I was writing music in 5-limit just intonation, and I discovered that the Pythagorean d7 (also called a GrGrGrd7 in 5-limit J.I. and tuned to 32768/19683 in either system) sounds the same as the 5-limit major sixth (tuned to 5/3). They only differ by like two cents. The difference is called the schisma, which is more formally an AcAcA0, justly tuned to 32805/32768. Since I can't hear the difference, I was curious what happens if we temper out the schisma, t(AcAcA0) = 1/1. Presumably, our 3D lattice for Just Intonation reduces by a dimension, and we get a 2D lattice (which might make a nice 2D keyboard layout).
To define the spacing of the other two dimensions, I had to pick tunings for two more intervals. I went with pure octaves, t(P8) = 2/1, and pure major thirds, t(M3) = 5/4. How do we figure out the tuned values for other rank-3 intervals using those three tuned values?
I don't know how the people on the Xenharmonic wiki do it, but here's how we do it with Cramer's rule.
Let's start with coordinates for some simple rank-3 intervals that we want to tune. I'll represent them in the (P8, P12, M17) basis, a.k.a. the 5-limit prime harmonic basis, which is justly tuned to (2/1, 3/1, 5/1).
P1 = (0, 0, 0) # 1/1
AcAcA0 = (-15, 8, 1) # 32805/32768
Ac1 = (-4, 4, -1) # 81/80
d2 = (7, 0, -3) # 128/125
A1 = (-3, -1, 2) # 25/24
Grm2 = (8, -5, 0) # 256/243
m2 = (4, -1, -1) # 16/15
M2 = (1, -2, 1) # 10/9
AcM2 = (-3, 2, 0) # 9/8
AcA2 = (-6, 1, 2) # 75/64
Grm3 = (5, -3, 0) # 32/27
m3 = (1, 1, -1) # 6/5
M3 = (-2, 0, 1) # 5/4
AcM3 = (-6, 4, 0) # 81/64
d4 = (5, 0, -2) # 32/25
P4 = (2, -1, 0) # 4/3
Ac4 = (-2, 3, -1) # 27/20
A4 = (-1, -2, 2) # 25/18
d5 = (2, 2, -2) # 36/25
Gr5 = (3, -3, 1) # 40/27
P5 = (-1, 1, 0) # 3/2
A5 = (-4, 0, 2) # 25/16
Grm6 = (7, -4, 0) # 128/81
m6 = (3, 0, -1) # 8/5
GrGrGrd7 = (15, -9, 0) # 32768/19683
M6 = (0, -1, 1) # 5/3
AcM6 = (-4, 3, 0) # 27/16
Grd7 = (7, -1, -2) # 128/75
Grm7 = (4, -2, 0) # 16/9
m7 = (0, 2, -1) # 9/5
M7 = (-3, 1, 1) # 15/8
d8 = (4, 1, -2) # 48/25
P8 = (1, 0, 0) # 2/1
After the "#" symbols I show the just tunings for those intervals, but now I want to detune some of them so that we can put 5-limit just intonation on a 2D lattice, and particularly a 2D lattice in which the schisma, which I can't hear, is tempered out.
Okay, so, our new basis is going to be (AcAcA0, P8, M3). In the (P8, P12, M17) basis, this basis is a matrix with coordinates [(-15, 8, 1), (1, 0, 0), (-2, 0, 1)]. This matrix isn't unimodular: it doesn't have determinant 1 or -1, so it can't represent just intonation intervals in integer coordinates. It happens to have determinant 8, and we'll see that the coordinates in this basis have denominators of 8.
