If you place a marker far from the sound hole so that 1/9 of the string is close to the tuning pegs and 8/9 of the string is free to vibrate when you pluck or bow or strum near the sound hole, the excited tone will have a frequency 9/8 over the frequency of the open string. If we put a marker there, I'll say that the marker is at 1/9 of the string length, meaning that 8/9 of the string is left free to vibrate.
The string length 8/9 and the frequency ratio 9/8 are reciprocal fractions, and this isn't a coincidence. Delightfully, the function that transforms 1/9 into 9/8 is also an involution, i.e. it can transform in both ways.
y = x / (x - 1)
frequency_ratio = (string_division) / (string_division - 1)
string_division = (frequency_ratio) / (frequency_ratio - 1)
If we divide the string into simple units, what simple frequency ratios do we get? Here's a little table
25.0 -> 25/24 _ 71c
24.5 -> 49/47 _ 72c
24.0 -> 24/23 _ 74c
23.5 -> 47/45 _ 75c
23.0 -> 23/22 _ 77c
22.5 -> 45/43 _ 79c
22.0 -> 22/21 _ 81c
21.5 -> 43/41 _ 82c
21.0 -> 21/20 _ 84c
20.5 -> 41/39 _ 87c
20.0 -> 20/19 _ 89c
19.8 -> 99/94 _ 90c
19.6 -> 98/93 _ 91c
19.5 -> 39/37 _ 91c
19.4 -> 97/92 _ 92c
19.2 -> 96/91 _ 93c
19.0 -> 19/18 _ 94c
18.8 -> 94/89 _ 95c
18.6 -> 93/88 _ 96c
18.5 -> 37/35 _ 96c
18.4 -> 92/87 _ 97c
18.2 -> 91/86 _ 98c
18.0 -> 18/17 _ 99c
17.8 -> 89/84 _ 100c
17.6 -> 88/83 _ 101c
17.5 -> 35/33 _ 102c
17.4 -> 87/82 _ 102c
17.2 -> 86/81 _ 104c
17.0 -> 17/16 _ 105c
16.8 -> 84/79 _ 106c
16.6 -> 83/78 _ 108c
16.5 -> 33/31 _ 108c
16.4 -> 82/77 _ 109c
16.2 -> 81/76 _ 110c
16.0 -> 16/15 _ 112c
15.8 -> 79/74 _ 113c
15.6 -> 78/73 _ 115c
15.5 -> 31/29 _ 115c
15.4 -> 77/72 _ 116c
15.2 -> 76/71 _ 118c
15.0 -> 15/14 _ 119c
14.8 -> 74/69 _ 121c
14.6 -> 73/68 _ 123c
14.5 -> 29/27 _ 124c
14.4 -> 72/67 _ 125c
14.2 -> 71/66 _ 126c
14.0 -> 14/13 _ 128c
13.8 -> 69/64 _ 130c
13.6 -> 68/63 _ 132c
13.5 -> 27/25 _ 133c
13.4 -> 67/62 _ 134c
13.2 -> 66/61 _ 136c
13.0 -> 13/12 _ 139c
12.8 -> 64/59 _ 141c
12.6 -> 63/58 _ 143c
12.5 -> 25/23 _ 144c
12.4 -> 62/57 _ 146c
12.2 -> 61/56 _ 148c
12.0 -> 12/11 _ 151c
11.8 -> 59/54 _ 153c
11.6 -> 58/53 _ 156c
11.5 -> 23/21 _ 157c
11.4 -> 57/52 _ 159c
11.2 -> 56/51 _ 162c
11.0 -> 11/10 _ 165c
10.8 -> 54/49 _ 168c
10.6 -> 53/48 _ 172c
10.5 -> 21/19 _ 173c
10.4 -> 52/47 _ 175c
10.2 -> 51/46 _ 179c
10.0 -> 10/9 _ 182c
9.9 -> 99/89 _ 184c
9.8 -> 49/44 _ 186c
9.7 -> 97/87 _ 188c
9.6 -> 48/43 _ 190c
9.5 -> 19/17 _ 193c
9.4 -> 47/42 _ 195c
9.3 -> 93/83 _ 197c
9.2 -> 46/41 _ 199c
9.1 -> 91/81 _ 202c
9.0 -> 9/8 _ 204c
8.9 -> 89/79 _ 206c
8.8 -> 44/39 _ 209c
8.7 -> 87/77 _ 211c
8.6 -> 43/38 _ 214c
8.5 -> 17/15 _ 217c
8.4 -> 42/37 _ 219c
8.3 -> 83/73 _ 222c
8.2 -> 41/36 _ 225c
8.1 -> 81/71 _ 228c
8.0 -> 8/7 _ 231c
7.9 -> 79/69 _ 234c
7.8 -> 39/34 _ 238c
7.7 -> 77/67 _ 241c
7.6 -> 38/33 _ 244c
7.5 -> 15/13 _ 248c
7.4 -> 37/32 _ 251c
7.3 -> 73/63 _ 255c
7.2 -> 36/31 _ 259c
7.1 -> 71/61 _ 263c
7.0 -> 7/6 _ 267c
6.9 -> 69/59 _ 271c
6.8 -> 34/29 _ 275c
6.7 -> 67/57 _ 280c
6.6 -> 33/28 _ 284c
6.5 -> 13/11 _ 289c
6.4 -> 32/27 _ 294c
