Chromatic Orders Of Rank-2 Intervals
The primitive rank-2 musical intervals, or the natural intervals, are these guys: (P1, m2, M2, m3, M3, P4, P5, m6, M6, m7, M7). The once-modified rank-2 intervals are these guys: (A1, A2, A3, A4, A5, A6, A7, d1, d2, d3, d4, d5, d6, d7).
In what order do different tuning systems put the modified intervals, relative to the natural ones? In turns out, there are multiple possibilities, even if we only look at rank-2 tuning systems with pure octaves, t(P8) = 2, that preserve the 12-TET order of the natural intervals.
Pythagorean tuning has this order:
[d1, 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]
This order
[d1, 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]
is shared by quarter-comma meantone (t(M3) = 5/4), third-comma meantone (t(m3) = 6/5), sixth-comma meantone (t(A4) = 45/32), septimal tuning (t(A6) = 7/4), and a tuning system I made up that tunes the minor second to its 5-limit value, (t(m2) = 16/15).
This order
[d1, 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]
is shared by the Tetracot tuning system (t(A5) = 3/2), another tuning system I made up that tunes the M2 to its 5-limit value, (t(M2) = 10/9), the 26-EDO generated by (t(ddd2) = 1), and the 33-EDO generated by (t(dddd2) = 1). This order separates the firsts from the seconds from the thirds, et cetera. No interleaving. That's a little neat.
After seeing 26-EDO and 33-EDO, I wondered if every EDO >= 26 would have the Tetracot order, but then I tried the 53-EDO generating by (t(ddddddd6) = 1) and found that it has the Pythagorean order.
So I've tried 12 tuning systems, and I've gotten three different orders of modified intervals. Neat. How many orders are there? How can we predict what order a tuning system will have? Let's find out!
I'm up to 5 orders now:
[d1, 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]: 26-EDO, 33-EDO, 40-EDO, 45-EDO, 47-EDO, 440-Floor, 5-limit M2, Tetracot
[d1, 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]: 31-EDO, 43-EDO, 50-EDO, 5-limit-m2, Quarter-comma meantone, Septimal, Sixth-comma meantone, Third-comma meantone
[d1, 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]: 17-EDO, 19-EDO, 29-EDO, 41-EDO, 46-EDO, 53-EDO, Pythagorean
[d1, d2, P1, m2, d3, A1, M2, m3, d4, A2, M3, P4, d5, A3, d6, A4, P5, m6, d7, A5, M6, m7, d8, A6, M7, P8, A7]: 39-EDO
[d1, d2, P1, m2, d3, A1, M2, m3, d4, A2, M3, P4, d5, d6, A3, A4, P5, m6, d7, A5, M6, m7, d8, A6, M7, P8, A7]: 27-EDO, 37-EDO, 42-EDO, 49-EDO, 52-EDO
and here are the definitions of the tuning systems:
t(A4) = 45/32 # Sixth-comma meantone
t(A5) = 3/2 # Tetracot
t(A6) = 7/4 # Septimal
t(M2) = 10/9 # 5-limit-M2
t(M3) = 5/4 # Quarter-comma meantone
t(P5) = 3/2 # Pythagorean
t(m2) = 16/15 # 5-limit-m2
t(m3) = 6/5 # Third-comma meantone
t(m3) = 67/55 # 440-Floor
t(dd3) = 1 # 17-EDO
t(dd2) = 1 # 19-EDO
t(ddd2) = 1 # 26-EDO
t(dddd5) = 1 # 27-EDO
t(dddd4) = 1 # 29-EDO
t(dddd3) = 1 # 31-EDO
t(dddd2) = 1 # 33-EDO
t(ddddd7) = 1 # 37-EDO
t(ddddd6) = 1 # 39-EDO
t(ddddd2) = 1 # 40-EDO
t(dddddd5) = 1 # 41-EDO
t(dddddd8) = 1 # 42-EDO
t(dddddd4) = 1 # 43-EDO
t(dddddd3) = 1 # 45-EDO
t(dddddd6) = 1 # 46-EDO
t(dddddd2) = 1 # 47-EDO
t(ddddddd8) = 1 # 49-EDO
t(ddddddd4) = 1 # 50-EDO
t(ddddddd10) = 1 # 52-EDO
t(ddddddd6) = 1 # 53-EDO
Time to get weird? I got these systems from ejlilley's Tuning.hs. I've never played with them before.