To get coordinates (x, y, z) in the (AcAcA0, P8, M3) basis for an interval with coordinates (m, n, o) in the (P8, P12, M17) basis, we're going to use Cramer's rule, just as ejlilley taught us to do for rank-2 intervals.
def rank_3_lilley_cramer_formula(B1, B2, B3, interval):
(m, n, o) = interval
(a, b, c) = B1
(d, e, f) = B2
(g, h, i) = B3
detA = a * (e * i - f * h) - b * (d * i - f * g) + c * (d * h - e * g)
x = (m * (e * i - f * h) - n * (d * i - f * g) + o * (d * h - e * g)) / detA
y = (a * (n * i - o * h) - b * (m * i - o * g) + c * (m * h - n * g)) / detA
z = (a * (e * o - f * n) - b * (d * o - f * m) + c * (d * n - e * m)) / detA
return (x, y, z)
So, for
B1 = (a, b, c) = AcAcA0 = (-15, 8, 1)
B2 = (d, e, f) = P8 = (1, 0, 0)
B3 = (g, h, i) = M3 = (-2, 0, 1)
we can see that M6 and GrGrGrd7 have coordinates in the (AcAcA0 , P1, M3) basis that only differ in the first component:
M6 = (-1/8, 3/8, 9/8)
GrGrGrd7 = (-9/8, 3/8, 9/8)
Since our schismatic temperament tunes the AcAcA0 to 1/1, it doesn't matter what coordinates we have for that component, since 1 raised to any real power is going to to be 1. In particular, both of those intervals are tuned to
(2)^(3/8) * (5/4)^(9/8) ~ 1.666901790204155
in this temperament. You might wonder if this simplifies at all when you expand out the fraction raised to a fractional power. It ends up being 5 * 10^(1/8) / 4. So not really. The 5-limit just value of M6 is 5/3 = 1.6666 repeating 6s, so we're doing well with representing our 5-limit fractions. How about our old 3-limit fractions? Well, this temperament tunes P5 to
(2)^(5/8) * (5/4)^(-1/8) = 1.4997884186649115
which is like 0.2 cents off from the just value of 3/2. Pretty bang up job, right? I think I've done a bang up job.
One down side is that a keyboard arranged in P8s and M3s would be hard to play, I think. And also the interval coordinates are fractional, so the grid might be a little slanty or something. But I think we can fix that: pick whatever intervals you think make a nice 2d arrangement for playing then tune them to the values of this great (AcAcA0, P8, M3) -> (1/1, 2/1, 5/4) temperament. Then the frequency ratios will be the same and the grid will be good as well. I think that's an option. You can define the temperament in different ways once you know the induced frequency ratios. You just need two points independent of AcAcA0. Maybe M2 and m2 would make a nice layout for example. To do those, you just need:
t(m2) = 4 * 10^(1/8) / 5
t(M2) = 5 * 10^(1/4) / 8
Here are a bunch of tuned values for simple intervals in this schismatic temperament with pure octaves and pure 5-limit major thirds:
P1 = (0, 0, 0) # (2)^(0) * (5/4)^(0) = 1/1
Ac1 = (-4, 4, -1) # (2)^(1/2) * (5/4)^(-3/2) ~ 1.0119288512538815
d2 = (7, 0, -3) # (2)^(1) * (5/4)^(-3) = 128/125
A1 = (-3, -1, 2) # (2)^(-5/8) * (5/4)^(17/8) ~ 1.0418136188775968
Grm2 = (8, -5, 0) # (2)^(-1/8) * (5/4)^(5/8) ~ 1.0542412585714556
m2 = (4, -1, -1) # (2)^(3/8) * (5/4)^(-7/8) ~ 1.0668171457306592
M2 = (1, -2, 1) # (2)^(-1/4) * (5/4)^(5/4) ~ 1.1114246312743268
AcM2 = (-3, 2, 0) # (2)^(1/4) * (5/4)^(-1/4) ~ 1.1246826503806981
Grm3 = (5, -3, 0) # (2)^(1/8) * (5/4)^(3/8) ~ 1.1856868528308278
m3 = (1, 1, -1) # (2)^(5/8) * (5/4)^(-9/8) ~ 1.1998307349319293
