Odd as it seems, abstractions of music are to be found in mathematics just as abstractions of triangles are to be found there. This seems to follow from the tradition of mathematics as an abstraction of real world experiences. There is an historical precedent for this assumption. For instance, as Richard Burton points out in The History of Mathematics, An Introduction:

In ancient Babylonia and Egypt, mathematics had been cultivated chiefly as a tool, either for immediate practical application or as part of the special knowledge befitting a privileged class of scribes. Greek mathematics, on the other hand, seems to have been a detached intellectual subject of the connoisseur.
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The Greeks’ habits of abstract thought distinguished them from previous thinkers; their concern was not with, say, triangular fields of grain but with 'triangles' and the characteristics that must accompany triangularity.

It is difficult to conceive that man could have ever made a direct leap to pure mathematics without passing through an era, like that of the Babylonians and Egyptians, of familiarization and experience with real world forms. Not too oddly, these sequent effects in thought seem to continue in a reciprocal fashion, for with the advent of abstractions and pure mathematics came new possibilities for real world applications that never would have been possible beforehand. Therefore it seems, at least in the majority of cases, that the major branches of mathematics are traceable to particular sectors of human experience: geometry can be traced to the human experience of shape and form; number theory can be traced to the human experience of a quantificational universe; calculus, the science of the infinite, can be seen to have evolved because of thousands of year’s of human inquiry and fascination with our solar system and the stars beyond. In spite of these precedents however, two important difficulties arise when trying assess the mathematical achievements of the ancients as they pertain to music.

The first of these difficulties is in properly discriminating the one science from the other. For instance, one of the earliest divisions of the curricula, which came to be known as the quadrivium and was originated by the Pythagoreans, held that music was merely a branch of mathematics and was strictly mathematical in nature (Garland, Kahn, 63). Yet we encounter in their writings on music many statements which by contemporary standards would classify the subject under acoustics and harmonics, not mathematics.

The second difficulty arises in trying to identify a cause-effect relationship between inventions in mathematics and inventions in music. Did musical experience bring about the initial interest in ratio and proportion, resulting in such discoveries as the harmonic mean, or did knowledge of these things come first from other experiences, as with spatial forms like pyramids, triangles, and circles, and then later get applied to encountered phenomena in music?

Solutions to these two difficulties are not easy. For the second one, there may never be a satisfactory answer. Two modern writers have said that, "We'll probably never know how much influence the development of a number system had on the music of a given culture, or vice versa," (Garland, Kahn, 37), and in light of the evidence one finds upon investigation, it is difficult to disagree with them. On the other hand, for the first difficulty there seems to be a good case for resolution. By the evidence that has come down to us, it appears safe to presume that a large part of that musical theory which the Pythagoreans and later writers, such as Euclid and Nicomachus, considered purely mathematical in nature, were indeed so, at least inasmuch as geometry might be considered similarly. Several factors help to support this discernment. For instance, in Euclid’s Sectio Canonis, for instance, "the task undertaken by the author is to show how the propositions of harmonics can be demonstrated as theorems within mathematics itself, given certain assumptions about the physical nature of the musical phenomena." (Barker, 7)

The Sectio is a work with a programme like that of the Elements: a series of tightly inferred propositions derived from a set of axiomatic assumptions. However, instead of triangles or circles taking the spotlight, we find in their place "intervals", abstractions like triangles or circles which have analogies in the real world but which strictly speaking do not exist as such outside of the mathematical system in which they are defined. Thus, propositions about these intervals end up being propositions about a generalization, not necessarily about musical intervals in particular.

Another factor in support of some of the early musical theory properly existing as a branch of mathematics was the existence of another tradition which began contemporaneously with the mathematical one, and which seems never to have been reconciled with it, even to the present day. This other tradition had its genesis with musicians, and the earliest complete treatise on the subject still surviving under this auspices comes to us from Aristoxenus, a musician from the end of the 4th century BC and whose work, Elementa Harmonica, dating from the same time as the Sectio, is equally impressive.

Various attempts through the ages have been made to synthesize the two traditions into a single subject, but whether such attempts have succeeded is a relative issue. For instance, in the 5th century AD, Roman philosopher and mathematician Boethius, wishing to perpetuate the Pythagorean tradition of "music as mathematics", gave the following definition for a musician:

That person is a musician, who, through careful rational contemplation, has gained the knowledge of making music, not through the slavery of labor, but through the sovereignty of reason. (Garland, Kahn, 64).

However, as writers Garland and Kahn point out in their book Math and Music, Harmonious Connections, "By today’s standards, this is a completely preposterous definition of a musician, since performers and composers aren’t even included." In spite of this, there is also evidence that a great deal of reconciliation has taken place. Today we often find a musician who is also a mathematician and vice versa, and in modern music theory classes the fundamental relationships accredited to Pythagoras himself of the intervallic ratios, such as the 2:1 ratio of the octave, are taught. Regardless of how this issue of reconciliation is viewed, the point is that the contrasting traditions of explaining musical phenomena help resolve the issue of how far we can consider mathematical achievements to be mathematical in spite of their bearing on other sciences such as physical acoustics or music.

