Suppose we have a space of intervals with just tunings that have only odd factors - no factors of 2. We can define 1-dimensional equal temperaments over such a space (equal divisions of a "decade" tuned to 3/1), or even unequal temperaments in higher dimensions that are still lower than the dimension of the interval space in question.
If we represent rank-2 intervals in the odd prime harmonic basis, then the elements are like exponents of 3 and 5 in the factorization of the justly associated frequency ratios. These ratios get large numerators and denominators very quickly, but here are a few easy ones:
[0, 0] # 1/1
[3, -2] # 27/25
[6, -4] # 729/625
[-7, 5] # 3125/2187
[-4, 3] # 125/81
[-1, 1] # 5/3
[2, -1] # 9/5
[5, -3] # 243/125
[8, -5] # 6561/3125
[-5, 4] # 625/243
[-2, 2] # 25/9
[1, 0] # 3/1
If we want to define an equal temperament over the decade, and EDD, we tune the decade purely and temper out some other interval X.
t(P12) = 3/1
t(X) = 1/1
This induces effects on the tuning of other intervals - and we can see this by expressing those intervals in the basis [[1, 0], X] instead of the rank-2 odd prime harmonic basis [[1, 0], [0, 1]]. I'll also call the interval X a comma or tempered comma.
To convert interval bases, we just "left multiply" or "vector-matrix multiply" the target interval we want to convert by a transformation matrix, which is simply the inverse of the matrix [[1, 0], X].
Suppose we've converted a target interval Y to have coordinates in this tempered comma basis, and we'll call the new coordinates [a, b]. Then we have this relation in interval space:
Y = a * P12 + b * X
and this has a parallel expression in frequency ratio space:
t(Y) = t(P12)^(a) * t(X)^(b)
where t(Y) is the tuning a interval Y in some tuning system. By substitution of the tunings For P12 and X in our tempered tuning system we have
t(Y) = (3/1)^(a) * (1/1)^b
which means that any target interval Y will be tuned to a frequency ratio of 3 to some rational number. I haven't proven to you that {a} will be rational, but there are only so many hours in a day.
For example, the frequency ratio 48828125/43046721 is quite small and would make a good comma. It has a prime factorization of
3^(-16) * 5 ^(11)
so the associated interval in the rank-2 odd prime harmonic basis is
[-16, 11]
Our transformation matrix is
inverse([[1, 0], [-16, 11]])
And you don't have to tune many target intervals with this to guess that it's producing frequency ratios of the form
3^(i/11)
i.e. this defines the "11 equal division of the decade" temperament, or 11-EDD. We can also verify this by seeing that the absolute determinant of the transformation matrix is 1/11.
There are actually many intervals you could temper out to produce 11-EDD. Another way to define 11-EDD is this: For a rank-N interval space, 11-EDD tunes each interval justly assocaited with an odd prime harmonic to the nearest frequency of the form 3^(i/11). If this had no mistuning, then for a prime P, we'd have
P = 3^(i/11)
or
i = 11 * log_3(P)
but usally there is mistuning, so we round {i} to the nearest integer:
i = round(11 * log_3(P))
Here are the nearest steps for 11-EDD for primes [3, 5, 7, 11, 13]
11-EDD: [11, 16, 19, 24, 26]
I'll call a sequence of harmonic EDD-steps like that a canonical definition for the EDD for a given rank, i.e. that's the canonical rank-5 definition of 11-EDD.
At some point down the line, as you might predict from the shape of the log function, multiple harmonics will be tuned to the same step. This has never been a problem for me. Like, 11-EDD has the same tuning for the 16th and 17th odd prime harmonic. But I don't use those? You don't need those harmonics for music making. The 16th and 17th harmonics, you need those. The 16th and 17th *prime* harmonics? No way, Jose.
Samller EDDs reach this point of ambiguity sooner, but you can just ... not use 5-EDD, or you can use it without considering how the steps might map to intervals with just tunings involving factors of 37 and 41. It's not hard. Or you can use just accept that 5-EDD tunes these identically. It's all fine.
It's really easy to use a canonical definition for an EDD to tune intervals in that EDD. Let's use 39-EDD and the interval [-5, 1, 2, 0] in odd-prime-harmonic coordinates as examples.
