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.

A Grammar Of Diminished Chords

 :: Intro

In my crash course on jazz piano, I described the grammatical/functional use of diatonic chords, modal chords, and augmented chords. I also talked at good length about diminished chords, but I didn't present a grammar for using them in relation to diatonic tetrads. I'm going to do that here and now.

:: Dim7 Chords In The Barry Harris System

In the Barry Harris jazz piano system, there's a diminished seventh chord that you can play before or after almost any other chord. 

Here are our main diatonic tetrads in C major for concreteness:

[C.maj6, C.maj7, D.m7, E.m7, F.maj7, G.7, A.m7, B.m7b5]

We'll target each of those with a .dim7 chord. In lots of jazz, C.maj6 and C.maj7 are played fairly interchangeably, but my understanding is that they're quite distinct in the Harris system, so we'll treat them separately in this section.

Let's write out the .dim7 options that alternate against diatonic tetrads in the key of C major.

C.maj6 ↔ B.dim7

C.maj7 ↔ F#.dim7

D.m7 ↔ E.dim7

E.m7 ↔ F#.dim7

F.maj7 ↔ B.dim7

G.7 ↔ C#.dim7 or B.dim7.

A.m7 ↔ B.dim7

B.m7b5 ↔ C#.dim7

If you've already learned about the Barry Harris system, here's a refresher on where these come from:

C.maj6 and B.dim7 are the two chords that alternate in the C major-diminished scale at the heart of the method.

Since A.m7 is an inversion of C.maj6, you play B.dim7, which comes from the C maj6-dim scale. 

Since D.m7 is an inversion of F.maj6, you play C#.dim7, which comes from the F maj6-dim scale.

Since E.m7 is an inversion of G.maj6, you play F#.dim7, which comes from the G maj6-dim scale.

Since F.maj9 is A.m7 over F, and we play B.dim7 against A.m7, we also play B.dim7 against F.maj7.

Since C.maj9 is E.m7 over C, and we play F#.dim7 against E.m7, we also play F#.dim7 against C.maj7.

Since B.m7b5 is an inversion of D.m6, you play C#.dim7, which comes from the D m6-dim scale.

G.7 is treated in the Barry Harris system as G.9 or G.7b9.

G.9 is B.m7b5 over G, and we play C#.dim7 against B.m7b5, so we also play C#.dim7 against G.9.

G.7b9 is B.dim7 over G, and B.dim7, so when you want a .dim7 chord that sounds good with G.7b9, obviously you can play B.dim7. But actually, since G.7b9 and B.dim7 have very similar sounds, you'd probably actually play C.maj6 against G.7b9. I haven't written that above because this post is just about diminished chords.

:: Dim7 Chords As Chromatic Approach Chords

Let's get away from the Barry Harris method now. Much like the idiom of approaching a note chromatically from below, there is an often used idiom of preceding a chord with a .dim7 chord whose root is a m2 below the root of the target chord. In the key of C major, that looks like

1. B.dim7 → [C.maj6 or C.maj7]

2. C#.dim7 → D.m7

3. D#.dim7 → E.m7

4. E.dim7 → F.maj7

5. F#.dim7 → G.7

6. G#.dim7 → A.m7

7. A#.dim7 → B.m7b5

Rule 7 here sounds a little worse than the others, but you can make it work. Rule 1 is partly familiar from Barry Harris (B.dim7 → C.maj6) and partly new (B.dim7 →  C.maj7). Rule 6 is also accounted for by the Harris system if you're working in a tuning such as 12-TET where G#.dim7 is enharmonic with B.dim7. The dim7 chords from rules 2, 4, and 7, namely (C#.dim7, E.dim7, and A#.dim7) are also enharmonic with each other. G.dim7 is also enharmonic with those three, so if you see a G.dim7 chord in a piece in C.maj, and you're struggling to analyze it, you might check if one of its enharmonic re-spellings makes more sense.

