Septimal Harmony

Some microtonal composers with perfect pitch will just tune chords to minimize beating and then know the just intervals names for the frequency ratios they've tuned by hand. 

I can't do that, but I'd still like to compose microtonal music with beautiful harmony. I probably should have tried using a computer model of acoustic beating, and I might do that next, but first I just listened to a bunch of tertian seventh chords and rated them "good", "okay", "bad". I recognize that all sorts of chords which sound bad in isolation can sound good through voice leading - there's definitely technical machinery for preparation and resolution of dissonance - but for I'm going to start by looking at consonance in isolation. Here are the first set of ratings I came up with:

[P1, Sbm3, P5, Sbm7], # 0. Good
[P1, Grm3, P5, Sbm7], # 1. Good
[P1, m3, P5, Sbm7], # 2. Good, but worse than previous
[P1, M3, P5, Sbm7], # 3. Good
[P1, AcM3, P5, Sbm7], # 4. Good, but a little worse than previous
[P1, Sbm3, P5, Grm7], # 5. Bad
[P1, Grm3, P5, Grm7], # 6. Okay
[P1, m3, P5, Grm7], # 7. Okay
[P1, M3, P5, Grm7], # 8. Good
[P1, AcM3, P5, Grm7], # 9. Good, but worse than previous
[P1, Sbm3, P5, m7], # 10. Bad
[P1, Grm3, P5, m7], # 11. Bad
[P1, m3, P5, m7], # 12. Good
[P1, M3, P5, m7], # 13. Okay
[P1, AcM3, P5, m7], # 14. Okay, but worse than previous
[P1, Sbm3, P5, Acm7], # 15. Bad
[P1, Grm3, P5, Acm7], # 16. Bad
[P1, m3, P5, Acm7], # 17. Okay
[P1, M3, P5, Acm7], # 18. Okay, but better than previous
[P1, AcM3, P5, Acm7], # 19. Bad  or Okay?
[P1, Sbm3, P5, GrM7], # 20. Bad
[P1, Grm3, P5, GrM7], # 21. Bad
[P1, m3, P5, GrM7], # 22. Bad
[P1, M3, P5, GrM7], # 23. Good // Really
[P1, AcM3, P5, GrM7], # 24. Okay 
[P1, Sbm3, P5, M7], # 25. Okay
[P1, Grm3, P5, M7], # 26. Bad
[P1, m3, P5, M7], # 27. Bad
[P1, M3, P5, M7], # 28. Okay
[P1, AcM3, P5, M7], # 29. Okay, but a little worse than previous
[P1, Sbm3, P5, AcM7], # 30. Bad
[P1, Grm3, P5, AcM7], # 31. Bad
[P1, m3, P5, AcM7], # 32. Bad
[P1, M3, P5, AcM7], # 33. Okay
[P1, AcM3, P5, AcM7], # 34. Okay

.

Next, I tried looking for patterns to explain/predict/compress my judgements. A few of the rating might have changed up or down a point on subsequent re-listenings as I investigated trends, but mostly those ratings were pretty stable for me. Here's what I came up with for explanatory trends:

First I looked at the relative-intervals between chord tones, instead of the absolute intervals in relation to the tonic, and rounded the just tunings to the nearest 20 cents. When I did that, I saw that chords with 400c or 440c as stepwise intervals almost always sound bad. I think this fact mostly boils down to the principle that one should not use anything but 5-limit m3 and 5-limit M3 for the second chord tone.

These two chords

[P1, Sbm3, P5, Sbm7], # Good

[P1, Grm3, P5, Sbm7], # Good

are exceptions to that rule and I don't understand them. Maybe the Sbm7 just sounds so good that it masks the badness of the large gap between the flat third and the pure fifth. Or maybe the 3rd and the 7th both being septimal has some merit and the Grm3 is approximating a Sbm3.

Within chords that don't have relative step intervals larger >= 400 cents, there's a trend that chords sound better with flatter 7ths (closer to the harmonic Sbm7) and sound worse with increasing size of the 7th.

This trend holds absolutely for chords with m3.

[P1, m3, Sbm7], # 2. Good
[P1, m3, Grm7], # 7. Good
[P1, m3, m7], # 12. Okay
[P1, m3, Acm7], # 17. Bad
[P1, m3, GrM7], # 22. Bad
[P1, m3, M7], # 27. Bad

Looking at only chords with M3, the trends is more bimodal:

[P1, M3, Sbm7], # 3. Good
[P1, M3, Grm7], # 8. Okay
[P1, M3, m7], # 13. Bad
[P1, M3, Acm7], # 18. Okay
[P1, M3, GrM7], # 23. Good
[P1, M3, M7], # 28. Good

So maybe Sbm7 and M7 are both good-in-principle when used with M3, and the Grm7 and GrM7 are acceptable approximations to those.

Now let's look at trends within the chords which have relative-intervals tuned to 400c or larger, which were mostly bad sounding.

The only good chords with 400c or 440c as rounded-tuned-relative steps are:

    [400, 300, 300] : [P1, AcM3, P5, Grm7], # 9. Good

    [400, 300, 380] : [P1, AcM3, P5, M7], # 29. Good

    [400, 300, 260] : [P1, AcM3, P5, Sbm7], # 4. Good

Which all have intervals of 400 and 300 in their first two members, because they all have the Pythagorean [P1, AcM3, P5] as their base triad.

In chords with a relative-step interval of 400 or 440, any chord with a M7 sounds good or okay. More likely just okay. The version of such a chord instead using AcM7 will be a little worse, but otherwise similarly good.

If we use Sbm3 or Grm3 for the second chord tone, the chord will generally sound bad, but they both sound okay when paired with M7 or Sbm7.

...

I looked for correspondences between the otonal and utonal spellings of the chords against my ratings of their consonance, and didn't find anything interesting. Just documenting that I did the fairly obvious thing.

...

Ben Johnston's Tricks

I really like the microtonal musical works of Ben Johnston. I'd like to make music that good. I don't have a lot of understanding of how he did his stuff, but I want to learn. In this post, I'll talk about some things he did.

He had a scale that was all made of octave-reduced overtones.

Here's how it looks when sorted by numerator:

P8: 2/1
P5: 3/2
M3: 5/4
m7: 7/4
M2: 9/8
A4: 11/8
m6: 13/8
M7: 15/8
A1: 17/16
m3: 19/16
P4: 21/16
M6: 27/16

And here it is by frequency ratio size, more compactly:

    [1/1, 17/16, 9/8, 19/16, 5/4, 21/16, 11/8, 3/2, 13/8, 27/16, 7/4, 15/8, 2/1]

Start yourself out with a scale like that and see how the sound of it change how you compose. That was one of Ben's tricks. He also used a scale made of the octave-complements of that scale, namely:

    [2/1, 32/17, 16/9, 32/19, 8/5, 32/21, 16/11, 4/3, 16/13, 32/27, 8/7, 16/15, 1/1]

Here's another one of his tricks: Take a bunch of super-particular ratios that multiplied together make an octave:

    [16/15 * 15/14 * 14/13 * 13/12 * 12/11 * 11/10 * 10/9 * 9/8] = 2/1

Now here's Ben's genius: do a cyclic permutation of the scale in thsi way:

    [12/11, 11/10, 10/9, 9/8, 16/15, 15/14, 14/13, 13/12]

And now when we accumulate these ratios multiplicatively, we get this scale:

    [1/1, 12/11, 6/5, 4/3, 3/2, 8/5, 12/7, 24/13, 2/1]

which has a justly tuned m3, P4, P5, and m6. It only has one (neutral) second, and one (minor) third, so it doesn't support chromaticism at the low end as well as the previous scale did, but you can still make some cool music out of it. I believe he called this scale "Eu15".

Ben also used utonality in addition to otonality, which is no big surprise since he was a student of Harry Partch. I should include some examples here.

Or, let's just explain the concepts first. You can define a chord like [4:5:6:7] as a short hand for (4/4, 5/4, 6/4, 7/4). Just divide everything through by the first element. Overtones, kind of. Divided by four, in this case. Otonality. Nice.

This chord is of course better known as

        (1/1, 5/4, 3/2, 7/4). 

a major chord with a 5-limit just third and a harmonic seventh. The relative steps of the chord are (5/4, 6/5, 7/6).

