Quartertone Harmony Over Maqamat

I was curious which arabic maqamat had some strong diatonic tetrads from the set in my recent Quartertone Harmony Chords post. I checked if for tetrads on (^1, ^2, ^5) and (^1, ^4, ^5) and a few others. From that, I think these maqamat lend themselves best to harmonizing in the style of Curt from Quartertone Harmony:

     'Iraq [0, 3, 7, 10, 13, 17, 21]

    Dalanshin (descends) [0, 4, 7, 10, 14, 18, 21]

    Husayni Ushayran [0, 3, 6, 10, 13, 16, 20]

    Jiharkah [0, 3, 7, 11, 14, 17, 21]

    Jiharkah_maqamworld [0, 3, 7, 11, 14, 17, 21]

    Nairuz [0, 4, 7, 10, 14, 17, 20]

    Rast (Upper Rast ending) [0, 4, 7, 10, 14, 18, 21]

    Sikah [0, 3, 7, 11, 14, 17, 21]

    Yakah [0, 4, 7, 10, 14, 17, 20]

If we just look at which maqamat have strong triads, then these maqamat have available diatonic chords on ^1, ^2, ^4, and ^5.

Lami [0, 2, 6, 10, 12, 16, 20]

Kurd [0, 2, 6, 10, 14, 16, 20]

Hijaz (Nahawand Ending) [0, 2, 8, 10, 14, 16, 20]

Husayni Ushayran [0, 3, 6, 10, 13, 16, 20]

Bayati (Nahawand Ending) [0, 3, 6, 10, 14, 16, 20]

Bayati (Rast Ending) [0, 3, 6, 10, 14, 17, 20]

'Iraq [0, 3, 7, 10, 13, 17, 21]

Jiharkah [0, 3, 7, 11, 14, 17, 21]

Sikah [0, 3, 7, 11, 14, 17, 21]

Nahawand (Kurd Ending) [0, 4, 6, 10, 14, 16, 20]

Nahawand (Hijaz Ending) [0, 4, 6, 10, 14, 16, 22]

Nawa Athar [0, 4, 6, 10, 14, 16, 22]

'Ushaq Masri [0, 4, 6, 10, 14, 17, 20]

Nikriz (descends) [0, 4, 6, 12, 14, 18, 20]

Suznak [0, 4, 7, 10, 14, 16, 22]

Nairuz [0, 4, 7, 10, 14, 17, 20]

Yakah [0, 4, 7, 10, 14, 17, 20]

Rast (Nahawand ending) [0, 4, 7, 10, 14, 18, 20]

Suzdalara (descends) [0, 4, 7, 10, 14, 18, 20]

Dalanshin (descends) [0, 4, 7, 10, 14, 18, 21]

Rast (Upper Rast ending) [0, 4, 7, 10, 14, 18, 21]

Mahur [0, 4, 7, 10, 14, 18, 22]

Shawq Afza [0, 4, 8, 10, 14, 16, 22]

'Ajam (Nahawand Ending) [0, 4, 8, 10, 14, 18, 20]

Jiharkah_wikipedia [0, 4, 8, 10, 14, 18, 21]

Jiharkah_maqamworld [0, 3, 7, 11, 14, 17, 21]

'Ajam (Upper Ajam Ending) [0, 4, 8, 10, 14, 18, 22]

So between triads and tetrads, we've got tons of options for harmonizing maqamat (in 24-EDO).

In the post on Quartertone Harmony Chords, the chords were all 24-EDO mistunings of just intonation chords. I wonder the just intonation forms of these 24-EDO diatonic maqamat chords have enough overlap that we can synthesize just intonation forms of the maqamat.

To start, I looked even more restrictively to see which maqamat had Curt-approved triad chords on every scale degree. There were quite a few:

Lami [0, 2, 6, 10, 12, 16, 20]

Kurd [0, 2, 6, 10, 14, 16, 20]

Husayni Ushayran [0, 3, 6, 10, 13, 16, 20]

Bayati (Nahawand Ending) [0, 3, 6, 10, 14, 16, 20]

Bayati (Rast Ending) [0, 3, 6, 10, 14, 17, 20]

'Iraq [0, 3, 7, 10, 13, 17, 21]

Sikah [0, 3, 7, 11, 14, 17, 21]

Nahawand (Kurd Ending) [0, 4, 6, 10, 14, 16, 20]

'Ushaq Masri [0, 4, 6, 10, 14, 17, 20]

Nairuz [0, 4, 7, 10, 14, 17, 20]

Yakah [0, 4, 7, 10, 14, 17, 20]

Rast (Nahawand ending) [0, 4, 7, 10, 14, 18, 20]

Suzdalara (descends) [0, 4, 7, 10, 14, 18, 20]

Dalanshin (descends) [0, 4, 7, 10, 14, 18, 21]

Rast (Upper Rast ending) [0, 4, 7, 10, 14, 18, 21]

'Ajam (Nahawand Ending) [0, 4, 8, 10, 14, 18, 20]

Jiharkah_wikipedia [0, 4, 8, 10, 14, 18, 21]

Jiharkah_maqamworld [0, 3, 7, 11, 14, 17, 21]

'Ajam (Upper Ajam Ending) [0, 4, 8, 10, 14, 18, 22]

A few of those are purely tonal. Here are the ones with odd steps:

Husayni Ushayran [0, 3, 6, 10, 13, 16, 20]

Bayati (Nahawand Ending) [0, 3, 6, 10, 14, 16, 20]

Bayati (Rast Ending) [0, 3, 6, 10, 14, 17, 20]

'Iraq [0, 3, 7, 10, 13, 17, 21]

Sikah [0, 3, 7, 11, 14, 17, 21]

'Ushaq Masri [0, 4, 6, 10, 14, 17, 20]

Nairuz [0, 4, 7, 10, 14, 17, 20]

Yakah [0, 4, 7, 10, 14, 17, 20]

Rast (Nahawand ending) [0, 4, 7, 10, 14, 18, 20]

Suzdalara (descends) [0, 4, 7, 10, 14, 18, 20]

Dalanshin (descends) [0, 4, 7, 10, 14, 18, 21]

Rast (Upper Rast ending) [0, 4, 7, 10, 14, 18, 21]

Jiharkah_wikipedia [0, 4, 8, 10, 14, 18, 21]

Jiharkah_maqamworld [0, 3, 7, 11, 14, 17, 21]

I'll probably go through them one by one by one offscreen trying to figure something out by hand. Mmmm, I hope one of them has an octave reduced 11th or 13th harmonic in one of its rotations. That would be cool.

...

Oh, I've got it. If you have tetrads on ^1, ^5, ^2, that specified an entire scale ^(1 3 5 7) ^(5 7 2 4) ^(2 4 6 1). So I just need those chords to be tuned justly in a way that one chord overlaps with the next. Also, if a maqam doesn't have tetrads on ^1, ^5, ^2, I could just as well use another sequence of three chords with tonics separated by fifths, like

    ^2, ^6, ^3
    ^3, ^7, ^4
    ^4, ^1, ^5
    ^5, ^2, ^6,
    ^6, ^3, ^7
    ^7, ^4, ^1

You just need three tetrads that line up with their relative intervals (the last interval of the chord on ^1 is the first interval on ^5, and the last interval of the chord on ^5 is the first interval of the chord on ^2) and also the whole thing adds up to on octave (or really two octaves you add up all 7 positionally distinct relative interval from the tertian chords).

Here's a spelling of Husayni 'Ushayran by 3rds: 

    [P1, Grm3, De5, PrDem7, DeAcM9, P11, m13, P15] :: [0, 6, 13, 20, 27, 34, 40, 48]

Here it is increasing by 2nd intervals, with its just tuning:

    [P1, DeAcM2, Grm3, P4, De5, m6, PrDem7, P8] # [1/1, 12/11, 32/27, 4/3, 16/11, 8/5, 39/22, 2/1]

There's no doubt in my mind that other detemperings would work, because the chords in 24-EDO steps have lots of compatible just determperings. This is just the first one I found. I think an Arab music theorist would find 8/5 a little bit of an unusual ratio to include since it's 5 limit, but the rest looks pretty plausible to me. And more than being plausible, this tuning of the scale supports Curt's notion of quartertone harmony.

Let's find one for maqam 'Iraq.

...

These three work, but they look pretty weird.

    [P1, AsGrm3, AsGrd5, AsGrm7, AsGrm9, PrAsGrd11, AsGrm13, P15]

    [P1, DeAcM3, De5, DeAcM7, DeAcM9, ReDeAcA11, DeAcM13, P15]

    [P1, Prm3, PrGrd5, PrGrm7, Prm9, PrPrd11, Prm13, P15]

I guess 'Iraq is alsways going to look weird in absolute intervals since it starts on a microtone. You're going to have all tones starting with "AsGrm" or "DeAcM" or "Prm" until you get back to the octave. Maybe a better test of the naturalness of these tunings is to rotate 'Iraq so that the tonic is C instead of Ed. That means rooting the scale on the 6th (or 13th).

Here they are respectively:

    [P1, AcM2, DeAcM3, P4, P5, Prm6, m7] # [1/1, 9/8, 27/22, 4/3, 3/2, 13/8, 9/5]

    [P1, AcM2, AsGrm3, P4, P5, ReM6, Grm7] # [1/1, 9/8, 11/9, 4/3, 3/2, 64/39, 16/9]

    [P1, M2, ReM3, P4, P5, Prm6, Grm7] # [1/1, 10/9, 16/13, 4/3, 3/2, 13/8, 16/9]

That first one looks amazingly arabic in the lower tetrachord and also has an octave reduced 13th harmonic. Just change the m7 to Grm7 and we've got it in terms of aesthetics of the absolute frequency ratios. Can we actually do that and still have nice tetrads on ^1, ^2, ^5, as well as (at least) good triads on the other scale degrees?

Sadly no. This scale (described relative to C still) has strong tetrads on ^1, ^2, ^4, strong triads on ^5 and ^7, and absolute garbage on ^3 and ^6.

Actually ^3 and ^6 are garbage regardless of whether we use m7 or Grm7. I think my procedure of "link of three chords with tonics separated by fifths and make sure it forms an octave" was insufficient. Oops. Probably my detemperings for Husayni 'Ushayran were wrong too.

Well, the good new is that I've done this by hand enough that I can automate it, go through all the possibilities, and check for validity.

The bad news is that I don't understand what I did wrong. If ^(1 3 5 7) is a good tetrad, why wouldn't ^(3 5 7) be a good triad? Maybe it is and I just don't have it in my library, since my library started by generating tetrads. Yeah. Yeah. Yeah. Probably yeah.

So like, if you remove the first note from a valid tetrad, it will still be according to Curt's rules. But the just tuning might be so complex that I would disregard it. But whether or not we keep them, I need to explicitly generate them and not just consider a chord valid if it appears in the first three notes of a valid tetrad. We want to look at both the first three and the last three notes.

Okay, here's the set of triads for Curt:

[P1, AcM3, P5] # [1/1, 81/64, 3/2]

[P1, AsGrm3, AsGrd5] # [1/1, 11/9, 22/15]

[P1, AsGrm3, AsSbGrd5] # [1/1, 11/9, 77/54]

[P1, AsGrm3, P5] # [1/1, 11/9, 3/2]

[P1, AsGrm3, PrGrd5] # [1/1, 11/9, 13/9]

[P1, DeAcM3, De5] # [1/1, 27/22, 16/11]

[P1, DeAcM3, DeAc5] # [1/1, 27/22, 81/55]

[P1, DeAcM3, DeSbAc5] # [1/1, 27/22, 63/44]

[P1, DeAcM3, P5] # [1/1, 27/22, 3/2]

[P1, Grm3, De5] # [1/1, 32/27, 16/11]

[P1, Grm3, Gr5] # [1/1, 32/27, 40/27]

[P1, Grm3, Grd5] # [1/1, 32/27, 64/45]

[P1, Grm3, P5] # [1/1, 32/27, 3/2]

[P1, Grm3, PrGrd5] # [1/1, 32/27, 13/9]

[P1, M3, Gr5] # [1/1, 5/4, 40/27]

[P1, M3, P5] # [1/1, 5/4, 3/2]

[P1, M3, PrDe5] # [1/1, 5/4, 65/44]

[P1, M3, Sb5] # [1/1, 5/4, 35/24]

[P1, PrDem3, De5] # [1/1, 13/11, 16/11]

[P1, PrDem3, P5] # [1/1, 13/11, 3/2]

[P1, PrDem3, PrDe5] # [1/1, 13/11, 65/44]

[P1, PrDem3, PrDeSbd5] # [1/1, 13/11, 91/66]

[P1, PrDem3, PrDed5] # [1/1, 13/11, 78/55]

[P1, PrDem3, PrGrd5] # [1/1, 13/11, 13/9]

[P1, Prm3, P5] # [1/1, 39/32, 3/2]

[P1, Prm3, PrGrd5] # [1/1, 39/32, 13/9]

[P1, Prm3, PrSbd5] # [1/1, 39/32, 91/64]

[P1, ReAsM3, P5] # [1/1, 33/26, 3/2]

[P1, ReAsM3, ReAsSb5] # [1/1, 33/26, 77/52]

[P1, ReM3, De5] # [1/1, 16/13, 16/11]

[P1, ReM3, P5] # [1/1, 16/13, 3/2]

[P1, ReM3, Re5] # [1/1, 16/13, 96/65]

[P1, ReM3, ReSb5] # [1/1, 16/13, 56/39]

[P1, Sbm3, AsSbGrd5] # [1/1, 7/6, 77/54]

[P1, Sbm3, DeSbAc5] # [1/1, 7/6, 63/44]

[P1, Sbm3, PrDeSbd5] # [1/1, 7/6, 91/66]

[P1, Sbm3, PrSbd5] # [1/1, 7/6, 91/64]

[P1, Sbm3, ReAsSb5] # [1/1, 7/6, 77/52]

[P1, Sbm3, ReSb5] # [1/1, 7/6, 56/39]

[P1, Sbm3, Sb5] # [1/1, 7/6, 35/24]

[P1, Sbm3, Sbd5] # [1/1, 7/6, 7/5]

[P1, m3, AsGrd5] # [1/1, 6/5, 22/15]

[P1, m3, DeAc5] # [1/1, 6/5, 81/55]

[P1, m3, Grd5] # [1/1, 6/5, 64/45]

[P1, m3, P5] # [1/1, 6/5, 3/2]

[P1, m3, PrDed5] # [1/1, 6/5, 78/55]

[P1, m3, Re5] # [1/1, 6/5, 96/65]

[P1, m3, Sbd5] # [1/1, 6/5, 7/5]

[P1, m3, d5] # [1/1, 6/5, 36/25]

I'm going to add these into the Quartertone Harmony Chords post too.

...

Ah, in a previous post, I shared microtonal chords that were compatible with Curt's Quartertone Harmony rules, but I removed all the ones that didn't have intervals tuned to odd steps in 24-EDO. That was the right thing to do for that post, but now my harmony analyzer doesn't think that a scale can have like a normal Pythagorean major or minor triad. That's probably why maqamat with the best harmonic options according to my analyzer were weird things like maqam 'Iraq with a half flat fifth. So I have to regenerate the full set of quartertone chords and not discard the tonal ones.

...

Okay, here are 268 Curt chords, triads and tetrads, neutral tones and non-neutral tones both.

[P1, AcM3, P5, AsGrm7] : [AcM3, Grm3, AsGrm3] _ [1/1, 81/64, 3/2, 11/6] :: [0, 8, 14, 21] # [8, 6, 7]

[P1, AcM3, P5, DeAcM7] : [AcM3, Grm3, DeAcM3] _ [1/1, 81/64, 3/2, 81/44] :: [0, 8, 14, 21] # [8, 6, 7]

[P1, AcM3, P5, Grm7] : [AcM3, Grm3, Grm3] _ [1/1, 81/64, 3/2, 16/9] :: [0, 8, 14, 20] # [8, 6, 6]

[P1, AcM3, P5, M7] : [AcM3, Grm3, M3] _ [1/1, 81/64, 3/2, 15/8] :: [0, 8, 14, 22] # [8, 6, 8]

[P1, AcM3, P5, PrDem7] : [AcM3, Grm3, PrDem3] _ [1/1, 81/64, 3/2, 39/22] :: [0, 8, 14, 20] # [8, 6, 6]

[P1, AcM3, P5, ReAsM7] : [AcM3, Grm3, ReAsM3] _ [1/1, 81/64, 3/2, 99/52] :: [0, 8, 14, 22] # [8, 6, 8]

[P1, AcM3, P5, ReM7] : [AcM3, Grm3, ReM3] _ [1/1, 81/64, 3/2, 24/13] :: [0, 8, 14, 21] # [8, 6, 7]

[P1, AcM3, P5, Sbm7] : [AcM3, Grm3, Sbm3] _ [1/1, 81/64, 3/2, 7/4] :: [0, 8, 14, 19] # [8, 6, 5]

[P1, AcM3, P5, m7] : [AcM3, Grm3, m3] _ [1/1, 81/64, 3/2, 9/5] :: [0, 8, 14, 20] # [8, 6, 6]

[P1, AsGrm3, AsGrd5, AsGrd7] : [AsGrm3, m3, m3] _ [1/1, 11/9, 22/15, 44/25] :: [0, 7, 13, 19] # [7, 6, 6]

[P1, AsGrm3, AsGrd5, AsGrm7] : [AsGrm3, m3, M3] _ [1/1, 11/9, 22/15, 11/6] :: [0, 7, 13, 21] # [7, 6, 8]

[P1, AsGrm3, AsGrd5, AsSbGrd7] : [AsGrm3, m3, Sbm3] _ [1/1, 11/9, 22/15, 77/45] :: [0, 7, 13, 18] # [7, 6, 5]

[P1, AsGrm3, AsGrd5, PrGrd7] : [AsGrm3, m3, PrDem3] _ [1/1, 11/9, 22/15, 26/15] :: [0, 7, 13, 19] # [7, 6, 6]

[P1, AsGrm3, AsGrd5, m7] : [AsGrm3, m3, DeAcM3] _ [1/1, 11/9, 22/15, 9/5] :: [0, 7, 13, 20] # [7, 6, 7]

[P1, AsGrm3, AsSbGrd5, AsSbGrd7] : [AsGrm3, Sbm3, m3] _ [1/1, 11/9, 77/54, 77/45] :: [0, 7, 12, 18] # [7, 5, 6]

[P1, AsGrm3, AsSbGrd5, PrSbGrd7] : [AsGrm3, Sbm3, PrDem3] _ [1/1, 11/9, 77/54, 91/54] :: [0, 7, 12, 18] # [7, 5, 6]

[P1, AsGrm3, AsSbGrd5, Sbm7] : [AsGrm3, Sbm3, DeAcM3] _ [1/1, 11/9, 77/54, 7/4] :: [0, 7, 12, 19] # [7, 5, 7]

[P1, AsGrm3, P5, AsGrm7] : [AsGrm3, DeAcM3, AsGrm3] _ [1/1, 11/9, 3/2, 11/6] :: [0, 7, 14, 21] # [7, 7, 7]

[P1, AsGrm3, P5, DeAcM7] : [AsGrm3, DeAcM3, DeAcM3] _ [1/1, 11/9, 3/2, 81/44] :: [0, 7, 14, 21] # [7, 7, 7]

[P1, AsGrm3, P5, Grm7] : [AsGrm3, DeAcM3, Grm3] _ [1/1, 11/9, 3/2, 16/9] :: [0, 7, 14, 20] # [7, 7, 6]

[P1, AsGrm3, P5, PrDem7] : [AsGrm3, DeAcM3, PrDem3] _ [1/1, 11/9, 3/2, 39/22] :: [0, 7, 14, 20] # [7, 7, 6]

[P1, AsGrm3, P5, ReM7] : [AsGrm3, DeAcM3, ReM3] _ [1/1, 11/9, 3/2, 24/13] :: [0, 7, 14, 21] # [7, 7, 7]

[P1, AsGrm3, P5, Sbm7] : [AsGrm3, DeAcM3, Sbm3] _ [1/1, 11/9, 3/2, 7/4] :: [0, 7, 14, 19] # [7, 7, 5]

[P1, AsGrm3, P5, m7] : [AsGrm3, DeAcM3, m3] _ [1/1, 11/9, 3/2, 9/5] :: [0, 7, 14, 20] # [7, 7, 6]

[P1, AsGrm3, PrGrd5, AsGrm7] : [AsGrm3, PrDem3, ReAsM3] _ [1/1, 11/9, 13/9, 11/6] :: [0, 7, 13, 21] # [7, 6, 8]

[P1, AsGrm3, PrGrd5, Grm7] : [AsGrm3, PrDem3, ReM3] _ [1/1, 11/9, 13/9, 16/9] :: [0, 7, 13, 20] # [7, 6, 7]

[P1, AsGrm3, PrGrd5, PrDem7] : [AsGrm3, PrDem3, DeAcM3] _ [1/1, 11/9, 13/9, 39/22] :: [0, 7, 13, 20] # [7, 6, 7]

[P1, AsGrm3, PrGrd5, PrGrd7] : [AsGrm3, PrDem3, m3] _ [1/1, 11/9, 13/9, 26/15] :: [0, 7, 13, 19] # [7, 6, 6]

[P1, AsGrm3, PrGrd5, PrGrm7] : [AsGrm3, PrDem3, M3] _ [1/1, 11/9, 13/9, 65/36] :: [0, 7, 13, 21] # [7, 6, 8]

[P1, AsGrm3, PrGrd5, PrSbGrd7] : [AsGrm3, PrDem3, Sbm3] _ [1/1, 11/9, 13/9, 91/54] :: [0, 7, 13, 18] # [7, 6, 5]

[P1, DeAcM3, De5, DeAcM7] : [DeAcM3, Grm3, AcM3] _ [1/1, 27/22, 16/11, 81/44] :: [0, 7, 13, 21] # [7, 6, 8]

[P1, DeAcM3, De5, DeM7] : [DeAcM3, Grm3, M3] _ [1/1, 27/22, 16/11, 20/11] :: [0, 7, 13, 21] # [7, 6, 8]

[P1, DeAcM3, De5, DeSbm7] : [DeAcM3, Grm3, Sbm3] _ [1/1, 27/22, 16/11, 56/33] :: [0, 7, 13, 18] # [7, 6, 5]

[P1, DeAcM3, De5, Dem7] : [DeAcM3, Grm3, m3] _ [1/1, 27/22, 16/11, 96/55] :: [0, 7, 13, 19] # [7, 6, 6]

[P1, DeAcM3, De5, Grm7] : [DeAcM3, Grm3, AsGrm3] _ [1/1, 27/22, 16/11, 16/9] :: [0, 7, 13, 20] # [7, 6, 7]

[P1, DeAcM3, De5, PrDem7] : [DeAcM3, Grm3, Prm3] _ [1/1, 27/22, 16/11, 39/22] :: [0, 7, 13, 20] # [7, 6, 7]

[P1, DeAcM3, De5, ReM7] : [DeAcM3, Grm3, ReAsM3] _ [1/1, 27/22, 16/11, 24/13] :: [0, 7, 13, 21] # [7, 6, 8]

[P1, DeAcM3, DeAc5, DeAcM7] : [DeAcM3, m3, M3] _ [1/1, 27/22, 81/55, 81/44] :: [0, 7, 13, 21] # [7, 6, 8]

[P1, DeAcM3, DeAc5, Dem7] : [DeAcM3, m3, Grm3] _ [1/1, 27/22, 81/55, 96/55] :: [0, 7, 13, 19] # [7, 6, 6]

[P1, DeAcM3, DeAc5, m7] : [DeAcM3, m3, AsGrm3] _ [1/1, 27/22, 81/55, 9/5] :: [0, 7, 13, 20] # [7, 6, 7]

[P1, DeAcM3, DeSbAc5, DeSbm7] : [DeAcM3, Sbm3, Grm3] _ [1/1, 27/22, 63/44, 56/33] :: [0, 7, 12, 18] # [7, 5, 6]

[P1, DeAcM3, DeSbAc5, Sbm7] : [DeAcM3, Sbm3, AsGrm3] _ [1/1, 27/22, 63/44, 7/4] :: [0, 7, 12, 19] # [7, 5, 7]

[P1, DeAcM3, P5, AsGrm7] : [DeAcM3, AsGrm3, AsGrm3] _ [1/1, 27/22, 3/2, 11/6] :: [0, 7, 14, 21] # [7, 7, 7]

[P1, DeAcM3, P5, DeAcM7] : [DeAcM3, AsGrm3, DeAcM3] _ [1/1, 27/22, 3/2, 81/44] :: [0, 7, 14, 21] # [7, 7, 7]

[P1, DeAcM3, P5, Grm7] : [DeAcM3, AsGrm3, Grm3] _ [1/1, 27/22, 3/2, 16/9] :: [0, 7, 14, 20] # [7, 7, 6]

[P1, DeAcM3, P5, PrDem7] : [DeAcM3, AsGrm3, PrDem3] _ [1/1, 27/22, 3/2, 39/22] :: [0, 7, 14, 20] # [7, 7, 6]

[P1, DeAcM3, P5, ReM7] : [DeAcM3, AsGrm3, ReM3] _ [1/1, 27/22, 3/2, 24/13] :: [0, 7, 14, 21] # [7, 7, 7]

[P1, DeAcM3, P5, Sbm7] : [DeAcM3, AsGrm3, Sbm3] _ [1/1, 27/22, 3/2, 7/4] :: [0, 7, 14, 19] # [7, 7, 5]

[P1, DeAcM3, P5, m7] : [DeAcM3, AsGrm3, m3] _ [1/1, 27/22, 3/2, 9/5] :: [0, 7, 14, 20] # [7, 7, 6]

[P1, Grm3, De5, DeSbm7] : [Grm3, DeAcM3, Sbm3] _ [1/1, 32/27, 16/11, 56/33] :: [0, 6, 13, 18] # [6, 7, 5]

[P1, Grm3, De5, Dem7] : [Grm3, DeAcM3, m3] _ [1/1, 32/27, 16/11, 96/55] :: [0, 6, 13, 19] # [6, 7, 6]

[P1, Grm3, De5, Grm7] : [Grm3, DeAcM3, AsGrm3] _ [1/1, 32/27, 16/11, 16/9] :: [0, 6, 13, 20] # [6, 7, 7]

[P1, Grm3, De5, PrDem7] : [Grm3, DeAcM3, Prm3] _ [1/1, 32/27, 16/11, 39/22] :: [0, 6, 13, 20] # [6, 7, 7]

[P1, Grm3, Gr5, GrM7] : [Grm3, M3, M3] _ [1/1, 32/27, 40/27, 50/27] :: [0, 6, 14, 22] # [6, 8, 8]

[P1, Grm3, Gr5, Grm7] : [Grm3, M3, m3] _ [1/1, 32/27, 40/27, 16/9] :: [0, 6, 14, 20] # [6, 8, 6]

[P1, Grm3, Gr5, M7] : [Grm3, M3, AcM3] _ [1/1, 32/27, 40/27, 15/8] :: [0, 6, 14, 22] # [6, 8, 8]

[P1, Grm3, Grd5, Dem7] : [Grm3, m3, DeAcM3] _ [1/1, 32/27, 64/45, 96/55] :: [0, 6, 12, 19] # [6, 6, 7]

[P1, Grm3, Grd5, Grm7] : [Grm3, m3, M3] _ [1/1, 32/27, 64/45, 16/9] :: [0, 6, 12, 20] # [6, 6, 8]

[P1, Grm3, Grd5, PrGrd7] : [Grm3, m3, Prm3] _ [1/1, 32/27, 64/45, 26/15] :: [0, 6, 12, 19] # [6, 6, 7]

[P1, Grm3, Grd5, m7] : [Grm3, m3, AcM3] _ [1/1, 32/27, 64/45, 9/5] :: [0, 6, 12, 20] # [6, 6, 8]

[P1, Grm3, P5, Grm7] : [Grm3, AcM3, Grm3] _ [1/1, 32/27, 3/2, 16/9] :: [0, 6, 14, 20] # [6, 8, 6]

[P1, Grm3, P5, M7] : [Grm3, AcM3, M3] _ [1/1, 32/27, 3/2, 15/8] :: [0, 6, 14, 22] # [6, 8, 8]

[P1, Grm3, P5, PrDem7] : [Grm3, AcM3, PrDem3] _ [1/1, 32/27, 3/2, 39/22] :: [0, 6, 14, 20] # [6, 8, 6]

[P1, Grm3, P5, ReAsM7] : [Grm3, AcM3, ReAsM3] _ [1/1, 32/27, 3/2, 99/52] :: [0, 6, 14, 22] # [6, 8, 8]

[P1, Grm3, P5, Sbm7] : [Grm3, AcM3, Sbm3] _ [1/1, 32/27, 3/2, 7/4] :: [0, 6, 14, 19] # [6, 8, 5]

[P1, Grm3, P5, m7] : [Grm3, AcM3, m3] _ [1/1, 32/27, 3/2, 9/5] :: [0, 6, 14, 20] # [6, 8, 6]

[P1, Grm3, PrGrd5, Grm7] : [Grm3, Prm3, ReM3] _ [1/1, 32/27, 13/9, 16/9] :: [0, 6, 13, 20] # [6, 7, 7]

[P1, Grm3, PrGrd5, PrDem7] : [Grm3, Prm3, DeAcM3] _ [1/1, 32/27, 13/9, 39/22] :: [0, 6, 13, 20] # [6, 7, 7]

[P1, Grm3, PrGrd5, PrGrd7] : [Grm3, Prm3, m3] _ [1/1, 32/27, 13/9, 26/15] :: [0, 6, 13, 19] # [6, 7, 6]

[P1, Grm3, PrGrd5, PrSbGrd7] : [Grm3, Prm3, Sbm3] _ [1/1, 32/27, 13/9, 91/54] :: [0, 6, 13, 18] # [6, 7, 5]

[P1, M3, A5, GrM7] : [M3, M3, Grm3] _ [1/1, 5/4, 25/16, 50/27] :: [0, 8, 16, 22] # [8, 8, 6]

