User:Overthink/13-limit interval flavors
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In this article, we will cover the various flavors of 13-limit intervals. We consider intervals that differ by a pythagorean interval to have the same flavor. The flavor of an interval depends on the primes higher than 3 in its prime factorization.
| D\N | 1/3/9 | 5 | 7 | 11 | 13 |
|---|---|---|---|---|---|
| 1/3/9 | 1/1 (wa) | 5/4 (yo) | 7/4 (zo) | 11/8 (ilo) | 13/8 (tho) |
| 5 | 8/5 (gu) | 1/1 | 7/5 (zogu) | 11/10 (logu) | 13/10 (thogu) |
| 7 | 8/7 (ru) | 10/7 (yoru) | 1/1 | 11/7 (loru) | 13/7 (thoru) |
| 11 | 16/11 (lu) | 20/11 (yolu) | 14/11 (zolu) | 1/1 | 13/11 (tholu) |
| 13 | 16/13 (thu) | 20/13 (yothu) | 14/13 (zothu) | 22/13 (lothu) | 1/1 |
The flavors of intervals
We first look at the pythagorean intervals:
| Cents | Ratio | FJS Name | Color name |
|---|---|---|---|
| 0.000 | 1/1 | P1 | wa 1sn |
| 90.225 | 256/243 | m2 | sawa 2nd |
| 203.910 | 9/8 | M2 | wa 2nd |
| 294.135 | 32/27 | m3 | wa 3rd |
| 407.820 | 81/64 | M3 | lawa 3rd |
| 498.045 | 4/3 | P4 | wa 4th |
| 588.270 | 1024/729 | d5 | sawa 5th |
| 611.730 | 729/512 | A4 | lawa 4th |
| 701.955 | 3/2 | P5 | wa 5th |
| 792.180 | 128/81 | m6 | sawa 6th |
| 905.865 | 27/16 | M6 | wa 6th |
| 996.090 | 16/9 | m7 | wa 7th |
| 1109.775 | 243/128 | M7 | lawa 7th |
| 1200.000 | 2/1 | P8 | wa 8ve |
We then look at intervals of 5:
| Cents | Ratio | FJS Name | Color name |
|---|---|---|---|
| 21.506 | 81/80 | P15 | gu 1sn |
| 111.731 | 16/15 | m25 | gu 2nd |
| 182.404 | 10/9 | M25 | yo 2nd |
| 315.641 | 6/5 | m35 | gu 3rd |
| 386.314 | 5/4 | M35 | yo 3rd |
| 519.551 | 27/20 | P45 | gu 4th |
| 590.224 | 45/32 | A45 | yo 4th |
| 609.776 | 64/45 | d55 | gu 5th |
| 680.449 | 40/27 | P55 | yo 5th |
| 813.686 | 8/5 | m65 | gu 6th |
| 884.359 | 5/3 | M65 | yo 6th |
| 1017.596 | 9/5 | m75 | gu 7th |
| 1088.269 | 15/8 | M75 | yo 7th |
| 1178.494 | 160/81 | P85 | yo 8ve |
And intervals of 7.
| Cents | Ratio | FJS Name | Color name |
|---|---|---|---|
| 27.264 | 64/63 | P17 | ru 1sn |
| 62.961 | 28/27 | m27 | zo 2nd |
| 231.174 | 8/7 | M27 | ru 2nd |
| 266.871 | 7/6 | m37 | zo 3rd |
| 435.084 | 9/7 | M37 | ru 3rd |
| 470.781 | 21/16 | P47 | zo 4th |
| 561.006 | 112/81 | d57 | zo 5th |
| 638.994 | 81/56 | A47 | ru 4th |
| 729.219 | 32/21 | P57 | yo 5th |
| 764.916 | 14/9 | m67 | gu 6th |
| 933.129 | 12/7 | M67 | yo 6th |
| 968.826 | 7/4 | m77 | gu 7th |
| 1137.039 | 27/14 | M77 | yo 7th |
| 1172.736 | 63/32 | P87 | yo 8ve |
Draft section
Content in this section includes ideas and constructions that are not yet complete. This section will remain until the page is completed.
Drafts below this line
From what I see here it seems like I should add A1 and possibly also A5 to the tables.
Flavor 11: 352/243, 88/81, 44/27, 11/9, 11/6, 11/8, 33/32, (99/64, 297/256, 891/512, 2673/2048)
Flavor /11: 243/176, 81/44, 27/22, 18/11, 12/11, 16/11, 64/33, (128/99, ...)
Flavor 13: (104/81,) 52/27*, 13/9, 13/12, 13/8, 39/32, 117/64, 351/256, 1053/1024, (3159/2048)
Flavor /13: (81/52,) 27/26*, 18/13, 24/13, 16/13, 64/39, 128/117, 512/351, 2048/1053, (4096/3159)
Flavor 7/5: (<- Marvelous 45/28,) 15/14*, 7/5, 21/20, 63/40, 189/160, 567/320, 1701/1280, 5103/2560, ...
Flavor 5/7: (<- Marvelous 56/45,) 28/15*, 10/7, 40/21, 80/63, 320/189, 640/567, 2560/1701, 5120/5103, ...
Flavor 11/5: (176/135,) 88/45, 22/15, 11/10, 33/20, 99/80, 297/160, 891/640, (2673/2560, ...)
Flavor 13/5: 208/135, 52/45, 26/15, 13/10, 39/20, 117/80
Flavor 11/7: