Math And Music.docx

The Brief Mathematics of Music To parallel what the Wikipedia article opens with, “Music… has no axiomatic foundation in modern mathematics,” meaning that music has no absolute and complete basis in math, however, the sound component of music has a large quantity of properties that coincide with mathematic theory. That being said, prior to elaborating upon the given sources, I wish to mention twelve-tone serialized music, which does, in fact, rely very heavily upon formulaic composition methods. Arnold Schoenberg developed this technique which derives its usage from utilization of all 12 notes of the chromatic scale. The goal is to never give any note more or less importance. The resulting music is often very formulaic and cannot be traditionally analyzed except by use of a matrix. (see attached 12-tone matrices examples) Continuing on with the given content, there are many ‘mathematical’ methods of analyzing traditional music. The first mentioned was the proportional relationship of pitch octaves. I find it fascinating and immensely satisfying that for each octave you ascend, the frequency of that pitch doubles (conversely, dropping octaves halves the frequency). For example, for the pitch A4, 440Hz is a common tuning frequency; the pitch A5 one octave higher resonates at 880Hz, double its lower octave relative. The same goes for all other pitches and their respective octave motion. Other pitch ratios include harmonics wherein notes can be built upon fundamental pitches. A2 at 110Hz can enharmonically create A3 (220Hz), E4 (330Hz), and so on to theoretical infinity. The downside to this mathematical method is that the resulting pitches that aren’t octave augmentations are usually very out of ‘tune,’ meaning they don’t sound pleasing to the ear in musical context. Mathematics do have a heavy influence on tuning system derivations. Equal temperament scale systems are created by dividing the octave into equal, logarithmic sections. This results in proportional and evenly divided scales, but the ratios of the frequencies are irrational and subsequently unproportioned. Conversely, just scales are built on rational frequency ratios, resulting in uneven scale divisions. Most western music schools of thought choose equal temperament over just in favor of equal sections within the octave, as it sounds most pleasing to the ear. Western music theory also used mathematic terminology to explain musical occurrences and relationships. This relates somewhat to the theory surrounding 12-tone music. To analyze such pieces, one usually starts with a set of given or composed tones. By applying transposition and inversion, as seen in 12-tone matrices, there are unique structures that can be observed. Perhaps the simplest form of music to mathematical relation is rhythm and meter. Note duration is incredibly proportional, especially in 4/4 or common time. 4/4 time consists of four beats to a measure, wherein a whole note lasting the entire measure is the ‘whole’ 4 beats. A half note is half of the measure, or 2 beats. A quarter note is a quarter of the measure or 1 beat. This proportional note division can be carried on infinitely but becomes rather impractical past 1/128 of the measure (which receives 1/32 of the beat). Lastly, a mention of Mozart and the Golden Ratio is fascinating. Mike May says, there is “considerable evidence [which] suggests that Mozart dabbled in mathematics.” Many of Mozart’s works seem to use the Golden Ratio as a means to compose the lengths of musical sections. The difficulty in confirming the validity of the conjecture, however, lies in the unfortunate fact that Mozart was not terribly consistent. In many works he appeared to start and stop using the Ratio as he pleased and when it fit his purposes. Though there are many parallels between music and mathematics, it is this writer’s firm opinion that it is just as well that music cannot entirely rely upon mathematics for its

foundations. As exemplified by 12-tone tonality, music built solely upon mathematic theory is very harsh on the ears and does not blend well with western practices. However, music that borrows here and there from such theory has been found to be very aurally pleasing for centuries.

Figure 1 (above): Common format for analysis and creation of dodecaphony (12-tone) This matrix is designed to be used in four directions (Prime, Inverted, Retrograde, and Inverted Retrograde)

Figure 2 (below): Completed 12-tone matrix on C#, notice that there are enough similarities linearly speaking to necessitate inverted, retrograde and inverted retrograde variants.