[P1, M3, A5, M7] : [M3, M3, m3] _ [1/1, 5/4, 25/16, 15/8] :: [0, 8, 16, 22] # [8, 8, 6]

[P1, M3, Gr5, DeM7] : [M3, Grm3, DeAcM3] _ [1/1, 5/4, 40/27, 20/11] :: [0, 8, 14, 21] # [8, 6, 7]

[P1, M3, Gr5, GrM7] : [M3, Grm3, M3] _ [1/1, 5/4, 40/27, 50/27] :: [0, 8, 14, 22] # [8, 6, 8]

[P1, M3, Gr5, Grm7] : [M3, Grm3, m3] _ [1/1, 5/4, 40/27, 16/9] :: [0, 8, 14, 20] # [8, 6, 6]

[P1, M3, Gr5, M7] : [M3, Grm3, AcM3] _ [1/1, 5/4, 40/27, 15/8] :: [0, 8, 14, 22] # [8, 6, 8]

[P1, M3, Gr5, PrGrm7] : [M3, Grm3, Prm3] _ [1/1, 5/4, 40/27, 65/36] :: [0, 8, 14, 21] # [8, 6, 7]

[P1, M3, P5, AsGrm7] : [M3, m3, AsGrm3] _ [1/1, 5/4, 3/2, 11/6] :: [0, 8, 14, 21] # [8, 6, 7]

[P1, M3, P5, DeAcM7] : [M3, m3, DeAcM3] _ [1/1, 5/4, 3/2, 81/44] :: [0, 8, 14, 21] # [8, 6, 7]

[P1, M3, P5, Grm7] : [M3, m3, Grm3] _ [1/1, 5/4, 3/2, 16/9] :: [0, 8, 14, 20] # [8, 6, 6]

[P1, M3, P5, M7] : [M3, m3, M3] _ [1/1, 5/4, 3/2, 15/8] :: [0, 8, 14, 22] # [8, 6, 8]

[P1, M3, P5, PrDem7] : [M3, m3, PrDem3] _ [1/1, 5/4, 3/2, 39/22] :: [0, 8, 14, 20] # [8, 6, 6]

[P1, M3, P5, ReAsM7] : [M3, m3, ReAsM3] _ [1/1, 5/4, 3/2, 99/52] :: [0, 8, 14, 22] # [8, 6, 8]

[P1, M3, P5, ReM7] : [M3, m3, ReM3] _ [1/1, 5/4, 3/2, 24/13] :: [0, 8, 14, 21] # [8, 6, 7]

[P1, M3, P5, Sbm7] : [M3, m3, Sbm3] _ [1/1, 5/4, 3/2, 7/4] :: [0, 8, 14, 19] # [8, 6, 5]

[P1, M3, P5, m7] : [M3, m3, m3] _ [1/1, 5/4, 3/2, 9/5] :: [0, 8, 14, 20] # [8, 6, 6]

[P1, M3, PrDe5, DeM7] : [M3, PrDem3, ReM3] _ [1/1, 5/4, 65/44, 20/11] :: [0, 8, 14, 21] # [8, 6, 7]

[P1, M3, PrDe5, M7] : [M3, PrDem3, ReAsM3] _ [1/1, 5/4, 65/44, 15/8] :: [0, 8, 14, 22] # [8, 6, 8]

[P1, M3, PrDe5, PrDem7] : [M3, PrDem3, m3] _ [1/1, 5/4, 65/44, 39/22] :: [0, 8, 14, 20] # [8, 6, 6]

[P1, M3, PrDe5, PrGrm7] : [M3, PrDem3, AsGrm3] _ [1/1, 5/4, 65/44, 65/36] :: [0, 8, 14, 21] # [8, 6, 7]

[P1, M3, Sb5, ReSbM7] : [M3, Sbm3, ReM3] _ [1/1, 5/4, 35/24, 70/39] :: [0, 8, 13, 20] # [8, 5, 7]

[P1, M3, Sb5, Sbm7] : [M3, Sbm3, m3] _ [1/1, 5/4, 35/24, 7/4] :: [0, 8, 13, 19] # [8, 5, 6]

[P1, PrDem3, De5, DeSbm7] : [PrDem3, ReM3, Sbm3] _ [1/1, 13/11, 16/11, 56/33] :: [0, 6, 13, 18] # [6, 7, 5]

[P1, PrDem3, De5, Dem7] : [PrDem3, ReM3, m3] _ [1/1, 13/11, 16/11, 96/55] :: [0, 6, 13, 19] # [6, 7, 6]

[P1, PrDem3, De5, Grm7] : [PrDem3, ReM3, AsGrm3] _ [1/1, 13/11, 16/11, 16/9] :: [0, 6, 13, 20] # [6, 7, 7]

[P1, PrDem3, De5, PrDem7] : [PrDem3, ReM3, Prm3] _ [1/1, 13/11, 16/11, 39/22] :: [0, 6, 13, 20] # [6, 7, 7]

[P1, PrDem3, P5, Grm7] : [PrDem3, ReAsM3, Grm3] _ [1/1, 13/11, 3/2, 16/9] :: [0, 6, 14, 20] # [6, 8, 6]

[P1, PrDem3, P5, M7] : [PrDem3, ReAsM3, M3] _ [1/1, 13/11, 3/2, 15/8] :: [0, 6, 14, 22] # [6, 8, 8]

[P1, PrDem3, P5, PrDem7] : [PrDem3, ReAsM3, PrDem3] _ [1/1, 13/11, 3/2, 39/22] :: [0, 6, 14, 20] # [6, 8, 6]

[P1, PrDem3, P5, ReAsM7] : [PrDem3, ReAsM3, ReAsM3] _ [1/1, 13/11, 3/2, 99/52] :: [0, 6, 14, 22] # [6, 8, 8]

[P1, PrDem3, P5, Sbm7] : [PrDem3, ReAsM3, Sbm3] _ [1/1, 13/11, 3/2, 7/4] :: [0, 6, 14, 19] # [6, 8, 5]

[P1, PrDem3, P5, m7] : [PrDem3, ReAsM3, m3] _ [1/1, 13/11, 3/2, 9/5] :: [0, 6, 14, 20] # [6, 8, 6]

[P1, PrDem3, PrDe5, M7] : [PrDem3, M3, ReAsM3] _ [1/1, 13/11, 65/44, 15/8] :: [0, 6, 14, 22] # [6, 8, 8]

[P1, PrDem3, PrDe5, PrDem7] : [PrDem3, M3, m3] _ [1/1, 13/11, 65/44, 39/22] :: [0, 6, 14, 20] # [6, 8, 6]

[P1, PrDem3, PrDeSbd5, DeSbm7] : [PrDem3, Sbm3, ReM3] _ [1/1, 13/11, 91/66, 56/33] :: [0, 6, 11, 18] # [6, 5, 7]

[P1, PrDem3, PrDeSbd5, PrDeSbd7] : [PrDem3, Sbm3, m3] _ [1/1, 13/11, 91/66, 91/55] :: [0, 6, 11, 17] # [6, 5, 6]

[P1, PrDem3, PrDeSbd5, PrSbGrd7] : [PrDem3, Sbm3, AsGrm3] _ [1/1, 13/11, 91/66, 91/54] :: [0, 6, 11, 18] # [6, 5, 7]

[P1, PrDem3, PrDeSbd5, Sbm7] : [PrDem3, Sbm3, ReAsM3] _ [1/1, 13/11, 91/66, 7/4] :: [0, 6, 11, 19] # [6, 5, 8]

[P1, PrDem3, PrDed5, Dem7] : [PrDem3, m3, ReM3] _ [1/1, 13/11, 78/55, 96/55] :: [0, 6, 12, 19] # [6, 6, 7]

[P1, PrDem3, PrDed5, PrDeSbd7] : [PrDem3, m3, Sbm3] _ [1/1, 13/11, 78/55, 91/55] :: [0, 6, 12, 17] # [6, 6, 5]

[P1, PrDem3, PrDed5, PrDem7] : [PrDem3, m3, M3] _ [1/1, 13/11, 78/55, 39/22] :: [0, 6, 12, 20] # [6, 6, 8]

[P1, PrDem3, PrDed5, PrGrd7] : [PrDem3, m3, AsGrm3] _ [1/1, 13/11, 78/55, 26/15] :: [0, 6, 12, 19] # [6, 6, 7]

[P1, PrDem3, PrDed5, m7] : [PrDem3, m3, ReAsM3] _ [1/1, 13/11, 78/55, 9/5] :: [0, 6, 12, 20] # [6, 6, 8]

[P1, PrDem3, PrGrd5, Grm7] : [PrDem3, AsGrm3, ReM3] _ [1/1, 13/11, 13/9, 16/9] :: [0, 6, 13, 20] # [6, 7, 7]

[P1, PrDem3, PrGrd5, PrDem7] : [PrDem3, AsGrm3, DeAcM3] _ [1/1, 13/11, 13/9, 39/22] :: [0, 6, 13, 20] # [6, 7, 7]

[P1, PrDem3, PrGrd5, PrGrd7] : [PrDem3, AsGrm3, m3] _ [1/1, 13/11, 13/9, 26/15] :: [0, 6, 13, 19] # [6, 7, 6]

[P1, PrDem3, PrGrd5, PrSbGrd7] : [PrDem3, AsGrm3, Sbm3] _ [1/1, 13/11, 13/9, 91/54] :: [0, 6, 13, 18] # [6, 7, 5]

[P1, Prm3, P5, AsGrm7] : [Prm3, ReM3, AsGrm3] _ [1/1, 39/32, 3/2, 11/6] :: [0, 7, 14, 21] # [7, 7, 7]

[P1, Prm3, P5, DeAcM7] : [Prm3, ReM3, DeAcM3] _ [1/1, 39/32, 3/2, 81/44] :: [0, 7, 14, 21] # [7, 7, 7]

[P1, Prm3, P5, Grm7] : [Prm3, ReM3, Grm3] _ [1/1, 39/32, 3/2, 16/9] :: [0, 7, 14, 20] # [7, 7, 6]

[P1, Prm3, P5, PrDem7] : [Prm3, ReM3, PrDem3] _ [1/1, 39/32, 3/2, 39/22] :: [0, 7, 14, 20] # [7, 7, 6]

[P1, Prm3, P5, ReM7] : [Prm3, ReM3, ReM3] _ [1/1, 39/32, 3/2, 24/13] :: [0, 7, 14, 21] # [7, 7, 7]

[P1, Prm3, P5, Sbm7] : [Prm3, ReM3, Sbm3] _ [1/1, 39/32, 3/2, 7/4] :: [0, 7, 14, 19] # [7, 7, 5]

[P1, Prm3, P5, m7] : [Prm3, ReM3, m3] _ [1/1, 39/32, 3/2, 9/5] :: [0, 7, 14, 20] # [7, 7, 6]

[P1, Prm3, PrGrd5, AsGrm7] : [Prm3, Grm3, ReAsM3] _ [1/1, 39/32, 13/9, 11/6] :: [0, 7, 13, 21] # [7, 6, 8]

[P1, Prm3, PrGrd5, Grm7] : [Prm3, Grm3, ReM3] _ [1/1, 39/32, 13/9, 16/9] :: [0, 7, 13, 20] # [7, 6, 7]

[P1, Prm3, PrGrd5, PrDem7] : [Prm3, Grm3, DeAcM3] _ [1/1, 39/32, 13/9, 39/22] :: [0, 7, 13, 20] # [7, 6, 7]

[P1, Prm3, PrGrd5, PrGrd7] : [Prm3, Grm3, m3] _ [1/1, 39/32, 13/9, 26/15] :: [0, 7, 13, 19] # [7, 6, 6]

[P1, Prm3, PrGrd5, PrGrm7] : [Prm3, Grm3, M3] _ [1/1, 39/32, 13/9, 65/36] :: [0, 7, 13, 21] # [7, 6, 8]

[P1, Prm3, PrGrd5, PrSbGrd7] : [Prm3, Grm3, Sbm3] _ [1/1, 39/32, 13/9, 91/54] :: [0, 7, 13, 18] # [7, 6, 5]

[P1, Prm3, PrSbd5, PrSbGrd7] : [Prm3, Sbm3, Grm3] _ [1/1, 39/32, 91/64, 91/54] :: [0, 7, 12, 18] # [7, 5, 6]

[P1, Prm3, PrSbd5, Sbm7] : [Prm3, Sbm3, ReM3] _ [1/1, 39/32, 91/64, 7/4] :: [0, 7, 12, 19] # [7, 5, 7]

[P1, ReAsM3, P5, AsGrm7] : [ReAsM3, PrDem3, AsGrm3] _ [1/1, 33/26, 3/2, 11/6] :: [0, 8, 14, 21] # [8, 6, 7]

[P1, ReAsM3, P5, DeAcM7] : [ReAsM3, PrDem3, DeAcM3] _ [1/1, 33/26, 3/2, 81/44] :: [0, 8, 14, 21] # [8, 6, 7]

[P1, ReAsM3, P5, Grm7] : [ReAsM3, PrDem3, Grm3] _ [1/1, 33/26, 3/2, 16/9] :: [0, 8, 14, 20] # [8, 6, 6]

[P1, ReAsM3, P5, M7] : [ReAsM3, PrDem3, M3] _ [1/1, 33/26, 3/2, 15/8] :: [0, 8, 14, 22] # [8, 6, 8]

[P1, ReAsM3, P5, PrDem7] : [ReAsM3, PrDem3, PrDem3] _ [1/1, 33/26, 3/2, 39/22] :: [0, 8, 14, 20] # [8, 6, 6]

[P1, ReAsM3, P5, ReAsM7] : [ReAsM3, PrDem3, ReAsM3] _ [1/1, 33/26, 3/2, 99/52] :: [0, 8, 14, 22] # [8, 6, 8]

[P1, ReAsM3, P5, ReM7] : [ReAsM3, PrDem3, ReM3] _ [1/1, 33/26, 3/2, 24/13] :: [0, 8, 14, 21] # [8, 6, 7]

[P1, ReAsM3, P5, Sbm7] : [ReAsM3, PrDem3, Sbm3] _ [1/1, 33/26, 3/2, 7/4] :: [0, 8, 14, 19] # [8, 6, 5]

[P1, ReAsM3, P5, m7] : [ReAsM3, PrDem3, m3] _ [1/1, 33/26, 3/2, 9/5] :: [0, 8, 14, 20] # [8, 6, 6]

[P1, ReAsM3, ReAs5, ReAsM7] : [ReAsM3, m3, M3] _ [1/1, 33/26, 99/65, 99/52] :: [0, 8, 14, 22] # [8, 6, 8]

[P1, ReAsM3, ReAs5, m7] : [ReAsM3, m3, PrDem3] _ [1/1, 33/26, 99/65, 9/5] :: [0, 8, 14, 20] # [8, 6, 6]

[P1, ReAsM3, ReAsSb5, Sbm7] : [ReAsM3, Sbm3, PrDem3] _ [1/1, 33/26, 77/52, 7/4] :: [0, 8, 13, 19] # [8, 5, 6]

[P1, ReM3, De5, DeAcM7] : [ReM3, PrDem3, AcM3] _ [1/1, 16/13, 16/11, 81/44] :: [0, 7, 13, 21] # [7, 6, 8]

[P1, ReM3, De5, DeM7] : [ReM3, PrDem3, M3] _ [1/1, 16/13, 16/11, 20/11] :: [0, 7, 13, 21] # [7, 6, 8]

[P1, ReM3, De5, DeSbm7] : [ReM3, PrDem3, Sbm3] _ [1/1, 16/13, 16/11, 56/33] :: [0, 7, 13, 18] # [7, 6, 5]

[P1, ReM3, De5, Dem7] : [ReM3, PrDem3, m3] _ [1/1, 16/13, 16/11, 96/55] :: [0, 7, 13, 19] # [7, 6, 6]

[P1, ReM3, De5, Grm7] : [ReM3, PrDem3, AsGrm3] _ [1/1, 16/13, 16/11, 16/9] :: [0, 7, 13, 20] # [7, 6, 7]

[P1, ReM3, De5, PrDem7] : [ReM3, PrDem3, Prm3] _ [1/1, 16/13, 16/11, 39/22] :: [0, 7, 13, 20] # [7, 6, 7]

[P1, ReM3, De5, ReM7] : [ReM3, PrDem3, ReAsM3] _ [1/1, 16/13, 16/11, 24/13] :: [0, 7, 13, 21] # [7, 6, 8]

[P1, ReM3, P5, AsGrm7] : [ReM3, Prm3, AsGrm3] _ [1/1, 16/13, 3/2, 11/6] :: [0, 7, 14, 21] # [7, 7, 7]

[P1, ReM3, P5, DeAcM7] : [ReM3, Prm3, DeAcM3] _ [1/1, 16/13, 3/2, 81/44] :: [0, 7, 14, 21] # [7, 7, 7]

[P1, ReM3, P5, Grm7] : [ReM3, Prm3, Grm3] _ [1/1, 16/13, 3/2, 16/9] :: [0, 7, 14, 20] # [7, 7, 6]

[P1, ReM3, P5, PrDem7] : [ReM3, Prm3, PrDem3] _ [1/1, 16/13, 3/2, 39/22] :: [0, 7, 14, 20] # [7, 7, 6]

[P1, ReM3, P5, ReM7] : [ReM3, Prm3, ReM3] _ [1/1, 16/13, 3/2, 24/13] :: [0, 7, 14, 21] # [7, 7, 7]

[P1, ReM3, P5, Sbm7] : [ReM3, Prm3, Sbm3] _ [1/1, 16/13, 3/2, 7/4] :: [0, 7, 14, 19] # [7, 7, 5]

[P1, ReM3, P5, m7] : [ReM3, Prm3, m3] _ [1/1, 16/13, 3/2, 9/5] :: [0, 7, 14, 20] # [7, 7, 6]

[P1, ReM3, Re5, Dem7] : [ReM3, m3, PrDem3] _ [1/1, 16/13, 96/65, 96/55] :: [0, 7, 13, 19] # [7, 6, 6]

[P1, ReM3, Re5, ReM7] : [ReM3, m3, M3] _ [1/1, 16/13, 96/65, 24/13] :: [0, 7, 13, 21] # [7, 6, 8]

[P1, ReM3, Re5, m7] : [ReM3, m3, Prm3] _ [1/1, 16/13, 96/65, 9/5] :: [0, 7, 13, 20] # [7, 6, 7]

[P1, ReM3, ReSb5, DeSbm7] : [ReM3, Sbm3, PrDem3] _ [1/1, 16/13, 56/39, 56/33] :: [0, 7, 12, 18] # [7, 5, 6]

[P1, ReM3, ReSb5, ReSbM7] : [ReM3, Sbm3, M3] _ [1/1, 16/13, 56/39, 70/39] :: [0, 7, 12, 20] # [7, 5, 8]

[P1, ReM3, ReSb5, Sbm7] : [ReM3, Sbm3, Prm3] _ [1/1, 16/13, 56/39, 7/4] :: [0, 7, 12, 19] # [7, 5, 7]

[P1, Sbm3, AsSbGrd5, AsSbGrd7] : [Sbm3, AsGrm3, m3] _ [1/1, 7/6, 77/54, 77/45] :: [0, 5, 12, 18] # [5, 7, 6]

[P1, Sbm3, AsSbGrd5, PrSbGrd7] : [Sbm3, AsGrm3, PrDem3] _ [1/1, 7/6, 77/54, 91/54] :: [0, 5, 12, 18] # [5, 7, 6]

[P1, Sbm3, AsSbGrd5, Sbm7] : [Sbm3, AsGrm3, DeAcM3] _ [1/1, 7/6, 77/54, 7/4] :: [0, 5, 12, 19] # [5, 7, 7]

[P1, Sbm3, DeSbAc5, DeSbm7] : [Sbm3, DeAcM3, Grm3] _ [1/1, 7/6, 63/44, 56/33] :: [0, 5, 12, 18] # [5, 7, 6]

[P1, Sbm3, DeSbAc5, Sbm7] : [Sbm3, DeAcM3, AsGrm3] _ [1/1, 7/6, 63/44, 7/4] :: [0, 5, 12, 19] # [5, 7, 7]

[P1, Sbm3, PrDeSbd5, DeSbm7] : [Sbm3, PrDem3, ReM3] _ [1/1, 7/6, 91/66, 56/33] :: [0, 5, 11, 18] # [5, 6, 7]

[P1, Sbm3, PrDeSbd5, PrDeSbd7] : [Sbm3, PrDem3, m3] _ [1/1, 7/6, 91/66, 91/55] :: [0, 5, 11, 17] # [5, 6, 6]

[P1, Sbm3, PrDeSbd5, PrSbGrd7] : [Sbm3, PrDem3, AsGrm3] _ [1/1, 7/6, 91/66, 91/54] :: [0, 5, 11, 18] # [5, 6, 7]

[P1, Sbm3, PrDeSbd5, Sbm7] : [Sbm3, PrDem3, ReAsM3] _ [1/1, 7/6, 91/66, 7/4] :: [0, 5, 11, 19] # [5, 6, 8]

[P1, Sbm3, PrSbd5, PrSbGrd7] : [Sbm3, Prm3, Grm3] _ [1/1, 7/6, 91/64, 91/54] :: [0, 5, 12, 18] # [5, 7, 6]

[P1, Sbm3, PrSbd5, Sbm7] : [Sbm3, Prm3, ReM3] _ [1/1, 7/6, 91/64, 7/4] :: [0, 5, 12, 19] # [5, 7, 7]

[P1, Sbm3, ReAsSb5, Sbm7] : [Sbm3, ReAsM3, PrDem3] _ [1/1, 7/6, 77/52, 7/4] :: [0, 5, 13, 19] # [5, 8, 6]

[P1, Sbm3, ReSb5, DeSbm7] : [Sbm3, ReM3, PrDem3] _ [1/1, 7/6, 56/39, 56/33] :: [0, 5, 12, 18] # [5, 7, 6]

[P1, Sbm3, ReSb5, Sbm7] : [Sbm3, ReM3, Prm3] _ [1/1, 7/6, 56/39, 7/4] :: [0, 5, 12, 19] # [5, 7, 7]

[P1, Sbm3, Sb5, Sbm7] : [Sbm3, M3, m3] _ [1/1, 7/6, 35/24, 7/4] :: [0, 5, 13, 19] # [5, 8, 6]

[P1, Sbm3, Sbd5, AsSbGrd7] : [Sbm3, m3, AsGrm3] _ [1/1, 7/6, 7/5, 77/45] :: [0, 5, 11, 18] # [5, 6, 7]

[P1, Sbm3, Sbd5, PrDeSbd7] : [Sbm3, m3, PrDem3] _ [1/1, 7/6, 7/5, 91/55] :: [0, 5, 11, 17] # [5, 6, 6]

[P1, Sbm3, Sbd5, SbSbd7] : [Sbm3, m3, Sbm3] _ [1/1, 7/6, 7/5, 49/30] :: [0, 5, 11, 16] # [5, 6, 5]

[P1, Sbm3, Sbd5, Sbd7] : [Sbm3, m3, m3] _ [1/1, 7/6, 7/5, 42/25] :: [0, 5, 11, 17] # [5, 6, 6]

[P1, Sbm3, Sbd5, Sbm7] : [Sbm3, m3, M3] _ [1/1, 7/6, 7/5, 7/4] :: [0, 5, 11, 19] # [5, 6, 8]

[P1, m3, AsGrd5, AsGrd7] : [m3, AsGrm3, m3] _ [1/1, 6/5, 22/15, 44/25] :: [0, 6, 13, 19] # [6, 7, 6]

[P1, m3, AsGrd5, AsSbGrd7] : [m3, AsGrm3, Sbm3] _ [1/1, 6/5, 22/15, 77/45] :: [0, 6, 13, 18] # [6, 7, 5]

[P1, m3, AsGrd5, PrGrd7] : [m3, AsGrm3, PrDem3] _ [1/1, 6/5, 22/15, 26/15] :: [0, 6, 13, 19] # [6, 7, 6]

[P1, m3, AsGrd5, m7] : [m3, AsGrm3, DeAcM3] _ [1/1, 6/5, 22/15, 9/5] :: [0, 6, 13, 20] # [6, 7, 7]

[P1, m3, DeAc5, Dem7] : [m3, DeAcM3, Grm3] _ [1/1, 6/5, 81/55, 96/55] :: [0, 6, 13, 19] # [6, 7, 6]

[P1, m3, DeAc5, m7] : [m3, DeAcM3, AsGrm3] _ [1/1, 6/5, 81/55, 9/5] :: [0, 6, 13, 20] # [6, 7, 7]

[P1, m3, Grd5, Dem7] : [m3, Grm3, DeAcM3] _ [1/1, 6/5, 64/45, 96/55] :: [0, 6, 12, 19] # [6, 6, 7]

[P1, m3, Grd5, Grm7] : [m3, Grm3, M3] _ [1/1, 6/5, 64/45, 16/9] :: [0, 6, 12, 20] # [6, 6, 8]

[P1, m3, Grd5, PrGrd7] : [m3, Grm3, Prm3] _ [1/1, 6/5, 64/45, 26/15] :: [0, 6, 12, 19] # [6, 6, 7]

[P1, m3, Grd5, m7] : [m3, Grm3, AcM3] _ [1/1, 6/5, 64/45, 9/5] :: [0, 6, 12, 20] # [6, 6, 8]

[P1, m3, P5, Grm7] : [m3, M3, Grm3] _ [1/1, 6/5, 3/2, 16/9] :: [0, 6, 14, 20] # [6, 8, 6]

[P1, m3, P5, M7] : [m3, M3, M3] _ [1/1, 6/5, 3/2, 15/8] :: [0, 6, 14, 22] # [6, 8, 8]

[P1, m3, P5, PrDem7] : [m3, M3, PrDem3] _ [1/1, 6/5, 3/2, 39/22] :: [0, 6, 14, 20] # [6, 8, 6]

[P1, m3, P5, ReAsM7] : [m3, M3, ReAsM3] _ [1/1, 6/5, 3/2, 99/52] :: [0, 6, 14, 22] # [6, 8, 8]

[P1, m3, P5, Sbm7] : [m3, M3, Sbm3] _ [1/1, 6/5, 3/2, 7/4] :: [0, 6, 14, 19] # [6, 8, 5]

[P1, m3, P5, m7] : [m3, M3, m3] _ [1/1, 6/5, 3/2, 9/5] :: [0, 6, 14, 20] # [6, 8, 6]

[P1, m3, PrDed5, Dem7] : [m3, PrDem3, ReM3] _ [1/1, 6/5, 78/55, 96/55] :: [0, 6, 12, 19] # [6, 6, 7]

[P1, m3, PrDed5, PrDeSbd7] : [m3, PrDem3, Sbm3] _ [1/1, 6/5, 78/55, 91/55] :: [0, 6, 12, 17] # [6, 6, 5]

[P1, m3, PrDed5, PrDem7] : [m3, PrDem3, M3] _ [1/1, 6/5, 78/55, 39/22] :: [0, 6, 12, 20] # [6, 6, 8]

[P1, m3, PrDed5, PrGrd7] : [m3, PrDem3, AsGrm3] _ [1/1, 6/5, 78/55, 26/15] :: [0, 6, 12, 19] # [6, 6, 7]

[P1, m3, PrDed5, m7] : [m3, PrDem3, ReAsM3] _ [1/1, 6/5, 78/55, 9/5] :: [0, 6, 12, 20] # [6, 6, 8]

[P1, m3, Re5, Dem7] : [m3, ReM3, PrDem3] _ [1/1, 6/5, 96/65, 96/55] :: [0, 6, 13, 19] # [6, 7, 6]

[P1, m3, Re5, m7] : [m3, ReM3, Prm3] _ [1/1, 6/5, 96/65, 9/5] :: [0, 6, 13, 20] # [6, 7, 7]

[P1, m3, ReAs5, ReAsM7] : [m3, ReAsM3, M3] _ [1/1, 6/5, 99/65, 99/52] :: [0, 6, 14, 22] # [6, 8, 8]

[P1, m3, ReAs5, m7] : [m3, ReAsM3, PrDem3] _ [1/1, 6/5, 99/65, 9/5] :: [0, 6, 14, 20] # [6, 8, 6]

[P1, m3, Sbd5, AsSbGrd7] : [m3, Sbm3, AsGrm3] _ [1/1, 6/5, 7/5, 77/45] :: [0, 6, 11, 18] # [6, 5, 7]

[P1, m3, Sbd5, PrDeSbd7] : [m3, Sbm3, PrDem3] _ [1/1, 6/5, 7/5, 91/55] :: [0, 6, 11, 17] # [6, 5, 6]

[P1, m3, Sbd5, Sbd7] : [m3, Sbm3, m3] _ [1/1, 6/5, 7/5, 42/25] :: [0, 6, 11, 17] # [6, 5, 6]

[P1, m3, Sbd5, Sbm7] : [m3, Sbm3, M3] _ [1/1, 6/5, 7/5, 7/4] :: [0, 6, 11, 19] # [6, 5, 8]

[P1, m3, d5, AsGrd7] : [m3, m3, AsGrm3] _ [1/1, 6/5, 36/25, 44/25] :: [0, 6, 12, 19] # [6, 6, 7]

[P1, m3, d5, Sbd7] : [m3, m3, Sbm3] _ [1/1, 6/5, 36/25, 42/25] :: [0, 6, 12, 17] # [6, 6, 5]

[P1, m3, d5, m7] : [m3, m3, M3] _ [1/1, 6/5, 36/25, 9/5] :: [0, 6, 12, 20] # [6, 6, 8]

[P1, PrDem3, PrGrd5] : [PrDem3, AsGrm3] _ [1/1, 13/11, 13/9] :: [0, 6, 13] # [6, 7]

[P1, m3, P5] : [m3, M3] _ [1/1, 6/5, 3/2] :: [0, 6, 14] # [6, 8]

[P1, ReAsM3, ReAs5] : [ReAsM3, m3] _ [1/1, 33/26, 99/65] :: [0, 8, 14] # [8, 6]

