On the Xenharmonic wiki, people constantly talk about tempered frequency ratios. Tempering an interval is a normal operation: it means tuning an interval, like a diminished second, to a frequency ratio of 1/1.
t(d2) = 1/1
Tuning systems perform that mapping: from intervals to frequency ratios.
"Tempering a frequency ratio" is a type error. You can't tune 5/3 to 1/1. Five thirds does not equal one.
This frustrated me for a long time, because many of the Xenharmonic people know more math than me and sometimes when I think I've come up with a new result in music theory, I'll find it on their wiki and feel sad that I got scooped by some crayon-eating group theory savant who thinks 5/3 = 1/1.
I'm not being too harsh. Look at this shit: "Tempering [441/440] out splits 11/10 into two even halves [and] equates 21/16 with 55/42."
I think I figured out some of the valid math that they're describing in an invalid way today, and it's not all that complex.
So, let's find "positive 7-limit frequency ratios that are tempered out by 53-EDO". We'll start by finding the approximate number of steps of 53-EDO we need to reach different prime harmonics up to the one with frequency ratio 7/1.
If
2^(i / 53) = (2/1)
then
i for 2/1 = 53 * log(2/1) / log(2)) = 53
We don't even have to round that one. But in general, we want an integer number of 53-EDO steps, so we will round that middle expression to the nearest integer. Here's the formula for the other primes:
i for 3/1 = round(53 * log(3/1) / log(2)) = 84
i for 5/1 = round(53 * log(5/1) / log(2)) = 123
i for 7/1 = round(53 * log(7/1) / log(2)) = 149
It doesn't matter so far if you know the name of the interval that is justly tuned to 7/1. Whatever the interval name is, you know how many steps it will be in 53-EDO. Or rather, we're defining a 53-EDO in which that interval is tuned to 149 steps. And it will be good enough, because 2^(149/53) = 7.01926881203, which is basically 7; don't quibble.
We've basically just defined a tuning system without knowing the basis intervals. We can use this half-assed monstrosity to find the 53-EDO frequency ratios for other intervals we don't know the names of: we just have to know how to build them from the prime basis intervals.
So, for example, there's a famous interval made from (whatever interval is justly tuned to 3/1) minus (whatever interval is justly tuned to 2/1). It happens to be the perfect fifth, but that's currently irrelevant. To tune this mystery difference intervals, we just do arithmetic with our EDO steps: 84 - 53 = 31. And, look,
2^(31/53) = 1.49994090308
is a fine approximation for the traditional just tuning of (the mystery difference interval), which is
3/2 = 1.5.
To find tempered out intervals, we just find coordinates (a, b, c, d) such that {steps = 0} in
steps = a * 53 + b * 84 + c * 123 + d * 149
Now you can do a search over coordinates, with {a} from, oh, -10 to 10, and {b} from whatever to whatever, and so on. Find any set of four coordinates that will give 0 steps. Then find the just frequency ratios for each of those sets of coordinates.
The actual frequency ratio for a given number of steps {i} in our 53-EDO is
2^(i/53)
but the just frequency ratio that we're approximating with 53-EDO is:
frequency_ratio = (2/1)^a * (3/1)^b * (5/1)^c * (7/1)^d
Do that for every tempered set of coordinates (a, b, c, d), and you'll get a list of justly tuned frequency ratios for unknown, unnamed intervals that are tempered out by 53-EDO:
1.0 = 1/1
1.0002286236854139 = 4375/4374
1.001129150390625 = 32805/32768
1.0013580322265625 = 65625/65536
1.0031020408163265 = 6144/6125
1.003331373701744 = 5120/5103
1.004234693877551 = 19683/19600
1.0044642857142858 = 225/224
1.0046939300411524 = 15625/15552
1.0075801749271136 = 1728/1715
1.0078105316200554 = 4000/3969
1.0087178844752187 = 177147/175616
1.0089485012755102 = 50625/50176
1.0091791708002646 = 390625/387072
1.0109368010242645 = 65536/64827
1.0120783007080383 = 2430/2401
1.0123096857790734 = 3125/3087
1.0154499117431228 = 51200/50421
1.0165965074076277 = 273375/268912
1.01682892544773 = 78125/76832
1.0197500312673413 = 839808/823543
1.0199831702776905 = 120000/117649
1.024536666573573 = 843750/823543
This list has famous fractions that people made stupid folksy names for because they didn't have a regular system for naming intervals. Like 4375/4374 is "ragisma". And 32805/32768 is "schisma". And 65625/65536 is "The Horwell comma". And 6144/6125 is "the porwell comma". And 15625/15552 is "kleisma" and it just goes on and on. Now you've got a wiki full of people saying shit like "sabric pote porkypine7 eris wa, hyper-Partchian monzo mos wedgie, unidecimal marvel[22] hobbit, keemic fourthward alpharabian sevond, oneirotonic semihard tamnams, zg32 zotrigu comma".