Undertones are not produced by harmonic instruments the way that overtones are, but people still make interesting music by inverting overtone chords. If we write descending integers, [7:6:5:4], that's shorthand for (7/7, 7/6, 7/5, 7/4), with relative intervals (7/6, 6/5, 5/4). We can represent that same chord with ascending integers by inverting all the fractions from the otonal chord

    (1/1, 4/5, 2/3, 4/7)

then reversing the order, and then multiplying through by the least common multiple of the denominators, which happens to be 105 for (7, 3, 5, 1):

     (4/7, 2/3, 4/5, 1/1) * 105 = [60:70:84:105]

So that's an ascending way of representing the otonal inverse of [4:5:6:7], but it sure hides all of the structure, so [7:6:5:4] is probably the better thing to use. 

When I say that Ben Johnston used utonality in addition to otonality, I mostly mean that he would play a chord like

    (1/1, 5/4, 3/2, 7/4)

and soon after also play a chord like

    (1/1, 7/6, 7/5, 7/4)

not necessarily on the same root. When you add a note to an otonal chord, you get a totally different utonal chord. For example, if we didn't have the harmonic seventh but just a plain 5-limit major chord, [4:5:6], then the utonal inverse would be (6/6, 6/5, 6/4), i.e.

    (1/1, 6/5, 3/2)

a just minor triad, which is nothing at all like the utonal chord we just saw with 7s in the numerators. But if you don't know how to use factors of 7 or 11 or whatever, you've got to start somewhere, and there are worse things to do than to make a scale with frequency ratios of the form {11/n}, for different values of {n} and noodle around in there.

Since Ben Johnston had overtone scales and undertone scales, I think it's likely he would just go down the undertone scale when he wanted undertone harmony, and go up overtone scale when he wanted overtone harmony. An easy recipe for cool music without having to think about the frequency ratios too much: you find what sounds good and then you just have to figure out post hoc how to notate it; Up from here, down from here, temporary tonicizations everywhere, all over frequency space, never fixed to a P1 of 440 hz or anything like that. That's my guess. For some of his works. He had different tricks for lots of different compositions.

When moving between two chords, Ben might have used voice leading based on super-particular ratios. Like it's okay to go from a D at 10/9 to a D with a bunch of weird high-prime accidentals at 260/243 for example, because they're related by (27/26). Or a weirdly inflected B at (140/81) to a weirdly inflected A at (400/243) is fine because they're related by 21/20. I'm saying he "might" have done this because super particular show up everywhere when you move by small amounts between just frequency ratios. You don't have to try very hard to find them. And all of his prime accidentals are tuned to super-particular frequency ratios, so anytime you move by a comma, like from like a B to a B-, you're getting free super-particularity. Or, like, even just looking at 5-limit frequency ratios, is it supposed to be impressive that a composer moved melodically by (Ac1, A1, m2, M2, AcM2, m3, M3, P4, or P5), which are all super particular? Good luck avoiding it, even while moving between crazy chords spaces. Between C utonal scale and C otonal scale, if you just move to the same letter name pitch or an adjacent letter name pitch, you have about a 1/2 chance of moving by a super-particular ratio. It's possible that Ben chose super-particular voice leading a lot more than that 1/2 chance. I haven't analyzed his pieces enough to know. It's a possibility and people have claimed it about his work.

...

If I wanted to make a 7-limit scale using overtones and undertones, I'd choose something that looked like this:

P1 : 1/1            P8 :  2/1
P5 : 3/2            P4 :  4/3
M3 :  5/4           m6 :  8/5
Sbm7 :  7/4         SpM2 :  8/7
AcM2 :  9/8         Grm7 :  16/9
M7 :  15/8          m2 :  16/15
SbAc4 :  21/16      SpGr5 :  32/21
A5 :  25/16         d4 :  32/25
AcM6 :  27/16       Grm3 :  32/27
AcA4 :  45/32       Grd5 :  64/45
SbAc8 :  63/32      SpGr1 :  64/63
AcA2 :  75/64       Grd7 : 128/75
AcM3 :  81/64       Grm6 : 128/81
A7 :  125/64        d2 : 128/125

It has 28 intervals. I bet you could do some cool things with it. If you think it's too limited, keep extending it down. If you think it's too expansive, crop it midway. Try composing harmony that only uses one side or the other. Or don't. I'm not the boss of you.

...

Dastgahs

The Persian/Iranian microtonal musical modes are called dastgahs. I was writing about them at length on microtonaltheory.com, but it got to be embarrassing how many sources I found that disagreed with each other about what the scales look like. I'm going to try figuring it all out and cleaning it all up here and then I'll repost an abbreviated form there.

There are only a few dastgahs, so this shouldn't be hard to codify. They are called:

    Shūr
    Māhūr
    Homāyūn
    Chahārgāh
    Segāh
    Navā
    Rāst-panjgāh
    Bayāt-i Iṣfahān
    Dashtī
    Abū ‛aṭā’
    Bayāt-i Tork
    Afshārī
    Bayāt-i Kord

They also have a concept of a scale derived from a dastgah, an "avaz", but I don't think they're really treated any different musically, so I'll just call them all dastgahs.

Supposedly a man named Ali-Naqi Vazir gave 24-EDO descriptions of the scales way back in 1913. I haven't found that. But I have found 24-EDO tetrachords / scale fragments attributed to him. There are only four tetrachords. Each tetrachord is called a "dang" and the dang-s are named after dastgahs. Below I show the name of a dang, the size of the relative intervals in cents, the size in steps of 24-EDO, and pitch classes rooted on C, using "d" as an accidental for half flats (and "t" as an accidental for half sharps, if that had come up).

Shur: [150, 150, 200] : [3, 3, 4] :: [C, Dd, Eb, F]
Chahargah: [150, 250, 100]: [3, 5, 2]  :: [C, Dd, E, F]
Dashti: [200, 100, 200] : [4, 2, 4]  :: [C, D, Eb, F]
Mahur: [200, 200, 100] : [4, 4, 2]  :: [C, D, E, F]

Here are descriptions of some scales in 24-EDO steps, described with absolute steps and then relative steps on each line:

    Shur/Nava : [0, 3, 6, 10, 14, 16, 20, 24] : [3, 3, 4, 4, 2, 4, 4]
    Homayun : [0, 3, 8, 10, 14, 16, 20, 24] : [3, 5, 2, 4, 2, 4, 4]
    Mahur/Rastpanjgah : [0, 4, 8, 10, 14, 18, 22, 24] : [4, 4, 2, 4, 4, 4, 2]
    Segah : [0, 3, 6, 10, 13, 16, 20, 24] : [3, 3, 4, 3, 3, 4, 4]
    Chahargah : [0, 3, 8, 10, 14, 17, 22, 24] : [3, 5, 2, 4, 3, 5, 2]

These are derived from "Transcultural Music" by Alireza Ostovar. Using Vaziri's tetrachords, we can see how the dang-s combine into these dastgahs:

Shur/Nava: Shur + Dashti + T.

Homayun: Chahargah + Dashti + T.

Mahur/Rastpanjgah: Mahur + T + Mahur.

Segah: Shur + Shur + T.

Chahargah: Chahargah + T + Chahargah.

Ostovara didn't give a 24-EDO analysis of dastgah Esfahan, but from other sources, it's clearly a cyclic permutation of dastgah Homayun, and we can give it as

Esfahan : Dashti + T + Chahargah.

or as

Esfahan : [0, 4, 6, 10, 14, 17, 22, 24] :: [4, 2, 4, 4, 3, 5, 2]

Which dastgahs are left to characterize in 24-EDO? Dashti, Abu 'Aṭa, Bayat-i Tork, Afshari, and Bayat-i Kord. We're told by Kees van den Doel of persianney.com that all of these have the same intervals as Shur, except that he doesn't mention Bayat-i Kord. So maybe we only have one left. But Darabi, Azimi, and Nojumi say that Bayat-i Kord shares its intervals with Shur as well, so everything sure is Shur.

Are we done? Not at all. Every source disagrees with every other source and I won't feel satisfied till I have some kind of framework for understanding what they're all smoking. So this is going to be our baseline and we'll try to align the works of others with it as much as we can, to figure out when and how they're deviating. Kees gives us precise intonation for the persian accidentals, which I shall render as:

    p = koron (60 cent flat)

    > = sori (40 cent sharp)

I like that they're not equal in magnitude but that they do sum to a 12-EDO A1 or m2.

