Intervals

Harmonic Sounds and Intervals

For mathy/physical reasons that I describe in detail elsewhere, pitched or harmonic sounds are representable as sums of integer multiples of pure tones (circular oscillations in air pressure). The frequency from which all constituents are formed as multiples is called the fundamental frequency; the frequencies formed as multiples are called harmonics.

When you have two sounds such that some multiples of their fundamental frequencies are equal (that is to say, their fundamental frequencies’ ratio is rational), some of their harmonics will coincide. For instance, consider a sound whose fundamental frequency is 1 and another whose fundamental frequency is 2. In this case, the harmonics of the first sound will just be all the (positive) natural numbers and the harmonics of the second sound will just be all the even numbers. In partiuclar, all of the frequencies comprising the second sound coincide with the first sound. The effect is that the sounds sound “smooth” together–that is called consonance, and the opposite situation (think whitenoise) is called dissonance.

Two sounds whose fundamental frequencies’ ratios is (very close to) simple will sound consonant. There is a fairly complicated–and somewhat speculative–mathematically rigorous notion (to do with the information content of the continued fraction representation of the number) of what it means for a rational number to be simple (and there is a tractable, but also somewhat convoluted, physiological/psychoacoustic explanation of what “very close to” means in terms of something called the critical band). Basically, rational numbers like 3/2 are simple and rational numbers like 355/113 are complicated.

Equal Temperament

Once again, for technical mathematical reasons (that the rationals are freely generated by the primes (and -1)), we can’t have nice things; a scale especially consonant in one key will dissonant in another. That limits our ability to change keys freely. Johan Sebastian (J.S.) Bach, famously popularized the natural solution to “split the difference” with his work: The Well-Tempered Clavier, with a Prelude and Fugue in each major and minor key.

There are 12 tones in an octave in standard western music–and they each have a constant ratio between them. Together with the knowledge that an octave is in ration 2:1, we can conclude that the common ratio is 2 ^ (1/12) ~ 1.05946309436, the twelfth root of two.

This raises an interesting question: what happens if you put a different number in there instead of 12? That’s a very good question! Indeed, you can do this, and it leads to interesting results. However, 12 equal divisions of the octave, or 12 EDO, has come to be ubiquitous for two main reasons:

• 12 EDO has very good approximations (less than 1% error) to simple ratios:
  • 2 ^ (2 / 12) ~ 1.122 ~ 1.125 = 9 / 8 = (3 ^ 2) / (2 ^ 3)
  • 2 ^ (3 / 12) ~ 1.19 ~ 1.2 = 6 / 5
  • 2 ^ (4 / 12) ~ 1.26 ~ 1.25 = 5 / 4
  • 2 ^ (5 / 12) ~ 1.334 ~ 1.333… = 4 / 3
  • Incidentally, the less-good approximations to less-simple fractions are simple with respect to sqrt(2):
    • 2 ^ (1/12) ~ sqrt(9 / 8) = 3 / 4 * sqrt(2)
    • 2 ^ (6 / 12) = sqrt(2) ~ 1.41 ~ 1.40 = 7 / 5
• 12 is still a fairly small number, all things considered; it is easy to make instruments for 12 EDO and it is easy to think about the 12 notes of music (once you become practiced)

Program

Without further ado, let us commence practice recognizing intervals with the tool I wrote:

EDO Setting (set this once):

Identify Interval Size (for feedback):