M3 = (-2, 0, 1) # (2)^(0) * (5/4)^(1) = 5/4
AcM3 = (-6, 4, 0) # (2)^(1/2) * (5/4)^(-1/2) ~ 1.2649110640673518
d4 = (5, 0, -2) # (2)^(1) * (5/4)^(-2) = 32/25
P4 = (2, -1, 0) # (2)^(3/8) * (5/4)^(1/8) ~ 1.333521432163324
Ac4 = (-2, 3, -1) # (2)^(7/8) * (5/4)^(-11/8) ~ 1.3494288109714632
A4 = (-1, -2, 2) # (2)^(-1/4) * (5/4)^(9/4) ~ 1.3892807890929082
d5 = (2, 2, -2) # (2)^(5/4) * (5/4)^(-9/4) ~ 1.4395937924872935
Gr5 = (3, -3, 1) # (2)^(1/8) * (5/4)^(11/8) ~ 1.4821085660385345
P5 = (-1, 1, 0) # (2)^(5/8) * (5/4)^(-1/8) ~ 1.4997884186649115
A5 = (-4, 0, 2) # (2)^(0) * (5/4)^(2) = 25/16
Grm6 = (7, -4, 0) # (2)^(1/2) * (5/4)^(1/2) ~ 1.5811388300841898
m6 = (3, 0, -1) # (2)^(1) * (5/4)^(-1) = 8/5
M6 = (0, -1, 1) # (2)^(3/8) * (5/4)^(9/8) ~ 1.666901790204155
AcM6 = (-4, 3, 0) # (2)^(7/8) * (5/4)^(-3/8) ~ 1.686786013714329
Grm7 = (4, -2, 0) # (2)^(3/4) * (5/4)^(1/4) ~ 1.7782794100389228
m7 = (0, 2, -1) # (2)^(5/4) * (5/4)^(-5/4) ~ 1.799492240609117
M7 = (-3, 1, 1) # (2)^(5/8) * (5/4)^(7/8) ~ 1.8747355233311396
d8 = (4, 1, -2) # (2)^(13/8) * (5/4)^(-17/8) ~ 1.9197291758910868
P8 = (1, 0, 0) # (2)^(1) * (5/4)^(0) = 2/1
I experimented with a bunch of bases/layouts, and I think this one is nice:
Meantone temperaments are named based on how they alter P5. For example, quarter comma meantone lowers the tuned value of the perfect fifth from the just value by a factor of (81/80)^(1/4). The base, 81/80, is the just value for the syntonic comma, and the exponent, 1/4, explains why the system is called quarter comma meantone. Here's a big set of meantone temperament definitions:
(Ac1, P5, P8), (1, 3/2, 2) # Pythagorean tuning (0-comma meantone)
t(Ac1, AcA4, P8) = (1, 45/32, 2) # 1/6-comma meantone
t(Ac1, M3, P8) = (1, 5/4, 2) # 1/4-comma meantone
t(Ac1, m3, P8) = (1, 6/5, 2) # 1/3-comma meantone
t(Ac1, M2, P8) = (1, 10/9, 2) # 1/2-comma meantone
t(Ac1, m2, P8) = (1, 16/15, 2) # 1/5-comma meantone
t(Ac1, A1, P8) = (1, 25/24, 2) # 2/7-comma meantone
The schismatic temperament I defined in this post tuned P5 to (2)^(5/8) * (5/4)^(-1/8), which simplifies to 10^(7/8)/5. I didn't want to solve for the exponent, but WolframAlpha assures me that this is flat of (3/2) by the eighth root of the tuned schisma:
(3/2) / (32805/32768)^(1/8) = 10^(7/8)/5
so I think I've defined the "1/8-comma schismatic temperament", where the comma is now the schisma instead of the acute unison.
Do you want to see what other schismatic temperaments we can define and what they shall be named? I know I do!
If we tune P5 purely, we get Pythagorean tuning again of course. We already know that tuning M3 to a pure 5/4 gives us 1/8-comma schismatic.
If we tune m3 to a pure 6/5, we get 1/9 comma schismatic, since:
(6/5)^(1/9) * 2^(5/9)= (3/2) / (32805/32768)^(1/9)
If we tune M2 to a pure (10/9), then we get 1/10-comma schismatic, since
(10/9)^(-1/10) * 2^(3/5) = (3/2) / (32805/32768)^(1/10)
If we tune m2 to a pure 16/15, then we get 1/7 schismatic, since
(16/15)^(1/7) * 2^(4/7) = (3/2) / (32805/32768)^(1/7)
It sure looks like the comma is in the exponent on the left hand side every time, and it's the exponent of the tuned value of the interval that we're altering between temperaments (not the 2/1 base for the octave).