Our target interval has a just tuning of
3^(-5) * 5^(1) * 7^(2) * 11^(0) = 245/243
In 39-EDD, the harmonics aren't exactly 3/1, 5/1, 7/1, they're nearby ratios of the form 3^(i/39) for integers {i}. And the integers are of course found in the canonical definition:
39-EDD: [39, 57, 69, 85] // rank-4
To find the number of steps of 39-EDD for the temepred tuning of the target interval [-5, 1, 2, 0], we just take the dot product.
[-5, 1, 2, 0] * [39, 57, 69, 85] = 0
This means that the target interval is tuned to zero steps by 39-EDD, i.e. it's tempered out or tuned to a frequency ratio of 1/1.
Some EDDs can't be defined over a rank-2 interval space: no matter what rank-2 interval you try to temper out, you'll get some other EDD. This is very easy to predict: if the first two values of the canonical definition for the EDD have a greatest common divisor of 1, you can give the EDD a definition in terms of a pure decade and a tempered comma. If the first *three* values have GCD = 1, then you can give a rank-3 definition in terms of a pure decade and *two* tempered commas. And so on. Here are a few EDDs, and the shortest canonical definition for each one that gives a GCD of 1:
10-EDD: [10, 15, 18] // rank-3
11-EDD: [11, 16] // rank-2
12-EDD: [12, 18, 21, 26] // rank-4
13-EDD: [13, 19] // rank-2
14-EDD: [14, 21, 25] // rank-3
15-EDD: [15, 22] // rank-2
16-EDD: [16, 23] // rank-2
17-EDD: [17, 25] // rank-2
18-EDD: [18, 26, 32, 39] // rank-4
19-EDD: [19, 28] // rank-2
20-EDD: [20, 29] // rank-2
21-EDD: [21, 31] // rank-2
22-EDD: [22, 32, 39] // rank-3
23-EDD: [23, 34] // rank-2
24-EDD: [24, 35] // rank-2
25-EDD: [25, 37] // rank-2
26-EDD: [26, 38, 46, 57] // rank-4
27-EDD: [27, 40] // rank-2
28-EDD: [28, 41] // rank-2
29-EDD: [29, 42] // rank-2
30-EDD: [30, 44, 53] // rank-3
31-EDD: [31, 45] // rank-2
32-EDD: [32, 47] // rank-2
33-EDD: [33, 48, 58] // rank-3
34-EDD: [34, 50, 60, 74, 79] // rank-5
35-EDD: [35, 51] // rank-2
36-EDD: [36, 53] // rank-2
37-EDD: [37, 54] // rank-2
38-EDD: [38, 56, 67] // rank-3
39-EDD: [39, 57, 69, 85] // rank-4
40-EDD: [40, 59] // rank-2
Using the canonical definitions, it's also really easy to figure out which rank-2 tempered comma you can use to define an EDD that is rank-2 definable. You can see that 11-EDDs has a rank-2 canonical definition of [11, 16]. If you reverse these numbers and flip the sign up one, i.e. temper out [-16, 11] or [16, -11], then you'll get 11-EDD. These two intervals are complementary: they sum to [0, 0], and equivalently, their just tunings will also have inverse frequency ratios. Only one of these ratios will be larger than 1/1, and that's the one I prefer to use in defining EDOs, but they're equivalent definitions, and tempering one out will also temper out the other.
If you give a non-canonical definition of an EDD, e.g. if you don't tune the prime harmonic interval associated with 5/1 to its nearest step, then you get different tempered commas; weird commas that don't have small just tunings near 1/1. If you make a transformation matrix from the decade and a non-canonical comma, you'll still map intervals to frequency ratios like 3^(i/11), but the mapping will be weird - things that should be nearby will be ripped up and sent far from each other, decades away. I once tried using multiple non-canonical definitions like this for an EDO to to generate interesting harmonies. It sounded pretty crazy.