Turning to the mathematics, among the earliest important recorded mathematical discoveries related to music are those accredited to Pythagoras himself. The discovery was that intervals between pitch were determined by numerical ratios. (Landels, 130). He found that the modern intervals of the octave, fourth and fifth (which today are still considered the most harmonious) corresponded to the ratios 2:1, 3:2, and 4:3, respectively. The Pythagorean's did not delve to deeply into the frequencies of pitch or the relationships between the length of a string and the frequencies produced by the string when strummed (Landels, 137). As a matter of fact, it wasn’t until the 17th century that mathematician Marin Mersenne showed the rules related to pitch, namely, that the frequency of a vibrating string is (a) inversely proportional to its length, (b) proportional to the square root of the tension applied to the string, (c), inversely proportional to its diameter, and (d), inversely proportional to the square root of its density. (Garland, Kahn, 142). Rather, the Pythagorean’s were content to find that, because the numbers in the ratios above agreed with the numbers 1 to 4, which went to make up the holy tetractys of the Pythagoreans (the triangular symbol consisting of 10 dots), these ratios were thought by the Pythagoreans to be the most concordant and harmonious in general, thus allowing them to make many other inferences like the “music of the spheres”.

Besides the discovery of the ratios and according to some modern accounts, the Pythagoreans’ final and most important contribution to mathematics from the phenomena of music seems to have been the theory of proportions and the arithmetic, geometric and harmonic means (Garland, Kahn, 51). As mentioned earlier, there is little evidence to support this cause-effect relationship directly, and the two authors cited do not give their sources. At the very least, it is clear that their is some relationship between the Pythagoreans' interest in musical phenomena and the arithmetic and harmonic progressions. That is, the lengths of the strings which form the most harmonious musical intervals of a first, fifth and eighth, taken in progression, form the harmonic progression 1, 2/3, 1/2, and the pitches the strings produced when strummed, taken in progression, form the arithmetic progression 1, 3/2, 2/1. According to another writer, however, the term “harmonic” in “harmonic progressions” is not to be attributed to this coincidence (Boyle, 2). Like the others, the writer gives no justification for the claim so, short of further information, this matter is inconclusive. (The handout from Dr. Zerla on proportions indicates that the former name for the harmonic progression by the Babylonians was the “subcontrary”, so it at least seems plausible that the name was changed to “harmonic” due to the discovery that it was found to occur in the harmonious progression I-V-VIII in music.)

Another major mathematical achievement related to music and following in the Pythagorean tradition was from no other than Euclid himself, embodied in his small treatise called the Sectio Canonis, mentioned earlier. (This work translated bears the name "Division of the Monochord".)

Among the proofs to be found in that work are mathematical correspondences to the musical operations of adding and subtracting intervals (Landels, 130). Euclid demonstrates that, though when you add intervals of say a fifth and a fourth, you get an octave, to get the same result mathematically you must multiply the ratio of the fifth with the ratio of the fourth to obtain the octave $\left(\frac{3}{2} \times \frac{4}{3} = \frac{2}{1}\right).$ Similarly, to find the difference of an octave and a fifth mathematically, you must divide their ratios rather than subtract them $\left(\frac{2}{1}\div\frac{3}{2}=\frac{4}{3}\right).$

The irrationality of certain numbers also come into play here because a third procedural difference between musical and mathematical treatments of intervals is in finding a note that falls exactly half way between two others. Mathematically, this is accomplished by taking the square root of the ratio of the outer notes, but if the two outer notes happen to be, for instance, in a 2:1 ratio (an octave), then to find the middle note mathematically it is necessary to find the square root of 2, an irrational number (Landels, 130).

All together, the Sectio is a collection of 20 propositions which have the purpose to demonstrate how a monochord may be divided to obtain a so-called "system of the diatonic genus". Translated into English, this "system" was a form of a scale which was widely used in ancient Greece and which was considered to be the basis for all other scales (though the Sectio does not attempt to prove as much.) As Andrew Barker indicates, "What is most original about it [the Sectio] is not what it asserts, nor even how each theorem is argued, but rather how the whole system of theorems is coordinated into a single scheme.” (Barker, 191)

Taken collectively, the achievements of the Pythagoreans, Euclid, and others following in the Pythagorean tradition, made important breakthroughs in mathematics which resulted from the musical experience, and it is clear that without the pioneering efforts of such mathematicians, the modern musical and mathematical landscapes as we know them would never have been possible.

Works Cited

Barker, Andrew. Greek Musical Writings, vol. II. Harmonic and Acoustic Theory. Cambridge University Press. 1989

Burton, David M. The History of Mathematics, An Introduction. USA: McGraw-Hill. 1997.

Garland, Trudi Hammel, and Charity Vaughan Kahn. Math and Music, Harmonious Connections. USA: Dale Seymor Publications. 1995.

Landels, John G. Music in Ancient Greece and Rome. New York: Routledge. 1999.

Lloyd, LL. S., and Hugh Boyle. Intervals, Scales and Temperaments. New York: St. Martin’s Press.1963.