Mahur is just the C major scale without any microtones, 

One deviation from the above that I suspect is true-to-practice is that Dastgahs normally just fill one octave, and below or above that octave you might have weird different notes as ornaments that you wouldn't expect from octave-repetition of the scale. So a dastgah might have both a B natural within its normal octave and a Bb outside the normal octave. Or something like that.

I believe there are also traditional variations within the octave, like if you'd play a B quarter flat in an ascending melodic line, there might be some good chance that you'd play it as a a B half-flat in a descending melodic line, for example. There are standardized deviations from the base scale.

We'll get to both kinds of ornaments in time, but first I just want to look at cases where sources seem to deviate totally on what the base scales are. We'll also look at sources on precise intonation of these scales, even though if you measure the frequency ratios of Persian musicians, there isn't very precise agreement.

...

I think the source that I found most regular and simple after the 24-EDo stuff was Kees van den Doel at persianney.com. Let's compare his stuff to the 24-EDO scales of Ostovar. Here's Mahur in terms of absolute and relative intervals:

    Mahur: [P1, M2, M3, P4, P5, M6, M7, P8] :: [M2, M2, m2] + M2 + [M2, M2, m2] // Also Rast-panjgah.

which matches the 24-EDO scale of Ostovar perfectly. 

    Mahur/Rastpanjgah : [0, 4, 8, 10, 14, 18, 22, 24] :: [4, 4, 2] + 4 + [4, 4, 2]

I'm not sure whether Persian scales are associated with definite tonics, as they are in Arabic music, such that a transposed scale would have a new name. Kees suggests that this isn't the case, and gives the dastgahs in multiple keys, which was great because the multiple keys mostly agreed with each other and this was good confirmation of what he thought the scales really were, without typos.

Here's a 2.3.7 justintervallic analysis of Kees's pitch classes for Homayoun:

    Homayoun: [P1, SbM2, M3, P4, P5, m6, m7, P8] :: [SbM2, SpM2, m2, M2, m2, M2, M2]

this is also consistent with the 24-EDO version from Ostovar,

    Homayun : [0, 3, 8, 10, 14, 16, 20, 24] :: [3, 5, 2] + 4 + [2, 4, 4]

If you can't see it, just look at the relative intervals after the double colon, and think of m2 as 2 steps of 24 edo, M2 as 4 steps, a sub-major second or super-minor second as 3 steps, and a super-major second or sub-minor third as 5 steps. The "Sub" flattens by a step and the "Super" raises by a step. The 5-step interval is, I think, fairly characteristically Persian. Not something you see in Arabic or Turkish intonation. I think. Still figuring this out.

Here's Esfahan from Kees:

    Esfahan: [P1, M2, m3, P4, P5, SbM6, M7, P8] :: [M2, m2, M2] + M2 + [SbM2, SpM2, m2]

which again has perfect agreement with the 24-EDO version from Ostovar:

    Esfahan : [0, 4, 6, 10, 14, 17, 22, 24] :: [4, 2, 4, 4, 3, 5, 2]

This is going swimmingly. Here's Chahargah from Kees:

    Chahargah: [P1, SbM2, M3, P4, P5, SbM6, M7, P8] : [SbM2, SpM2, m2, M2, SbM2, SpM2, m2]

Which perfectly matches the 24-EDO version from Ostovar:

    Chahargah : [0, 3, 8, 10, 14, 17, 22, 24] : [3, 5, 2] + 4 + [3, 5, 2]

I knew I liked this guy for a reason.

But! Kees's dastgah Segah looks like this, in a 2.3.7 just interval analysis:

    Segah: [P1, Spm2, Spm3, P4, Spd5, Spm6, Spm7, P8] : [Spm2, M2, SbM2, Spm2, M2, M2, SbM2]

This at first looks starkly different from the 24-EDO version of Ostovar. If we tune Kees's Segah in 24-EDO, we get these for absolute and relative steps:

    Segah : [0, 3, 7, 10, 13, 17, 21, 24] :: [3, 4, 3, 3, 4, 4, 3]

In comparison, here's the version in 24-EDO from Ostovar: 

    Segah : [0, 3, 6, 10, 13, 16, 20, 24] :: [3, 3, 4, 3, 3, 4, 4]

We can now see they're cyclic permutations of each other! Look at the relative intervals in Kees' version of Segah, and shift the last 3 to the start. Now they're equal. I'm tempted to prefer Ostovar's version, since it can be constructed from Vaziri's tetrachordal dang-s. Although maybe there's a way that both can be right. Maybe the tonic center of the Shur dang isn't at the bottom of the tetrachord. Then Kees could be right about where the tonic of the scale is and Ostovar could be right in presenting the scales such that the underlying tetrachord structure is clear and contiguous.

Kees's dastgah Shur can be rendered in 2.3.7 just intervals as:

    Shur: [P1, SbM2, m3, P4, P5, m6, m7, P8] :: [SbM2, Spm2, M2] + [M2, m2, M2] + M2 //  [Shur + Dashti + T]

or

    Shur: [P1, SbM2, m3, P4, Sb5, m6, m7, P8] :: [SbM2, Spm2, M2] + [SbM2, Spm2, M2] + M2  //  [Shur + Shur + T]

According to Kees, dastgah Shur has an optional half flat on the 5th scale degree. How peculiar not not hit P5! To be clear, we're describing the Shur dastgah (scale) in terms of components dang-s (tetrachords), which include a dang called Shur.

Here's the dastgah Shur of Ostovar for comparison: 

    Shur/Nava : [0, 3, 6, 10, 14, 16, 20, 24] :: [3, 3, 4] + [4, 2, 4] + [4]

Which is the [Shur + Dashti + T] form, not the [Shur + Shur + T] form. This second form is actually Ostovar's Segah. So Shur with a half-flat 5th is a lot like Segah.

Remember how almost everything sure looks like Shur? It's a little confusing that Kees has two forms for dastgah Shur, because that also introduces uncertainty about (Nava, Dashti, Abu 'Aṭa, Bayat-i Tork, Afshari, and Bayat-i Kord), which supposedly have the same pitch classes as dastgah Shur. Is a septimal sub perfect fifth an option for all of them? I dunno. If it were, then most of the Persian scales, in addition to basically being Shur, would also be basically Segah. I wonder why they have so many names if they're all the same. They're probably not all the same.

Looking at lots of other sources that all have their own characteristic notations, ornaments, and perhaps typos, if I had to reconstruct the correct spelling of Segah from the ?misspellings, I'd write it as

    [Ed, F, G, Ad, Bb, C, D, Ed]

This form was given exactly by Ella Zonis, and was given like this but with quarter-flats instead of half-flats by Navid Goldrick, and wikipedia relates this same form but with an option of a half flat on the D (the 7th scale degree) and attributes this to Mirza Abdollah.

Some other sources are quite different. I kind of glossed over that Kees Segah is ...I presented what I thought was a cleaned up version of his data, which wasn't self consistent.

He gives

First position: [Ed F G Ad Bb(d) C D Eb]

Second position: [Bd C D Ed Fb(t) G A Bd]

but these aren't equivalent through transposition. Like the first position doesn't reach an octave and the optional accidental on the fifth scale degree changes in a way that it shouldn't. But if we ignore Kees's ornametn on the 5th scale degree and Mirza's ornament on the seventh scale degree, and replace Navid's quarter-flats with half-flats, then it seems everyone things Segah is something like:

        [Ed, F, G, Ad, Bb, C, D, Ed]

But this is the version of Segah I originally presented with Kees, not the cyclic permuted version that agrees with Ostovar! So it seems like multiple sources are disagreeing with Ostovar. Or maybe that the Shur dang doesn't have its tonal center at its bottom, but then that should also change the form of other dastgahs so that other sources disagree with Ostovar.

I think Navid's notation of a quarter flat E for the root of Segah is quite interesting. Using the intonation of Kees, the koron accidental is 60 cents flat (relative to ... Pythagorean, 12-EDO, meantone, whatever - no one specified), not 25 cents. And 35 cents difference between musicians is kind of a lot.

...

Rank-3 EDO Distinction and Diatonicity

I've written at length in the past about defining EDOs in terms of pure octaves and a tempered rank-2 interval, and I've given conditions for these to be diatonic in the sense of ordering the natural (rank-2) intervals in the natural way:

    [P1, m2, M2, m3, M3, P4, P5, m6, M6, m7, M7, P8]

I've also argued at some length that rank-3 intervals and higher should be justly tuned so that the natural impure intervals 

    [m2, M2, m3, M3, m6, M6, m7, M7]

have 5-limit frequency ratios, rather than 3-limit (Pythagorean) frequency ratios.

