Music Theory for Conworlders: Simple Scales from the Overtone Series

Previously I described the overtone series. Let's use it to build our first scale!

For ease of calculation, we'll start at 50hz: 

We notice that for each octave we go up, the number of tones increases:

50 100 ← first octave
100 150 200 ← second octave
200 250 300 350 400 ← third octave
400 450 500 550 600 650 700 750 800 ← fourth octave

the overtones of a 50hz tone, given in hertz, presented in successive octaves

1 2
2 3 4
4 5 6 7 8
8 9 10 11 12 13 14 15 16 

 the multiples that give the overtones, presented as successive octaves 

We pick the 400-800 (or 8-16) octave simply because it's the first octave in which there's 'enough' tones for a reasonable scale (four-note scales for some reason generally aren't considered much in scale design - it seems most scales you find in human cultures consist of five or more). We use that as a model for every octave from now on, so we end up having this:

50 56.25 62.5 ...

100 112.5 125 ...

200 225 250 ...

Since the difference now actually is a constant number of hertz, we get a scale represented by adding a constant to the enumerator of a ratio:

8/8 9/8 10/8 11/8 12/8 13/8 14/8 15/8 16/8

overtones from the eighth to the sixteenth, tuned down until they're in the first octave 

And the intervals between subsequent notes will shrink:

9/8, 10/9, 11/10, 12/11, 13/12 ...

In fact, the first step is almost twice the size of the last step. Here's a short two-voice composition in this scale, although from C to C rather than G400 to G800, 'Overtone ditty':

So, how would this kind of scale appear in real life? Consider the string or the tube of air in organs, flutes etc. If we construct frets linearly, so that there's one fret at roughly half the string, and the rest of the frets are equidistantly along the neck, you'll approximate overtone tuning fairly well. With tubes of air, the same roughly applies: half the length gives an octave, then make pipes that are equal divisions of that length. Overtone singers also easily can produce overtone scales, and overtone-instruments such as overtone flutes.

Of course, we needn't use 1/1 as our 'root', we can use some other pitch as our 'root': we might start from, say, 550 - and get 550 600 700 750 800 900 1000 ... in which case the 'steps' will not just shrink - we get a large step between 800 and 900, from which a new sequence of shrinkage appears for a while. The scale will start out as a series along the lines of 11/11 12/11 13/11 14/11 15/11 16/11 18/11 20/11 22/11.

Here we go on using Overtone Ditty, but I tune down the fundamental so that the root for the melody keeps being C. However, the starting point - the 8/8 - from which the overtone series is calculated is reduced by one each iteration (except the last that returns us to C as both the root and fundamental):

The first iteration was the same piece as previously; in the second part, whenever the first part went 8/8 9/8 10/8 ... the new one went 9/9 10/9 11/9; in the third you got 10/10 11/10 12/10; the fourth even went as far as 11/11, 12/11, 13/11 ...; It was a bit tedious to retune the parts, so I didn't go any further than that, I hope the different sounds of the four parts illustrate pretty well what's going on.

I might later compose a short ditty illustrating some of the next few ideas.

We could of course build a similar rational scale using any denominator, and not be restricted to exactly 8 (or even 2ⁿ, n ∈ Z) pitches - any number, really, could work. Let's look at the set we get with the series from 7 to 7*2:

7/7 8/7 9/7 10/7 11/7 12/7 13/7 14/7

This would correspond to starting out by halving the length of a string or pipe, and then placing frets or producing new pipes (or adding holes) at equal distances between the full length and the half-length.

For each such scale, we can also investigate and use the scales we get from using some other tone than the original N/N as our starting point and introducing a slight "discontinuity" in the sizes of the intervals, such as:

8/7  → (1/1)' (i.e. 8/7 is now the new 'standard' for the previously given scale; 'new' is marked by an apostrophe in the ad hoc notation I now am using)

... 7/8) 8/8 9/8 5/4 11/8 3/2 13/8 7/4 2/1

(You can see how the interval size consecutively shrinks until you hit the distance from 7/4 to 2/1. Repeated shifting along this procedure will of course move the discontinuity rightwards.)

A few examples then, would be 7/7 to 14/7, 5/5 to 10/5, 6/6 to 12/6. I've thrown in 16/16 to 32/16 as well to give an example of a rather crowded scale.

To understand what happens with our scale (given as a series of numbers), it's simply that we multiply each interval by the inverse of the number we want as our new root (so, if in the series from 7/7 to 14/7, we now want 8/7 as our root, we multiple each with 7/8), and those that now are less than 1/1, those that drop 'under' our new root, will reappear from an upper octave. Mathematically the same result is obtained by multiplying by two any interval that is now lower than one - 7/8 becomes 7/4, 8/9 becomes 16/9, and thus remains in the scale. We could do the same for 5/4 and the whole 7/7-to-14/7 series:

5/4  → (1/1)'

... 4/5 9/10) 5/5 11/10 6/5 13/10 7/5 8/5 18/10 10/5 

Another thing we can do is the omission of some notes. We could take our eight tones:

1/1 9/8 5/4 11/8 3/2 13/8 7/4 15/8 2/1

and pick some way of removing a few. Maybe we want to have a highest 'factor', i.e. any ratio involving larger primes is omitted. Omitting 11 and upwards would give

1/1 9/8 5/4 3/2 7/4 15/8 2/1

Which is almost our pentatonic scale - it has a somewhat sharp sixth, and a major seventh to boot that extend it, though. (Why I call those intervals by those names, I'll explain later).

We could of course do some other approach too: only pick primes and composite numbers which are square and composite numbers where at least one factor is 2:

1/1 9/8 5/4 11/8 3/2 13/8 14/8 2/1

This is a weird variety of mixolydian (and I'll get to that later). We needn't even be particularly logical about this - we can just exclude some intervals because we think they sound bad in combination with the others.

"Proper" overtone scales appear in throat singing, and part of the overtone series is close to parts of the western diatonic scale. Intervals do recur - if you go high enough up. But for each recurrence of an interval, there'll be twice as many new intervals appearing if you ascend the scale one note at a time.

Although scales based on the overtone series aren't necessarily what you'll want to use in your conculture, overtone series are a good thing to be familiar with. A lot of what we'll be doing with scales that are much more like the diatonic scale will be based on the overtone series in one way or another.