Quartertone Harmony

Quartertone Harmony Chords

Quartertone Harmony is the youtube channel of a 24-EDO music theorist and composer. Curt's his name. 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. I'm pretty sure that Curt allows you one note to bring more than two other notes to the party - i.e. the chain of friends is branched, not a straight line. It's not clear to me if the chain has to form a circle or if the ends can be disconnected. When I started coding up his methods, I went with a single-linked disconnected chain of friends for generating chords in his style and it worked *really well*.

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, neutral 3rd, M3] -> [5, 6, 7, 8]

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 to me at all. 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 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 9 quarter tones or 15 quarter tones. Since I used a single-linked chain and no two of my chain-of-friend intervals can sum to form 1 or 9, I don't have to look out for enemies of that type. To me, this just means that we can't have consecutive relative intervals of size (8, 7)\24 or (7, 8)\24. Easy.

Rule 3. No crowding. No note can have more than one other note that is closer than a major second. This is super easy for me since I'm using a single linked chain of friends. Even if we had used Curt's harmonic second, SpM2, instead of the Sbm3, 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. And the resulsts are *really good*.

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. 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 hope you like my chords, Curt. I made them for you. In the video where Curt shares these rules, he also shares his names for some chords that adhere to the rules.

Curt describes the "minor harmonic seventh" chord, and I think 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 he notates as [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" or "the 19th, the 15th chord". 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],

If the Ct in the original 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.

Thanks for your chord construction technique, Curt. Good guy, Curt.

To close out, here are some valid triads in the system of Quartertone Harmony:

         [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.

I should note that while I checked that these chords don't have notes separated by enemy intervals in root position, I haven't checked all the inversions. My guess is that if I like the sound of a chord in root position, then I'll also like the inversions, but using these inverted might not accord with Curt's method.