagoreans had also observed that the numbers 1, 2, 3, and 4 added up to 10, and the coincidence of this with the musical relationship just described proved irresistible. The "tetrad," or first four numbers, became a cornerstone of pythagorean theory, and with that the number of consonances was fixed." Later pythagoreans broadened this classification of intervals by recognizing that epimores, even when made of numbers higher than 3:4, were still as a class simpler than epimeres. This principle was stated much later by Ptolemy, who called epimore intervals "emmelic," that is, suitable for melodic progressions. He used them as much as possible for the construction of scales-in preference to "ecmelic" or unmelodic intervals with epimere ratios.12But 193 the same principle (as we will see later) is evident in interval calculations that go back to Archytas in 400 B.C.; already by that time pythagoreans had begun to distinguish between various kinds of non-consonant intervals according to whether their ratios were epimores or epimeres. The intervals in question are major and minor thirds, whole-tones, and fractions of whole-tones. In musical practice these intervals existed in various sizes: the major third, for example, could vary between the ratios 64:80 (4:5) and 64:81, the first size being-to our ears-flatter and sweeter. Early pythagoreans used the size 64:81, for reasons we will discover further on. Later theorists, however, preferred the size 64:80, which is the epimore 4:5, rejecting 64:81 as a rather complex epimere in which the larger "side" exceeded the smaller by 1/6 4ths. Similarly, epimore ratios such as 7:8, 8:9, 9:10, 10:11 were preferred for whole-tones; 15:16, 16:17, 17:18 for semitones; 27:28, 35:36, 38:39 for quarter-tones. This may seem to us acoustical quibbling. It seemed that way to Aristoxenus, who around 320 B.C. tried to deal with all intervals smaller than a fourth in terms of their function rather than their acoustical size.13 Indeed, what practical basis do we have for judging the whole-tone 800:900 (an epimore) to be simpler, more "melodic" than, say, 805:900 (an epimere)? As long as this interval sounds like a whole-tone and behaves like one, is its quality significantly related to the type of ratio it represents? Perhaps the answer is "no." Still, there are strong arithmetic reasons for arguing the other way. If the earlier terms of the epimore series are consonant-and they inshould the series dubitably are-why abruptly become dissonant after 4, or 5, or x terms? And if an interval like 4:5 or 9:10 is in some subtle sense "consonant" by virtue of being an epimore, then it must be more "consonant" than the epimeres (and irrational values) that surround it. Clearly the difficulty arises from the fact that as epimores get more complex their consonant property is harder to perceive, and as a result they seem less different from their dissonant neighbors. In any given century musical taste and style draw a hazy bound-

RICHARD ary between intervals simple enough to be consonant and those complex enough to be dissonant. As time goes by, the consonant group tends to include higher and higher epimores. It may be of interest to note that twentieth-century style is rapidly forcing the acceptance of new consonances, one of which, lying between a major second and a minor third, could easily have the size 6:7 -even though orthodox theory has up to now admitted as consonant no interval smaller than 5:6.14 While the epimore series leads to ever smaller intervals, the multiple series leads in the opposite direction to larger and larger ones. At the beginning of this series stands the octave, whose status has never been questioned. Beyond the octave lie other multiple intervals that also have a claim to consonance, even though they, too, diminish in consonance as they depart from unity. These intervals are the twelfth (1:3), double-octave (1:4), double-octave-and-third (1:5), and so on. While most people would grant the status of consonance to these three, they would probably baulk at the double-octave-and-seventh (1:7); but the argument that applies to epimores applies here too. Ptolemy asked why the ratio 1:5 could not be included among the consonances.15 This objection, a reasonable corollary to the admission of the higher epimores as "emmelic," attacks the limit L. CROCKER principle as a tool for explaining the phenomenon of musical consonance. Taken all in all, the six-fold classification of ratios seems to me to give a remarkably true picture of the