6.3 -> 63/53 _ 299c
6.2 -> 31/26 _ 305c
6.1 -> 61/51 _ 310c
6.0 -> 6/5 _ 316c
5.9 -> 59/49 _ 322c
5.8 -> 29/24 _ 328c
5.7 -> 57/47 _ 334c
5.6 -> 28/23 _ 341c
5.5 -> 11/9 _ 347c
5.4 -> 27/22 _ 355c
5.3 -> 53/43 _ 362c
5.2 -> 26/21 _ 370c
5.1 -> 51/41 _ 378c
5.0 -> 5/4 _ 386c
4.9 -> 49/39 _ 395c
4.8 -> 24/19 _ 404c
4.7 -> 47/37 _ 414c
4.6 -> 23/18 _ 424c
4.5 -> 9/7 _ 435c
4.4 -> 22/17 _ 446c
4.3 -> 43/33 _ 458c
4.2 -> 21/16 _ 471c
4.1 -> 41/31 _ 484c
4.0 -> 4/3 _ 498c
3.9 -> 39/29 _ 513c
3.8 -> 19/14 _ 529c
3.7 -> 37/27 _ 545c
3.6 -> 18/13 _ 563c
3.5 -> 7/5 _ 583c
3.4 -> 17/12 _ 603c
3.3 -> 33/23 _ 625c
3.2 -> 16/11 _ 649c
3.1 -> 31/21 _ 674c
3.0 -> 3/2 _ 702c
2.9 -> 29/19 _ 732c
2.8 -> 14/9 _ 765c
2.7 -> 27/17 _ 801c
2.6 -> 13/8 _ 841c
2.5 -> 5/3 _ 884c
2.4 -> 12/7 _ 933c
2.3 -> 23/13 _ 988c
2.2 -> 11/6 _ 1049c
2.1 -> 21/11 _ 1119c
2.0 -> 2/1 _ 1200c
Now, there's no real reason to use decimal as string divisors. You could just as easily place a marker at 3.1 of a the string length as you could at 22/7 of the string length, but I wanted to see how this looked. I've also hidden any frequency ratios with numerators more than 99.
There are some conspicuous absences on this list. Like the justly tuned minor sixth, 8/5, or the justly tuned major seventh, 15/18, and the justly tuned minor seventh, 9/5. Our involution function certainly has less precision near the octave, so we might try using another decimal digit at the high end:
3.0 -> 3/2 _ 702c
2.96 -> 74/49 _ 714c
2.95 -> 59/39 _ 717c
2.92 -> 73/48 _ 726c
2.9 -> 29/19 _ 732c
2.88 -> 72/47 _ 738c
2.85 -> 57/37 _ 748c
2.84 -> 71/46 _ 751c
2.8 -> 14/9 _ 765c
2.76 -> 69/44 _ 779c
2.75 -> 11/7 _ 782c
2.72 -> 68/43 _ 793c
2.7 -> 27/17 _ 801c
2.68 -> 67/42 _ 809c
2.65 -> 53/33 _ 820c
2.64 -> 66/41 _ 824c
2.6 -> 13/8 _ 841c
2.56 -> 64/39 _ 858c
2.55 -> 51/31 _ 862c
2.52 -> 63/38 _ 875c
2.5 -> 5/3 _ 884c
2.48 -> 62/37 _ 894c
2.45 -> 49/29 _ 908c
2.44 -> 61/36 _ 913c
2.4 -> 12/7 _ 933c
2.36 -> 59/34 _ 954c
2.35 -> 47/27 _ 960c
2.32 -> 58/33 _ 976c
2.3 -> 23/13 _ 988c
2.28 -> 57/32 _ 999c
2.25 -> 9/5 _ 1018c
2.24 -> 56/31 _ 1024c
2.2 -> 11/6 _ 1049c
2.16 -> 54/29 _ 1076c
2.15 -> 43/23 _ 1083c
2.12 -> 53/28 _ 1105c
2.1 -> 21/11 _ 1119c
2.08 -> 52/27 _ 1135c
2.05 -> 41/21 _ 1158c
2.04 -> 51/26 _ 1166c
2.0 -> 2 _ 1200c
This got us our minor seventh, but not the other two chromatic 5-limit frequency ratios. It turns out those don't have finite decimal representations: A string divisor of 15/7 gives us a frequency ratio of 15/18, and a string divisor of 8/3 gives us a frequency ratio of 8/5.
I kind of like this? We've found a procedure which privileges a different set of ratios and intervals compared to normal just intonation.
It's not much of a difference: the 2.68 divisor gives us a frequency ratio of 67/42, which is a perceptually indistinguishable 5 cents flat of 8/5. The sounds are still there. But I still think it's a neat sound space. Maybe don't use the high precision divisors if you want a really distinct sound?
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