By Nicholas D. Lewis
The philosopher Pythagoras (6th century B.C) and his disciples are credited with the discovery of the numerical relationships governing the basic intervals of music. Pythagoras was concerned with numbers, and it is with him that the idea of “harmony of the spheres” was born. Although many people don’t realize it, this is the very reason that astronomy and music were linked and even codified later by Plato and his followers.
Plato considered harmony to be its own branch of physics. This alone suggests the kinship of harmony and mathematics. On one micro-level, sound itself is governed by the laws of mathematics. Frequency, amplitude, and loudness – to name a few wave characteristics – all depend upon the structure and speed of a given wave in terms of oscillation and size. Rhythm is intrinsically mathematical in nature. Beats flow along time and are subdivided. Rhythms fill those pulses. Even tuning systems are based upon the divisions between certain frequencies.
Most important, perhaps, is that music has patterns. Mathematics, almost by definition (and certainly by connotation) deals with patterns. Since mathematics deals with patterns, and one of the most fundamental concepts of most world music is patterns, it stands to reason that mathematics is a good way to describe music. Nearly conversely, it seems to stand to reason that people could use mathematics to create a musical composition. And moving into the 20th century, that is exactly what composers began to think.
Set theory, a branch of mathematics that deals primary with collections of objects, is a very convenient modern way of showing the relationship between different musical sounds. Although musical set theory and mathematical set theory are different in many ways, they share a number of similarities. While musical set theory does not deal with various sizes of infinitely large sets, it does serve to achieve the same basic goal: to organize elements. Although there are many types of musical set theories, the most well known is probably pitch-class set theory. When given pitch classes are introduced as elements into a given set, they can be related by techniques such as transposition, inversion, and complementation. So why does this matter?
Although the music created by people using mathematics a medium for composition is inherently structured, it rarely sounds good to most people. This is because it does not meet all (and sometimes none) of Dmitri Tymoczko ”Five things that make music sound good”. Those five things are the following:
“1. Conjunct melodic motion. Melodies tend to move by short distances from note to
note. Large leaps sound inherently unmelodic.
2. Harmonic consistency. The chords in a passage of music, whatever they may be,
tend to be structurally similar to one another.
3. Acoustic consonance. Some chords sound intrinsically good or pleasing. These
are said to be consonant.
4. Scales. Over small spans of musical time (say 30 seconds or so), most musical
styles tend to use just a few types of notes, between 5 to 8.
5. Centricity. Over moderate spans of musical time, one tonic note is heard as being
more prominent than the others, appearing more frequently and serving as a goal
of musical motion.”
When one of these rules is broken, people may perceive the music as “random” or “discordant” or any of the other infinite gradation of inaccuracies that I often hear about 20th century art music.
But Tymoczko’s claims in the “Geometry of Music”, while wonderful in their own right, are far too subjective to be taken as a universal truth. In Western Art Music during the common practice period, his theory holds largely true. Most art music does in fact make conform to the five rudimentary points of his theory. But, a great number of people like Xenakis. A great number of people think Stockhausen, Boulez, Carter, Ustvolskaya, Cage, Ligeti, Babbit, and other avant-garde composers sound good.
I know I do. Do they share similarities that make their music sound good to me? And indeed, what are those similarities?
I will deal with a relatively famous piece by a relatively famous composer: Metastaseis, by Iannis Xenakis. This work, scored for 61 players, has nobody playing the same part. Here is a quote from the Wikipedia article about the piece:
“As Newtonian views of time show it flowing linearly, Einsteinian views show it as a function of matter and energy; change one of those quantities and time too is changed. Xenakis attempted to make this distinction in his music. While most traditional compositions depend on strictly measured time for the progress of the line, using an unvarying tempo, time signature, or phrase length, Metastasis changes intensity, register, and density of scoring, as the musical analogues of mass and energy. It is by these changes that the piece propels itself forward: the first and third movements of the work do not have even a melodic theme or motive to hold them together, but rather depend on the strength of this conceptualization of time.”
This is interesting for several reasons. One, it shows a new approach to the concept of time in music. Every time I am in a car I am reminded of Xenakis’ music by analogy. Whenever two turn signals are on, but click at different speeds, it captures my imagination. Every so often they click at the same time, but most of the time they are flashing at two different speeds… Abstractly ,they are two different universes. Another thing that is interesting about this quote is that it shows, though not outrightly, that Xenakis was relatively careful about macro-phrasing. And that brings me to the purpose of this article.
Almost all music created by sentient beings has a phrase. In fact, upon analysis of Xenakis’ work, one will find that it follows many of the same rules, though perhaps more abstractly, that the classical sonatas of Mozart’s time followed. Or more accurately, that the composers followed. Metastaseis follows all of the same overarching rules. Although the tonality and approach are different, one aspect is the same: contours intentionally lead to phrases (or at least macro-phrases). There are, at least abstractly, cadences, phrases, and even melodies in the music. Does this suggest that the phrase is the most important of all elements of music? It is essentially the only thing that almost all music, and all non-chance music, shares. And indeed, I must ask a question that seems to have an obvious answer at first: is that all that’s important in music? Do think about it.