So if we define a tuning system by
t(AcAcA0, AcA4, P8) = (1, 45/32, 2)
we get a tuned value of
(45/32)^(-1/6) * 2^(2/3)
for P5.
I bet we'll find that this is 1/6-comma schismatic. Let's check.
(45/32)^(-1/6) * 2^(2/3) = (3/2) / (32805/32768)^(1/6)
WolframAlpha says True! Nice.
If we define a schismatic temperament in which we purely tune A1 to 25/24, we get a t(P5) of
(25/24)^(-1/17) * 2^(10/17)
Do we therefore get 1/17-comma schismatic? Nope! We get 2/17-comma schismatic
(25/24)^(-1/17) * 2^(10/17) = (3/2) / (32805/32768)^(2/17)
So I guess I don't know the rule. But I'll figure it out. In the meantime, these are still nice names for the temperaments.
I'm going to do a few more. Still tempering out the schisma, still tuning the octave purely, let's just name the last interval that's tuned purely. Here's a big condensed table, arranged by increasing denominators in the fractional comma:
P5 -> 3/2: 0-comma schismatic (Pythagorean tuning)
Grd4 -> 512/405: 1/4-comma schismatic.
AcA4 -> 45/32: 1/6-comma schismatic.
m2 -> 16/15: 1/7-comma schismatic.
M3 -> 5/4: 1/8-comma schismatic. Also d2 -> 128/125. Also d4 -> 32/25.
m3 -> 6/5: 1/9-comma schismatic. Also A2 -> 125/108. Also A4 -> 25/18.
M2 -> 10/9: 1/10-comma schismatic. Also GrM3 -> 100/81.
Ac4 -> 27/20: 1/11-comma schismatic
Ac1 -> 81/80: 1/12-comma schismatic.
Gr4 -> 320/243: 1/13-comma schismatic.
AcA2 -> 75/64: 2/15-comma schismatic.
A1 -> 25/24: 2/17-comma schismatic.
Acm2 -> 27/25: 2/19-comma schismatic.
Acm3 -> 243/200: 2/21-comma schismatic.
A3 -> 125/96: 3/25-comma schismatic.
d3 -> 144/125: 3/26-comma schismatic.
GrA1 -> 250/243: 3/29-comma schismatic.
GrA3 -> 625/486: 4/37-comma schismatic.
GrA2 -> 2500/2187: 4/39-comma schismatic.
I'm pretty intrigued by what's showing up so far and what's not. But I don't know how to use these things musically, so maybe it's best that I stop monkeying around.
...
Or I could monkey around with the meantone temperaments a little? For all of these tuning systems, I'll tune Ac1 to 1/1 and tune P8 to 2/1, and I'll just list the third interval that's tuned purely and what system results:
A3 -> 125/96: 3/11-comma meantone.
A4 -> 25/18: 1/3-comma meantone again.
Ac4 -> 27/20: 1/1-comma meantone.
AcA2 -> 75/64: 2/9-comma meantone.
Acd1 -> 243/250: 3/8-comma meantone.
Acd2 -> 648/625: 1/3-comma meantone again.
Acm2 -> 27/25: 2/5-comma meantone.
Acm3 -> 243/200: 2/3-comma meantone.
Gr4 -> 320/243: negative 1/1-comma meantone?
GrA1 -> 250/243: 3/7-comma meantone.
GrA2 -> 2500/2187: 4/9-comma meantone.
GrA3 -> 625/486: 4/11-comma meantone.
GrA4 -> 1000/729: 1/2-comma meantone again.
GrM2 -> 800/729: 1/1-comma meantone again.
GrM3 -> 100/81: 1/2-comma meantone again.
Grd2 -> 2048/2025: 1/6-comma meantone again.
Grd3 -> 256/225: 1/5-comma meantone again
Grd4 -> 512/405: 1/4-comma meantone again
d3 -> 144/125: 3/10-comma meantone.
d4 -> 32/25: 1/4-comma meantone again.
AcA1 -> 135/128: 1/7-comma meantone.
...
And here they all are sorted by increasing comma fraction denominators:
P5 -> 3/2: Pythagorean tuning (0-comma meantone)
Gr4 -> 320/243: negative 1/1-comma meantone?