The obvious next step is to give conditions for when an EDO is diatonic over rank-3 intervals, in the sense of putting the rank-3 natural intervals in the same usual order that we know and love from 12-TET.

Surprisingly, there's only one EDO that badly violates rank-3 diatonicity. When we tune 

    [P1, m2, M2, m3, M3, P4, P5, m6, M6, m7, M7, P8]

in 11-EDO we get these steps:

    [0, 1, 3, 2, 4, 5, 6, 7, 9, 8, 10, 11]

in which (3 comes before 2) and (9 comes before 8). All of the other EDOs between 5-EDO and 500-EDO are non-decreasing from left to right in their tunings of the natural rank-3 intervals.

Also a little surprising to me is that most EDOs tune the natural rank-3 intervals to distinct steps. Obviously everything below 11 has too few steps to put all 12 natural intervals on differing steps, so those EDOs are indistinct over rank-3 natural intervals: 

 

    5-EDO: [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5]

    6-EDO: [0, 0, 0, 2, 2, 2, 4, 4, 4, 6, 6, 6]

    7-EDO: [0, 1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 7]

    8-EDO: [0, 0, 1, 2, 3, 3, 5, 5, 6, 7, 8, 8]

    9-EDO: [0, 1, 2, 2, 3, 4, 5, 6, 7, 7, 8, 9]

    10-EDO: [0, 1, 1, 3, 3, 4, 6, 7, 7, 9, 9, 10]

The only other indistinct EDOs are:

    13-EDO: [0, 1, 1, 4, 4, 5, 8, 9, 9, 12, 12, 13]

    14-EDO: [0, 1, 3, 3, 5, 6, 8, 9, 11, 11, 13, 14]

    17-EDO: [0, 2, 2, 5, 5, 7, 10, 12, 12, 15, 15, 17]

    20-EDO: [0, 2, 2, 6, 6, 8, 12, 14, 14, 18, 18, 20]

Seeing that 17-EDO tunes the rank-3 minor second and major second together, as well as the minor third with the major third, was a bit of a surprise. With rank-2 intervals, 17-EDO is very well behaved, giving us these steps for the natural intervals:

    17: [0, 1, 3, 4, 6, 7, 10, 11, 13, 14, 16, 17]

So here's a condition for rank-3 lax diatonicity: "Don't be 11-EDO".

And here's a condition for rank-3 distinction, which we might require for "strict diatonicity": "Be larger than 20-EDO or find yourself in the set (12, 15, 16, 18, 19)".

I can't help but feel that the rank-2 conditions were a lot harder to figure out, a lot mathier. Still, I'm happy to have made some progress.

Nothing I've said ensures that these well behaved EDOs don't collapse to a smaller EDO: for example, 24-EDO only tunes rank-3 intervals to even steps, equivalent to 12-EDO. I'm still figuring out the conditions for ... full occupancy of the EDO at a given rank.

Maybe I could figure out some rule that lets us see why [13, 14, 17, 20]-EDO are indistinct next.

It the EDO tunes the rank-3 A1 to 0 steps, then the scale is indistinct, but that's not necessary for it to be indistinct. If the EDO tunes the rank-3 d2 to -1 steps, then the scale is indistinct, but again, not necessary. If AcA1 is tuned to zero steps and d2 is tuned to zero steps, then it's indistinct. That literally only described 9-EDO but it's the last edge case. Man, I don't know.

...

I propose that these guys:

    [9, 11, 14, 16, 21, 23, 28, 33, 35, 40, 47, 52, 64]-EDO

are secretly not so well behaved over rank-3 intervals. Those are the EDOs which tune the Acute Unison to a negative number of steps. Like a bunch of a noobs.

...

Let's talk about tuned orders of the once-modified rank-3 intervals that are induced by different EDOs.

There are very few EDOs that tune the augmented unison to zero steps: the full set between 5 and 100 is [6, 7, 10, 13, 17, 20]-EDO.

6-EDO, over the rank-3 intervals, reduced to 3 edo, and has this order:

    [A1=A2=M2=P1=d2=m2, A3=A4=M3=P4=d3=d4=m3, A5=A6=M6=P5=d5=d6=m6, A7=M7=P8=d7=d8=m7]

The other members of the set all have this order: 

    [A1=P1, A2=M2=d2=m2, A3=M3=d3=m3, A4=P4=d4, A5=P5=d5, A6=M6=d6=m6, A7=M7=d7=m7, P8=d8]

Within an equivalence set connected by "="s, I have things sorted alphabetically, but it might be easier to see that e.g.

    dd2 = d2 = m2 = M2 = A2 = AA2

The number of augmentations and diminutions doesn't affect the tuning of that interval since A1 is tempered out.

From 5-EDO up to 100-EDO, there are 22 different orders of the intervals [d2, P1, m2, A1, d3, M2, m3, A2, d4, M3, P4, A3, d5, A4, d6, P5, m6, A5, d7, M6, m7, A6, d8, M7, P8, A7], which I'll call the once modified intervals, although we could also modify intervals with acuteness and gravity, instead of just augmentation and diminution. Most of these are slightly degenerate orders in which two intervals are tuned to the same step. The only intervals that are not at all degenerate over the once-modified:

    1. [P1, A1, d2, m2, M2, A2, d3, m3, M3, A3, d4, P4, A4, d5, P5, A5, d6, m6, M6, A6, d7, m7, M7, A7, d8, P8]

    2. [P1, d2, A1, m2, M2, A2, d3, m3, M3, d4, A3, P4, A4, d5, P5, d6, A5, m6, M6, A6, d7, m7, M7, d8, A7, P8]

    3. [P1, d2, A1, m2, M2, d3, A2, m3, M3, d4, A3, P4, A4, d5, P5, d6, A5, m6, M6, d7, A6, m7, M7, d8, A7, P8]

...

Order 1 is shared by [26, 29, 32, 45, 48, 51, 54]-EDO.

Order 2 is shared by [31, 43, 46, 47, 50, 55, 58, 61, 62, 65, 67, 69, 70, 73, 74, 77, 80, 81, 84, 86, 88, 89, 92, 93, 95, 96, 98, 99, 100]-EDO.

Order 3 is shared by [37, 56, 59, 71, 75, 78, 90, 94, 97]-EDO.

Now, Order 1 is the same as what I called tetracot ordering in the post on rank-2 orderings of once modified intervals. Order 2 is the meantone ordering. And order 3 is new, but it's really really close to the meantone ordering. It only swaps (d3 with A2) and (d7 with A6).

All three of these orders are strict in the sense of being ordered by ">". If we reinterpret them as being laxly ordered by ">=", what other EDOs can we describe as falling into these orders, and are there any other orders we'll need to describe the EDOs that are degenerate in the tuned orders they induce over rank-3 once modified intervals?

There's a great degenerate order I've notice

[P1, A1=d2, m2, M2, A2, d3, m3, M3, A3=d4, P4, A4, d5, P5, A5=d6, m6, M6, A6, d7, m7, M7, A7=d8, P8]

that's shared by [22, 25, 41, 44, 60, 63, 66, 79, 82, 85]-EDO. I think is is consistent with both the lax version of order1 and the lax version of order 2, in that this degenerate order equates all of the pairwise swaps from order 1 to order 2.

So now I'm curious if there's an EDO order that equates (d3 with A2) and (d7 with A6), which would of course be described by both lax order 2 and lax order 3.

Past 12 divisions, no EDO tuned multiple once-modified intervals to the same step.

I did this in kind of lazy way with text editing instead of programming, but these guys with degenerate orders over once modified rank-3 intervals are consistent with lax order 1:

    [19, 22, 25, 38, 41, 44, 57, 60, 63, 66, 76, 79, 82, 85]-EDO

and maybe others as well. It's possible for a list to be consistent with multiple lax orders. These guys are consistent with lax order 2:

    [15, 18, 30]-EDO

at least.

And these guys are consistent with lax order 3 at least:

    [12, 24, 27, 28, 36, 39, 40, 42, 52, 64]

Doing that same cheap textual analysis in a slightly different way also tells me that these guys are consistent with lax order 2:

    [22, 25, 41, 44, 60, 63, 66, 79, 82, 85]-EDO

and these guys are consistent with lax order 3:

    [19, 35, 38, 57, 76]-EDO

.

So these EDOs are both ordered by lax order 1 and lax order 3:

    [19, 38, 57, 76]-EDO

And maybe others should be here and maybe some of these are also lax order 3.

These guys are ordered by lax order 1 and lax order 2, at least:

    [22, 25, 41, 44, 60, 63, 66, 79, 82, 85]

...

Rank-3 Schismatic And Syntonic Temperaments

I was writing music in 5-limit just intonation, and I discovered that the Pythagorean d7 (also called a GrGrGrd7  in 5-limit J.I. and tuned to 32768/19683 in either system) sounds the same as the 5-limit major sixth (tuned to 5/3). They only differ by like two cents. The difference is called the schisma, which is more formally an AcAcA0, justly tuned to 32805/32768. Since I can't hear the difference, I was curious what happens if we temper out the schisma, t(AcAcA0) = 1/1. Presumably, our 3D lattice for Just Intonation reduces by a dimension, and we get a 2D lattice (which might make a nice 2D keyboard layout). 

To define the spacing of the other two dimensions, I had to pick tunings for two more intervals. I went with pure octaves, t(P8) = 2/1, and pure major thirds, t(M3) = 5/4. How do we figure out the tuned values for other rank-3 intervals using those three tuned values?

I don't know how the people on the Xenharmonic wiki do it, but here's how we do it with Cramer's rule.

Let's start with coordinates for some simple rank-3 intervals that we want to tune. I'll represent them in the (P8, P12, M17) basis, a.k.a. the 5-limit prime harmonic basis, which is justly tuned to (2/1, 3/1, 5/1).

P1 = (0, 0, 0) # 1/1

AcAcA0 = (-15, 8, 1) # 32805/32768

Ac1 = (-4, 4, -1) # 81/80

d2 = (7, 0, -3) # 128/125

A1 = (-3, -1, 2) # 25/24

Grm2 = (8, -5, 0) # 256/243

m2 = (4, -1, -1) # 16/15

M2 = (1, -2, 1) # 10/9

AcM2 = (-3, 2, 0) # 9/8

AcA2 = (-6, 1, 2) # 75/64

Grm3 = (5, -3, 0) # 32/27

m3 = (1, 1, -1) # 6/5

M3 = (-2, 0, 1) # 5/4

AcM3 = (-6, 4, 0) # 81/64

d4 = (5, 0, -2) # 32/25

P4 = (2, -1, 0) # 4/3

Ac4 = (-2, 3, -1) # 27/20

A4 = (-1, -2, 2) # 25/18

d5 = (2, 2, -2) # 36/25

Gr5 = (3, -3, 1) # 40/27

P5 = (-1, 1, 0) # 3/2

A5 = (-4, 0, 2) # 25/16

Grm6 = (7, -4, 0) # 128/81

m6 = (3, 0, -1) # 8/5

GrGrGrd7 = (15, -9, 0) # 32768/19683

M6 = (0, -1, 1) # 5/3

AcM6 = (-4, 3, 0) # 27/16

Grd7 = (7, -1, -2) # 128/75

Grm7 = (4, -2, 0) # 16/9

m7 = (0, 2, -1) # 9/5

M7 = (-3, 1, 1) # 15/8

d8 = (4, 1, -2) # 48/25

P8 = (1, 0, 0) # 2/1

After the "#" symbols I show the just tunings for those intervals, but now I want to detune some of them so that we can put 5-limit just intonation on a 2D lattice, and particularly a 2D lattice in which the schisma, which I can't hear, is tempered out.

Okay, so, our new basis is going to be (AcAcA0, P8, M3). In the (P8, P12, M17) basis, this basis is a matrix with coordinates [(-15, 8, 1), (1, 0, 0), (-2, 0, 1)]. This matrix isn't unimodular: it doesn't have determinant 1 or -1, so it can't represent just intonation intervals in integer coordinates. It happens to have determinant 8, and we'll see that the coordinates in this basis have denominators of 8.

To get coordinates (x, y, z) in the (AcAcA0, P8, M3) basis for an interval with coordinates (m, n, o) in the (P8, P12, M17) basis, we're going to use Cramer's rule, just as ejlilley taught us to do for rank-2 intervals.

def rank_3_lilley_cramer_formula(B1, B2, B3, interval):

(m, n, o) = interval

(a, b, c) = B1

(d, e, f) = B2

(g, h, i) = B3

detA = a * (e * i - f * h) - b * (d * i - f * g) + c * (d * h - e * g)

x = (m * (e * i - f * h) - n * (d * i - f * g) + o * (d * h - e * g)) / detA

y = (a * (n * i - o * h) - b * (m * i - o * g) + c * (m * h - n * g)) / detA

z = (a * (e * o - f * n) - b * (d * o - f * m) + c * (d * n - e * m)) / detA

return (x, y, z)

So, for

    B1 = (a, b, c) = AcAcA0 = (-15, 8, 1)

    B2 = (d, e, f) = P8 =  (1, 0, 0)

    B3 = (g, h, i) = M3 =  (-2, 0, 1)

we can see that M6 and GrGrGrd7 have coordinates in  the (AcAcA0 , P1, M3) basis that only differ in the first component:

    M6 = (-1/8, 3/8, 9/8)

    GrGrGrd7 = (-9/8, 3/8, 9/8)

Since our schismatic temperament tunes the AcAcA0 to 1/1, it doesn't matter what coordinates we have for that component, since 1 raised to any real power is going to to be 1. In particular, both of those intervals are tuned to

    (2)^(3/8) * (5/4)^(9/8) ~ 1.666901790204155

in this temperament. You might wonder if this simplifies at all when you expand out the fraction raised to a fractional power. It ends up being 5 * 10^(1/8) / 4. So not really. The 5-limit just value of M6 is 5/3 = 1.6666 repeating 6s, so we're doing well with representing our 5-limit fractions. How about our old 3-limit fractions? Well, this temperament tunes P5 to 

    (2)^(5/8) * (5/4)^(-1/8) = 1.4997884186649115

which is like 0.2 cents off from the just value of 3/2. Pretty bang up job, right? I think I've done a bang up job.

One down side is that a keyboard arranged in P8s and M3s would be hard to play, I think. And also the interval coordinates are fractional, so the grid might be a little slanty or something. But I think we can fix that: pick whatever intervals you think make a nice 2d arrangement for playing then tune them to the values of this great (AcAcA0, P8, M3) -> (1/1, 2/1, 5/4) temperament. Then the frequency ratios will be the same and the grid will be good as well. I think that's an option. You can define the temperament in different ways once you know the induced frequency ratios. You just need two points independent of AcAcA0. Maybe M2 and m2 would make a nice layout for example. To do those, you just need:

t(m2) = 4 * 10^(1/8) / 5

t(M2) = 5 * 10^(1/4) / 8

Here are a bunch of tuned values for simple intervals in this schismatic temperament with pure octaves and pure 5-limit major thirds:

     P1 = (0, 0, 0) # (2)^(0) * (5/4)^(0) = 1/1

Ac1 = (-4, 4, -1) # (2)^(1/2) * (5/4)^(-3/2) ~ 1.0119288512538815

d2 = (7, 0, -3) # (2)^(1) * (5/4)^(-3) = 128/125

A1 = (-3, -1, 2) # (2)^(-5/8) * (5/4)^(17/8) ~ 1.0418136188775968

Grm2 = (8, -5, 0) # (2)^(-1/8) * (5/4)^(5/8) ~ 1.0542412585714556

m2 = (4, -1, -1) # (2)^(3/8) * (5/4)^(-7/8) ~ 1.0668171457306592

M2 = (1, -2, 1) # (2)^(-1/4) * (5/4)^(5/4) ~ 1.1114246312743268

AcM2 = (-3, 2, 0) # (2)^(1/4) * (5/4)^(-1/4) ~ 1.1246826503806981

Grm3 = (5, -3, 0) # (2)^(1/8) * (5/4)^(3/8) ~ 1.1856868528308278

m3 = (1, 1, -1) # (2)^(5/8) * (5/4)^(-9/8) ~ 1.1998307349319293

M3 = (-2, 0, 1) # (2)^(0) * (5/4)^(1) = 5/4

AcM3 = (-6, 4, 0) # (2)^(1/2) * (5/4)^(-1/2) ~ 1.2649110640673518

d4 = (5, 0, -2) # (2)^(1) * (5/4)^(-2) = 32/25

P4 = (2, -1, 0) # (2)^(3/8) * (5/4)^(1/8) ~ 1.333521432163324

Ac4 = (-2, 3, -1) # (2)^(7/8) * (5/4)^(-11/8) ~ 1.3494288109714632

A4 = (-1, -2, 2) # (2)^(-1/4) * (5/4)^(9/4) ~ 1.3892807890929082

d5 = (2, 2, -2) # (2)^(5/4) * (5/4)^(-9/4) ~ 1.4395937924872935

Gr5 = (3, -3, 1) # (2)^(1/8) * (5/4)^(11/8) ~ 1.4821085660385345

P5 = (-1, 1, 0) # (2)^(5/8) * (5/4)^(-1/8) ~ 1.4997884186649115

A5 = (-4, 0, 2) # (2)^(0) * (5/4)^(2) = 25/16

Grm6 = (7, -4, 0) # (2)^(1/2) * (5/4)^(1/2) ~ 1.5811388300841898

m6 = (3, 0, -1) # (2)^(1) * (5/4)^(-1) = 8/5

M6 = (0, -1, 1) # (2)^(3/8) * (5/4)^(9/8) ~ 1.666901790204155

AcM6 = (-4, 3, 0) # (2)^(7/8) * (5/4)^(-3/8) ~ 1.686786013714329

Grm7 = (4, -2, 0) # (2)^(3/4) * (5/4)^(1/4) ~ 1.7782794100389228

m7 = (0, 2, -1) # (2)^(5/4) * (5/4)^(-5/4) ~ 1.799492240609117

M7 = (-3, 1, 1) # (2)^(5/8) * (5/4)^(7/8) ~ 1.8747355233311396

d8 = (4, 1, -2) # (2)^(13/8) * (5/4)^(-17/8) ~ 1.9197291758910868

P8 = (1, 0, 0) # (2)^(1) * (5/4)^(0) = 2/1

I experimented with a bunch of bases/layouts, and I think this one is nice:

Meantone temperaments are named based on how they alter P5. For example, quarter comma meantone lowers the tuned value of the perfect fifth from the just value by a factor of (81/80)^(1/4). The base, 81/80, is the just value for the syntonic comma, and the exponent, 1/4, explains why the system is called quarter comma meantone. Here's a big set of meantone temperament definitions:

    (Ac1, P5, P8), (1, 3/2, 2) # Pythagorean tuning (0-comma meantone)

    t(Ac1, AcA4, P8) = (1, 45/32, 2) # 1/6-comma meantone

    t(Ac1, M3, P8) = (1, 5/4, 2) # 1/4-comma meantone

    t(Ac1, m3, P8) = (1, 6/5, 2) # 1/3-comma meantone 

    t(Ac1, M2, P8) = (1, 10/9, 2) # 1/2-comma meantone

    t(Ac1, m2, P8) = (1, 16/15, 2) # 1/5-comma meantone

    t(Ac1, A1, P8) = (1, 25/24, 2) # 2/7-comma meantone

The schismatic temperament I defined in this post tuned P5 to (2)^(5/8) * (5/4)^(-1/8), which simplifies to 10^(7/8)/5. I didn't want to solve for the exponent, but WolframAlpha assures me that this is flat of (3/2) by the eighth root of the tuned schisma:

    (3/2) / (32805/32768)^(1/8) = 10^(7/8)/5

so I think I've defined the "1/8-comma schismatic temperament", where the comma is now the schisma instead of the acute unison.

Do you want to see what other schismatic temperaments we can define and what they shall be named? I know I do!

If we tune P5 purely, we get Pythagorean tuning again of course. We already know that tuning M3 to a pure 5/4 gives us 1/8-comma schismatic. 

If we tune m3 to a pure 6/5, we get 1/9 comma schismatic, since:

    (6/5)^(1/9) * 2^(5/9)= (3/2) / (32805/32768)^(1/9)

If we tune M2 to a pure (10/9), then we get 1/10-comma schismatic, since

    (10/9)^(-1/10) * 2^(3/5) = (3/2) / (32805/32768)^(1/10)

If we tune m2 to a pure 16/15, then we get 1/7 schismatic, since

    (16/15)^(1/7) * 2^(4/7) = (3/2) / (32805/32768)^(1/7) 

It sure looks like the comma is in the exponent on the left  hand side every time, and it's the exponent of the tuned value of the interval that we're altering between temperaments (not the 2/1 base for the octave). 

So if we define a tuning system by 

    t(AcAcA0, AcA4, P8) = (1, 45/32, 2)

we get a tuned value of  

    (45/32)^(-1/6) * 2^(2/3)

for P5.

I bet we'll find that this is 1/6-comma schismatic. Let's check.

    (45/32)^(-1/6) * 2^(2/3) = (3/2) / (32805/32768)^(1/6)

WolframAlpha says True! Nice.

If we define a schismatic temperament in which we purely tune A1 to 25/24, we get a t(P5) of

    (25/24)^(-1/17) * 2^(10/17)

Do we therefore get 1/17-comma schismatic? Nope! We get 2/17-comma schismatic

     (25/24)^(-1/17) * 2^(10/17) = (3/2) / (32805/32768)^(2/17)

So I guess I don't know the rule. But I'll figure it out. In the meantime, these are still nice names for the temperaments.

I'm going to do a few more. Still tempering out the schisma, still tuning the octave purely, let's just name the last interval that's tuned purely. Here's a big condensed table, arranged by increasing denominators in the fractional comma:

P5 -> 3/2: 0-comma schismatic (Pythagorean tuning)

Grd4 -> 512/405: 1/4-comma schismatic.

AcA4 -> 45/32: 1/6-comma schismatic.

m2 -> 16/15: 1/7-comma schismatic.

M3 -> 5/4: 1/8-comma schismatic. Also d2 -> 128/125. Also d4 -> 32/25.

m3 -> 6/5: 1/9-comma schismatic. Also A2 -> 125/108. Also A4 -> 25/18.

M2 -> 10/9: 1/10-comma schismatic. Also GrM3 -> 100/81.

Ac4 -> 27/20: 1/11-comma schismatic

Ac1 -> 81/80: 1/12-comma schismatic.

Gr4 -> 320/243: 1/13-comma schismatic.

AcA2 -> 75/64: 2/15-comma schismatic.

A1 -> 25/24: 2/17-comma schismatic.

Acm2 -> 27/25: 2/19-comma schismatic.

Acm3 -> 243/200: 2/21-comma schismatic.

A3 -> 125/96: 3/25-comma schismatic.

d3 -> 144/125: 3/26-comma schismatic.

GrA1 -> 250/243: 3/29-comma schismatic.

GrA3 -> 625/486: 4/37-comma schismatic.

GrA2 -> 2500/2187: 4/39-comma schismatic.

I'm pretty intrigued by what's showing up so far and what's not. But I don't know how to use these things musically, so maybe it's best that I stop monkeying around.

...

Or I could monkey around with the meantone temperaments a little? For all of these tuning systems, I'll tune Ac1 to 1/1 and tune P8 to 2/1, and I'll just list the third interval that's tuned purely and what system results:

    A3 -> 125/96: 3/11-comma meantone.

    A4 -> 25/18: 1/3-comma meantone again.

    Ac4 -> 27/20: 1/1-comma meantone.

    AcA2 -> 75/64: 2/9-comma meantone.

    Acd1 -> 243/250: 3/8-comma meantone.

    Acd2 -> 648/625: 1/3-comma meantone again.

    Acm2 -> 27/25: 2/5-comma meantone.

    Acm3 -> 243/200: 2/3-comma meantone.

    Gr4 -> 320/243: negative 1/1-comma meantone?

    GrA1 -> 250/243: 3/7-comma meantone.

    GrA2 -> 2500/2187: 4/9-comma meantone.

    GrA3 -> 625/486: 4/11-comma meantone.

    GrA4 -> 1000/729: 1/2-comma meantone again.

    GrM2 -> 800/729: 1/1-comma meantone again.

    GrM3 -> 100/81: 1/2-comma meantone again.

    Grd2 -> 2048/2025: 1/6-comma meantone again.

    Grd3 -> 256/225: 1/5-comma meantone again

    Grd4 -> 512/405: 1/4-comma meantone again

    d3 -> 144/125: 3/10-comma meantone.

    d4 -> 32/25: 1/4-comma meantone again.