t(8 * P4) = t(d25) = 10 # Schismatic
t(3 * M3) = t(A7) = 2 # Augmented
t(4 * P5) = t(M17) = 24/5 # Pelogic
t(8 * M3) = t(AAAA17) = 6 # Wuerschmidt
t(4 * m3) = t(d9) = 2 # Dimipent
t(5 * A1) = t(AAAAA1) = 6/5 # Sycamore
t(9 * M3) = t(AAAAA19) = 16384/2187 # Escapade
t(7 * A1) = t(AAAAAAA1) = 4/3 # Vishnuzmic
Schismatic has the Pythagorean order. Augmented, Wuerschmidt, Vishnuzmic, and Escapade have the Meantone order. Sycamore has the Tetracot order. Pelogic doesn't even have the 12-TET order over natural intervals. It's also crazy. The tuned major seventh in Pelogic is t(M7).= 6/25 * 10^(3/4) * 3^(1/4). Dimipent has the same pitches as 12-TET, so it doesn't distinguish between the modified intervals and it doesn't get an ordering on this page. Or maybe we should say pure octave tuning systems with t(P5) = 2^(7/12) have a degenerate order over once-modified rank-2 intervals.
Next, let's check more generally whether tuning systems with the same orders have similar P5s.
Good news!
1.4863324494754246 : 440-Floor - Tetracot order
1.4877378261644905 : Tetracot - Tetracot order
1.4887216904814808 : 54-EDO - Tetracot order
1.4891283272685385 : 47-EDO - Tetracot order
1.4896774631227023 : 40-EDO - Tetracot order
1.4904599153098523 : 33-EDO - Tetracot order
1.49071198499986 : 5-limit M2 - Tetracot order
1.4909906251234766 : 59-EDO - Tetracot order
1.4916644904914018 : 26-EDO - Tetracot order
1.492548464309911 : 45-EDO - Tetracot order
1.4937553085226773 : Sycamore - Tetracot order
1.4937589616544857 : 19-EDO - Tetracot order
1.4938015821857216 : Third-comma meantone - Meantone order
1.4947443156280251 : Vishnuzmic - Meantone order
1.4948492486349383 : 50-EDO - Meantone order
1.4953487812212205 : Quarter-comma meantone - Meantone order
1.4955178823482085 : 31-EDO - Meantone order
1.495576529604238 : Escapade - Meantone order
1.4956115235716425 : Septimal - Meantone order
1.4956577455914317 : Wuerschmidt - Meantone order
1.4962778697388446 : 5-limit m2 - Meantone order
1.4962957394862462 : 43-EDO - Meantone order
1.4968975827619546 : Sixth-comma meantone - Meantone order
1.4983070768766817 : Augmented - Meantone order
1.4997884186649117 : Schismatic - Pythagorean order
1.4999409030781112 : 53-EDO - Pythagorean order
1.5 : Pythagorean - Pythagorean order
1.5004194330574077 : 41-EDO - Pythagorean order
1.5012943823463352 : 29-EDO - Pythagorean order
1.5020746584842646 : 46-EDO - Pythagorean order
1.5034066538560549 : 17-EDO - Pythagorean order
1.5049792436093965 : 39-EDO - 39-EDO order
1.5071643388112317 : 49-EDO - 27-EDO order
1.5079541804033867 : 27-EDO - 27-EDO order
1.5100481939178911 : 37-EDO - 27-EDO order
1.5107218870420942 : 42-EDO - 27-EDO order
1.5116811224148876 : 52-EDO - 27-EDO order
We can predict the order of once-modified intervals in a tuning system from the size of its tuned perfect fifth interval (the number on the far left). Now I just have to figure out where the five families live. And also, maybe there are more than five families if you go to higher precision or go out farther on the ends toward 2^(4/7) on the low end and 2^(3/5) on the high end (The details of where these come from are presented in the previous blog post).
Ok, there still appear to be only five families. We transition from Tetracot order to Meantone order right at t(P5) = 2^(11/19). That kind of makes sense. There's a 19-EDO with the 12-TET order over natural intervals and the only value it has in the range (2^(4/7), 2^(3/5)) is the 11/19 thing. On that basis, we might expect to see 2^x for x in [10/17, 11/19, 13/22, 15/26, 16/27, 17/29, 18/31, ...] as other transition points.
Oh! And also 2^(7/12) from 12-TET. That's where Meantone order switches to Pythagorean order. Then we switch from Pythagorean order to the 39-EDO order at 2^(10/17) and we switch from the 39-EDO order to the 27-EDO order at t(P5) = 2^(13/22).
All together now:
Tuning systems with t(P5) between 2^(4/7) and 2^(11/19) have the Tetracot order over once-modified intervals.
