Music Theory for Conworlders: Chords as Building Blocks for Scales, Temperaments pt 1

So, we recently saw how a major chord is built, and looked at some other chords as well. Let us build a scale from these chords!

We could stack them in some way - i.e. we build a chord, let's call it M. We index its notes M1, M2, M3. Once we reach M3, we also call that note N1, and build a new chord N1, N2, N3. So, using the major chord outlined earlier and arbitrary 1/1, we have 4/4, 5/4, 6/4. We stack a new similar series on top of 6/4, giving us 6/4, 15/8, 9/4. If we assume we have octave equivalence, we can replace 9/4 by 9/8, getting us this scale: 4/4 9/8 5/4 3/2 15/8. We now repeat this one last time on top of 9/8 - getting us 45/32 and 27/16. So, our full set now is

F	G	A	B	C	D	E
8/8	9/8	10/8	45/32	12/8	27/16	15/8

If we were to further remap the composition to this set (so that the starting tone is remapped to C, rather than F, instead of the previously used approaches), we get this result:

If we further approach it, mostly without any clearly thought-out approach, but adding backing chords basically by picking a chord for which the melody over some span shares a significant number of notes with that chord, we get this, which already has turned our rather weird overtone thing into a rather pleasant, albeit not very complicated thing:

The distribution of step-sizes here is:

F

G

A

B

C

D

E

9/8

10/9

9/8

16/15

9/8

10/9

16/15

204c

182c

204c

111c

204c

182c

111c

[Note: I've rounded the cents values to integers.]
The underlined ones form the first triplet, the italic ones form the second triplet, the bolded ones form the third triplet. Notice that there are tones that belong to two triplets. If we instead of stacking twice, instead had stacked 'downwards' as well as 'upwards', we'd have in front of us the major scale. Now we have a rotation of the major scale instead. (By 'stacking downwards', I mean building a triplet L1,L2,L3 such that its L3 = M1, where M1 is our starting point; had we started at M1, built M1,2,3, L1,2,3 and N1,2,3, we'd have the regular major scale.) We're currently looking at the same thing, but starting at the wrong tone in it - the thing in front of us right now is called a lydian scale. This is why I opted to call the first tone F instead of C. We could go about similarly with the minor chords as well, and build the natural minor scale by stacking [6/6,6/5,6/4] the same way. If we replace the last of them with a major triad, we get the harmonic minor, and if we replace the triad we build 'downwards' with a major triad, we get melodic minor ascending.

We could of course go and add 7/4 to the major triad too - so we would have chords of 4/4 - 5/4 - 6/4 - 7/4, but stack them at 6/4. (or at 7/4 if we'd like, but I'm not going to do that right now)

1/1

9/8     7/6

5/4

21/16

45/32

6/4

27/16 7/4

15/8

204c 266c

386c

470cc

586c

702c

906c  966c

1090c

This scale would be a bit of a curiosity, although who knows, it might be possible to compose nice music it in. Here's a really shoddy attempt:

We are not restricted to 4:5:6(:7:...) either. We could pick, say, 3:4:5, 5:6:7 or 6:7:8 or something that has a different number of notes, or gaps in it. 4:5:7 gives an interesting scale - one that obviously is a subset of the one given above. Nothing stops us at two iterations either - but we probably want a manageable number of notes. And of course, we can alternate what kinds of sets we stack on top of something, just to see what happens.

We'll go back to the one presented above that just had seven distinct notes. We might notice that there's basically three sizes of steps in it - and that two of these sizes are fairly close together, whereas the third is a bit further off. Could we perhaps utilize that somehow? Actually, one thing we could do is call the 204 cent intervals 'L', the 182 cent intervals 'M' (for middle), and the 111c intervals 's', for small. This, in fact, is a useful notation: when we describe a scales shape, we sometimes actually benefit from ignoring the exact sizes we have to deal with, and instead just consider the scale as having the form LMsLMLs. We will later on encounter scales (and parts of scales) that are only described as strings of L:s, M:s, s:s, and possibly other letters, and possibly only a framing interval given explicitly or even implicitly.

For now, our framing interval is the octave. Our plan is to 'equalize' the intervals in some sense, to drop some of the exact tuning in favour of the convenience of a more 'simple' system.
We then have the following thing to solve: 3L + 2M + 2s = 1200 (since there's three L, two M, and two s in the scale). This can be solved in several ways; we can go to extremes, and conflate them all, L=M=s, in which case we get L=M=s=171 cents. It is useful, however, if M,L and s all are multiples of some shared unit - so e.g.

3L + 2M + 2s = 1200 cents | our basic setup
 L = M | we decide to conflate 'large' and 'middle' steps→ 5M + 2s = 1200 cents
 M = 2s | we decide we want the small steps to be half of the large steps → 12s = 1200
→ M = 200 cents, s = 100 cents

We now end up with

Overtune, now in 12-tone equal temperament.

Another thing we could do is:

3L + 2M + 2s = 1200 cents | ...
L = M | we decide to conflate large and middle againM = 3/2s | we decide we want the large only to exceed the small by half the size of the small→ u = 1/2 s (u for 'unit') | we decide to introduce the 'unit' that corresponds to the smallest necessary division for this to work out→ 5M + 2s = 1200 c
→ 5*(3/2)s + 4s = 1200 c
→ 15u + 4u = 1200 c
→ u = 1200c/19

This would give us a tuning that sounds like this:

That same tune in 19 tones to the octave.

The first example is one way of ending up at modern equal temperament. However, the same mathematical approach coupled with some different decisions will give other nice temperaments that have many similar properties - but may have other, dissimilar properties as well. The second example gave 19-tone equal temperament, a temperament which very well could be a dead-ringer for 12-tone equal temperament, especially with regards to most of the repertoire that doesn't modulate wildly. Why this is the case, we will look at a bit later when we get into meantone. (There are more in tune meantones than 19, but 19 is conveniently small - basically, 12, 19 and 31 are pretty good meantones - but 31 is rather big).

We will look how this same method can be used to find other temperaments with similar properties - or temperaments that give us 'twisted' tunings - tunings where things from 12-tone equal temperament get turned on their heads. (One hint, though: imagine what happens if we substitutes M→s, s→M, and solved for that? 5s + 2M, distributed as ssMsssM actually produces a weird yet attested in real life cultures scale. The same type of inversion can be done for any scale!)

We'll also see how this can be used more generally to consider the results of other building blocks than 4:5:6. But before that, we'll go and look at Pythagorean tuning, the Meantone tuning it gave way to, and our modern very moderate meantone tuning, viz. 12-tone equal temperament. We'll see a few other approaches that we may take to reach 12-tone equal temperament as well (but also other points to which those approaches could have taken us).