Ben Johnston was one of the best microtonal composers ever. He wrote in just intonation with high prime limits. To write this, he had a staff notation with accidentals for pitches, and there were accidentals associated with all of the primes up to 31. He also wrote about the use of these accidentals: how they transform familiar frequency ratios into related ones with higher prime factors. I wrote a program to try to find similar relations. I'm going to call them Johnston relations.
Here's an example: the 5-limit m7 interval is tuned to 9/5. The reduced 7th harmonic is tuned to 7/4. The ratio of these is a superparticular ratio:
9/5 / 7/4 = 36/35
and indeed, 36/35 and 35/36 were the frequency ratios associated with Johnston's accidentals for raising and lowering pitches by a septimal comma. We might say that the function of 36/35 is to translate between 9/5 and 7/4.
There are some other relations we can find that involve a 5-limit frequency ratio and the reduced 7th harmonic producing a super-particular ratio:
7/4 / 5/3 = 21/20
7/4 / 3/2 = 7/6
15/8 / 7/4 = 15/14
16/9 / 7/4 = 64/63
7/4 / 27/16 = 28/27
The 64/63 one is used in HEJI staff notation and converts from a Pythagorean tuned m7.
Here are some Johnston relations for the reduced 11th harmonic:
25/18 / 11/8 = 100/99
3/2 / 11/8 = 12/11
11/8 / 4/3 = 33/32
11/8 / 5/4 = 11/10
Ben used 33/32 in his staff notation.
These involve the reduced 13th harmonic:
5/3 / 13/8 = 40/39
27/16 / 13/8 = 27/26
13/8 / 25/16 = 26/25
13/8 / 8/5 = 65/64
13/8 / 3/2 = 13/12
Johnston used 65/64.
These involve the reduced 17th harmonic:
17/16 / 25/24 = 51/50
16/15 / 17/16 = 256/255
Johnston used 51/50.
These involve the reduced 19th harmonic:
6/5 / 19/16 = 96/95
5/4 / 19/16 = 20/19
19/16 / 32/27 = 513/512
Johnston used 96/95.
These use the reduced 23rd harmonic:
3/2 / 23/16 = 24/23
36/25 / 23/16 = 576/575
but Johnston didn't use either of these! He used 46/45, with this given relation:
(23/16) / (45/32) = 46/45
I will freely admit that I did not think to include 45/32, the justly tuned acute augmented fourth, in my program's dictionary of simple 5-limit JI fractions.
These involve the reduced 29th harmonic:
15/8 / 29/16 = 30/29
29/16 / 9/5 = 145/144
Johnston used 145/144.
And this one involves the reduced 31st harmonic:
31/16 / 15/8 = 31/30
and Johnston used 31/16 for his 31-limit comma.
I think this is a rousing success! I had wondered in the past how Johnston chose the frequency ratios for his commas. This programs narrows down the space of options by a lot.
I think that having commas that don't lay right on top of each other would be something of a consideration, but then I'm less sure when I actually look at the ones he used. Like 81/80 and 65/64 only differ by 5 cents, and 36/35 and 33/32 only differ by 5 cents, and 31/30 and 33/32 only differ by 4 cents, and ... A lot of these guys lay right on top of each other. I feel like the reduced harmonic being close to the explanatory target is also important, but I don't think he's just selecting the closest one.
Let's go a little higher. I mean, 31-limit is plenty in practice, but just out of mathematical curiosity, can we find some Johnston relations up to, say, 47-limit?
A few, yeah!
25/24 / 37/36 = 75/74
41/36 / 10/9 = 41/40
41/36 / 9/8 = 82/81
43/36 / 32/27 = 129/128
6/5 / 43/36 = 216/215
4/3 / 47/36 = 48/47
Although the ones for the reduced 43rd harmonic are pretty ugly. What if we expand our set of 5-limit frequency ratios a little to make that one shine? No luck. But I found a few more for 37, 41, and 47:
250/243 / 37/36 = 1000/999
41/36 / 256/225 = 1025/1024
47/36 / 125/96 = 376/375
Maybe (129/128) for a 43-limit accidental isn't so bad. Another option is
43/36 / 7/6 = 43/42
which uses 7/6, the septimal sub-minor third, as an explanatory target rather than a 5-limit ratio. I love 7-limit JI; this appeals to me. But any other small super-particular ratios with 43 in their factors introduce even higher primes in the explanatory targets. Like 44/43 has a factor of 11. And 86/85 would introduce a factor of (85/5=) 17. And 87/86 has a factor of (87/3=) 29. No bueno.
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