[P1, Grm3, De5] : [Grm3, DeAcM3] _ [1/1, 32/27, 16/11] :: [0, 6, 13] # [6, 7]

[P1, ReM3, Re5] : [ReM3, m3] _ [1/1, 16/13, 96/65] :: [0, 7, 13] # [7, 6]

[P1, M3, A5] : [M3, M3] _ [1/1, 5/4, 25/16] :: [0, 8, 16] # [8, 8]

[P1, Grm3, P5] : [Grm3, AcM3] _ [1/1, 32/27, 3/2] :: [0, 6, 14] # [6, 8]

[P1, Sbm3, ReAsSb5] : [Sbm3, ReAsM3] _ [1/1, 7/6, 77/52] :: [0, 5, 13] # [5, 8]

[P1, ReM3, De5] : [ReM3, PrDem3] _ [1/1, 16/13, 16/11] :: [0, 7, 13] # [7, 6]

[P1, m3, AsGrd5] : [m3, AsGrm3] _ [1/1, 6/5, 22/15] :: [0, 6, 13] # [6, 7]

[P1, m3, DeAc5] : [m3, DeAcM3] _ [1/1, 6/5, 81/55] :: [0, 6, 13] # [6, 7]

[P1, ReM3, ReSb5] : [ReM3, Sbm3] _ [1/1, 16/13, 56/39] :: [0, 7, 12] # [7, 5]

[P1, PrDem3, P5] : [PrDem3, ReAsM3] _ [1/1, 13/11, 3/2] :: [0, 6, 14] # [6, 8]

[P1, AsGrm3, AsGrd5] : [AsGrm3, m3] _ [1/1, 11/9, 22/15] :: [0, 7, 13] # [7, 6]

[P1, m3, Re5] : [m3, ReM3] _ [1/1, 6/5, 96/65] :: [0, 6, 13] # [6, 7]

[P1, m3, ReAs5] : [m3, ReAsM3] _ [1/1, 6/5, 99/65] :: [0, 6, 14] # [6, 8]

[P1, Sbm3, Sbd5] : [Sbm3, m3] _ [1/1, 7/6, 7/5] :: [0, 5, 11] # [5, 6]

[P1, m3, Sbd5] : [m3, Sbm3] _ [1/1, 6/5, 7/5] :: [0, 6, 11] # [6, 5]

[P1, Sbm3, ReSb5] : [Sbm3, ReM3] _ [1/1, 7/6, 56/39] :: [0, 5, 12] # [5, 7]

[P1, M3, P5] : [M3, m3] _ [1/1, 5/4, 3/2] :: [0, 8, 14] # [8, 6]

[P1, Prm3, PrSbd5] : [Prm3, Sbm3] _ [1/1, 39/32, 91/64] :: [0, 7, 12] # [7, 5]

[P1, ReM3, P5] : [ReM3, Prm3] _ [1/1, 16/13, 3/2] :: [0, 7, 14] # [7, 7]

[P1, m3, Grd5] : [m3, Grm3] _ [1/1, 6/5, 64/45] :: [0, 6, 12] # [6, 6]

[P1, M3, Gr5] : [M3, Grm3] _ [1/1, 5/4, 40/27] :: [0, 8, 14] # [8, 6]

[P1, Prm3, P5] : [Prm3, ReM3] _ [1/1, 39/32, 3/2] :: [0, 7, 14] # [7, 7]

[P1, AcM3, P5] : [AcM3, Grm3] _ [1/1, 81/64, 3/2] :: [0, 8, 14] # [8, 6]

[P1, M3, PrDe5] : [M3, PrDem3] _ [1/1, 5/4, 65/44] :: [0, 8, 14] # [8, 6]

[P1, PrDem3, De5] : [PrDem3, ReM3] _ [1/1, 13/11, 16/11] :: [0, 6, 13] # [6, 7]

[P1, DeAcM3, DeSbAc5] : [DeAcM3, Sbm3] _ [1/1, 27/22, 63/44] :: [0, 7, 12] # [7, 5]

[P1, AsGrm3, AsSbGrd5] : [AsGrm3, Sbm3] _ [1/1, 11/9, 77/54] :: [0, 7, 12] # [7, 5]

[P1, Grm3, Gr5] : [Grm3, M3] _ [1/1, 32/27, 40/27] :: [0, 6, 14] # [6, 8]

[P1, M3, Sb5] : [M3, Sbm3] _ [1/1, 5/4, 35/24] :: [0, 8, 13] # [8, 5]

[P1, AsGrm3, P5] : [AsGrm3, DeAcM3] _ [1/1, 11/9, 3/2] :: [0, 7, 14] # [7, 7]

[P1, Sbm3, PrSbd5] : [Sbm3, Prm3] _ [1/1, 7/6, 91/64] :: [0, 5, 12] # [5, 7]

[P1, DeAcM3, P5] : [DeAcM3, AsGrm3] _ [1/1, 27/22, 3/2] :: [0, 7, 14] # [7, 7]

[P1, PrDem3, PrDeSbd5] : [PrDem3, Sbm3] _ [1/1, 13/11, 91/66] :: [0, 6, 11] # [6, 5]

[P1, Sbm3, AsSbGrd5] : [Sbm3, AsGrm3] _ [1/1, 7/6, 77/54] :: [0, 5, 12] # [5, 7]

[P1, AsGrm3, PrGrd5] : [AsGrm3, PrDem3] _ [1/1, 11/9, 13/9] :: [0, 7, 13] # [7, 6]

[P1, Grm3, Grd5] : [Grm3, m3] _ [1/1, 32/27, 64/45] :: [0, 6, 12] # [6, 6]

[P1, DeAcM3, DeAc5] : [DeAcM3, m3] _ [1/1, 27/22, 81/55] :: [0, 7, 13] # [7, 6]

[P1, DeAcM3, De5] : [DeAcM3, Grm3] _ [1/1, 27/22, 16/11] :: [0, 7, 13] # [7, 6]

[P1, m3, PrDed5] : [m3, PrDem3] _ [1/1, 6/5, 78/55] :: [0, 6, 12] # [6, 6]

[P1, Sbm3, Sb5] : [Sbm3, M3] _ [1/1, 7/6, 35/24] :: [0, 5, 13] # [5, 8]

[P1, Prm3, PrGrd5] : [Prm3, Grm3] _ [1/1, 39/32, 13/9] :: [0, 7, 13] # [7, 6]

[P1, Sbm3, DeSbAc5] : [Sbm3, DeAcM3] _ [1/1, 7/6, 63/44] :: [0, 5, 12] # [5, 7]

[P1, PrDem3, PrDe5] : [PrDem3, M3] _ [1/1, 13/11, 65/44] :: [0, 6, 14] # [6, 8]

[P1, PrDem3, PrDed5] : [PrDem3, m3] _ [1/1, 13/11, 78/55] :: [0, 6, 12] # [6, 6]

[P1, Grm3, PrGrd5] : [Grm3, Prm3] _ [1/1, 32/27, 13/9] :: [0, 6, 13] # [6, 7]

[P1, m3, d5] : [m3, m3] _ [1/1, 6/5, 36/25] :: [0, 6, 12] # [6, 6]

[P1, ReAsM3, P5] : [ReAsM3, PrDem3] _ [1/1, 33/26, 3/2] :: [0, 8, 14] # [8, 6]

[P1, ReAsM3, ReAsSb5] : [ReAsM3, Sbm3] _ [1/1, 33/26, 77/52] :: [0, 8, 13] # [8, 5]

[P1, Sbm3, PrDeSbd5] : [Sbm3, PrDem3] _ [1/1, 7/6, 91/66] :: [0, 5, 11] # [5, 6]

These actually only fall into 49 distinct chords when tuned in 24-EDO:

[0, 8, 14, 21]

[0, 8, 14, 20]

[0, 8, 14, 22]

[0, 8, 14, 19]

[0, 7, 13, 19]

[0, 7, 13, 21]

[0, 7, 13, 18]

[0, 7, 13, 20]

[0, 7, 12, 18]

[0, 7, 12, 19]

[0, 7, 14, 21]

[0, 7, 14, 20]

[0, 7, 14, 19]

[0, 6, 13, 18]

[0, 6, 13, 19]

[0, 6, 13, 20]

[0, 6, 14, 22]

[0, 6, 14, 20]

[0, 6, 12, 19]

[0, 6, 12, 20]

[0, 6, 14, 19]

[0, 8, 16, 22]

[0, 8, 13, 20]

[0, 8, 13, 19]

[0, 6, 11, 18]

[0, 6, 11, 17]

[0, 6, 11, 19]

[0, 6, 12, 17]

[0, 7, 12, 20]

[0, 5, 12, 18]

[0, 5, 12, 19]

[0, 5, 11, 18]

[0, 5, 11, 17]

[0, 5, 11, 19]

[0, 5, 13, 19]

[0, 5, 11, 16]

[0, 6, 13]

[0, 6, 14]

[0, 8, 14]

[0, 7, 13]

[0, 8, 16]

[0, 5, 13]

[0, 7, 12]

[0, 5, 11]

[0, 6, 11]

[0, 5, 12]

[0, 7, 14]

[0, 6, 12]

[0, 8, 13]

....

Oh, I should also note that I don't check cyclic permutations of chords to look for enemy intervals (from Curt's rule fro chains of friends for constructing good quarter tone chords). I just build them  up by thirds, and if they look good in that form, then I feel free to rotate them. I'm not sure if Curt would like that. Probably not. But I liked how the chords sounded in root position enough that I'm going to keep using them to try to harmonize maqamat. I still have to option to use them in root position, or to only use inversions or voicings that don't have enemy intervals.

...

Fantastic! When I add tonal chords in, all of these maqamat have good tetrads on scale degrees 1 through 7:

'Ajam (Nahawand Ending): [0, 4, 8, 10, 14, 18, 20]

'Ajam (Upper Ajam Ending): [0, 4, 8, 10, 14, 18, 22]

'Iraq: [0, 3, 7, 10, 13, 17, 21]

'Ushaq Masri: [0, 4, 6, 10, 14, 17, 20]

Bayati (Nahawand Ending): [0, 3, 6, 10, 14, 16, 20]

Bayati (Rast Ending): [0, 3, 6, 10, 14, 17, 20]

Dalanshin (descends): [0, 4, 7, 10, 14, 18, 21]

Husayni Ushayran: [0, 3, 6, 10, 13, 16, 20] 

Jiharkah_maqamworld: [0, 3, 7, 11, 14, 17, 21]

Jiharkah_wikipedia: [0, 4, 8, 10, 14, 18, 21]

Kurd: [0, 2, 6, 10, 14, 16, 20]

Lami: [0, 2, 6, 10, 12, 16, 20]

Musta'ar: [0, 5, 7, 11, 13, 17, 21]

Nahawand (Kurd Ending): [0, 4, 6, 10, 14, 16, 20]

Nairuz: [0, 4, 7, 10, 14, 17, 20]

Rast (Nahawand ending): [0, 4, 7, 10, 14, 18, 20]

Rast (Upper Rast ending): [0, 4, 7, 10, 14, 18, 21]

Sikah: [0, 3, 7, 11, 14, 17, 21]

Suzdalara (descends): [0, 4, 7, 10, 14, 18, 20]

Yakah: [0, 4, 7, 10, 14, 17, 20]

...

Ah, I found a typo in my transcription of maqam Jiharkah from MaqamWorld. The B pitch was notated Bb_down, but I transcribed it as Bd instead of Bb or Bbd. Navid from Oud For Guitarists says that both the A and Bb are played flat (relative to 12-TET), with A about 15 cents flat and Bb about 35 to 40 cents flat. Also the Ed is 60 cents flat relative to E, rather than 50. But he calls the pitch Bb, so I guess I will too. That means Bb is a perfect fourth above the tonic of F, or 10\24. The corrected scale, starting on the tonic F instead of the approach note Ed below the tonic, is

     [0, 4, 8, 10, 14, 18, 21] :: [4, 4, 2, 4, 4, 3, 3] # Maqam Jiharkah from Maqam Wolrd

which is the same as on Wikipedia. I'll have to find out who made the scores and audio files on Wikipedia and send some positive reinforcement their way.

...

New plan: I'm just going to find just intonation versions of these maqamat:

    [0, 4, 7, 10, 14, 18, 20]: Rast (Nahawand ending)

    [0, 4, 7, 10, 14, 18, 21]: Rast (Upper Rast ending)

    [0, 3, 6, 10, 14, 16, 20]: Bayati (Nahawand Ending)

    [0, 3, 6, 10, 14, 17, 20]: Bayati (Rast Ending)

    [0, 4, 7, 10, 14, 17, 20]: Nairuz

    [0, 3, 7, 11, 14, 17, 21]: Sikah

    [0, 3, 7, 10, 13, 17, 21]: 'Iraq

that are compatible with Curt Chords. And I'm also just going to use Pythagorean intonation for the even steps.

[P1, 3\24, 7\24, 11\24, P5, 17\24, AcM7]: Sikah

[P1, 3\24, 7\24, P4, 13\24, 17\24, AcM7]: 'Iraq

[P1, 3\24, Grm3, P4, P5, Grm6, Grm7]: Bayati (Nahawand Ending)

[P1, 3\24, Grm3, P4, P5, 17\24, Grm7]: Bayati (Rast Ending)

[P1, AcM2, 7\24, P4, P5, 17\24, Grm7]: Nairuz

[P1, AcM2, 7\24, P4, P5, AcM6, AcM7]: Rast (Upper Rast ending)

[P1, AcM2, 7\24, P4, P5, AcM6, Grm7]: Rast (Nahawand ending)

So we need intonations for

    3\24

    7\24

    11\24

    13\24

    17\24

that make good tetrads on all scale degrees. We can do this. Here's an obvious first guess:

    3\24 -> DeAcM2 # 12/11

    7\24 -> DeAcM3 # 27/22

    11\24 -> As4 # 11/8

    13\24 ->DeAcM6 # 18/11

    17\24 -> DeAcM7 # 81/44

Sadly with this intonation, none of the maqamat have valid triads or tetrads on every scale degree. Bayati (Nahawand ending) has 6/7 valid, or 4/7 valid with the Rast ending. Maqam Rast has 6/7 valid either way, but can't muster a triad on scale degree ^4. Nairuz has three validchords, Sikah only has two, and 'Iraq has none. I should have treated Sikah and 'Iraq differently since they start on neutral tones. The first 3\24 sized interval is the Grm3 complement of the 3\24 sized intervals in maqamat with tonal tonics, and that changes all the absolute intervals that come after. Oops.

...

Let's look at ^4 for Rast and see why we can't get a Curt chord to fit there.

...