qualities of rational intervals. It divides these intervals up in a way that is simple and clear, yet manifold enough to reflect sonorous reality. This classification makes it possible to distinguish clearly between consonances of the multiple series (octave, twelfth, etc.) from those of the epimore series (fifth, fourth, etc.). The one series gets larger, the other smaller; both depart from the octave (1:2), since this ratio is the first in the multiple series-and also, in a curious way, the root of the epimore series. The ratio 1:2 is not an epimore, of course; still, its two terms differ by 1, which is a characteristic of epimores in their lowest terms. Occurring immediately before the first epimore 2:3, the ratio 1:2 has something of its form. In the same way, octave and fifth are both perfect consonances, only the octave is more so. Like everything else connected with the integer series, this classification of intervals becomes more diffuse as it moves further from unity; even so, the distinction of epimores and epimeres is not without musical significance. The reader can check this against his own experience, for which purpose it is helpful to arrange the six classes in the following way. Example 7 equal-- - epimoremultiple the early pythagoreans placed on the group of consonances, while accepting the principle that consonance was a function of multiple or epimore ratios. Then Ptolemy further objected that the octave-and-fourth (the eleventh) should be considered consonant on the grounds that octavecompounds of consonances must also be consonant. Since the ratio of the octave-andfourth is 3:8 (a multiple-epimere), it was never included in the pythagorean class of consonances. Here Ptolemy was attacking the principle itself, and here again there is much that could be said in defense of this p epimere multiple-epimore multiple-epimere It is perhaps of interest to the modern reader to observe that the multiple ratios occur-as a group and in order-between a fundamental and each successive overtone; the epimores occur-as a group and in order-between successive pairs of adjacent overtones; the remaining classes occur between non-adjacent pairs of overtones. This is only to say that the overtone series is a sounding image of the integer series. From a pythagorean point of view, such a phenomenon, while interesting in itself, does not make the results any more true or the consonant intervals any more consonant.

For the pythagoreans, intervals are consonant because they sound consonant and because their ratios are simple. It is easy to underestimate the significance of these conclusions, particularly in the light of later developments; we have to remember that the pythagoreans came to their conclusions before there were any later developments. There are, of course, other ways to deal with musical intervals. Aristoxenus in 320 B.C. dealt with intervals at length in a way no less rigorous than that of the pythagoreans, yet without a single reference to arithmetic ratios. But the point is, the pythagoreans came first-and coming first, used a mathematics that counted things simply by means of units. There was no reason to invent a more sophisticated mathematics until this first, most obvious kind had solved all the problems it could and revealed the nature of those it could not solve. In this respect too, the pythagoreans' theory of consonance, dealing as it did with the simplest intervals, was the musical corollary of their arithmetic. II Once musical intervals were expressed as arithmetic ratios, they could be combined by using arithmetic operations. Here, however, it seems that it was music that provided the method for arithmetic-at least in the beginning.'6 The notion of compounding ratios was a difficult one to express in Greek mathematics, both in arithmetic and in geometry. Modern musical theory tells us that to compound two intervals we multiply their ratios (2/ x 3/4 = V1), and to double an interval, or compound it with itself, we raise its ratio to a power, ([2/3]2 = %4). The Greeks did not express powers as exponents but rather as geometrical shapes like squares and cubes.17 And although the pythagoreans were capable of carrying out a multiplication of ratios-indeed of very extended calculations18-it is not clear that the early pythagoreans compounded ratios in this way. Even if they did, it seems certain that they would have regarded such multiplication as a purely mechanical operation: it is difficult to see what theoretical significance they could have