    AcA1 -> 135/128: 1/7-comma meantone.

...

And here they all are sorted by increasing comma fraction denominators:

P5 -> 3/2: Pythagorean tuning (0-comma meantone)

Gr4 -> 320/243: negative 1/1-comma meantone?

Ac4 -> 27/20: 1/1-comma meantone.

GrM2 -> 800/729: 1/1-comma meantone again.

M2 -> 10/9: 1/2-comma meantone.

GrM3 -> 100/81: 1/2-comma meantone again.

GrA4 -> 1000/729: 1/2-comma meantone again.

m3 -> 6/5: 1/3-comma meantone.

Acd2 -> 648/625: 1/3-comma meantone again.

A4 -> 25/18: 1/3-comma meantone again.

Acm3 -> 243/200: 2/3-comma meantone.

M3 -> 5/4: 1/4-comma meantone.

d4 -> 32/25: 1/4-comma meantone again.

Grd4 -> 512/405: 1/4-comma meantone again.

m2 -> 16/15: 1/5-comma meantone.

Grd3 -> 256/225: 1/5-comma meantone again.

Acm2 -> 27/25: 2/5-comma meantone.

AcA4 -> 45/32: 1/6-comma meantone.

Grd2 -> 2048/2025: 1/6-comma meantone again.

AcA1 -> 135/128: 1/7-comma meantone.

A1 -> 25/24: 2/7-comma meantone.

GrA1 -> 250/243: 3/7-comma meantone.

Acd1 -> 243/250: 3/8-comma meantone.

AcA2 -> 75/64: 2/9-comma meantone.

GrA2 -> 2500/2187: 4/9-comma meantone.

d3 -> 144/125: 3/10-comma meantone.

A3 -> 125/96: 3/11-comma meantone.

GrA3 -> 625/486: 4/11-comma meantone.

I have no idea what to make of this. Neat though, right?

Questions for people who are better at math than me: Are 1/8-comma meantone or 2/11-comma meantone possible? Is 1/14-comma schismatic possible? How about 2/23-comma schismatic or 3/23-schismatic?

...

Oh, good. I figured out how to derive the comma fractions from the tuning system. Now I don't have to wait for WolframAlpha to solve them. And that means I can iterate over weird temperament tuning systems automatically to find weird fractional commas.

Basically, if we have a basis (B1, B2, B3) tuned (t(B1), t(B2), t(B3)), with B1 equal to the tempered comma, B2 an interval of interest that we'll tune justly, and B3 equal to the octave, also tuned justly, then if the tempered coordinates for P5 are (m, n, o), we have empirically

     t(B2)^ n * t(B3)^o = t(P5) / t(B1)^(x)

in frequency space, which corresponds to

     B2 * n + B3 * o = P5 - B1 * (x)

in interval space. With a little rearrangement, 

    x = (P5 - (B2 * n + B3 * o)) / B1

. 

For example, if we define a temperament by t(Ac1, M3, P8) = (1/1, 5/4, 2/1), then this equation becomes:

    x =  (-1/1, 1/1, -1/4) / (-4, 4, -1)

Now, if you divide the vectors elementwise, you get (1/4, 1/4, 1/4) confirming that this is quarter-comma meantone. I don't know any reason why this should always work (or always work when B2 is independent of each B1 and B3), in the sense that the three entries of x are always equal, but it works empirically and I'll continue on investigating in this manner.

Okay, here are some meantone/Ac1 temperaments defined by the fractional power of the justly tuned Ac1 that you flatten P5 on the left and the interval that you tune purely on the right:

-1/2 Ac1 temperament: AcAcM2

-1/1 Ac1 temperament: Gr4

0/1 Ac1 temperament: AcM2, AcM3, Grm2, Grm3

1/1 Ac1 temperament: GrM2

1/2 Ac1 temperament: AcAcd4, GrM3, M2

1/3 Ac1 temperament: A2, Acd2, Acdd4, Acddd3, GrAAA1, m3

2/3 Ac1 temperament: Acm3

1/4 Ac1 temperament: M3, d2, d4

3/4 Ac1 temperament: GrGrM3

1/5 Ac1 temperament: Grd3, m2

2/5 Ac1 temperament: Acd3, Acm2

3/5 Ac1 temperament: AcAcm2

1/7 Ac1 temperament: AcA1

2/7 Ac1 temperament: A1, AA1

3/7 Ac1 temperament: GrA1

3/8 Ac1 temperament: Acd4, GrAA2

2/9 Ac1 temperament: AcA2

4/9 Ac1 temperament: GrA2

3/10 Ac1 temperament: d3, ddd5

3/11 Ac1 temperament: A3

4/11 Ac1 temperament: GrA3

5/12 Ac1 temperament: AcAcd2

5/14 Ac1 temperament: GrAA1

4/15 Ac1 temperament: dd4

5/16 Ac1 temperament: AA2

5/17 Ac1 temperament: dd3

6/17 Ac1 temperament: Acdd3

6/19 Ac1 temperament: GrAA0

7/22 Ac1 temperament: Acddd4

7/23 Ac1 temperament: AAA2

.

And here are some schismatic/AcAcA0 temperament definitions:

0/1 AcAcA0 temperament: AcM2, AcM3, Grm2, Grm3

1/5 AcAcA0 temperament: AcA1

1/7 AcAcA0 temperament: Grd3, m2

1/8 AcAcA0 temperament: M3, d2, d4

1/9 AcAcA0 temperament: A2, Acd2, Acdd4, Acddd3, GrAAA1, m3

1/10 AcAcA0 temperament: AcAcd4, GrM3, M2

1/11 AcAcA0 temperament: GrM2

1/13 AcAcA0 temperament: Gr4

1/14 AcAcA0 temperament: AcAcM2

2/15 AcAcA0 temperament: AcA2

2/17 AcAcA0 temperament: A1, AA1

2/19 AcAcA0 temperament: Acd3, Acm2

2/21 AcAcA0 temperament: Acm3

3/25 AcAcA0 temperament: A3

3/26 AcAcA0 temperament: d3, ddd5

3/28 AcAcA0 temperament: Acd4, GrAA2

3/29 AcAcA0 temperament: GrA1

3/31 AcAcA0 temperament: AcAcm2

3/32 AcAcA0 temperament: GrGrM3

4/33 AcAcA0 temperament: dd4

4/37 AcAcA0 temperament: GrA3

4/39 AcAcA0 temperament: GrA2

5/43 AcAcA0 temperament: dd3

5/44 AcAcA0 temperament: AA2

5/46 AcAcA0 temperament: GrAA1

5/48 AcAcA0 temperament: AcAcd2

6/53 AcAcA0 temperament: GrAA0

6/55 AcAcA0 temperament: Acdd3

7/61 AcAcA0 temperament: AAA2

7/62 AcAcA0 temperament: Acddd4

.

...

I just had the best idea! I'll sort the temperaments within a family by the size of their P5s, and sort them alongside EDOs that temper out the (syntonic, schismatic)-commas as well. That way we can be like, "this two dimensional temperament has a very close P5 to the one-dimensional 45-EDO" or whatever. And maybe that will help me to figure out range limits on the fractional commas! Like, if you temper out the syntonic comma and keep octaves pure, then tuning a third octave purely seems to put pretty tight constraints on how P5 gets tuned, and maybe we can say that tempered Ac1 and pure P8 means that P5 has to fall between, oh, 5-EDO's P5 and 7-EDO's P5, or something like that!

...

Ah! From preliminary investigation, it seems that the flattest P5 you get with schismatic temperaments comes with 1/2-comma schismatic, which tunes GrGrd3 purely. GrGrd3 is tuned justly to 32768/32805, which is the inverse of the schisma, the AcAcA0, justly tuned to 32805/32768. That I did not expect. The sharpest we go with P5 for schismatic temperaments is a frequency ratio of 3/2 in 0-comma schismatic, i.e. Pythagorean tuning. And all the fractional commas fall between 1/2 and 0/1.

Maybe that's not really the case though? Because when I investigate syntonic temperaments, the range seems to go quite a bit wider in both directions. For example, a syntonic temperament defined by tuning GrGrGrGrM2 purely produces 5/2-comma syntonic, which is obviously more than 1 comma flat. On the other side of a just P5, by purely tuning GrGrGrGr4, we get negative4-comma syntonic, which is sharper than P5 by four acute unisons, i.e. 86ish cents. And I wouldn't be at all surprised if these bounds kept increasing as I tried defining weirder temperaments from weirder purely tuned intervals.