Tuning systems with t(P5) between 2^(11/19) and 2^(7/12) have the Meantone order.
Tuning systems with t(P5) exactly 2^(7/12) have a degenerate order over once-modified intervals.
Tuning systems with t(P5) between 2^(7/12) and 2^(10/17) have the Pythagorean order.
Tuning systems with t(P5) between 2^(10/17) and 2^(13/22) have the 39-EDO order.
Tuning systems with t(P5) between 2^(13/22) and 2^(3/5) have the 27-EDO order.
Nice.
So what about 2^(15/26 )and 2^(16/27) and higher? Maybe those points distinguish between orders of twice-modified intervals like dd6 and AA2.
Which order is the best? Pelogic. Nah, probably the Meantone one that's also 31-EDO and septimal, but I don't have an argument for why. Or, here's an argument: 1) The best ordering has to be the Pythagorean or the Meantone, because those two straddle the 12-TET order at t(P5) = 2^(7/12), and 2) it's not the Pythagorean order. The Meantone ordering is also what you get when you name rank-3 intervals according to ejlilley's rank-3 interval algebra and then tune them in 5-limit just intonation, although I wouldn't be surprised if he built his algebra to create that ordering, given that he's a man of good taste who appreciates the finer things in life (meantone temperaments).
Let's compare the Pythagorean ordering to the Meantone ordering.
[d1, 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] # Meantone
[d1, 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]: # Pythagorean
Relative to Meantone, Pythagoras swaps [P1 with d2], [A1 with m2], [M2, d3], [A2, m3], [M3, d4], [A3, P4], [A4, d5], [P5, d6], [A5, m6], [M6, d7], [A6, m7], [M7, d8], and [A7, P8]. What effect does that swap have on melodies and harmonies?
...
I don't know, but I looked at some orders of twice modified intervals, and 40-EDO is cool:
[dd1, d1, P1, A1, AA1, dd2, d2, m2, M2, A2, AA2, dd3, d3, m3, M3, A3, AA3, dd4, d4, P4, A4, AA4, dd5, d5, P5, A5, AA5, dd6, d6, m6, M6, A6, AA6, dd7, d7, m7, M7, A7, AA7, dd8, d8, P8, A8, AA8]
There's still no interleaving! All the firsts come first, then the seconds, then the thirds, and so on.
Back to the once-modified intervals, I wanted to be able to say more about the regularities that hold across all five of orders. We've already seen, in Pythagorean order versus Meantone order, that t(P1) is larger and sometimes t(d2) is larger, depending on the tuning system. I'm going to notate that incomparability as
P1 >< d2
meaning that the order of the two is not consistently distinguished in pure octave tuning systems that have the 12-TET order over natural intervals. "Not less than equal to or greater than". I also considered using ~ or <!=!> or <≠> or ≮ ≠ ≯ or ⊥.
Besides the incomparability relations already listed when we compared the Pythagorean order and the Meantone order, there's also incomparability between A(n) and d(n+1). e.g.
A6 >< d7
and between A(n) and d(n+1), .e.g.
A4 >< d6
and finally between A3 >< d6, which is a weird lone example of A(n) >< dn(n + 2). What gives? It looks like A3 is way below d6 in Tetracot, but they're a little closer in Meantone, a little closer in Pythagorean, a little closer in 39-EDO, and then they swap position in 27-EDO. Bug in my code? Let's double-check the tuned values in 27-EDO:
t(d6) = 2^(13/27) = 1.3961766429234026
t(A3) = 2^(14/27) = 1.4324834970826286
No bug.
So if you're trying to compose microtonal counterpoint, and you're examining a situation where one of your voices might move melodically by a d6 and one might move melodically by an A3, I don't think there's any fact of the matter of whether that's contrary motion or voice crossing until you pick a tuning system. They're octave complements, so it shouldn't be oblique or motion or similar motion, and you'll only have parallel motion in a tuning system that equates the two.
This contrapuntal ambiguity doesn't just happen for A3 >< d6, but for all of the incomparability relations, like P1 >< d2 and A6 >< d7 and A4 >< d6.
I think I need to make a math diagram that summarizes the relationships, inequalities and incomparabilities, between all of all the once-modified intervals next. A Hasse diagram, right? Which means I need to find a transitive reduction. I think I can do that. I've never done it before, but how hard can it be?