Anyway, the most important EDD is 13-EDD, which is definable with rank-2 intervals. Here's its rank-2 canonical harmonic prefix:
13-EDD: [13, 19] // rank-2
So we could define it by tempering out
[-19, 13] # 3^(-19) * 5^(13) = 1220703125/1162261467
But it's more common to define it over rank-3 interval space. It happens to be definable over rank-3 interval space in terms of a pure decade and tempered commas that have just tunigns of 245/243 and 3087/3125.
Finding a rank-2 tempered comma was really easy, but I don't have a short procedure I can explain to you here for finding two commas that generate an equal temperament over rank-3 interval space. I mean, you could do a simialr thing to the rank-2 case by re-ordering and negating some numbers from the canonical definition, but they're going to be really ugly intervals with high complexity just tunings. I gess you don't really need to have low complexity ratios to define an EDD, but it's nice to give a short simple definition for a temperament that way and I'd feel dirty if I did anything less.
When I first got into microtonality, I would just tune a bunch of intervals in an EDO, e.g. using the canonical definition, then see which ones were tempered out to 0 steps, and look for the ones with the lowest complexity frequency ratios in that set, or search through linear combinations of my found tempered commas to find even lower complexity frequency ratios. Then I'd find the determinant of the transformation matrix to verify that my lowest complexity commas were sufficient to define the EDO. The same thing would work with EDDs, I'm sure.
Later on I starting using the Lenstra–Lenstra–Lovasz lattice basis reduction algorithm to find tempered commas of EDOs. I could probably adapt the code to work for EDDs without much work.
Anyway, if you divide a 13-EDD steps into three parts, you instead get 39-EDD. This does an equally good job of representing ratios with factors of 3, 5, 7, and a much better job of representing ratios with factors of 11. So if we want to write music that sounds like 11-odd-limit just intonation using only a finite set of pitches that supports free modulation, then 39-EDD is a good choice. We could also look at detempering 39-EDD in order to play around with pure 11-limit harmony.
Let's do it! We'll start by finding 39-EDDs tempered commas. It's rank-4 so we know that we'll need ratios involving factors of 11. I've never thought about this before, but look at the rank-4 canonical definition:
39-EDD: [39, 57, 69, 85] // rank-4
The first and last harmonic steps, 39 and 85, have a GCD of 1. So maybe we could define 39-EDD in the 3.11 just intonation subgroup? That could be cool. But I'll just start with 3.5.7.11.
Here are some intervals that are tempered out by 39-EDD, along with their just tunings:
[-5, 1, 2, 0] # 245/243
[-3, 0, -2, 3] # 1331/1323
[-2, 5, -3, 0] # 3125/3087
[1, 5, -1, -3] # 9375/9317
[-2, 1, 4, -3] # 12005/11979
[-7, 6, -1, 0] # 15625/15309
[3, 4, -5, 0] # 16875/16807
[-10, 2, 4, 0] # 60025/59049
[-8, 1, 0, 3] # 6655/6561
Let's just check if the three lowest complexity commas (in terms of numerator magnitude), plus the decade, have an absolute determinant of 39. This works surprisingly often.
abs(determinant of [[1, 0, 0, 0], [-5, 1, 2, 0], [-3, 0, -2, 3], [-2, 5, -3, 0]]) = 39
Woo! That is a minimal definition of 39-EDD. For different notions of comma complexity, you could have different minimal definitions, but I think this one is pretty good.
If you wanted to explore some good non-equal odd-limit temperaments, you could temper out any 1 or 2 of these commas.
There is a little more complexity to defining an unequal temperament than choosing commas though. For example, if you had an interval space that was justly tuned in the 3.5.7 JI subgroup, you could temper out a 7-limit comma and keep the harmonics 3 and 7 pure, and that would give you an interesting thing. But if you tempered out the same 7-limit comma and kept 3 and 5 pure, then your temperament would just look like just intonation in the 3.5 subgroup. Said a little differently: if you temper out a 7-limit comma, you need to keep the 7th harmonic pure, or you're simply defining a tuning system that forgets all the 7-limit ratios. If you temper out some 7-ness, you'll also want to retain some 7-ness. There might be something you could tune the 7th harmonic to besides purity that would keep the interval space from collapsing like this, but I wouldn't. To me, the point of an unequal temperament is to reduce the size of the interval space a little bit for manageability or playability while still maintaining as much harmonic purity as you can.
