There was a Greek music theorist named Archytas. He came up with some unusual musical scales. I haven't read translations of the primary sources, but my understanding is that he was enamored with super-particular ratios (those of the form {(n) / (n - 1)}) and the mathematical operation known as the harmonic mean.
The inverse of the harmonic mean of {n} numbers is the average of the inverses of the {n} numbers. Weird operation to define, right? For two numbers {a} and {b}, it has a simple form:
H(a, b) = (2 * a * b) / (a + b)
Archytas, supposedly, would divide a frequency ratio {f} into smaller parts by taking the harmonic mean of {f} and {1/1}. The ratio of {f} to the harmonic mean of {f} and {1/1} was also of interest to him. Let's call it the complementary ratio. The complementary ratio is actually the arithmetic mean of {f} with 1/1. Anyway, let's call "harmonic mean with {f} and 1/1" the Archytas divisor of {f} to save space.
And now let's just do a few. The Archytas divisor of {2/1} is {4/3}:
H((2/1), (1/1)) = (4/3)
The complementary (arithmetic) ratio is {3/2} because
(2/1) / (4/3) = (3/2)
or because
((2/1) + (1/1)) / 2 = (3/2)
The Archytas divisor of (4/3) is (8/7) with a complement of (7/6).
The Archytas divisor of (3/2) is (6/5) with a complement of (5/4).
You can see that these are all super-particular ratios. Also, for a super particular ratio with numerator {n}, the Archytas divisor will have numerator {2 * n} and the denominator will be {2 * n - 1}. Further, the complement of the divisor will be {n - 1 / n - 2}.
A little algebra will show that the product of the Archytas divisor and its complement reproduces the original ratio:
((2n) / (2n - 1)) * ((2n - 1) / (2n - 2)) = n / (n - 1)
.
Here's a table of some Archytas divisors:
Arch(4/3) = 8/7 & 7/6
Arch(8/7) = 16/15 & 15/14
Arch(16/15) = 32/31 & 31/30
Arch(15/14) = 30/29 & 29/28
Arch(7/6) = 14/13 & 13/12
Arch(14/13) = 28/27 & 27/26
Arch(13/12 v (1/1) = 26/25 & 25/24
Arch(3/2) = 6/5 & 5/4
Arch(6/5) = 12/11 & 11/10
Arch(12/11) = 24/23 & 23/22
Arch(11/10) = 22/21 & 21/20
Arch(5/4) = 10/9 & 9/8
Arch(10/9) = 20/19 & 19/18
Arch(9/8) = 18/17 & 17/16
Arch(18/17) = 36/35 & 35/34
Arch(17/16) = 34/33 & 33/32
I don't really know why, but Archytas liked the ratio (28/27) and used it in his scales a lot. I think it's interesting to imagine a world where Archytas had more influence and we used septimal frequency ratios like (8/7) and (7/6) and (28/27) everywhere in our music.
Here's a thought experiment: if we think of 8/7 and 7/6 as halves of 4/3, perhaps a small half and a large half, then what ratios are the one-quarters and three-quarters of 4/3?
The smaller half, 8/7, divides into 16/15 and 15/14. The large half, 7/6, divides into 14/13 and 13/12. Those are the four Archytas quarters of 4/3.
If we try to recombine the four Archytas quarters, we get these as our derived halves:
(16/15) * (15/14) = 8/7 # 231 cents
(16/15) * (14/13) = 224/195 # 240 cents
(16/15) * (13/12) = 52/45 # 250 cents
(15/14) * (14/13) = 15/13 # 248 cents
(15/14) * (13/12) = 65/56 # 258 cents
(14/13) * (13/12) = 7/6 # 267 cents
Since 4/3 is 498 cents, the actual half of it is 249 cents, which is really close to both 15/13 and 52/45. These two also produce 4/3 exactly when multiplied. So they're also very good halves of 4/3.
The first of these, 15/13, is called the justly tuned value for a "recessed acute augmented second", ReAcA2, in my rank-6 Lilley-Johnston system for naming intervals, because it's one tridecimal comma smaller than the acute augmented second:
t(Pr1) = (65/64)
t(AcA2) = (75/64)
AcA2 - Pr1 = ReAcA2
(75/64) / (65/64) = (15/13)
and prominent pairs with recessed for describing rank-6 intervals, such as those that are justly tuned to tridecimal frequency ratios.
The second one, 52/45, is a a justly tuned "prominent grave diminished third", PrGrd3, since a grave diminished third is justly tuned to (256/225).
If we multiply the Archytas halves (8/7, 7/6) of 4/3 by the quarters (16/15, 15/14, 14/13, 13/12), we get a bunch of three-quarters:
(8/7 * 16/15) = (128/105) # 343 cents
(8/7 * 15/14) = (60/49) # 351 cents
(7/6 * 14/13) = (49/39) # 395 cents
(7/6 * 13/12) = (91/72) # 405 cents
(7/6 * 16/15) = (56/45) # 379 cents
(7/6 * 15/14) = (5/4) # 386 cents
(8/7 * 14/13) = (16/13) # 360 cents
(8/7 * 13/12) = (26/21) # 370 cents
but only the last four of these can be gotten by dividing (4/3) by the quarters:
(4/3) / (16/15) = 5/4 # 386 cents
(4/3) / (15/14) = 56/45 # 379 cents
(4/3) / (14/13) = 26/21 # 370 cents
(4/3) / (13/12) = 16/13 # 360 cents
so I think of them as the truer three-quarters of 4/3 perhaps. The weird ones can probably be used to get weird flavors of not-quite 4/3, if that's something you might like.
If we take Archytas means and complements of the eighths of 4/3, namely (13/12, 14/13, 15/14, 16/15), then we get super particular ratios from (25/24, 26/25, 27/26, ... all the way up to ..., 31/30, 32/31).
I don't know that I want to figure out cents for all of them, but the middle ones multiplied together give
(28/27) * (29/28) = 29/27
which we might think of as a very good non-Archytas fourth of 4/3, just as 15/13 and 52/45 were very good non-Archytas halves.
If we try breaking 15/13 and 52/45 into Archytas means and complements, we get:
15/14 # 119 cents // Archytas mean of 15/13
104/97 # 121 cents // Archytas mean of 52/45
14/13 # 128 cents // Archytas complement of 15/13
97/90 # 130 cents // Archytas complement of 52/45
If we take the middle two and recombine them, we get even better approximations for the actual frequency-space half of (4/3), and the combination is 112/97.
I wonder if any of these are mathematically optimal rational halves for 4/3 in any sense, like truncations of the continued fraction expansion of sqrt(4/3). The continued fraction is [1, 6, 2, 6, 2, 6, 2, ...].
When I did that calculation, I messed up and got sad and tried something else. Since
sqrt(4/3) = 2/3 * sqrt(3)
I just looked up truncations of the C.F. for sqrt(3) and multiplied them by 2/3. Possibly mathematically equivalent? I don't know. But it worked. These guys approximate sqrt(3):
7/4, 26/15, 97/56, ...
and scaled by 2/3 we get
7/6, 52/45, 97/84, ...
And the first two of those have already made an appearance. Which just goes to show that sometimes when you calculate rational approximations to sqrt(4/3) by one method, you get some overlap with approximations to sqrt(4/3) produced by a second method.
Since 97 is quite a large prime as far as audible harmonics are concerned, I don't have much use for it in extended just intonation. And the next term in the series has a prime factor of 607, which is even worse. But I'm glad we looked all the same.
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