Ac4 -> 27/20: 1/1-comma meantone.
GrM2 -> 800/729: 1/1-comma meantone again.
M2 -> 10/9: 1/2-comma meantone.
GrM3 -> 100/81: 1/2-comma meantone again.
GrA4 -> 1000/729: 1/2-comma meantone again.
m3 -> 6/5: 1/3-comma meantone.
Acd2 -> 648/625: 1/3-comma meantone again.
A4 -> 25/18: 1/3-comma meantone again.
Acm3 -> 243/200: 2/3-comma meantone.
M3 -> 5/4: 1/4-comma meantone.
d4 -> 32/25: 1/4-comma meantone again.
Grd4 -> 512/405: 1/4-comma meantone again.
m2 -> 16/15: 1/5-comma meantone.
Grd3 -> 256/225: 1/5-comma meantone again.
Acm2 -> 27/25: 2/5-comma meantone.
AcA4 -> 45/32: 1/6-comma meantone.
Grd2 -> 2048/2025: 1/6-comma meantone again.
AcA1 -> 135/128: 1/7-comma meantone.
A1 -> 25/24: 2/7-comma meantone.
GrA1 -> 250/243: 3/7-comma meantone.
Acd1 -> 243/250: 3/8-comma meantone.
AcA2 -> 75/64: 2/9-comma meantone.
GrA2 -> 2500/2187: 4/9-comma meantone.
d3 -> 144/125: 3/10-comma meantone.
A3 -> 125/96: 3/11-comma meantone.
GrA3 -> 625/486: 4/11-comma meantone.
I have no idea what to make of this. Neat though, right?
Questions for people who are better at math than me: Are 1/8-comma meantone or 2/11-comma meantone possible? Is 1/14-comma schismatic possible? How about 2/23-comma schismatic or 3/23-schismatic?
...
Oh, good. I figured out how to derive the comma fractions from the tuning system. Now I don't have to wait for WolframAlpha to solve them. And that means I can iterate over weird temperament tuning systems automatically to find weird fractional commas.
Basically, if we have a basis (B1, B2, B3) tuned (t(B1), t(B2), t(B3)), with B1 equal to the tempered comma, B2 an interval of interest that we'll tune justly, and B3 equal to the octave, also tuned justly, then if the tempered coordinates for P5 are (m, n, o), we have empirically
t(B2)^ n * t(B3)^o = t(P5) / t(B1)^(x)
in frequency space, which corresponds to
B2 * n + B3 * o = P5 - B1 * (x)
in interval space. With a little rearrangement,
x = (P5 - (B2 * n + B3 * o)) / B1
.
For example, if we define a temperament by t(Ac1, M3, P8) = (1/1, 5/4, 2/1), then this equation becomes:
x = (-1/1, 1/1, -1/4) / (-4, 4, -1)
Now, if you divide the vectors elementwise, you get (1/4, 1/4, 1/4) confirming that this is quarter-comma meantone. I don't know any reason why this should always work (or always work when B2 is independent of each B1 and B3), in the sense that the three entries of x are always equal, but it works empirically and I'll continue on investigating in this manner.
Okay, here are some meantone/Ac1 temperaments defined by the fractional power of the justly tuned Ac1 that you flatten P5 on the left and the interval that you tune purely on the right:
-1/2 Ac1 temperament: AcAcM2
-1/1 Ac1 temperament: Gr4
0/1 Ac1 temperament: AcM2, AcM3, Grm2, Grm3
1/1 Ac1 temperament: GrM2
1/2 Ac1 temperament: AcAcd4, GrM3, M2
1/3 Ac1 temperament: A2, Acd2, Acdd4, Acddd3, GrAAA1, m3
2/3 Ac1 temperament: Acm3
1/4 Ac1 temperament: M3, d2, d4
3/4 Ac1 temperament: GrGrM3
1/5 Ac1 temperament: Grd3, m2
2/5 Ac1 temperament: Acd3, Acm2
3/5 Ac1 temperament: AcAcm2
1/7 Ac1 temperament: AcA1
2/7 Ac1 temperament: A1, AA1
3/7 Ac1 temperament: GrA1
3/8 Ac1 temperament: Acd4, GrAA2
2/9 Ac1 temperament: AcA2
4/9 Ac1 temperament: GrA2
3/10 Ac1 temperament: d3, ddd5
3/11 Ac1 temperament: A3
4/11 Ac1 temperament: GrA3
5/12 Ac1 temperament: AcAcd2
5/14 Ac1 temperament: GrAA1
4/15 Ac1 temperament: dd4
5/16 Ac1 temperament: AA2
5/17 Ac1 temperament: dd3
6/17 Ac1 temperament: Acdd3
6/19 Ac1 temperament: GrAA0
7/22 Ac1 temperament: Acddd4
7/23 Ac1 temperament: AAA2
.