attached to it. 195 On the other hand, when expressed in terms of musical intervals, the compounding of ratios is so clear as to be self-evident.19 A fourth and a fifth "add up" to an octave just by being placed next to one another. Once it was established that their respective ratios were 3:4 and 2:3, the method of compounding these ratios was equally selfevident: they "added up" to the ratio 1:2 by virtue of the series 2:3:4. This operation (at least in the case just described) is presumably as old as Greek mathematics, if not older. It is not hard to imagine it as the model for further combination of ratios. Upon closer inspection, the intervals of fourth and fifth can be added up so easily only because their ratios happen to be continuous, that is, share a common term. Because of this, no multiplication is necessary; in fact, multiplication is irrelevant, the result being explicit in the number series itself. When the two ratios are not continuous, however, their compounding is not self-evident and some further operation is needed. To compound the ratios 3:4 and 8:9, for example, we must change one or the other ratio to make the two continuous, Clearly it is the smaller that must be changed, the larger being incapable of reduction to the terms of the smaller. The smaller ratio is easily expressed as 6:8, whence the continuous series 6:8:9, and the ratio 6:9 or 2:3 as the resulting compound. The Greeks were apparently willing and able to attack such problems by simple trial; out of such trials could have come the purely mechanical operation of multiplying ratios. The tetrad also contained a model for compounding a ratio with itself, 1:2:4. Stated differently, this is a special case of a continuous series in which every ratio is equal to every other. This kind of series is "in proportion," ana logon,20 and is called an analogia. When compounding a fifth (2:3) with itself, we are forced as before to express the ratios in higher terms in order to make them continuous: 4:6:9. But unlike the series for octaves, 1:2:4, this series for fifths comes to an end; in order to compound three fifths, we must raise to still higher terms. This operation, an important one in pythagorean arithmetic, leads to the following super-series.

RICHARD 3 6 9 Example 8 8 16 32 12 24 48 18 36 72 27 54 108 81 162 243 ... ... ... ... ... ... Each column represents a continuous series of fifths, in the ratio 2:3. It is interesting to observe that the gnomon of each column is given by the preceding column. Even more interesting, the top of the second column is the square of the first, the top of the third column the cube of the first, and so on; the same is true of the bottom of each column. Stated more generally, the tops and bottoms of the columns form two continuous proportions, emanating from the terms of the original ratio, and are linked vertically by other continuous proportions in the ratio 2:3.21 Once the principle is discerned, any size interval may be compounded with itself by the same method. Thus the arithmetician finds general solutions for the specific problems encountered in combining musical intervals. Confronted with the result, however, the theorist of music is moved to carry out yet another operation that would not occur to an arithmetician. In music the ratio 1:2 has a peculiar auditory force: the notes of the octave are much more alike to the ear than the terms of its ratio are to the eye. The theorist of music, therefore, is inclined to use the octave as a measuring stick for other large intervals, such as the sum of several fifths or fourths. Looking at the series of two fifths (example 8, second column), we can see that a note an octave from 4 would be 8, which note would form with the note two fifths away from 4 the small interval 8:9. Skipping over to the column headed "16," a similar calculation reveals the small interval 64:81 as the difference between four fifths and two octaves; logically enough, 64:81 is double the interval 8:9. In the same way, two fourths (9:12:16) fall short of an octave by 16:18, or 8:9 again. These small discrepancies came to be very important to the pythagoreans, who found in them a way to handle certain op- L. CROCKER erations that gracelessly refused to come out even. One remarkable feature of the sixfold classification of ratios is that no ratio ever belongs to more than one class. This has the corollary that if any ratio is compounded with itself any number of times, the result will never equal any other such compound of any other