And if syntonic temperaments behave that way, then maybe schismatic ones do too, way out in the dark waters.

...

Okay, I promised EDOs and I'm going to do EDOs. There seem to be finitely many EDOs that temper out the syntonic comma, Ac1. They are: [5, 7, 12, 19, 24, 26, 31, 36, 38, 43, 45, 50, 55, 57, 62, 67, 69, 74, 76, 81, 86, 88, 93, 98, 100, 105, 117, 129]-EDO. Nothing else between 5-EDO and 5000-EDO. This list is small enough, I feel we can look at all the P5s:

     1.4859942891369484 : 7-EDO

1.4916644904914018 : 26-EDO

1.492548464309911 : 45-EDO

1.4937589616544857 : 19-EDO

1.4937589616544857 : 38-EDO

1.4937589616544857 : 57-EDO

1.4937589616544857 : 76-EDO

1.4943783453027453 : 88-EDO

1.494548945312803 : 69-EDO

1.4948492486349383 : 100-EDO

1.4948492486349383 : 50-EDO

1.495105110169352 : 81-EDO

1.4955178823482085 : 31-EDO

1.4955178823482085 : 62-EDO

1.4955178823482085 : 93-EDO

1.4958363844631488 : 105-EDO

1.4959698311839842 : 74-EDO

1.4960896011977585 : 117-EDO

1.4962957394862462 : 129-EDO

1.4962957394862462 : 43-EDO

1.4962957394862462 : 86-EDO

1.4965418805580937 : 98-EDO

1.496734346325667 : 55-EDO

1.4970159080002896 : 67-EDO

1.4983070768766815 : 12-EDO

1.4983070768766815 : 24-EDO

1.4983070768766815 : 36-EDO

1.515716566510398 : 5-EDO

So only 5-EDO's P5 is sharper than the just P5 at 3/2. 

The P5 of 7-EDO falls between the P5s of 4/5-comma syntonic and 3/4-comma syntonic:

     1.4851668043517086 4/5 Ac1 temperament : AcAcAcm2

1.4859942891369484 : 7 -EDO

1.4860895666142713 3/4 Ac1 temperament : GrGrM3

And 5-EDO's tuned P5 falls between the tuned P5s of -2/3-comma syntonic and -1-comma syntonic.

The EDOs that temper out the schisma also seem to be finite, but it's a much larger list: [12, 17, 24, 29, 36, 41, 53, 65, 77, 82, 89, 94, 101, 106, 118, 130, 135, 142, 147, 154, 159, 171, 183, 195, 200, 207, 212, 219, 224, 236, 248, 253, 260, 265, 272, 277, 289, 301, 313, 318, 325, 330, 342, 354, 366, 371, 378, 383, 390, 395, 407, 419, 424, 431, 436, 443, 448, 460, 472, 484, 489, 496, 501, 508, 513, 525, 537, 542, 549, 554, 561, 566, 578, 590, 602, 607, 614, 619, 626, 631, 643, 655, 660, 667, 672, 679, 684, 696, 708, 720, 725, 732, 737, 744, 749, 761, 773, 778, 785, 790, 797, 802, 814, 826, 838, 843, 850, 855, 862, 867, 879, 891, 896, 903, 908, 915, 920, 932, 944, 956, 961, 968, 973, 985, 997, 1009, 1014, 1021, 1026, 1033, 1038, 1050, 1062, 1067, 1074, 1079, 1086, 1091, 1103, 1115, 1127, 1132, 1139, 1144, 1151, 1156, 1168, 1180, 1185, 1192, 1197, 1204, 1209, 1221, 1233, 1245, 1250, 1257, 1262, 1269, 1274, 1286, 1298, 1303, 1310, 1315, 1322, 1327, 1339, 1351, 1363, 1368, 1375, 1380, 1387, 1392, 1404, 1416, 1421, 1428, 1433, 1440, 1445, 1457, 1469, 1481, 1486, 1493, 1498, 1505, 1510, 1522, 1534, 1539, 1546, 1551, 1558, 1563, 1575, 1587, 1599, 1604, 1611, 1616, 1628, 1640, 1652, 1657, 1664, 1669, 1676, 1681, 1693, 1705, 1710, 1717, 1722, 1729, 1734, 1746, 1758, 1770, 1775, 1782, 1787, 1794, 1799, 1811, 1823, 1828, 1835, 1840, 1847, 1852, 1864, 1876, 1888, 1893, 1900, 1905, 1912, 1917, 1929, 1941, 1946, 1953, 1958, 1965, 1970, 1982, 1994, 2006, 2011, 2018, 2023, 2030, 2035, 2047, 2059, 2064, 2071, 2076, 2083, 2088, 2100, 2112, 2124, 2129, 2136, 2141, 2148, 2153, 2165, 2177, 2182, 2189, 2194, 2206, 2218, 2230, 2242, 2247, 2259, 2271, 2283, 2295, 2300, 2312, 2324, 2336, 2348, 2353, 2365, 2377, 2389, 2401, 2418, 2430, 2442, 2454, 2471, 2483, 2495, 2536, 2548, 2589, 2601, 2654, 2707]. Nothing else between 5-EDO and 5000-EDO.

The flattest P5 in an EDO that tempers out the schisma comes from 12-EDO and the sharpest such P5 comes from 17-EDO.

The 12-EDO tuned P5 is even slightly flatter than the 1/2-comma schismatic P5, and the 17-EDO tuned P5 is even slightly sharper than the tuned P5 of 0-comma schismatic, so these feel like good bounds.

The possibly non-existent bounds on the P5 of syntonic temperaments still befuddle me a little. I'll just have to wade deeper out in the water to investigate. Or content myself that temperaments with P5 outside of [7-EDO's P5, 5-EDO's P5] are non-diatonic and not worth too much of my time.

...

When I look up other commas that people have used to define temperaments, I see a lot of diminished seconds:

    Grd2 = (11, -4, -2) # 2048/2025

    d2 = (7, 0, -3) # 128/125

    Acd2 = (3, 4, -4) # 648/625

 

If you're feeling sassy, analyzing those might be fun.

I hadn't realized that finitely many EDOs temper out a given comma either. That's another ... source of data that could be catalogued and analyzed.

If you want a rank-4 interval to temper out that's justly associated with a small 7-limit frequency ratio, we've got some options:

4375/4374 at 0.4 cents, "ragisma"

2401/2400 at 0.7, "breedsma"

5120/5103 at 5.8c, "hemifamity"

225/224 at 7.7 cents cents, "marvel"

1029/1024 at 8.4 cents, "gamelisma"

126/125 at 13.8 cents, "starling"

245/243 at 14.2 cents, "sensamagic"

I've also listed the silly names for these fractions as they are known on the Xenharmonic wiki for some reason. The first ratio has a nice interpretation as

    (25/24) / ((36/35) * (81/80))

i.e. it's the difference between a grave augmented unison, justly tuned to 250/243, and the septimal super unison of Ben Johnston, 36/35. And a temperament which tunes 

a complicated rank-3 interval to the same frequency ratio as a simpler rank-4 which was perceptually indistinguishable under just tuning is a very good temperament.

The 2401/2400 does not have this property, but 5120/5103 does; it's just

    (36/35) / (81/80)^2

I also like 225/224 at 7.7 cents for this, which can be explained as:

    (36/35) / (128/125)

The 1029/1024 does not have a tidy rank-3 to rank-4 relationship.

The 14-cent 126/125 is a more complicated version of the 225/224 relationship:

    (36/35) / ((128/125) * (81/80))

and significantly more perceptible, so I'm not very impressed by that one.

The last just septimal comma, 245/243, is kind of cool:

    (245/243) = (16/15) / (36/35)^2

It's also too wide for my liking, but this just nicely shows how two septimal commas produce something like a minor second.

To summarize, I'd be friends with anyone who thought that the intervals associated with these ratios:

    ragisma: 4375/4374 at 0.4 cents # (25/24) / ((36/35) * (81/80))

    hemifamity: 5120/5103 at 5.8c # (36/35) / (81/80)^2

    marvel: 225/224 at 7.7 cents cents # (36/35) / (128/125)

were cool things to temper out to reduce rank-4 intervals by a dimension.

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.