And we know that 39-EDD already has good accuracy over the 3.5.7.11 subgroup, so these are good commas to temper out.
Let's look at a detempering of 39-EDD. Among all the intervals tuned to each step, I show the just tuning of the one with the simplest just tuning here:
0\39 # 1/1
1\39 # 77/75
2\39 # 35/33
3\39 # 27/25
4\39 # 55/49
5\39 # 63/55
6\39 # 25/21
7\39 # 11/9
8\39 # 125/99
9\39 # 9/7
10\39 # 33/25
11\39 # 15/11
12\39 # 7/5
13\39 # 175/121
14\39 # 49/33
15\39 # 75/49
16\39 # 11/7
17\39 # 121/75
18\39 # 5/3
19\39 # 77/45
20\39 # 135/77
21\39 # 9/5
22\39 # 225/121
23\39 # 21/11
24\39 # 49/25
25\39 # 55/27
26\39 # 343/165
27\39 # 15/7
28\39 # 11/5
29\39 # 25/11
30\39 # 7/3
31\39 # 297/125
32\39 # 27/11
33\39 # 63/25
34\39 # 55/21
35\39 # 121/45
36\39 # 25/9
37\39 # 77/27
38\39 # 225/77
39\39 # 3/1
Let's see if that's symmetric under decade-complementation.
Original scale: 1/1, 77/75, 35/33, 27/25, 55/49, 63/55, 25/21, 11/9, 125/99, 9/7, 33/25, 15/11, 7/5, 175/121, 49/33, 75/49, 11/7, 121/75, 5/3, 77/45, 135/77, 9/5, 225/121, 21/11, 49/25, 55/27, 343/165, 15/7, 11/5, 25/11, 7/3, 297/125, 27/11, 63/25, 55/21, 121/45, 25/9, 77/27, 225/77, 3/1
Decade Complement: 1/1, 77/75, 81/77, 27/25, 135/121, 63/55, 25/21, 11/9, 125/99, 9/7, 33/25, 15/11, 7/5, 495/343, 81/55, 75/49, 11/7, 121/75, 5/3, 77/45, 135/77, 9/5, 225/121, 21/11, 49/25, 99/49, 363/175, 15/7, 11/5, 25/11, 7/3, 297/125, 27/11, 63/25, 55/21, 147/55, 25/9, 99/35, 225/77, 3/1
Not quite. I count 16 ratios that only show up once, so I we're differing on 8 scale degrees.
Let's just use the original scale instead of deciding which ratios we want to use for each of those scale degrees. It'll be fine.
What simple triad chords show up in this scale? Well, we get a lot of 7-limit chords that we saw in Odd Limit Harmony In Bohlen Pierce.
We also get some new 11-limit chords.
[P1, SpM3, AsGr5] # [1/1, 9/7, 55/36]
[P1, DeAcM3, DeA5] # [1/1, 27/22, 50/33]
[P1, m3, DeA5] # [1/1, 6/5, 50/33]
[P1, Sbm3, AsGr5] # [1/1, 7/6, 55/36]
[P1, SpM3, DeAc5] # [1/1, 9/7, 81/55]
[P1, M3, DeAc5] # [1/1, 5/4, 81/55]
[P1, AsGrm3, Gr5] # [1/1, 11/9, 40/27]
[P1, M3, DeA5] # [1/1, 5/4, 50/33]
[P1, Sbm3, DeAc5] # [1/1, 7/6, 81/55]
[P1, M3, AsGr5] # [1/1, 5/4, 55/36]
[P1, DeM3, d5] # [1/1, 40/33, 36/25]
[P1, DeM3, DeA5] # [1/1, 40/33, 50/33]
These don't look so good. We don't have a single perfect fifth interval. I wondered if they had nice otonal representations. A few of them are okay?
[27, 33, 40] : [P1, AsGrm3, Gr5]
[33, 40, 50] : [P1, DeM3, DeA5]
[36, 42, 55] : [P1, Sbm3, AsGr5]
[36, 45, 55] : [P1, M3, AsGr5]
I don't think this is a very good scale. Now we know.
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