ratio. Thus no number of fifths, or of thirds, or of any other interval, ever adds up to any number of octaves.22 Sometimes, to be sure, two different series come within a hair's breadth of coinciding. These extremely small differences have always attracted the attention of theorists. The pythagoreans, who discussed them first, called them "commas," and one comma in particular is still called the "pythagorean comma." It is usually described as the difference between 12 fifths and 7 octaves, but while the pythagoreans could certainly calculate it that way, they probably arrived at the result differently. When Aristoxenus indicated that the octave could be divided into six equal whole-tones, he was taken to task by an anonymous writer23 who demonstrated what every pythagorean knew, that the sum of six tones each 8:9 missed being an octave by the rather small quantity 524288:531441. Here is the demonstration. 8 64 512 9 72 576 81 648 729 Example 9 4096 32768 262144 (X 2 = 4608 36864 294912 5184 41472 331776 5832 46656 373248 6561 52488 419904 59049 472392 531441 524288) It is important to notice that pythagorean tunings are not "out-of-tune" by the pythagorean comma-a common misunderstanding. It could never occur to a pythagorean to tamper with the octave, the interval closest to unity. The demonstration was originally carried out, as we just saw, to show that six whole-tones did not exactly divide an octave. The pythagoreans drew from this the conclusion that the octave would have to be divided in some other way. The early pythagoreans did not, then, construct their scale by a "cycle of fifths"; instead, they sought both the principle and

the unit for a scale in the conclusions already reached regarding the small-number ratios and the musical intervals they engendered. Contemplating the series 2:3:4 the pythagoreans-perhaps Pythagoras himself-observed that there sprang from this nucleus of intervals the small interval 8:9. This interval, which we have already met as the excess of two fifths over an octave, or of the octave over two fourths, is also the difference between fifth and fourth. Being everywhere amidst the principal consonances as the measure of their difference, this interval 8:9 easily became the basic unit of the scale, the "tone" of music. The early pythagoreans projected this tone inside the fourth; it went twice and a little over, this latter quantity being called the "limma" or remainder. Its actual size was determined in the usual way, by constructing a series of two tones and comparing the result with a fourth. tone/ (8 Example 10 64 192 [9 8 72 216 9 81 243 3 256 4}limma 256 4 Each fourth, or "tetrachord," contains two such tones and a limma. Since the octave contains two tetrachords and a tone, it has in all five tones and two limmas. (These were arranged in practice like the white keys on the piano descending from E to Eonly the Greek whole-tones were a little larger and the limma a little smaller than the irrational intervals on the piano.) This division of the octave, obtained not by a "cycle of fifths" but by projecting the tone 8:9 inside the fourth, is perhaps the oldest one used by the pythagoreans24 and in some ways the most characteristic. It uses the principles inherent in the beginning of the integer series more economically than any other. For if we reflect on the matter, we see that in some sense the fourth itself is a limma, left over from the projection of the fifth back into the octave. The fifth, in its own way, is a limma, left over from the projection of the octave forward into the twelfth (1:3). Only the octave seems to remain aloof from this process, being gen- 197 erated in some more mysterious way directly from the womb of unity itself. The division of the octave by the tone 8:9 proved to be a durable one. It survived the decline of ancient civilization, becoming virtually the only division used in the West during the Middle Ages-partly because it was easy to demonstrate on the monochord and worked well for chant, partly because it had the theoretical advantage of being built on the four simplest integers, an advantage that loomed large in the early Middle Ages when men looked for strong, simple solutions in music as in other aspects of culture. This division of the octave went out of style only in the Renaissance; indeed, this "pythagorean" division seemed more popular with the Franks than with the Greeks themselves, for already by 400 B.C. other divisions had been proposed, and soon there was a host of rival scales, each