Rank-3 Chords

 In rank-3 pitch space, the intervals between successive steps of the major scale are these: 

    [P1, M2, AcM2, m2, AcM2, M2, AcM2, m2]

If we commit ourselves to C major being the major scale with natural pitch classes, then other scales necessarily get new alterations relative to their rank-2 spellings. For example, a G major scale has to be:

    [G, A, B, C, D+, E, F#+, G]

where a "+" is the accidental that indicates raising by an acute unison, a.k.a. a syntonic comma.

If we build diatonic chords by third from a major scale, we get these 13th chords on the scale degrees of the major scale:

I: [P1, M3, P5, M7, M9, P11, M13]

II: [P1, m3, P5, m7, AcM9, Ac11, AcM13]

III: [P1, m3, P5, Grm7, m9, P11, m13]

IV: [P1, M3, P5, M7, AcM9, AcA11, M13]

V: [P1, M3, Gr5, Grm7, M9, P11, M13]

VI: [P1, m3, P5, m7, AcM9, P11, m13]

VII: [P1, Grm3, Grd5, Grm7, m9, P11, m13]

That takes care of how to spell diatonic chords in rank-3 space. But what about non-diatonic chords? For example, how should one spell the diminished seventh chord, which in rank-2 space had been [P1, m3, d5, d7]? We can see from the VII diatonic chord that a half-diminished seventh chord, a.ka. a "m7b5" chord is now spelled [P1, Grm3, Grd5, Grm7]. So we might expect that we'd have to lower Grm7 by an augmented unison. It's also common to analyze a dim7 chord as a rootless 7b9 chord, with an implied root on the fifth scale degree of the major scale. So maybe we should start with the diatonic dominant ninth chord, [P1, M3, P5, M7, AcM9], lower the ninth by an augmented unison, and drop the root.

I tried maybe 15 to 20 different variations like that, and I think [P1, Grm3, Grd5, GrGrd7] sounds the best when tuned in 5-limit just intonation. I don't know how to explain it. This post is for figuring out why there's a GrGrd7 at the top of that chord that sounds so good, and maybe along the way we'll figure out principles for making other non-diatonic rank-3, 5-limit just intonation chords sound good.

...

After some experimentation, I've found three dim7-like chords that sound even better than [P1, Grm3, Grd5, GrGrd7]. Here are the four presented together:

[0, 294, 588, 882] # [P1, Grm3, GrGrd5, GrGrGrd7]

[0, 294, 590, 884] # [P1, Grm3, AcA4, M6]

[0, 294, 610, 904] # [P1, Grm3, Grd5, GrGrd7] // Not as good

[0, 296, 590, 906] # [P1, AcAcA2, AcA4, AcM6]

We can see that AcAcA2 only differs from Grm3 by 2 cents. Likewise M6 only differs from GrGrGrd7 by 2 cents. Likewise GrGrd7 only differs from AcM6 by 2 cents. So the chord I'm grappling toward has a second absolute interval around 294 to 296 cents. The third absolute interval is around 588 to 590 cents. And then my ear is embarrassingly liberal in the choice of the last interval, since it can differ by a syntonic comma without my preference changing. I can't help but wonder if my ear is just really used to 12-TET and the chord I'm grappling toward is just [0, 300, 600, 900] cents. But I like the chords with a third degree of 588 to 590 cents more than the old one with a third degree of 610 cents, so that vaguely suggests that I'm ...not complete garbage. After listening repeatedly, I also think that the fourth interval is better around 882 to 884 than around 904 to 906.

So if this is going to be spelled correctly by thirds, then the dim7 chord has to be [P1, Grm3, GrGrd5, GrGrGrd7]. It's weird that the 7th is such a complicated interval just to get us to something perceptually indistinguishable from a 5-limit major sixth.

This dim7 chord is made up of three Grm3 intervals though, so that's nice and regular. And it's also not 12-TET, which is good. I'd once heard someone say that it never sounds good when you stack two identical intervals in 5-limit just intonation, and I'm here to report that it does so. You can stack an interval three times and get a lovely dim7 chord. Although the Grm3 is actually 3-limit, so maybe that's why it works. It's just the Pyjthagorean m3, and three of them make a Pythagorean d7.

Alternatively, maybe I found a chord spelled by thirds that looks like a dim7, but since I don't know the sound of 12-TET dim7 all that well, I'm perhaps fooling myself, and I've found a very nice sounding some-other-chord that happens to be spelled like a dim7. Like, maybe the top interval is a M6 and I've reconstructed some permutation of a different chord. You know how there are like four ways to interpret every dim7 chord in 12-TET? Maybe I've got ...a dim7 chord but it's permuted.

If I interpret the GrGrGrd7 as a M6 and drop it an octave so it becomes the root, then the new intervals relative to the root are: 

    [0, 316, 610, 904] # [P1, m3, Grd5, GrGrd7] 

I have to admit that this sounds different but also good. Not quite as good, but definitely pretty good. This one has relative intervals of [m3, Grm3, Grm3].