24 EDO Arabic Maqamat

 I've shared and analyzed 24-EDO arabic maqamat that were on wikipedia, and done a little joint 24-EDO and 53-EDO determpering analysis of some maqamat from Mohamed Alsiadi, and I've shared pitch classes for maqamat from MaqamWorld, but I don't think I've ever given a 24-EDO analysis of this last set. A few of those maqamat are too confusing for me, but the majority of them are here:

[0, 2, 6, 10, 12, 16, 20, 24]: "Lami",
[0, 2, 6, 10, 14, 16, 20, 24]: "Kurd",
[0, 2, 6, 12, 14, 16, 22, 24]: "Athar Kurd",
[0, 2, 6, 8, 14, 16, 20, 24]: "Saba Zamzam ('Ajam ending)",
[0, 2, 8, 10, 14, 16, 20, 24]: "Hijaz (Nahawand Ending)",
[0, 2, 8, 10, 14, 16, 22, 24]: "Hijazkar (or Shadd 'Araban) (descends) ",
[0, 2, 8, 10, 14, 17, 20, 24]: "Hijaz (Rast Ending)",
[0, 2, 8, 10, 14, 18, 20, 24]: "Zanjaran (descends)",
[0, 3, 6, 10, 12, 18, 20, 24]: "Bayati Shuri",
[0, 3, 6, 10, 14, 16, 20, 24]: "Bayati (Nahawand Ending)",
[0, 3, 6, 10, 14, 17, 20, 24]: "Bayati (Rast Ending)",
[0, 3, 6, 8, 14, 16, 20, 24]: "Saba ('Ajam ending)",
[0, 3, 7, 10, 12, 14, 17, 21, 24]: "Sikah Baladi (descends)",
[0, 3, 7, 10, 13, 17, 21, 24]: "'Iraq",
[0, 3, 7, 11, 14, 17, 21, 24]: "Jiharkah",
[0, 3, 7, 11, 14, 17, 21, 24]: "Sikah",
[0, 3, 7, 9, 15, 17, 21, 24]: "Huzam",
[0, 3, 7, 9, 15, 17, 23, 24]: "Awj ‘Iraq (descends)",
[0, 4, 6, 10, 12, 18, 20, 24]: "Nahawand Murassa'",
[0, 4, 6, 10, 14, 16, 20, 24]: "Nahawand (Kurd Ending)",
[0, 4, 6, 10, 14, 16, 22, 24]: "Nahawand (Hijaz Ending)",
[0, 4, 6, 10, 14, 16, 22, 24]: "Nawa Athar",
[0, 4, 6, 10, 14, 17, 20, 24]: "'Ushaq Masri",
[0, 4, 6, 12, 14, 18, 20, 24]: "Nikriz (descends)",
[0, 4, 7, 10, 14, 16, 22, 24]: "Suznak",
[0, 4, 7, 10, 14, 17, 20, 24]: "Nairuz",
[0, 4, 7, 10, 14, 17, 20, 24]: "Yakah",
[0, 4, 7, 10, 14, 18, 20, 24]: "Rast (Nahawand ending)",
[0, 4, 7, 10, 14, 18, 20, 24]: "Suzdalara (descends)",
[0, 4, 7, 10, 14, 18, 21, 24]: "Dalanshin (descends)",
[0, 4, 7, 10, 14, 18, 21, 24]: "Rast (Upper Rast ending)",
[0, 4, 7, 10, 14, 18, 22, 24]: "Mahur",
[0, 4, 8, 10, 14, 16, 22, 24]: "Shawq Afza",
[0, 4, 8, 10, 14, 18, 20, 24]: "'Ajam (Nahawand Ending)",
[0, 4, 8, 10, 14, 18, 22, 24]: "'Ajam 'Ushayran (descends)",
[0, 4, 8, 10, 14, 18, 22, 24]: "'Ajam (Upper Ajam Ending)",
[0, 5, 7, 11, 13, 17, 21, 24]: "Musta'ar",
[0, 6, 7, 10, 14, 18, 21, 24]: "Sazkar (descends)",

And here are the few from wikipedia for comparison:

[0, 2, 8, 10, 14, 16, 20, 24]: "Hijaz",

[0, 4, 6, 12, 14, 16, 22, 24]: "Nawa Athar",

[0, 2, 8, 10, 14, 16, 22, 24]: "Shad Araban",

[0, 3, 6, 10, 14, 16, 20, 24]: "Bayati",

[0, 4, 8, 10, 14, 18, 21, 24]: "Jiharkah",

[0, 3, 7, 9, 15, 17, 21, 24]: "Huzam",

[0, 3, 7, 9, 15, 17, 21, 24]: "Rahat al-Arwah",

[0, 3, 6, 8, 14, 16, 20, 24]: "Saba",

[0, 4, 7, 10, 14, 18, 21, 24]: "Rast",

[0, 3, 6, 10, 13, 16, 20, 24]: "Husayni Ushayran",

Husayni Ushayran and Rahat al-Arwah only appear in the second set. And the second set has a differnt Jiharkah:

[0, 4, 8, 10, 14, 18, 21, 24]: "Jiharkah_wikipedia",

[0, 3, 7, 11, 14, 17, 21, 24]: "Jiharkah_maqamworld",

And that might be my fault with transcription, I'll have to look into it.

Other than that, the sources agree.

The Belt 7 Breakdown

Formation: Square

Moves to explain:

<Turn As A Couple>: Promenade position, lark walks forward, robin walks backward, go halfway round.

<Courtesy Turn> Lark catches robin by the right hand, left hand the robin’s back, lark turns counterclockwise and brings the robin around.

Look in _,

Look out _, Larks roll your corners  

_ _, Look in _, 

Look out _, Larks roll your corners

_ _, Look in _, 

Look out _, Robins roll your corners  

_ _, Look in _, 

Look out _, Larks to the middle, Drop 

Hands with your partner, Larks turn the tent  

_ _, Turn as a couple,

Halfway ‘round, Robins turn the tent, 

_ _, Turn as a couple,

Halfway ‘round, Larks turn the tent, 

_ _, Turn as a couple,

Halfway ‘round, Robins turn the tent

_ _ Catch your, robins by the right and

Courtesy home, Look in, _ 

Quartertone Harmony Chords

Quartertone Harmony is the youtube channel of a 24-EDO music theorist and composer. Curt's his name. He's so cute. He has a bunny named Poopy. I've wanted to do a deep dive into his methods for a long time. We're going to start with his first video.

He talks about a method for building 24-edo chords. Here are the rules.

Rule 1. Every chord has to have a chain of friends connecting every note in the chord. It's not totally clear to me if you can have three friends hanging off one one friend - the chain has branches - of if the chain is a straight line. It's also not clear to me if the chain has to form a circle, or if the ends can be disconnected. I went with a single-linked disconnected chain when generating chords in his style.

Curt says that friends are connected by a major third, minor third, neutral third, or "harmonic second", which is supposed the be the interval between G natural and A half sharp. He also says that the harmonic second is the inverse of the harmonic seventh. To me the inverse of an interval X is

    P1 - X

but he's talking abou the octave complement

    P8 - X

And that's fine. In my notation, the harmonic seventh is a sub-minor seventh, Sbm7, with a just tuning of 7/4, and its complement is a super major second, SpM2, with a just tuning of 8/7.

The intervals that can connect friends in Curt's system have these 24-EDO tunings:

    [SpM2, m3, n3, M3] -> [5, 6, 7, 8]

Unfortunately, Curt equivocates a little bit between EDO steps and intervals. For example, he says that the 11th harmonic and the 13th harmonic are separated by a minor third. In my notation, the interval between the 13th and 11th harmonics is a prominent descendent minor third, PrDem3, with a just frequency ratio of 13/11. It's true that this has "m3" at the end, so it's a kind of minor third, and they're both tuned by 24-EDO to 6\24 steps, but they're not the same. So when Curt says that two intervals can be friends if they're separated by a minor third, I don't interpret that intervallically: he just means they're friends if they're separated by 6 steps of 24-EDO.

Because of this, and because I like tertian chords spelled by thirds, I use Sbm3 instead of SpM2 for a 5\24-sized interval when I'm building up chords in Curt's style. Maybe I should use both, but I don't.

In fact, I use a bunch of third intervals that I think Curt wouldn't mind, since they have the right size in 24-EDO.

5\24 - Sbm3 # 7/6

6\24 - Grm3 # 32/27

6\24 - m3 # 6/5

6\24 - PrDem3 # 13/11

8\24 - ReAsM3 # 33/26

7\24 - AsGrm3 # 11/9

7\24 - DeAcM3 # 27/22

7\24 - Prm3 # 39/32

7\24 - ReM3 # 16/13

8\24 - M3 # 5/4

8\24 - AcM3 # 81/64 

It's really odd to me that Curt doesn't use a 9-step super-major third, SpM3, with a just tuning of 9/7. The sub-minor third and the super-major third are like the two core sounds of 7-limit just intonation, and I would never have thought to make a 13-limit interpretation of 24-EDO harmony that didn't include that sound. But this is Curt's method, mostly, and we shall continue in this vein.

Rule 2. No note can have an enemy. An enemy is a note separate from a target note by 1 quarter tone or (sometimes) 9 quarter tones or 15 quarter tones. I don't know what he means by "sometimes", so I just ruled out all chords that had notes separated by 9 or 15 quartone intervals. No two of my chain-of-friend intervals can sum to form 1 or 9, so that's not a problem. So really this just means that we can't have consecutive relative intervals of size (8, 7)\24 or (7, 8)\24. If I'd had a branching chain of friends, the 1\24 and 9\24 steps might have been a problem.

Rule 3. No crowding. No note can have more than one other (?note that is?) closer than a major second. This rule is another reason why I think Curt might be okay with a branched-chain of friends - if there's a single-linked chain of friends all connected by thirds of size 5\24 or more, you're never going to have crowding, so there wouldn't be a reason to mention this. Even if we use Curt's harmonic second, SpM2, that's not smaller than a major second, so you could in principle have a note with notes on either side surrounded by SpM2. I confess that I mishead this one when I first starded coding up Curt's method: I thought he considered 5\24-sized intervals on either side of a note to be crowded, so I was deleting any chord I generated with consecutive (5, 5)\24 sized intervals. Which is just more restricive, it's not really a problem.

Those are all the rules. Well, he also says that you should use a string timbre, but that's not a principle of chord construction. And I particularly like using clarinet, tuba, and oud for my microtonal works, so there. Let's see what we can make from these rules! When generating chords, I also remove any chord that has an interval with a just tuning with three digits or more in the numerator. I also remove any chords that only have even steps in their 24-EDO tuning, because I think they'll sound too much like 12-TET.

I came up with 161 chords. They *all* sound good. I must be really starved for microtones. How can they all sound so good? Some of them have the same 24-EDO tunings, but many of those still have very distinct sonic characteristics in their just tunings. I'm really pretty shocked about how good these sound. Like, I have tried to make 11-limit and 13-limit music using similar principles and totally failed to find anything this nice.

[0, 5, 11, 16] _ [P1, Sbm3, Sbd5, SbSbd7] - [1/1, 7/6, 7/5, 49/30]

[0, 5, 11, 17] _ [P1, Sbm3, PrDeSbd5, PrDeSbd7] - [1/1, 7/6, 91/66, 91/55]

[0, 5, 11, 17] _ [P1, Sbm3, Sbd5, PrDeSbd7] - [1/1, 7/6, 7/5, 91/55]

[0, 5, 11, 17] _ [P1, Sbm3, Sbd5, Sbd7] - [1/1, 7/6, 7/5, 42/25]

[0, 5, 11, 18] _ [P1, Sbm3, PrDeSbd5, DeSbm7] - [1/1, 7/6, 91/66, 56/33]

[0, 5, 11, 18] _ [P1, Sbm3, PrDeSbd5, PrSbGrd7] - [1/1, 7/6, 91/66, 91/54]

[0, 5, 11, 18] _ [P1, Sbm3, Sbd5, AsSbGrd7] - [1/1, 7/6, 7/5, 77/45]

[0, 5, 11, 19] _ [P1, Sbm3, PrDeSbd5, Sbm7] - [1/1, 7/6, 91/66, 7/4]

[0, 5, 11, 19] _ [P1, Sbm3, Sbd5, Sbm7] - [1/1, 7/6, 7/5, 7/4]

[0, 5, 12, 18] _ [P1, Sbm3, AsSbGrd5, AsSbGrd7] - [1/1, 7/6, 77/54, 77/45]

[0, 5, 12, 18] _ [P1, Sbm3, AsSbGrd5, PrSbGrd7] - [1/1, 7/6, 77/54, 91/54]

[0, 5, 12, 18] _ [P1, Sbm3, DeSbAc5, DeSbm7] - [1/1, 7/6, 63/44, 56/33]

[0, 5, 12, 18] _ [P1, Sbm3, PrSbd5, PrSbGrd7] - [1/1, 7/6, 91/64, 91/54]

[0, 5, 12, 18] _ [P1, Sbm3, ReSb5, DeSbm7] - [1/1, 7/6, 56/39, 56/33]

[0, 5, 12, 19] _ [P1, Sbm3, AsSbGrd5, Sbm7] - [1/1, 7/6, 77/54, 7/4]

[0, 5, 12, 19] _ [P1, Sbm3, DeSbAc5, Sbm7] - [1/1, 7/6, 63/44, 7/4]

[0, 5, 12, 19] _ [P1, Sbm3, PrSbd5, Sbm7] - [1/1, 7/6, 91/64, 7/4]

[0, 5, 12, 19] _ [P1, Sbm3, ReSb5, Sbm7] - [1/1, 7/6, 56/39, 7/4]

[0, 5, 13, 19] _ [P1, Sbm3, ReAsSb5, Sbm7] - [1/1, 7/6, 77/52, 7/4]

[0, 5, 13, 19] _ [P1, Sbm3, Sb5, Sbm7] - [1/1, 7/6, 35/24, 7/4]

[0, 6, 11, 17] _ [P1, PrDem3, PrDeSbd5, PrDeSbd7] - [1/1, 13/11, 91/66, 91/55]

[0, 6, 11, 17] _ [P1, m3, Sbd5, PrDeSbd7] - [1/1, 6/5, 7/5, 91/55]

[0, 6, 11, 17] _ [P1, m3, Sbd5, Sbd7] - [1/1, 6/5, 7/5, 42/25]

[0, 6, 11, 18] _ [P1, PrDem3, PrDeSbd5, DeSbm7] - [1/1, 13/11, 91/66, 56/33]

[0, 6, 11, 18] _ [P1, PrDem3, PrDeSbd5, PrSbGrd7] - [1/1, 13/11, 91/66, 91/54]

[0, 6, 11, 18] _ [P1, m3, Sbd5, AsSbGrd7] - [1/1, 6/5, 7/5, 77/45]

[0, 6, 11, 19] _ [P1, PrDem3, PrDeSbd5, Sbm7] - [1/1, 13/11, 91/66, 7/4]

[0, 6, 11, 19] _ [P1, m3, Sbd5, Sbm7] - [1/1, 6/5, 7/5, 7/4]

[0, 6, 12, 17] _ [P1, PrDem3, PrDed5, PrDeSbd7] - [1/1, 13/11, 78/55, 91/55]

[0, 6, 12, 17] _ [P1, m3, PrDed5, PrDeSbd7] - [1/1, 6/5, 78/55, 91/55]

[0, 6, 12, 17] _ [P1, m3, d5, Sbd7] - [1/1, 6/5, 36/25, 42/25]

[0, 6, 12, 19] _ [P1, Grm3, Grd5, Dem7] - [1/1, 32/27, 64/45, 96/55]