:: Half Diminished Chords As Secondary Dominants

Next let's talk about half diminished chords as secondary dominants. Since V.7 and VII.m7b5 both have dominant functions, we can talk about secondary VII.m7b5 chords just as much as secondary V.7 chords. In the key of C major, this gives us the following relations:

B.m7b5 → C.maj

C#.m7b5 → D.m7

D#.m7b5 → E.m7

E.m7b5 → F.maj7

F#.m7b5 → G.7

G#.m7b5 → A.m7

A#.m7b5 → B.m7b5

You'll notice these are all just half-diminished versions of the full diminished chord rules in the previous section. Easy. No sweat. Actually, we could also consider these chords to be rootless V.9 secondary dominants.

:: Dim7 Chords As Descending Chromatic Line Cliches

In addition to preceding a diatonic chord with a .dim7 approach from below, you can often follow a minor chord on a root with a .dim7 chord that has the same root, for a sound that is something like a descending chromatic line cliche. Here are some examples:

D.m7 → D.dim7 → C.maj7

F.maj7 → F.m7 → F.dim7 → ?(G.7) → C.maj7

A.m7 → A.dim7 → ?

E.m7 → E.dim7 → ?

The question marks are there as placeholders until I sit down at a piano and verify what my memory is telling me about these.

:: Dim7 Chords On Flat Three

Another pretty common use for .dim7 chords that you see in jazz is the [I → bIII] transition. Here's a common way to continue that:

C.maj7 → Eb.dim7 → D.m7

From all our previous experience with diminished chords, we'd expect that here, before D.m7, that we'd see an E.dim7 in the Harris system or C#.dim7 in the chromatic approach idiom, or C#.m7b5 as a half-diminished secondary dominant. But Eb.dim7 isn't any of those. Eb.dim7 happens to be enharmonic in 12-TET with chords such as D#.dim7, Gb.dim7, F#.dim7, A.dim7, and C.dim7. No match, no cigar.

Maybe the bIII dim7 chord is a rootless secondary dominant! You can add one of these notes [B, D, F, G#] to Eb.dim7 and get one of these dominant chords, [B.7b9, D.7b9, F.7b9, or G#.7b9], respectively, which means that Eb.dim7 can act like a rootless version of any of those. But none of those are secondary dominants to D.m7.

You could analyze this progression by calling the first transition a chromatic mediant relation between C and Eb, but those usually have chromatic mediant motion usually involves major and minor chord sonorities, not diminished.

Here's what I think is going on. My key insight was that this also sounds good:

    C.maj7 → Eb.dim7 → G.7

Here Eb.dim7, with notes [Eb, Gb, A, C], is a acting as an alteration of D.7, with notes [D, F#, A, C], which is a secondary dominant of G.7. You could also call it rootless D.7b9. And then

    C.maj7 → D.7 → D.m7

is like a chromatically descending like cliche, maybe.

There are probably other uses of .dim7 chords as alterations of secondary dominants or descending line cliches like these, but I haven't put in the leg work to find them yet.

...

I guess .dim7 chords as rootless V.7b9 chords would look like this:

Ab.dim7 → C.maj7

Bb.dim7 → D.m7

C.dim7 → E.m7

Db.dim7 → F.maj7

Eb.dim7 → G.7

F.dim7 → A.m7

...

The last transition, F.dim7 → A.m7, sounds a little weak, but the rest are very good.

We already saw .m7b5 as rootless secondary V.9 dominants of target chords. These .dim7 transitions are rootless secondary V.7b9 dominants of target chords. 

If the secondary V.7 of a chord has a root a perfect fifth up from the original chord, then this .dim7 chord, that arises as a rootless .7b9, has a tonic P5 + m2 = m6 above the original chord (or a major third below).

:: Outro

If you know any other uses of .dim7 or .m7b5 chords that aren't accounted for by the above rules, or if you have a different analysis of I.maj7 → bIII.dim7, please do let me know.