I couldn't quickly find an explanation online of how to do it, but this is what I came up with: suppose we have an interval i1, and its "small set" is the set of intervals smaller than it. Suppose the smallset of interval i1 contains an interval i2. Here's the algorithm: draw an arrow from i1 to i2 only if i2 does not appear in the smallsets (smallersets) of any of the other intervals in i1's smallset. I did that and it looks amazing:
It's a ladder! There had to be one big long line because we required the natural intervals to be ordered like 12-TET, but the two lines and the triangles and stuff, that's something inherent to the structure of rank-2 musical intervals. And the horizonal pairs are the ones that were switched in Pythagoras relative to Meantone! And the rungs are fairly regular, except they switch direction in the middle and that's why A3 and d6 are incomparable. Because of the direction switch! This graph makes sense of it all. So I think I did my graph transitive reduction correctly! Nice.
Also, you can see that all of the augmented and major intervals are on one side while all the diminished and minor intervals are on th'other. It's a good graph, am I right?
Oh! What do you think it looks like with twice modified intervals included? Are there FOUR rungs maybe?!?! I have to find all the twice modified interval orderings first, but it's going to happen.
Okay! I found them. There are 11 of them, not counting some degenerate orders that are exactly on the transition points like t(P5) = 2^(7/12) or t(P5) = 2^(19/32). It feels a little spammy to post all of the twice-modified interval orderings, but the transition points are 2^(x) for x in:
[(4/7), (19/33), (15/26), (11/19), (18/31), (7/12), (17/29), (10/17), (13/22), (16/27), (19/32), (3/5)]
which are the t(P5)s for EDOs of increasing size that have the 12-TET ordering, just as I predicted. I didn't know how many there would be though! Eleven. Maybe it's a soluble question how many orders you'll have for once-modified, twice-modified, n-th modified intervals. We'll solve it another day.
Now for the Hasse diagram? Now for the Hasse diagram!
Super Kabbalistic. Probably the most surprising thing here is that AA2 can be larger than dd7. That happens in the 11th order of modified intervals, with 2^(19/32) < t(P5) < 2^(3/5). The lowest EDO in which this happens is 37, where
t(dd7) = 2^(18/37) = 1.401028677549888
t(AA2) = 2^(19/37) = 1.4275225283022686
.
The two are exactly equal in 32-EDO (which is the degenerate order on the low transition point of the range).
I thought this was a really good post at the time, and I still do, but coming back to it after a few months, it needs a closer. Here it comes:
Lots of cool tuning systems have the same order of natural intervals as our beloved 12-TET. Most of them also have cool modified (augmented and diminished) intervals, whereas 12-TET collapses those down to the natural intervals. The once-modified intervals of different tuning systems that respect the 12-TET order of natural intervals, they come in just a few different possible orders - five of them, plus a degenerate order for 12-TET.
Is there a fast way to predict which order of modified intervals a pure-octave syntonic tuning system will have? Sure as shit! Here's what I found: When we look at the frequency ratios assigned by all of our favorite tuning systems to the interval P5, we see immediately that tuning systems with similar t(P5)s generally have the same order of modified intervals. What's more, the modified-interval orders switch from one order to the next as t(P5) increases. To keep the 12-TET order, all of these t(P5)s have to be between 2^(4/7) and 2^(3/5), which was the topic of the previous post. These values are the tuned frequency ratios for the perfect fifth in 7-EDO and 5-EDO, respectively. It turns out that all of the cut-off points at which the modified intervals change from one order to another are also P5s of different EDO tuning systems. Two posts ago, in "EDO Generators", we found out that for all but finitely many cases, EDOs of any number of divisions can be made that respect the 12-TET order of natural intervals. The workable ones begin [5, 7, 12, 17, 19, 22, 26, 27, 29, 31, 32, ....], and the t(P5)s of the first few of these EDOs are precisely where the divisions occur between our five families of modified interval orders.
The normal orders appear on dashes:
[2^(4/7) - 2^(11/19) - 2^(7/12) - 2^(10/17) - 2^(13/22) - 2^(3/5)]
.
For the sake of rigor, I should probably say something about what happens on the boundary points. Just looking back over this post and not doing any calculations, I can already see a mistake. In one spot I said that 19-EDO has the Pythagorean order and in another spot, I say it has Tetracot order. It's definitely not Pythagorean and I don't know why I ever said that. Oops.
Even worse, it doesn't have the Tetracot order, either, not exactly. Tetracot looks like this:
[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]
.