And here are some schismatic/AcAcA0 temperament definitions:
0/1 AcAcA0 temperament: AcM2, AcM3, Grm2, Grm3
1/5 AcAcA0 temperament: AcA1
1/7 AcAcA0 temperament: Grd3, m2
1/8 AcAcA0 temperament: M3, d2, d4
1/9 AcAcA0 temperament: A2, Acd2, Acdd4, Acddd3, GrAAA1, m3
1/10 AcAcA0 temperament: AcAcd4, GrM3, M2
1/11 AcAcA0 temperament: GrM2
1/13 AcAcA0 temperament: Gr4
1/14 AcAcA0 temperament: AcAcM2
2/15 AcAcA0 temperament: AcA2
2/17 AcAcA0 temperament: A1, AA1
2/19 AcAcA0 temperament: Acd3, Acm2
2/21 AcAcA0 temperament: Acm3
3/25 AcAcA0 temperament: A3
3/26 AcAcA0 temperament: d3, ddd5
3/28 AcAcA0 temperament: Acd4, GrAA2
3/29 AcAcA0 temperament: GrA1
3/31 AcAcA0 temperament: AcAcm2
3/32 AcAcA0 temperament: GrGrM3
4/33 AcAcA0 temperament: dd4
4/37 AcAcA0 temperament: GrA3
4/39 AcAcA0 temperament: GrA2
5/43 AcAcA0 temperament: dd3
5/44 AcAcA0 temperament: AA2
5/46 AcAcA0 temperament: GrAA1
5/48 AcAcA0 temperament: AcAcd2
6/53 AcAcA0 temperament: GrAA0
6/55 AcAcA0 temperament: Acdd3
7/61 AcAcA0 temperament: AAA2
7/62 AcAcA0 temperament: Acddd4
.
...
I just had the best idea! I'll sort the temperaments within a family by the size of their P5s, and sort them alongside EDOs that temper out the (syntonic, schismatic)-commas as well. That way we can be like, "this two dimensional temperament has a very close P5 to the one-dimensional 45-EDO" or whatever. And maybe that will help me to figure out range limits on the fractional commas! Like, if you temper out the syntonic comma and keep octaves pure, then tuning a third octave purely seems to put pretty tight constraints on how P5 gets tuned, and maybe we can say that tempered Ac1 and pure P8 means that P5 has to fall between, oh, 5-EDO's P5 and 7-EDO's P5, or something like that!
...
Ah! From preliminary investigation, it seems that the flattest P5 you get with schismatic temperaments comes with 1/2-comma schismatic, which tunes GrGrd3 purely. GrGrd3 is tuned justly to 32768/32805, which is the inverse of the schisma, the AcAcA0, justly tuned to 32805/32768. That I did not expect. The sharpest we go with P5 for schismatic temperaments is a frequency ratio of 3/2 in 0-comma schismatic, i.e. Pythagorean tuning. And all the fractional commas fall between 1/2 and 0/1.
Maybe that's not really the case though? Because when I investigate syntonic temperaments, the range seems to go quite a bit wider in both directions. For example, a syntonic temperament defined by tuning GrGrGrGrM2 purely produces 5/2-comma syntonic, which is obviously more than 1 comma flat. On the other side of a just P5, by purely tuning GrGrGrGr4, we get negative4-comma syntonic, which is sharper than P5 by four acute unisons, i.e. 86ish cents. And I wouldn't be at all surprised if these bounds kept increasing as I tried defining weirder temperaments from weirder purely tuned intervals.