[0, 6, 12, 19] _ [P1, Grm3, Grd5, PrGrd7] - [1/1, 32/27, 64/45, 26/15]

[0, 6, 12, 19] _ [P1, PrDem3, PrDed5, Dem7] - [1/1, 13/11, 78/55, 96/55]

[0, 6, 12, 19] _ [P1, PrDem3, PrDed5, PrGrd7] - [1/1, 13/11, 78/55, 26/15]

[0, 6, 12, 19] _ [P1, m3, Grd5, Dem7] - [1/1, 6/5, 64/45, 96/55]

[0, 6, 12, 19] _ [P1, m3, Grd5, PrGrd7] - [1/1, 6/5, 64/45, 26/15]

[0, 6, 12, 19] _ [P1, m3, PrDed5, Dem7] - [1/1, 6/5, 78/55, 96/55]

[0, 6, 12, 19] _ [P1, m3, PrDed5, PrGrd7] - [1/1, 6/5, 78/55, 26/15]

[0, 6, 12, 19] _ [P1, m3, d5, AsGrd7] - [1/1, 6/5, 36/25, 44/25]

[0, 6, 13, 18] _ [P1, Grm3, De5, DeSbm7] - [1/1, 32/27, 16/11, 56/33]

[0, 6, 13, 18] _ [P1, Grm3, PrGrd5, PrSbGrd7] - [1/1, 32/27, 13/9, 91/54]

[0, 6, 13, 18] _ [P1, PrDem3, De5, DeSbm7] - [1/1, 13/11, 16/11, 56/33]

[0, 6, 13, 18] _ [P1, PrDem3, PrGrd5, PrSbGrd7] - [1/1, 13/11, 13/9, 91/54]

[0, 6, 13, 18] _ [P1, m3, AsGrd5, AsSbGrd7] - [1/1, 6/5, 22/15, 77/45]

[0, 6, 13, 19] _ [P1, Grm3, De5, Dem7] - [1/1, 32/27, 16/11, 96/55]

[0, 6, 13, 19] _ [P1, Grm3, PrGrd5, PrGrd7] - [1/1, 32/27, 13/9, 26/15]

[0, 6, 13, 19] _ [P1, PrDem3, De5, Dem7] - [1/1, 13/11, 16/11, 96/55]

[0, 6, 13, 19] _ [P1, PrDem3, PrGrd5, PrGrd7] - [1/1, 13/11, 13/9, 26/15]

[0, 6, 13, 19] _ [P1, m3, AsGrd5, AsGrd7] - [1/1, 6/5, 22/15, 44/25]

[0, 6, 13, 19] _ [P1, m3, AsGrd5, PrGrd7] - [1/1, 6/5, 22/15, 26/15]

[0, 6, 13, 19] _ [P1, m3, DeAc5, Dem7] - [1/1, 6/5, 81/55, 96/55]

[0, 6, 13, 19] _ [P1, m3, Re5, Dem7] - [1/1, 6/5, 96/65, 96/55]

[0, 6, 13, 20] _ [P1, Grm3, De5, Grm7] - [1/1, 32/27, 16/11, 16/9]

[0, 6, 13, 20] _ [P1, Grm3, De5, PrDem7] - [1/1, 32/27, 16/11, 39/22]

[0, 6, 13, 20] _ [P1, Grm3, PrGrd5, Grm7] - [1/1, 32/27, 13/9, 16/9]

[0, 6, 13, 20] _ [P1, Grm3, PrGrd5, PrDem7] - [1/1, 32/27, 13/9, 39/22]

[0, 6, 13, 20] _ [P1, PrDem3, De5, Grm7] - [1/1, 13/11, 16/11, 16/9]

[0, 6, 13, 20] _ [P1, PrDem3, De5, PrDem7] - [1/1, 13/11, 16/11, 39/22]

[0, 6, 13, 20] _ [P1, PrDem3, PrGrd5, Grm7] - [1/1, 13/11, 13/9, 16/9]

[0, 6, 13, 20] _ [P1, PrDem3, PrGrd5, PrDem7] - [1/1, 13/11, 13/9, 39/22]

[0, 6, 13, 20] _ [P1, m3, AsGrd5, m7] - [1/1, 6/5, 22/15, 9/5]

[0, 6, 13, 20] _ [P1, m3, DeAc5, m7] - [1/1, 6/5, 81/55, 9/5]

[0, 6, 13, 20] _ [P1, m3, Re5, m7] - [1/1, 6/5, 96/65, 9/5]

[0, 6, 14, 19] _ [P1, Grm3, P5, Sbm7] - [1/1, 32/27, 3/2, 7/4]

[0, 6, 14, 19] _ [P1, PrDem3, P5, Sbm7] - [1/1, 13/11, 3/2, 7/4]

[0, 6, 14, 19] _ [P1, m3, P5, Sbm7] - [1/1, 6/5, 3/2, 7/4]

[0, 7, 12, 18] _ [P1, AsGrm3, AsSbGrd5, AsSbGrd7] - [1/1, 11/9, 77/54, 77/45]

[0, 7, 12, 18] _ [P1, AsGrm3, AsSbGrd5, PrSbGrd7] - [1/1, 11/9, 77/54, 91/54]

[0, 7, 12, 18] _ [P1, DeAcM3, DeSbAc5, DeSbm7] - [1/1, 27/22, 63/44, 56/33]

[0, 7, 12, 18] _ [P1, Prm3, PrSbd5, PrSbGrd7] - [1/1, 39/32, 91/64, 91/54]

[0, 7, 12, 18] _ [P1, ReM3, ReSb5, DeSbm7] - [1/1, 16/13, 56/39, 56/33]

[0, 7, 12, 19] _ [P1, AsGrm3, AsSbGrd5, Sbm7] - [1/1, 11/9, 77/54, 7/4]

[0, 7, 12, 19] _ [P1, DeAcM3, DeSbAc5, Sbm7] - [1/1, 27/22, 63/44, 7/4]

[0, 7, 12, 19] _ [P1, Prm3, PrSbd5, Sbm7] - [1/1, 39/32, 91/64, 7/4]

[0, 7, 12, 19] _ [P1, ReM3, ReSb5, Sbm7] - [1/1, 16/13, 56/39, 7/4]

[0, 7, 12, 20] _ [P1, ReM3, ReSb5, ReSbM7] - [1/1, 16/13, 56/39, 70/39]

[0, 7, 13, 18] _ [P1, AsGrm3, AsGrd5, AsSbGrd7] - [1/1, 11/9, 22/15, 77/45]

[0, 7, 13, 18] _ [P1, AsGrm3, PrGrd5, PrSbGrd7] - [1/1, 11/9, 13/9, 91/54]

[0, 7, 13, 18] _ [P1, DeAcM3, De5, DeSbm7] - [1/1, 27/22, 16/11, 56/33]

[0, 7, 13, 18] _ [P1, Prm3, PrGrd5, PrSbGrd7] - [1/1, 39/32, 13/9, 91/54]

[0, 7, 13, 18] _ [P1, ReM3, De5, DeSbm7] - [1/1, 16/13, 16/11, 56/33]

[0, 7, 13, 19] _ [P1, AsGrm3, AsGrd5, AsGrd7] - [1/1, 11/9, 22/15, 44/25]

[0, 7, 13, 19] _ [P1, AsGrm3, AsGrd5, PrGrd7] - [1/1, 11/9, 22/15, 26/15]

[0, 7, 13, 19] _ [P1, AsGrm3, PrGrd5, PrGrd7] - [1/1, 11/9, 13/9, 26/15]

[0, 7, 13, 19] _ [P1, DeAcM3, De5, Dem7] - [1/1, 27/22, 16/11, 96/55]

[0, 7, 13, 19] _ [P1, DeAcM3, DeAc5, Dem7] - [1/1, 27/22, 81/55, 96/55]

[0, 7, 13, 19] _ [P1, Prm3, PrGrd5, PrGrd7] - [1/1, 39/32, 13/9, 26/15]

[0, 7, 13, 19] _ [P1, ReM3, De5, Dem7] - [1/1, 16/13, 16/11, 96/55]

[0, 7, 13, 19] _ [P1, ReM3, Re5, Dem7] - [1/1, 16/13, 96/65, 96/55]

[0, 7, 13, 20] _ [P1, AsGrm3, AsGrd5, m7] - [1/1, 11/9, 22/15, 9/5]

[0, 7, 13, 20] _ [P1, AsGrm3, PrGrd5, Grm7] - [1/1, 11/9, 13/9, 16/9]

[0, 7, 13, 20] _ [P1, AsGrm3, PrGrd5, PrDem7] - [1/1, 11/9, 13/9, 39/22]

[0, 7, 13, 20] _ [P1, DeAcM3, De5, Grm7] - [1/1, 27/22, 16/11, 16/9]

[0, 7, 13, 20] _ [P1, DeAcM3, De5, PrDem7] - [1/1, 27/22, 16/11, 39/22]

[0, 7, 13, 20] _ [P1, DeAcM3, DeAc5, m7] - [1/1, 27/22, 81/55, 9/5]

[0, 7, 13, 20] _ [P1, Prm3, PrGrd5, Grm7] - [1/1, 39/32, 13/9, 16/9]

[0, 7, 13, 20] _ [P1, Prm3, PrGrd5, PrDem7] - [1/1, 39/32, 13/9, 39/22]

[0, 7, 13, 20] _ [P1, ReM3, De5, Grm7] - [1/1, 16/13, 16/11, 16/9]

[0, 7, 13, 20] _ [P1, ReM3, De5, PrDem7] - [1/1, 16/13, 16/11, 39/22]

[0, 7, 13, 20] _ [P1, ReM3, Re5, m7] - [1/1, 16/13, 96/65, 9/5]

[0, 7, 13, 21] _ [P1, AsGrm3, AsGrd5, AsGrm7] - [1/1, 11/9, 22/15, 11/6]

[0, 7, 13, 21] _ [P1, AsGrm3, PrGrd5, AsGrm7] - [1/1, 11/9, 13/9, 11/6]

[0, 7, 13, 21] _ [P1, AsGrm3, PrGrd5, PrGrm7] - [1/1, 11/9, 13/9, 65/36]

[0, 7, 13, 21] _ [P1, DeAcM3, De5, DeAcM7] - [1/1, 27/22, 16/11, 81/44]

[0, 7, 13, 21] _ [P1, DeAcM3, De5, DeM7] - [1/1, 27/22, 16/11, 20/11]

[0, 7, 13, 21] _ [P1, DeAcM3, De5, ReM7] - [1/1, 27/22, 16/11, 24/13]

[0, 7, 13, 21] _ [P1, DeAcM3, DeAc5, DeAcM7] - [1/1, 27/22, 81/55, 81/44]

[0, 7, 13, 21] _ [P1, Prm3, PrGrd5, AsGrm7] - [1/1, 39/32, 13/9, 11/6]

[0, 7, 13, 21] _ [P1, Prm3, PrGrd5, PrGrm7] - [1/1, 39/32, 13/9, 65/36]

[0, 7, 13, 21] _ [P1, ReM3, De5, DeAcM7] - [1/1, 16/13, 16/11, 81/44]

[0, 7, 13, 21] _ [P1, ReM3, De5, DeM7] - [1/1, 16/13, 16/11, 20/11]

[0, 7, 13, 21] _ [P1, ReM3, De5, ReM7] - [1/1, 16/13, 16/11, 24/13]

[0, 7, 13, 21] _ [P1, ReM3, Re5, ReM7] - [1/1, 16/13, 96/65, 24/13]

[0, 7, 14, 19] _ [P1, AsGrm3, P5, Sbm7] - [1/1, 11/9, 3/2, 7/4]

[0, 7, 14, 19] _ [P1, DeAcM3, P5, Sbm7] - [1/1, 27/22, 3/2, 7/4]

[0, 7, 14, 19] _ [P1, Prm3, P5, Sbm7] - [1/1, 39/32, 3/2, 7/4]

[0, 7, 14, 19] _ [P1, ReM3, P5, Sbm7] - [1/1, 16/13, 3/2, 7/4]

[0, 7, 14, 20] _ [P1, AsGrm3, P5, Grm7] - [1/1, 11/9, 3/2, 16/9]

[0, 7, 14, 20] _ [P1, AsGrm3, P5, PrDem7] - [1/1, 11/9, 3/2, 39/22]

[0, 7, 14, 20] _ [P1, AsGrm3, P5, m7] - [1/1, 11/9, 3/2, 9/5]

[0, 7, 14, 20] _ [P1, DeAcM3, P5, Grm7] - [1/1, 27/22, 3/2, 16/9]

[0, 7, 14, 20] _ [P1, DeAcM3, P5, PrDem7] - [1/1, 27/22, 3/2, 39/22]

[0, 7, 14, 20] _ [P1, DeAcM3, P5, m7] - [1/1, 27/22, 3/2, 9/5]

[0, 7, 14, 20] _ [P1, Prm3, P5, Grm7] - [1/1, 39/32, 3/2, 16/9]

[0, 7, 14, 20] _ [P1, Prm3, P5, PrDem7] - [1/1, 39/32, 3/2, 39/22]

[0, 7, 14, 20] _ [P1, Prm3, P5, m7] - [1/1, 39/32, 3/2, 9/5]

[0, 7, 14, 20] _ [P1, ReM3, P5, Grm7] - [1/1, 16/13, 3/2, 16/9]

[0, 7, 14, 20] _ [P1, ReM3, P5, PrDem7] - [1/1, 16/13, 3/2, 39/22]

[0, 7, 14, 20] _ [P1, ReM3, P5, m7] - [1/1, 16/13, 3/2, 9/5]

[0, 7, 14, 21] _ [P1, AsGrm3, P5, AsGrm7] - [1/1, 11/9, 3/2, 11/6]

[0, 7, 14, 21] _ [P1, AsGrm3, P5, DeAcM7] - [1/1, 11/9, 3/2, 81/44]

[0, 7, 14, 21] _ [P1, AsGrm3, P5, ReM7] - [1/1, 11/9, 3/2, 24/13]

[0, 7, 14, 21] _ [P1, DeAcM3, P5, AsGrm7] - [1/1, 27/22, 3/2, 11/6]

[0, 7, 14, 21] _ [P1, DeAcM3, P5, DeAcM7] - [1/1, 27/22, 3/2, 81/44]

[0, 7, 14, 21] _ [P1, DeAcM3, P5, ReM7] - [1/1, 27/22, 3/2, 24/13]

[0, 7, 14, 21] _ [P1, Prm3, P5, AsGrm7] - [1/1, 39/32, 3/2, 11/6]

[0, 7, 14, 21] _ [P1, Prm3, P5, DeAcM7] - [1/1, 39/32, 3/2, 81/44]

[0, 7, 14, 21] _ [P1, Prm3, P5, ReM7] - [1/1, 39/32, 3/2, 24/13]

[0, 7, 14, 21] _ [P1, ReM3, P5, AsGrm7] - [1/1, 16/13, 3/2, 11/6]

[0, 7, 14, 21] _ [P1, ReM3, P5, DeAcM7] - [1/1, 16/13, 3/2, 81/44]

