Music Theory for Conworlders: Intervals

For most humans, the important thing when listening to music is not the absolute pitches. A song with A440hz, C528hz, D586hz, E660hz and G782hz is not identifiable on account of those pitches being at those particular hertz. A melody that goes ACACDED ACACDEGa aGaGEDCA is not recognizable for the reason that it plays those particular frequencies in that particular order, it is recognizable due to other reasons.

The thing that makes it recognizable is the relations between the notes. (Music is a bit tricky though, and it seems the rhythm of a melody is even more important for recognition - a piece of advice an old fiddler once gave me was that it's more okay to play a wrong tone at the correct time than the right tone too early or late.) Of course, our hearing has some leeway, but roughly speaking, the ratio between the involved pitches is the interesting thing. Not the absolute difference in frequency - a difference of a hundred hertz will sound very different in a motif that goes 100hz - 200hz - 100hz - 200hz (basically that's a disco bass octave thingy), or one that goes 800hz - 900hz - 800hz - 900hz (that's basically one part of the Rudolph the Red-Nosed Reindeer's main melody, although the sheet music I found for that puts it closer to ~330 - 371 - 330 - 371).

So, the distance between 100hz and 200hz is comparable to the distance between 800hz and 1600hz. The distance between 800hz and 900hz is comparable to that between 100hz and 112.5hz.

The scale I provided in the first paragraph - a version of the pentatonic minor scale - is basically this shape:

A		C		D		E		G
1/1		6/5		4/3		3/2		16/9
	6/5		10/9		9/8		32/27		9/8

The last ratio in the lower line is the remainder needed to get to the next 'a' at 2 times the frequency of the previous A.

Here, the upper series gives each note as its relation to our starting point, A. We could basically pick any frequency we like for A, at this point that does not matter at all. The second list is the intervals between each neighbouring pair of notes.

How do we combine the distance from A to C with the distance from C to D to calculate the distance from A to C? We multiply them! A/B * C/D = (A*C)/(B*D), and as it turns out, 6*10/5*9 = 60/45 = 4/3.

Multiplication is somewhat complicated, and it gets difficult to compare ratios at a glance - 32/27 and 6/5 are actually pretty similar intervals, but this is hard to spot. For this reason, the unit 'cents' has been invented. One cent is a hundredth of a semitone, but that doesn't tell us much of its mathematical properties. The cent is 1200 * log₂A, where A is the interval we're looking at.

If you don't know logarithms, logarithms basically 'shift the gear' of the numbers we're considering in such a way that log(A*B) = log(A) + log(B). In log₂ , if we deal with a doubling, we just add 1; in 1200*log₂ we add 1200. log₂(5/2) = log₂(5/4) + 1; since we're dealing with the weird situation where we have a factor of 1200 everywhere, 1200log₂(5/2) = 1200log₂(5/4) + 1200. Stated simply, logarithms change gears so that multiplication turns into addition.

This gives us one further way of representing the scale given above (here rounded to integer cents):

A		C		D		E		G
0		316		498		702		996
	316		182		204		294		204

It is now easier to compare the sizes of the intervals. The size of the cent is picked to reflect the 12 tone equal temperament in a clear and simple way - each semitone is a hundred cents, and each tone of the western scale therefore is an integer multiple thereof. It is a relative measure, so it doesn't make sense to say that any particular tone is 0 cents - although we can decide for some context to use a certain tone as the starting point.

Another convenient fact is that an equal temperament of N tones to the octave will have steps that are 1200/N cents. This simplifies calculations a lot - if we live in the regular, non-logarithmic world, we need to take N:th roots, which is way more cumbersome.

Finally, any positive number is an interval: 11 is an interval, as is e⁵ or 36π/32. Human hearing stretches from about 20hz to somewhere a bit shy of 20 000hz, so intervals wider than 10 000 probably are not all that useful, since even if we pick the lowest possible point as one end of the interval, the other end will be outside our hearing range. Our hearing is not too precise, so differences in intervals of much less than a cent are probably not very useful either and therefore it might not make sense to distinguish 5 and 5 + 10^-12. And finally, due to the octave equivalence we've previously seen, we will probably only really want to deal with intervals "inside" of the range of [1, 2] and use those to fill out a reasonable chunk of the audible space. We will, however, look at some other approaches as well.

Next installment: more on intervals, Pythagorean tuning