I don't really hate these people. I know that I'm basically one of them. But I want them to use real words. Weird music, standard math, real words; I swear, it's a good combination.
In summary, you can define an EDO tuning system by finding a number of EDO steps that approximates each prime harmonic up to a desired limit, and then use the intervals that are justly associated with the prime harmonics as an unnamed, unspoken basis for all of your intervals, and then you can even find new intervals that are tempered out by the half-constructed EDO tuning system, and give their just frequency ratios new horrible unsystematic names. It's fun! It's also pretty fast. Want to see 11-limit just frequency ratios for some unnamed intervals that are tempered out by, oh, 31-EDO?
The steps for the 11-limit basis intervals are found like this:
round(31 * log(2/1) / log(2)) = 31
round(31 * log(3/1) / log(2)) = 49
round(31 * log(5/1) / log(2)) = 72
round(31 * log(7/1) / log(2)) = 87
round(31 * log(11/1) / log(2)) = 107
So the number of 31-EDO steps for an arbitrary interval in the 11-limit prime basis will be
steps = a * 31 + b * 49 + c * 72 + d * 87 + e * 107
Here are 11-limit justly tuned frequency ratios for some rank-5 intervals that are tempered out by the 31-EDO that we constructed from the harmonics, along with coordinates in parentheses from the prime harmonic interval basis:
1/1 (0, 0, 0, 0, 0)
3025/3024 (-4, -3, 2, -1, 2)
2401/2400 (-5, -1, -2, 4, 0)
540/539 (2, 3, 1, -2, -1)
1375/1372 (-2, 0, 3, -3, 1)
441/440 (-3, 2, -1, 2, -1)
385/384 (-7, -1, 1, 1, 1)
3136/3125 (6, 0, -5, 2, 0)
3388/3375 (2, -3, -3, 1, 2)
243/242 (-1, 5, 0, 0, -2)
225/224 (-5, 2, 2, -1, 0)
6912/6875 (8, 3, -4, 0, -1)
176/175 (4, 0, -2, -1, 1)
1331/1323 (0, -3, 0, -2, 3)
3773/3750 (-1, -1, -4, 3, 1)
2835/2816 (-8, 4, 1, 1, -1)
1728/1715 (6, 3, -1, -3, 0)
2420/2401 (2, 0, 1, -4, 2)
126/125 (1, 2, -3, 1, 0)
121/120 (-3, -1, -1, 0, 2)
1944/1925 (3, 5, -2, -1, -1)
99/98 (-1, 2, 0, -2, 1)
1617/1600 (-6, 1, -2, 2, 1)
2430/2401 (1, 5, 1, -4, 0)
81/80 (-4, 4, -1, 0, 0)
1815/1792 (-8, 1, 1, -1, 2)
3168/3125 (5, 2, -5, 0, 1)
2662/2625 (1, -1, -3, -1, 3)
2187/2156 (-2, 7, 0, -2, -1)
8712/8575 (3, 2, -2, -3, 2)
2541/2500 (-2, 1, -4, 1, 2)
891/875 (0, 4, -3, -1, 1)
3993/3920 (-4, 1, -1, -2, 3)
8748/8575 (2, 7, -2, -3, 0)
9801/9604 (-2, 4, 0, -4, 2)
5103/5000 (-3, 6, -4, 1, 0)
3267/3200 (-7, 3, -2, 0, 2)
8019/7840 (-5, 6, -1, -2, 1)
6561/6400 (-8, 8, -2, 0, 0)
The 3025/3024 ratio is called the "lehmerisma". The 2401/2400 is "breedsma". The 540/539 is "swetisma". The 441/440 is... "Werckmeister's undecimal septenarian schisma". 3136/3125 is the "hemimean comma". A lot of these are superparticular ratios. They're kind of pretty if you don't look up the Xenharmonic names.