offering its own musical or arithmetic advantages. In dividing up the octave, the pythagoreans encountered problems different from those we have already seen; at the same time they found opportunities to use other operations which-even though perhaps older-were becoming popular around 400 B.C. These operations involved placing a third term-a "mean"-between the two terms of a given ratio. The various kinds of means are even more intimately related to music than the general arithmetic operations studied so far-may, in fact, have been developed as solutions to special musical problems; their discussion properly belongs in another chapter. (To be concluded in the Spring issue.) 1Plato und die sogenannten Pythagoreer (Halle, 1923). 2 See G. S. Kirk and J. E. Raven, The Presocratic Philosophers (Cambridge, 1960) pp. 217 if., for a conservative evaluation. 3See I. Henderson, "The History of Greek Music," in The New Oxford History of Music, I: Ancient and Oriental Music, ed. E. Wellesz (London, 1957), p. 376 ff.; R. P. Winnington-Ingram, "Greek Music (Ancient)" in Grove's Dictionary of Music and Musicians, ed. E. Blom (5th edition, London 1954), 4Pythagorean arithmetic is expounded at length in three very late writers, Nichomachus of Gerasa

(lst century A.D.), Theon of Smyrna (2nd century A.D.), and Iamblichus (3rd-4th century A.D.). My immediate source for information concerning pythagorean arithmetic is a comprehensive volume by P. H. Michel, De Pythagore a Euclide (Paris, 1950), from which I have drawn the materials necessary for this discussion. For point-numbers, see Michel, pp. 295 ff. 6 Michel, pp. 304 ff. 6 Michel, pp. 311 if. 7Aristotle comments upon the "same" and the "different" as elements of pythagorean arithmetic in the Physics; see Kirk and Raven, op cit., p. 243. 8Michel, pp. 348 ff. I have omitted the "subclasses" or inversions, and slightly rearranged those given in order to present a clearer picture. 9This classification is attributed to the "older" pythagoreans by Ptolemy in his Harmonika, ed. I. During (Goteborg, 1930), trans. During in Ptolemaios und Porphyrios fiber die Musik (Goteborg, 1934), p. 29. 0 Diiring, p. 31. See Kirk and Raven, op. cit., pp. 229 ff. la During, pp. 32 if. 18Aristoxenus disposes of the pythagorean approach in one sentence of his Harmonics, ed. and trans. H. S. Macran (Oxford, 1902), p. 189. 14 Cf. the speculations advanced on this subject by D. Kraehenbuehl and C. Schmidt, "On the Development of Musical Systems," Journal of Music Theory, VI (1962), 32. 15 Diring, op. cit., p. 31. 6 Here I am following an idea put forward by P. Tannery in a very interesting article, "Du role de la musique grecque dans le developpement de la mathematique pure," Memoires scientifiques, III (Paris, 1915), 68 if.; the article first appeared in 1902. Tannery felt that Euclid's-that is, Eudoxus'-ap- RICHARD L. CROCKER proach to the operation of compounding ratios (Euclid's Elements VI, proposition 23) suggested an origin in music, on the grounds of certain peculiarities of terminology that made no sense in arithmetic but were self-evident in music. It seems to me, however, that similar peculiarities are apparent in pythagorean arithmetic itself-as preserved in Euclid. One can compare proposition 4 of Book VIII as the arithmetic equivalent of the geometric operation described in VI, 23. 17 See Tannery,p. 71. 8 See T. L. Heath, The Thirteen Books of Euclid's Elements, trans. and commentary (Cambridge, 1908), II, 119, for an example. 19 Tannery, p. 72. 20See Heath, II, 129, 117 ff., 292 ff. 21See Michel, pp. 360 ff. 22The same principle is expressed differently by J. Yasser, A Theory of Evolving Tonality (New York, 1932), pp. 117 ff. 2. The Sectio canonis, formerly attributed to Euclid, ed. C. Jahn, Musici scriptores graeci (Leipsig, 1895), p. 113; French trans. Ch. Em. Ruelle, L'Introduction harmonique de Cleonide; La division du Canon d'Euclide le geometre; Canons harmoniques de Florence, in Collection des Auteurs grecs relatifs td la musique, III (Paris, 1894), 50 ff. 24R. P. Winnington-Ingram, "Aristoxenos and the Intervals of Greek Music," Classical Quarterly, XXVI (1932), 200, admits that pre-Platonic theorists used the tone 8:9 to construct a scale, but for some reason is reluctant to attribute this construction to the pythagoreans. The fact that Plato describes this procedure when reporting on pythagorean mathematics in the Timaeus seems to me -in the absence of contrary indication-sufficient evidence for considering it to be pythagorean