[0, 7, 14, 21] _ [P1, ReM3, P5, ReM7] - [1/1, 16/13, 3/2, 24/13]

[0, 8, 13, 19] _ [P1, M3, Sb5, Sbm7] - [1/1, 5/4, 35/24, 7/4]

[0, 8, 13, 19] _ [P1, ReAsM3, ReAsSb5, Sbm7] - [1/1, 33/26, 77/52, 7/4]

[0, 8, 13, 20] _ [P1, M3, Sb5, ReSbM7] - [1/1, 5/4, 35/24, 70/39]

[0, 8, 14, 19] _ [P1, AcM3, P5, Sbm7] - [1/1, 81/64, 3/2, 7/4]

[0, 8, 14, 19] _ [P1, M3, P5, Sbm7] - [1/1, 5/4, 3/2, 7/4]

[0, 8, 14, 19] _ [P1, ReAsM3, P5, Sbm7] - [1/1, 33/26, 3/2, 7/4]

[0, 8, 14, 21] _ [P1, AcM3, P5, AsGrm7] - [1/1, 81/64, 3/2, 11/6]

[0, 8, 14, 21] _ [P1, AcM3, P5, DeAcM7] - [1/1, 81/64, 3/2, 81/44]

[0, 8, 14, 21] _ [P1, AcM3, P5, ReM7] - [1/1, 81/64, 3/2, 24/13]

[0, 8, 14, 21] _ [P1, M3, Gr5, DeM7] - [1/1, 5/4, 40/27, 20/11]

[0, 8, 14, 21] _ [P1, M3, Gr5, PrGrm7] - [1/1, 5/4, 40/27, 65/36]

[0, 8, 14, 21] _ [P1, M3, P5, AsGrm7] - [1/1, 5/4, 3/2, 11/6]

[0, 8, 14, 21] _ [P1, M3, P5, DeAcM7] - [1/1, 5/4, 3/2, 81/44]

[0, 8, 14, 21] _ [P1, M3, P5, ReM7] - [1/1, 5/4, 3/2, 24/13]

[0, 8, 14, 21] _ [P1, M3, PrDe5, DeM7] - [1/1, 5/4, 65/44, 20/11]

[0, 8, 14, 21] _ [P1, M3, PrDe5, PrGrm7] - [1/1, 5/4, 65/44, 65/36]

[0, 8, 14, 21] _ [P1, ReAsM3, P5, AsGrm7] - [1/1, 33/26, 3/2, 11/6]

[0, 8, 14, 21] _ [P1, ReAsM3, P5, DeAcM7] - [1/1, 33/26, 3/2, 81/44]

[0, 8, 14, 21] _ [P1, ReAsM3, P5, ReM7] - [1/1, 33/26, 3/2, 24/13]

Hats off to Curt, I guess. I hope you like my chords, Curt. I made them for you. Okay, time to look at more of his videos and see what I did wrong. His next step is to describe some chords that follow his rules.

Here is the chord Curt describes the "minor harmonic seventh" chord, with pitches [G, Bb, D, Fb_up]. I don't know his notation - it's probably just HEJI, but I refuse to learn HEJI. It seems odd to call a subminor seventh F flat_up when it's lower than Fb. Maybe it's a typo, or maybe it's odd notation. Either way, presumably he's talking about this:

[0, 6, 14, 19] _ [P1, m3, P5, Sbm7] - [1/1, 6/5, 3/2, 7/4]

which was among my 161 chords. Nice.

Curt next plays the "neutral 13th chord", which is [G, Bb, Ct, Ed]. Ct is C half sharp, Ed is E half flat. I think in 24-EDO steps that would be [0, 6, 11, 17]. This isn't spelled by thirds - though it could be if it were inverted / cyclically permuted to be rooted on C. But let's continue anyway. The octave reduced 13th harmonic is a prominent minor sixth in my naming system, Prm6, with just tuning of 13/8, and a 24-edo tuning of 17\24. So clearly when Curt calls this chord "neutral 13th", part of what he means is that it includes the 13th harmonic, which is a half-flat / neutral tone. The octave-reduced 11th harmonic is the ascendant fourth, As4, with a just frequency ratio of 11/8 and a 24-EDO tuning of 11\24 steps. I'm pretty sure Curt is thinking of this chord as

    [0, 6, 11, 17] _ [P1, m3, As4, Prm6] # [1/1, 6/5, 11/8, 13/8]

If we move the bottom two notes up by an octave, and then subtract As4 from everything (or divide all the frequency ratios by 11/8), then we get this chord from my set of 161 chords:

    [0, 6, 13, 19] _ [P1, PrDem3, De5, Dem7] - [1/1, 13/11, 16/11, 96/55]

That's the rooted tertian spelling of Curt's neutral 13th chord. Great! I'm very pleased that our chords are alignable so far, if not identical. If you're curious, De5 in its just tuning is flat of P5 by about 53 cents,

    (3/2) / (16/11)  = 33/32

    1200 * log_2(33/32) ~ 53.2 cents

The next chord Curt presents is the "19th to 15th" chord. Is he going to use the 19th harmonic? He had only mentioned 13-limit ratios up to this point. If so, this guy's on another level. The pitches given are [F#, A, Ct, Eb_up]. In 24-EDO steps, I think that would be [0, 6, 11, 19].

My set of 161 chords had two intervallic chords with that same 24-EDO tuning, namely

[0, 6, 11, 19] _ [P1, PrDem3, PrDeSbd5, Sbm7] - [1/1, 13/11, 91/66, 7/4],

[0, 6, 11, 19] _ [P1, m3, Sbd5, Sbm7] - [1/1, 6/5, 7/5, 7/4],

But I don't know any reason you'd call either of those a "19th to 15th" chord. I don't think "19th to 15th" is referring to a sequence of prime harmonics - this chord is a long way from the utonal 19:17:16:15, for example. And for otonal chords, 15:16:17:18:19 would be 

    [1/1, 16/15, 17/15, 6/5, 19/15]

which is also way off from the 24-EDO tuning of Curt's chord. A 19th interval, octave reduced, is a fifth, and a 15th interval, octave reduced, is a unison. That's probably not what he means either. I just don't know what the name means. But I've got a chord that matches his chord, so that's something.

After listening again, I think he's saying, "The 19th, the 15th chord". I still don't know what that means.

He talks about "9-15" chords in a later video called "The Magic Chord". So maybe I misheard him every time? But I just relistened, and it really sounds like "the 19th to 15th" or "the 19th, the 15th". Anyway, in "The Magic Chord", Curt gives [A, C, Ed, Gd] as a 9-15 chord on A, which would be

    [0, 6, 13, 19]\24

I think. But that's the rooted tertian spelling of Curt's neutral 13th chord. So confused. Ah, but then he says that he goofed, the chord spelled that way is not a 9-15 chord, it is instead "nine fifteen fo(u)r flat seven(th)". So lost.

If this Ct is the same one that was 11/4 over G, then Curt might be thinking of this chord as

    [P1, m3, AsGrd5, Sbm7] # [1/1, 6/5, 22/15, 7/4]

i.e. we're widening the 11/4 by a 5-limit minor second to represent the gap (Ct - F#) instead of (Ct - G).

My program didn't find that chord because between the chord degrees ^5 and ^7 there's an unusual interval of DeSbAcM3 with a just tuning of 105/88. On the other hand, DeSbAcM3 is tuned to 6 steps of 24-EDO, so maybe Curt just thinks of it as a minor third.

The next chord Curt gives is the "added 13th minor", with pitches of [G, Bb, D, Eb_up]. Curt's probably thinking of this as

    [P1, m3, P5, Prm6]

which we can invert to get a tertian chord. Move the top three notes up an octave and reduce/rebase/re-root. That gives us

    [0, 7, 13, 21] _ [P1, ReM3, Re5, ReM7] # [1/1, 16/13, 96/65, 24/13]

which was one of my 161 tertian intervallic chords. Nice.

He also plays the minor harmonic 11th chord, for which the accidentals look a little weird. It looks like [G, Bbt, C, Fb_up], but most people wouldn't use both flat "b" and half-sharp "t" on one note unless they were thinking intervallically, and I ... didn't think that Curt was? I thought he would just write Bd for "half flat" instead of "flatten by one comma and raise by another". Anyway, in steps of 24-EDO, if Fb_up is supposed to be F three quarters flat, then this is

    [0, 7, 14, 19]

in 24-EDO steps. I had four chords with this tuning in my set of 161 intervallic chords, namely

[0, 7, 14, 19] _ [P1, AsGrm3, P5, Sbm7] - [1/1, 11/9, 3/2, 7/4]

[0, 7, 14, 19] _ [P1, DeAcM3, P5, Sbm7] - [1/1, 27/22, 3/2, 7/4]

[0, 7, 14, 19] _ [P1, Prm3, P5, Sbm7] - [1/1, 39/32, 3/2, 7/4]

[0, 7, 14, 19] _ [P1, ReM3, P5, Sbm7] - [1/1, 16/13, 3/2, 7/4]

And I think either of the first two (11-limit) chords is a fine just intonation for Curt's neutral 11 chord. 

He also shares chords called the neutral 11th, the major harmonic 11th, the neutral triad, the neutral harmonic seventh, the neutral dominant seventh, the harmonic diminished seventh, and "two stacked harmonic seconds".

He also plays some nice sounding chords that break his rule about not allowing 9-step and 15-step intervals. These are the subminor triad, the sub-minor harmonic seventh, the sub-minor dominant seventh, and three stacked harmonic seconds.  

I might write those out and analyze them eventually, but in the meantime, I think we're doing fine. We've mostly found the same chords as him. Good job, us.

I started writing about his content in other videos, but it sounded kind of judgey, so I'll stop here. Thanks for your chord construction technique, Curt. Good guy, Curt.

Here is a rendering of all 161 chords in order. I won't pretend that 4 minutes of microtonal chords on the same tonic is easy listening, but I hope you'll agree that the chords individually aren't too dissonant.

Edit: Here are valid triads:

         [0, 5, 11] _ [P1, Sbm3, PrDeSbd5] # [1/1, 7/6, 91/66]

         [0, 5, 11] _ [P1, Sbm3, Sbd5] # [1/1, 7/6, 7/5]

         [0, 5, 12] _ [P1, Sbm3, AsSbGrd5] # [1/1, 7/6, 77/54]

         [0, 5, 12] _ [P1, Sbm3, DeSbAc5] # [1/1, 7/6, 63/44]

         [0, 5, 12] _ [P1, Sbm3, PrSbd5] # [1/1, 7/6, 91/64]

         [0, 5, 12] _ [P1, Sbm3, ReSb5] # [1/1, 7/6, 56/39]

         [0, 5, 13] _ [P1, Sbm3, ReAsSb5] # [1/1, 7/6, 77/52]

         [0, 5, 13] _ [P1, Sbm3, Sb5] # [1/1, 7/6, 35/24]

         [0, 6, 11] _ [P1, PrDem3, PrDeSbd5] # [1/1, 13/11, 91/66]

         [0, 6, 11] _ [P1, m3, Sbd5] # [1/1, 6/5, 7/5]

         [0, 6, 12] _ [P1, Grm3, Grd5] # [1/1, 32/27, 64/45]

         [0, 6, 12] _ [P1, PrDem3, PrDed5] # [1/1, 13/11, 78/55]

         [0, 6, 12] _ [P1, m3, Grd5] # [1/1, 6/5, 64/45]

         [0, 6, 12] _ [P1, m3, PrDed5] # [1/1, 6/5, 78/55]

         [0, 6, 12] _ [P1, m3, d5] # [1/1, 6/5, 36/25]

         [0, 6, 13] _ [P1, Grm3, De5] # [1/1, 32/27, 16/11]

         [0, 6, 13] _ [P1, Grm3, PrGrd5] # [1/1, 32/27, 13/9]

         [0, 6, 13] _ [P1, PrDem3, De5] # [1/1, 13/11, 16/11]

         [0, 6, 13] _ [P1, PrDem3, PrGrd5] # [1/1, 13/11, 13/9]

         [0, 6, 13] _ [P1, m3, AsGrd5] # [1/1, 6/5, 22/15]

         [0, 6, 13] _ [P1, m3, DeAc5] # [1/1, 6/5, 81/55]

         [0, 6, 13] _ [P1, m3, Re5] # [1/1, 6/5, 96/65]

         [0, 6, 14] _ [P1, Grm3, Gr5] # [1/1, 32/27, 40/27]

         [0, 6, 14] _ [P1, Grm3, P5] # [1/1, 32/27, 3/2]

         [0, 6, 14] _ [P1, PrDem3, P5] # [1/1, 13/11, 3/2]

         [0, 6, 14] _ [P1, PrDem3, PrDe5] # [1/1, 13/11, 65/44]

         [0, 6, 14] _ [P1, m3, P5] # [1/1, 6/5, 3/2]

         [0, 7, 12] _ [P1, AsGrm3, AsSbGrd5] # [1/1, 11/9, 77/54]

         [0, 7, 12] _ [P1, DeAcM3, DeSbAc5] # [1/1, 27/22, 63/44]

         [0, 7, 12] _ [P1, Prm3, PrSbd5] # [1/1, 39/32, 91/64]

         [0, 7, 12] _ [P1, ReM3, ReSb5] # [1/1, 16/13, 56/39]

         [0, 7, 13] _ [P1, AsGrm3, AsGrd5] # [1/1, 11/9, 22/15]

         [0, 7, 13] _ [P1, AsGrm3, PrGrd5] # [1/1, 11/9, 13/9]

         [0, 7, 13] _ [P1, DeAcM3, De5] # [1/1, 27/22, 16/11]

         [0, 7, 13] _ [P1, DeAcM3, DeAc5] # [1/1, 27/22, 81/55]

         [0, 7, 13] _ [P1, Prm3, PrGrd5] # [1/1, 39/32, 13/9]

         [0, 7, 13] _ [P1, ReM3, De5] # [1/1, 16/13, 16/11]

         [0, 7, 13] _ [P1, ReM3, Re5] # [1/1, 16/13, 96/65]

         [0, 7, 14] _ [P1, AsGrm3, P5] # [1/1, 11/9, 3/2]

         [0, 7, 14] _ [P1, DeAcM3, P5] # [1/1, 27/22, 3/2]

         [0, 7, 14] _ [P1, Prm3, P5] # [1/1, 39/32, 3/2]

         [0, 7, 14] _ [P1, ReM3, P5] # [1/1, 16/13, 3/2]

         [0, 8, 13] _ [P1, M3, Sb5] # [1/1, 5/4, 35/24]

         [0, 8, 13] _ [P1, ReAsM3, ReAsSb5] # [1/1, 33/26, 77/52]

         [0, 8, 14] _ [P1, AcM3, P5] # [1/1, 81/64, 3/2]

         [0, 8, 14] _ [P1, M3, Gr5] # [1/1, 5/4, 40/27]

         [0, 8, 14] _ [P1, M3, P5] # [1/1, 5/4, 3/2]

         [0, 8, 14] _ [P1, M3, PrDe5] # [1/1, 5/4, 65/44]

         [0, 8, 14] _ [P1, ReAsM3, P5] # [1/1, 33/26, 3/2]

Nice.