Of the commas, I thought that these four were not constructible from sums of the others:
[-4, -3, 2, -1, 2]: 3025/3024
[-5, -1, -2, 4, 0]: 2401/2400
[2, 3, 1, -2, -1]: 540/539
[6, 0, -5, 2, 0]: 3136/3125
and that sums of those four could be used to make all of the others. But when I do a larger search over value ranges for the components, (a, b, c, d, e), my procedure for generating that smell summary set gives different values, so maybe it's nonsense. It's not an intrinsic fact of the EDO.
I did a search for a fifth vector that can be combined with those four to make a unimodular matrix, and I found all of these:
[-5, 1, 0, 0, 1] # 33/32
[2, 2, -1, -1, 0] # 36/35
[-2, 2, 1, 0, -1] # 45/44
[-4, -1, 0, 2, 0] # 49/48
[1, 0, 2, -2, 0] # 50/49
[-1, -3, 1, 0, 1] # 55/54
[3, 0, -1, 1, -1] # 56/55
[0, -1, -2, 1, 1] # 77/75
[2, -2, 2, 0, -1] # 100/99
[0, -5, 1, 2, 0] # 245/243
[5, 0, 0, -3, 1] # 352/343
[3, 4, -4, 0, 0] # 648/625
And many more complex ones. These are all tuned to one step of 31-EDO.
I did the same procedure for rank-5 coordinates with 19-EDO. These are "the four independent commas" which are probably just noise induced by the extent of my search of parameter space:
[6, 0, -5, 2, 0] = 3136/3125
[-1, -7, 4, 1, 0] = 4375/4374
[2, 3, 1, -2, -1] = 540/539
[-6, 6, -3, 0, 1] = 8019/8000
I searched for a coordinates of a fifth interval that would make the total matrix unimodular, and found all of these:
[-2, 1, -1, 1, 0] # 21/20
[-3, -1, 2, 0, 0] # 25/24
[2, -3, 0, 1, 0] # 28/27
[-5, 1, 0, 0, 1] # 33/32
[0, -1, 1, 1, -1] # 35/33
[2, 2, -1, -1, 0] # 36/35
[1, 0, 2, -2, 0] # 50/49
[-1, -3, 1, 0, 1] # 55/54
[0, -1, -2, 1, 1] # 77/75
[4, 0, 1, -1, -1] # 80/77
[0, 4, 0, -1, -1] # 81/77
[-1, 2, 0, -2, 1] # 99/98
[4, 0, -2, -1, 1] # 176/175
[-4, 3, 0, 1, -1] # 189/176
And many more complex ones. These are all coordinates for rank-5 intervals in the-basis-of-unnamed-prime-harmonic-intervals that are tuned to one step of 19-EDO in 19-EDO.
Here are some intervals tempered out by 53-EDO:
[-1, 2, 0, -2, 1] # 99/98
[-3, -1, -1, 0, 2] # 121/120
[4, 0, -2, -1, 1] # 176/175
[-5, 2, 2, -1, 0] # 225/224
[-7, -1, 1, 1, 1] # 385/384
[2, 3, 1, -2, -1] # 540/539
And here are coordinates for some simple intervals that are tuned to one step of 53-EDO along with their just tunings:
[1, 0, 2, -2, 0] # 50/49
[-1, -3, 1, 0, 1] # 55/54
[6, -2, 0, -1, 0] # 64/63
[-4, 4, -1, 0, 0] # 81/80
[2, -2, 2, 0, -1] # 100/99
[1, 2, -3, 1, 0] # 126/125
[-1, 5, 0, 0, -2] # 243/242
[0, -5, 1, 2, 0] # 245/243
[5, 0, 0, -3, 1] # 352/343
.
Cool.
No comments:
Post a Comment