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.

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