Tuning Theory for Conworlders: On With 11 - Chords

Now, we've previously considered the intervals of 12-tone equal temperament as approximations of a specific set of just intonation intervals, viz. something along the lines of the following:

1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 16/9 15/8 just intonation major scale intervals bolded, other intervals of some kind of chromatic just intonation scale intervals inbetween

For the record, 16/15 ≃ 17/16, 8/5 ≃ 25/16, 6/5 ≃ 64/54, 16/9 ≃ 9/5, etc.
What just intonation intervals do we approximate now?

We have previously noted how an interval in a temperament is an approximation of another interval - such as four steps in 12-edo approximating 5/4. However, a thing I never said explicitly is that the tempered interval really approximates multiple just intonation intervals. C-E thus approximates 5/4, 81/64, 65536/50625, etc. Some of these approximations may have really big errors, however - and the error might grow as we stack them - even if C-E approximates 5/4, this doesn't mean C-G# (which is C→E→G#, that is, we repeat the same interval twice) is a very good approximation of (5/4)². Now, one thing we might notice is that 11-edo's minor second is pretty close to that of 12-edo, so they probably approximate the very same intervals - we can probably claim all of 16/15, 17/16, 18/17 and even maybe 15/14 as intervals it approximates. Of course, we also have intervals further up the overtone series we could claim - 181/170 very probably could be included, for one. (Try calculating the cents value for these intervals and compare to the cent value obtained for 11-edo above.) (In fact, a simple way of showing that any interval can approximate any number of intervals is to decide on a permissible error, then find a ratio within that error boundary, increase the denominator and enumerator by some factor, then reduce or increase one of them just a little towards the 'center' of the permitted range. We can repeat this procedure an infinite number of times.)

When looking at the major second, we get a much wider gap. We're somewhere between 9/8 and 8/7. So we just go ahead and sort of claim both as being represented by the second.

The minor third is worse. Is it a rather wide 6/5? A really narrow 11/9? Is it 17/14? Note - multiple options may all work for us.

The major third is somewhat close to 9/7, but also to 14/11 and 22/17. This is a good time to note certain things: if we want to build chords, we may want to find reasonably simple series in this. Major chords, as noted, consist of the very elegant triplet 4:5:6. We won't find something that simple in 11-edo, because our 3/2 (6:4) is totally missing. However, to make it easier to find series, we might want to try and find a) things with similar denominators, or b) things with similar enumerators. If we find a/a, b/a, c/a, however, we'll also find c/c. c/b, c/a. So we only need to look for either the same denominators or the same enumerators.

We go on: the perfect fourth is pretty close to 11/8, but also 15/11. The "perfect fifth", of course, is 16/11 (inverse of 11/8), or 22/15 (inverse of 15/11).

Minor sixth - 11/7, 14/9, 17/11.
Major sixth - 5/3, 18/11, 28/17
Minor seventh - 9/5, 16/9, 14/8, 30/17
Major seventh - 15/8, 17/9, 28/15, 32/17

So, given in the form above, we get something like (and note that each number may be replaced by other options):

1/1 17/16 8/7 17/14 14/11 11/8 16/11 11/7 18/11 14/8 17/9

 I won't arrange them as a table, however, arranging them by tables such that a column represents a shared denominator is a good idea. Recall that e.g. 7/4 = 14/8, so you may want to reproduce any number with relatively simple denominators in terms of prime factorization in other columns, so .e.g 7/4 also appears as 21/12, 14/8, etc. Anyways, some series I have found in this are:

8:11:14:(17) (so, starting at 8/8 we add 3 to the enumerator at each step!)
7:9:11 (13/7 and 13/9 are reasonably well approximated as well, so that might be an option - 13/11 is badly approximated, though - but of course, the precision you may want is up to you)
17:22:28 (17+5, 22+6)

I like having a common constant added at each step: a/a, (a+b)/a, (a+2b)/a, (a+3b)/a ..., but this is not strictly speaking a necessity. 8:9:12 chords sound fairly good and are popular in some western music, as are 6:8:9 chords.

These differ from the 4:5:6 triplet in a few interesting ways:
8:11:14:17 starts with a wide fourth, on top of which there's a major third; these add up to a minor seventh (unlike the major sixth we'd get in 12-edo by stacking the intervals that most closely correspond over there). The chord thus covers the octave quite differently from our major triplet: root-fourth-seventh, rather than root-third-fifth. Even if we compared the most similar inversions, we get a slight difference: 8:11:14 -> root-fourth-seventh, 4:5:6 -> root-fourth-sixth.

Let's try some rather unrestricted chord progressions using chords of these kinds. My samples here are not in 11-tet, but rather use (almost) just intonation and lots of inversions and "smooth voice leading", i.e. I try to move voices by small steps and preferrably as few changes when going from one chord to the next:

Now, the chords given above are almost perfectly in tune compared to the overtone series; in 11-edo, this is a luxury we don't have. On the other hand, we gain another benefit: we won't keep drifting by tiny bits of a semitone for (almost) every chord change we do.

Another thing, though - we have not yet considered how these chords fit into any scale - the above progressions are just unstructured movements around pitch space, which can be all nice and dandy, but it's hard to compose with such an open-ended pitch-world. Let's try building scales with this chord as our building block. Our chord consists of the tones at indexes {0, 5, 9}. Let us build an identical chord from 9: {9, 3, 7}. We now have {0,3,5,7,9} for our scale. A reasonable pentatonic scale. It also gives us two instances of the "minor" version of our chord - {5,9,3}, {7,0,5}. The scale we now have obtained goes by the name machine[5] in xenharmonic circles. Usually, it is constructed as a stack of major seconds - starting at the tone we now indexed '3'. We could repeat this operation, by stacking a new chord on top of 7: {7, 1, 5}. We're in luck - we only add one tone to the entire scale, giving us {0,1,3,5,7,9}, which is machine[6], starting at index 1. Much like each major scale (meantone[7]) contains three pentatonic major scales (meantone[5]) (CDEFGABC contains CDEGA, FGACD and GABCD), machine[6] contains two machine[5] - {1,3,5,7,9,11} contains both {0,3,5,7,9} and {9,1,3,5,7}. This can be of some melodic interest, possibly, or of some use for transposition and the like?

We could try stacking our chords in a different manner. {0,5,9}, and a new identical stack on top of 5: {5,10,3}, and maybe even once more on top of 10: {10, 4, 8}. We now have {0,3,4,5,8,9,10}. This does not look all that great - lots of huge stacks of minor seconds, coupled with large empty swathes.

How about one chord on top of both member tones of the original chord:
{0,5,9}, [5,10,3}, {9,3,7}. We get {0,3,5,7,10} - a slightly uneven pentatonic scale. Might be useful though.

What about the utonal chord? {0,4,9}. Repeated stacking on top of the topmost note gives {0,4,9}, {9,2,7}, {7,0,5} which gives {0,2,4,5,7,9} which is a rotation of the {0, 1, 3, 5 ,7, 9} scale we already looked at once.