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? Get out of my face.
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.
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.
...
No comments:
Post a Comment