And if syntonic temperaments behave that way, then maybe schismatic ones do too, way out in the dark waters.
...
Okay, I promised EDOs and I'm going to do EDOs. There seem to be finitely many EDOs that temper out the syntonic comma, Ac1. They are: [5, 7, 12, 19, 24, 26, 31, 36, 38, 43, 45, 50, 55, 57, 62, 67, 69, 74, 76, 81, 86, 88, 93, 98, 100, 105, 117, 129]-EDO. Nothing else between 5-EDO and 5000-EDO. This list is small enough, I feel we can look at all the P5s:
1.4859942891369484 : 7-EDO
1.4916644904914018 : 26-EDO
1.492548464309911 : 45-EDO
1.4937589616544857 : 19-EDO
1.4937589616544857 : 38-EDO
1.4937589616544857 : 57-EDO
1.4937589616544857 : 76-EDO
1.4943783453027453 : 88-EDO
1.494548945312803 : 69-EDO
1.4948492486349383 : 100-EDO
1.4948492486349383 : 50-EDO
1.495105110169352 : 81-EDO
1.4955178823482085 : 31-EDO
1.4955178823482085 : 62-EDO
1.4955178823482085 : 93-EDO
1.4958363844631488 : 105-EDO
1.4959698311839842 : 74-EDO
1.4960896011977585 : 117-EDO
1.4962957394862462 : 129-EDO
1.4962957394862462 : 43-EDO
1.4962957394862462 : 86-EDO
1.4965418805580937 : 98-EDO
1.496734346325667 : 55-EDO
1.4970159080002896 : 67-EDO
1.4983070768766815 : 12-EDO
1.4983070768766815 : 24-EDO
1.4983070768766815 : 36-EDO
1.515716566510398 : 5-EDO
So only 5-EDO's P5 is sharper than the just P5 at 3/2.
The P5 of 7-EDO falls between the P5s of 4/5-comma syntonic and 3/4-comma syntonic:
1.4851668043517086 4/5 Ac1 temperament : AcAcAcm2
1.4859942891369484 : 7 -EDO
1.4860895666142713 3/4 Ac1 temperament : GrGrM3
And 5-EDO's tuned P5 falls between the tuned P5s of -2/3-comma syntonic and -1-comma syntonic.
The EDOs that temper out the schisma also seem to be finite, but it's a much larger list: [12, 17, 24, 29, 36, 41, 53, 65, 77, 82, 89, 94, 101, 106, 118, 130, 135, 142, 147, 154, 159, 171, 183, 195, 200, 207, 212, 219, 224, 236, 248, 253, 260, 265, 272, 277, 289, 301, 313, 318, 325, 330, 342, 354, 366, 371, 378, 383, 390, 395, 407, 419, 424, 431, 436, 443, 448, 460, 472, 484, 489, 496, 501, 508, 513, 525, 537, 542, 549, 554, 561, 566, 578, 590, 602, 607, 614, 619, 626, 631, 643, 655, 660, 667, 672, 679, 684, 696, 708, 720, 725, 732, 737, 744, 749, 761, 773, 778, 785, 790, 797, 802, 814, 826, 838, 843, 850, 855, 862, 867, 879, 891, 896, 903, 908, 915, 920, 932, 944, 956, 961, 968, 973, 985, 997, 1009, 1014, 1021, 1026, 1033, 1038, 1050, 1062, 1067, 1074, 1079, 1086, 1091, 1103, 1115, 1127, 1132, 1139, 1144, 1151, 1156, 1168, 1180, 1185, 1192, 1197, 1204, 1209, 1221, 1233, 1245, 1250, 1257, 1262, 1269, 1274, 1286, 1298, 1303, 1310, 1315, 1322, 1327, 1339, 1351, 1363, 1368, 1375, 1380, 1387, 1392, 1404, 1416, 1421, 1428, 1433, 1440, 1445, 1457, 1469, 1481, 1486, 1493, 1498, 1505, 1510, 1522, 1534, 1539, 1546, 1551, 1558, 1563, 1575, 1587, 1599, 1604, 1611, 1616, 1628, 1640, 1652, 1657, 1664, 1669, 1676, 1681, 1693, 1705, 1710, 1717, 1722, 1729, 1734, 1746, 1758, 1770, 1775, 1782, 1787, 1794, 1799, 1811, 1823, 1828, 1835, 1840, 1847, 1852, 1864, 1876, 1888, 1893, 1900, 1905, 1912, 1917, 1929, 1941, 1946, 1953, 1958, 1965, 1970, 1982, 1994, 2006, 2011, 2018, 2023, 2030, 2035, 2047, 2059, 2064, 2071, 2076, 2083, 2088, 2100, 2112, 2124, 2129, 2136, 2141, 2148, 2153, 2165, 2177, 2182, 2189, 2194, 2206, 2218, 2230, 2242, 2247, 2259, 2271, 2283, 2295, 2300, 2312, 2324, 2336, 2348, 2353, 2365, 2377, 2389, 2401, 2418, 2430, 2442, 2454, 2471, 2483, 2495, 2536, 2548, 2589, 2601, 2654, 2707]. Nothing else between 5-EDO and 5000-EDO.
