Ben Johnston's Tricks

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

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

Here's how it looks when sorted by numerator:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

This chord is of course better known as

        (1/1, 5/4, 3/2, 7/4). 

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

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

    (1/1, 4/5, 2/3, 4/7)

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

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

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

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

    (1/1, 5/4, 3/2, 7/4)

and soon after also play a chord like

    (1/1, 7/6, 7/5, 7/4)

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

    (1/1, 6/5, 3/2)

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

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

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

...

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

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

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

...

Dastgahs

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

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

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

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

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

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

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

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

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

Shur/Nava: Shur + Dashti + T.

Homayun: Chahargah + Dashti + T.

Mahur/Rastpanjgah: Mahur + T + Mahur.

Segah: Shur + Shur + T.

Chahargah: Chahargah + T + Chahargah.

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

Esfahan : Dashti + T + Chahargah.

or as

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

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

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

    p = koron (60 cent flat)

    > = sori (40 cent sharp)

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

Mahur is just the C major scale without any microtones, 

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

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

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

...

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

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

which matches the 24-EDO scale of Ostovar perfectly. 

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

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

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

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

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

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

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

Here's Esfahan from Kees:

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

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

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

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

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

Which perfectly matches the 24-EDO version from Ostovar:

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

I knew I liked this guy for a reason.

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

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

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

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

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

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

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

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

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

or

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

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

Here's the dastgah Shur of Ostovar for comparison: 

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

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

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

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

    [Ed, F, G, Ad, Bb, C, D, Ed]

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

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

He gives

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

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

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

        [Ed, F, G, Ad, Bb, C, D, Ed]

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

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

...

Rank-3 EDO Distinction and Diatonicity

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

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

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

    [m2, M2, m3, M3, m6, M6, m7, M7]

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

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

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

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

in 11-EDO we get these steps:

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

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

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

 

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

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

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

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

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

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

The only other indistinct EDOs are:

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

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

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

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

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

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

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

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

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

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

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

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

...

I propose that these guys:

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

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

...

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

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

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

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

The other members of the set all have this order: 

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

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

    dd2 = d2 = m2 = M2 = A2 = AA2

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

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

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

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

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

...

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

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

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

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

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

There's a great degenerate order I've notice

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

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

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

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

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

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

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

    [15, 18, 30]-EDO

at least.

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

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

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

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

and these guys are consistent with lax order 3:

    [19, 35, 38, 57, 76]-EDO

.

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

    [19, 38, 57, 76]-EDO

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

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

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

...