It's entirely possible that different dim7 chords sound good rooted on different scale degrees relative to the tonic. I should probably try alternating a major chord against different dim7 chords with lots of different root pitches to examine that.

My first impression is that the [m3, Grm3, Grm3] sounds better rooted a M6 over the tonic and the [Grm3, Grm3, Grm3] sounds better rooted on the tonic or a M7 over the tonic. But I haven't actually tested dozens of chords at all three of those positions to be sure. I wish I had a tunable keyboard so I could iterate this stuff more rapidly.

...

Okay, I'm going to try ten different dim7 variants that are all spelled correctly / made of some kind of minor thirds. Rooted on the m2, the best sounding 5-limit dim7 variants, when played alternately against  a 5-limit major chord, are these (defined by relative/adjacent intervals):

[Grm3, Grm3, Grm3] // best

[Grm3, Grm3, m3] // decent

[Grm3, m3, Grm3] // just okay

.

For M2, I thought the best variants were

[Grm3, Grm3, Grm3],

[m3, Grm3, Grm3],

.

For a root on m3, the only one that sounded good was:

    [Grm3, Grm3, Grm3]

and I'm starting to think that's going to work pretty well everywhere.

I've noticed an ambiguity in music theory texts about the use of the dim7 chord. Diminished seventh chords work especially well as insertions between chords whose roots are moving up by a major second, or sometimes up a major third. In either case, the dim7 insertion is rooted a "semitone" below the root of the upper/final/target/postfix chord. But sometimes this "semitone below" is notated as an augmented unison below and sometimes as a minor second below. These are both tuned to one step of 12-EDO, so they're both semitones, but the intervals are tuned differently in other other tuning systems, and it's time we settled which way it should be in systems that distinguish them.

For example, is it 

     (I.maj, IIb.dim7, II.min)

or 

     (I.maj, I#.dim7, II.min)

?

And is it

(IV.maj, Vb.dim7, V.maj)

or

(IV.maj, IV#.dim7, V.maj)

?

My guess is you diminish the upper one, but I'll have a listen and find out. And also, in 5-limit JI, it might be some number of syntonic commas away from A1 or m2.

So! a dim7 rooted A4 over P1 sounds way better than a dim7 rooted a d5 over P1. So, e.g. 

    (F.maj7, F#.dim7, G.7, C.maj)

sound way better than

    (F.maj7, Gb.dim7, G.7, C.maj)

.

And A4 is an Acm2 below P5 (the tone we're approaching relative to P1).

Also, GrA4 and AcA4 and Acd5 sound bad. But Grd5 actually sounds good too, alongside A4! 

    (F.maj7, Gb-.dim7, G.7, C.maj)

And that's obviously an Ac1 below P5. I'm not yet sure which one I like more. A friend says that the A4 one is less jarring, but the Grd5 one might have a more satisfying dissonance and resolution thing going on. I think I agree, but let's have a look at the two progressions I liked:

[498, 884, 1200, 1586]: F.maj7

[569, 884, 1178, 1473]: F#.dim7

[702, 1088, 1382, 1698]: G.7

[0, 386, 702]: C.maj

versus

[498, 884, 1200, 1586]: F.maj7

[610, 925, 1220, 1514]: Gb-.dim7

[702, 1088, 1382, 1698]: G.7

[0, 386, 702]: C.maj

.

The numbers are cents over C natural. You can see that in the first progression, the F#.dim7 share its second atone with the F.maj7. In the second chord progression, the 1200 cents to 1220 cent melodic jump is close enough to be an interesting near equivalence.

I want to examine the melodic voice-leading intervals in those chord progressions now! Maybe that's part of the key to figuring out rank-3 chords. 

...

Maybe there's a thing where melodic steps of these sizes: [0, 20, 22, 41, 71, 73, 92, 112, 114, 133, 163, 184, 204, 225] in cents (which showed up in the voice leading of two good dim7 progressions, rooted on A4 and Grd5) are mostly okay, and melodic steps including some number of these melodic steps: [30, 49, 51, 63, 84, 120, 141, 155, 247] in cents (which showed up in voice leading of bad progressions where the dim7 was rooted on d5 or GrA4 or Acd5) are mostly bad. If I had to guess, I'd say that my ear is protesting against 24-EDO quarter tones, i.e. intervals of size (n*100 + 50) cents (for {n} an integer) and that the [30, 63, 84, 120, 141] intervals are less grating than the [49, 51, 155, 247] intervals. Although I really like middle eastern music with neutral tones / quarter tones. But that's almost exclusively not polyphonic, so who knows. Maybe the voice leading intervals don't matter at all. Maybe the principle is something other than quarter-tone proximity.

On further review, a dim7 on AcA4 over P1 is also decent in sound. The voice-leading intervals also check out (as coming from the same set as A4 and Grd5). After one more listen through, I'll stand by the claim that a dim7 chord rooted on A4 is better than on Grd5, which is better than rooting it on AcA4, but they're all decent in a (F.maj7, <?>, G.7) progression.