The flattest P5 in an EDO that tempers out the schisma comes from 12-EDO and the sharpest such P5 comes from 17-EDO.
The 12-EDO tuned P5 is even slightly flatter than the 1/2-comma schismatic P5, and the 17-EDO tuned P5 is even slightly sharper than the tuned P5 of 0-comma schismatic, so these feel like good bounds.
The possibly non-existent bounds on the P5 of syntonic temperaments still befuddle me a little. I'll just have to wade deeper out in the water to investigate. Or content myself that temperaments with P5 outside of [7-EDO's P5, 5-EDO's P5] are non-diatonic and not worth too much of my time.
...
When I look up other commas that people have used to define temperaments, I see a lot of diminished seconds:
Grd2 = (11, -4, -2) # 2048/2025
d2 = (7, 0, -3) # 128/125
Acd2 = (3, 4, -4) # 648/625
If you're feeling sassy, analyzing those might be fun.
I hadn't realized that finitely many EDOs temper out a given comma either. That's another ... source of data that could be catalogued and analyzed.
If you want a rank-4 interval to temper out that's justly associated with a small 7-limit frequency ratio, we've got some options:
4375/4374 at 0.4 cents, "ragisma"
2401/2400 at 0.7, "breedsma"
5120/5103 at 5.8c, "hemifamity"
225/224 at 7.7 cents cents, "marvel"
1029/1024 at 8.4 cents, "gamelisma"
126/125 at 13.8 cents, "starling"
245/243 at 14.2 cents, "sensamagic"
I've also listed the silly names for these fractions as they are known on the Xenharmonic wiki for some reason. The first ratio has a nice interpretation as
(25/24) / ((36/35) * (81/80))
i.e. it's the difference between a grave augmented unison, justly tuned to 250/243, and the septimal super unison of Ben Johnston, 36/35. And a temperament which tunes
a complicated rank-3 interval to the same frequency ratio as a simpler rank-4 which was perceptually indistinguishable under just tuning is a very good temperament.
The 2401/2400 does not have this property, but 5120/5103 does; it's just
(36/35) / (81/80)^2
I also like 225/224 at 7.7 cents for this, which can be explained as:
(36/35) / (128/125)
The 1029/1024 does not have a tidy rank-3 to rank-4 relationship.
The 14-cent 126/125 is a more complicated version of the 225/224 relationship:
(36/35) / ((128/125) * (81/80))
and significantly more perceptible, so I'm not very impressed by that one.
The last just septimal comma, 245/243, is kind of cool:
(245/243) = (16/15) / (36/35)^2
It's also too wide for my liking, but this just nicely shows how two septimal commas produce something like a minor second.
To summarize, I'd be friends with anyone who thought that the intervals associated with these ratios:
ragisma: 4375/4374 at 0.4 cents # (25/24) / ((36/35) * (81/80))
hemifamity: 5120/5103 at 5.8c # (36/35) / (81/80)^2
marvel: 225/224 at 7.7 cents cents # (36/35) / (128/125)
were cool things to temper out to reduce rank-4 intervals by a dimension.
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