There are more than 19 intervals there, so 19-EDO can't have a distinguishable order for all of them. In particular, 19-EDO has:
[P1, A1=d2, m2, M2, A2=d3, m3, M3, A3=d4, P4, A4, d5, P5, A5=d6, m6, M6, A6=d7, m7, M7, d8]
. and I just figured out why there are five families of true orders that fully distinguish the once-modified intervals! EDOs with divisions of [5, 7, 12, 17, 19, 22] don't have enough distinct pitches in an octave to separate all the once modified intervals. It's not until 26-EDO that we can fit them all. Every transition point is a degenerate order and the true orders live strictly between the transition points.
When we look at orders of twice-modified intervals, there are now 11 non-degenerate orders instead of five.
[2^(4/7) - 2^(19/33) - 2^(15/26) - 2^(11/19) - 2^(18/31) - 2^(7/12) - 2^(17/29) - 2^(10/17) - 2^(13/22) - 2^(16/27) - 2^(19/32) - 2^(3/5)]
.
These transition points are still t(P5)s of EDOs. And why are there 11 families? Because....wait, shouldn't there also be a transition point for 37-EDO? I might have ... failed to include intervals like AA0 and d-1 when I was doing my numeric code. Oops. Let's fix it!
First, up, 37-EDO collapses AA0=dd6 and dd9=AA3. Also, I think 2^(4/37) and 2^(33/37) correspond to thrice-modified intervals, which is kind of crazy, because the twice-modified intervals didn't all get their own spots.
Next, 39-EDO doesn't seem to collapse any once- or twice-modified intervals. I think the frequency ratios of 2^(1/39) and 2^(28/39) must correspond to thrice modified intervals. Quite interesting. All together, in the range [P1, P8) there are 37 intervals that are natural, once modified, or twice modified. I think I can say that any non-degenerate order that includes twice modified intervals will also have 37 intervals in that 1 octave range. Consider, e.g. if d2 is smaller than P1 and it falls out the bottom. Then d9 becomes smaller than P8 and falls in from the top. It's only when intervals collapse in on each other than the number changes. So says my cursory reasoning. But it's late and I have to take out the bins for the binmen and go to bed. Another time, perhaps?
Next time we'll find out if there are actually 12 orders of twice-modified intervals. That's my guess. The upper range probably looks like [... 2^(19/32) - 2^(22/37) - 2^(3/5)].
Since there are only finitely many EDOs that don't have the 12-tet ordering and they come early, things should get more regular as we get into high degrees of modified intervals. Gosh, I never figured out why there are finitely many exceptions, did I? Another thing to do!
....
One reason that I investigated the things in this post is that I wanted to compose classical counterpoint algorithmically, and I wanted to do it correctly, with musical intervals instead of, like, pretending that sharps and flats are the same thing and then finding semitone separations between notes by subtracting midi numbers. That's bullshit and I'm so much better than that.
But when you see a work of classical music, there usually isn't a big note that says "Oh and by the way, tune your t(d25) = 10/1 or you're dead to me". So we expect that there's a way to compose counterpoint so that the tuning system doesn't matter all that much, or else it would be notated reliably. But if you actually look at different tuning systems, even just pure-octave syntonic tuning system with 12-TET order over natural intervals, one generally doesn't put modified intervals in the same order as another! In principle, two people could play the same song, and one would go low to high and one would go high to low. It's crazy! So I had to figure out what's consistent between tuning systems - what can we rely on to be true for all of them, and then we can compose around those constraints so that our music doesn't flip directions depending on the performer.
That's where we get into partial orders. I'm using partial orders to express the fact that some pairs of intervals can change relative position, from one tuning system to another, and some pairs of intervals can't, e.g. P5 tunes to a frequency ratio larger than both A4 and d5 in all pure-octave syntonic tuning system that respect the 12-TET order over natural intervals. You can always count on that, whether you're composing classically inspired microtonal music in, say, quarter-comma meantone because you're shocked and appalled and confused and embarrassed that no one today can compose as well a bunch of guys who died 250 years ago, even when we have their music to work off of, or whether you're doing some crazy sci-fi shit in 31-EDO because you want something that pushes you out of your comfort zone into new realms of complex beauty. The achievement of this post is that I've shown what interval relationships you can rely on for composing across microtonal tuning systems, whatever your motivations.
I'm not alone in this by the way: most music theory, and in particular the historic rules of counterpoint, is/are phrased in interval space, not pitch space. I'm hoping that if we can rewrite the historic rules to allow for an expanded set of consonant intervals, then we can adapt the old machinery to parse and generate all sorts of music, from Chopin to Bill Evan to Ben Johnston to Philipp Gerschlauer and Sintel and Zheanna Erose and beyond. One day, I'm going to teach a computer to compose in interval space. This post is a small but hopefully important step toward doing that systematically.