Music Theory for Conworlders: Investigating that Just-Intonation Scale a bit more Thoroughly

Investigating the scales we obtain may yield interesting finds, and I will provide you with some examples of that:

8/8 9/8 10/8 4/3 12/8 5/3 15/8 2/1 (9/4 10/4 8/3 12/4 10/3 15/4 4/1)

We recall we have three chords in there: one that western music calls the 'tonic' [8/8, 10/8, 12/8], we have the subdominant [4/3, 5/3, 6/3] and the dominant [12/8, 15/8, 18/8]. However, we may have other chords present here! So let us have a look.

We have earlier noticed that a minor chord is a major chord with its 'inner intervals' switching place. Since 12/8 is to 10/8 what 6/5 is to 6/6, and since we have a 4:5:6 structure running in parallel, we should be able to build a minor chord there:

4 : 5 : 6
          4 : 5 : 6

(Each member of the upper line has a 6/4-relation to a member in the lower line)

 C E G
        G B D

{E,G,B} - 10/8, 12/8, 15/8; we divide each by 10/8, and obtain 10/10, 12/10, 15/10, which we've previously established is a minor chord. The same thing can be done one more time, in fact!

F A C
       C E G
               G B D

{A,C,E} gives us the very same structure! If you happen to know your chords and scales, you might wonder why I haven't mentioned D minor yet - and that's because D minor is somewhat exceptional. Canonically, D minor consists of D, F and A.
D to F gives us (4/3) / (9/8) = (4*8) / (3*9) = 32/27; 32/27 just cannot be rewritten as 6/5. Further, A is D's fifth, and we can also see a similar problem there: (5/3) / (9/8) = 40/27. That is about a tenth of a tone flat of a regular fifth, and thus it's a wolf. 'Correcting' A by 20 cents would make [FAC] somewhat odd, with a near neutral third.

We can look at one final thing, the chord starting at B; [BD] is obviously 6/5 (because [GBD] is 4:5:6). [DF] we have already had a look at and found wanting, er found to be 32/27. 32/27 is very close to the equal tempered minor third we often use, so it's not all that impossibly bad, and the way the chord we'd build on top of B normally is used anyway is expected to sound dissonant - it's basically that horror-movie sonority.

If we used 6/6: 6/5 : 6/4 as our building block, obtaining

F Ab C
          C Eb G
                    G Bb D

A similar problem would occur for our Bb major chord; we would have good Ab major and Eb major. If we build scales in this manner, we cannot have three perfect major and three perfect minor chords in the same scale.

How could we make the scale more 'general'? One way, of course, is equal temperaments. Another is extended just intonation - which simply means we keep adding notes so that we have notes for each chord we might want to use. Illustrating the notes of such a scale can easily be done by various geometrical means, consider for instance upwards = 3, downwards = 1/3; leftwards = 1/5, rightwards = 5

	...			A (27/16)
		...	Bb 9/5	D (9/8)	F# (45/32)	A# (225/128)	...
			Eb (6/5)	G (3/2)	B (15/8)	D# (75/64)	F## (375/256)	A## (...)
Bbbb	Dbb (128/125)	Fb (32/25)	Ab (8/5)	C (1/1)	E (5/4)	G# (25/16)	B# (125/64)	D## (625/512)
			C# 16/15	F (4/3)	A (5/3)	...
		...		Bb (16/9)	...
				Eb		...
	...			Ab

A type of chord, a melody or a type of scale would all have a 'shape' in this system, and moving the shape around would transpose it - thus marking all tones currently slightly to the left and up from the currently bolded ones would give Eb minor. Notice, however, that we have two A of rather similar pitch, but with dissimilar function - one the fifth of D, the other the third of F.

For, say, small percussive instruments that are struck, a layout along a matrix like this makes sense. Keyboard layouts along these lines have been successfully tested, and in some sense, the layout on some accordeons is a tempered version of the same idea, although with other intervals as dimensions.