User:Overthink/The 7-limit in 53edo
- Development of this page is paused indefinitely. 171edo is much more interesting.
In 53edo, the 7-limit is well-approximated, and especially the 5- and 3-limits. On this page, we will analyze the approximations and structures of 53edo in the 7-limit.
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 1 | 22.642 | 531441/524288, 81/80, 64/63, 50/49 |
| 2 | 45.283 | 36/35, 49/48, 128/125, 250/243 |
| 3 | 67.925 | 28/27, 25/24 |
| 4 | 90.566 | 256/243, 135/128, 21/20 |
| 5 | 113.208 | 16/15, 15/14, 2187/2048 |
| 6 | 135.849 | 27/25 |
| 7 | 158.491 | 35/32 |
| 8 | 181.132 | 10/9 |
| 9 | 203.774 | 9/8 |
| 10 | 226.415 | 8/7 |
| 11 | 249.057 | 81/70, 125/108, 144/125, 147/128 |
| 12 | 271.698 | 7/6, 75/64 |
| 13 | 294.340 | 32/27 |
| 14 | 316.981 | 6/5 |
| 15 | 339.623 | 105/64, 243/200 |
| 16 | 362.264 | 100/81, 315/256 |
| 17 | 384.906 | 5/4 |
| 18 | 407.547 | 81/64 |
| 19 | 430.189 | 9/7, 32/25 |
| 20 | 452.830 | 64/49, 35/27 |
| 21 | 475.472 | 21/16 |
| 22 | 498.113 | 4/3 |
| 23 | 520.755 | 27/20 |
| 24 | 543.396 | 48/35 |
| 25 | 566.038 | 1/1 |
| 26 | 588.679 | 1024/729, 7/5, 45/32 |
| 27 | 611.321 | 729/512, 10/7, 64/45 |
| 28 | 633.962 | 1/1 |
| 29 | 656.604 | 1/1 |
| 30 | 679.245 | 1/1 |
| 31 | 701.887 | 3/2 |
| 32 | 724.528 | 1/1 |
| 33 | 747.170 | 1/1 |
| 34 | 769.811 | 1/1 |
| 35 | 792.453 | 1/1 |
| 36 | 815.094 | 1/1 |
| 37 | 837.736 | 1/1 |
| 38 | 860.377 | 1/1 |
| 39 | 883.019 | 1/1 |
| 40 | 905.660 | 1/1 |
| 41 | 928.302 | 1/1 |
| 42 | 950.943 | 1/1 |
| 43 | 973.585 | 1/1 |
| 44 | 996.226 | 1/1 |
| 45 | 1018.868 | 1/1 |
| 46 | 1041.509 | 1/1 |
| 47 | 1064.151 | 1/1 |
| 48 | 1086.792 | 1/1 |
| 49 | 1109.434 | 1/1 |
| 50 | 1132.075 | 1/1 |
| 51 | 1154.717 | 1/1 |
| 52 | 1177.358 | 1/1 |
| 53 | 1200.000 | 2/1 |
The 81/80 and 64/63 commas translate pythagorean intervals into nearby pental and septimal intervals respectively. Considering them seperately is too complex, so we conflate them into one comma step, tempering out 5120/5103. Here's a table of intervals organized using tempering of 5120/5103. Each interval is a fifth above the interval to the left of it, and a comma above the interval below it. Not all ratios are shown, or else the table will be too complex.
| 256/175 | 256/245,
729/700 |
384/245 | 512/343 | |||||||||||
| 48/25 | 36/25 | 27/25 | 81/50 | 128/105,
243/200 |
64/35 | 48/35 | 36/35 | 54/35 | 81/70, | 243/140,
256/147 |
64/49 | 96/49 | 72/49 | 54/49 |
| 135/128 | 64/45 | 16/15 | 8/5 | 6/5 | 9/5 | 27/20 | 64/63,
81/80 |
32/21 | 8/7 | 12/7 | 9/7 | 27/14 | 81/56 | 243/224 |
| 28/15,
4096/2187 |
7/5,
1024/729 |
21/20,
256/243 |
63/40,
128/81 |
32/27 | 16/9 | 4/3 | 1/1 | 3/2 | 9/8 | 27/16 | 80/63,
81/64 |
40/21,
243/128 |
10/7,
729/512 |
15/14,
2187/2048 |
| 448/243 | 112/81 | 28/27 | 14/9 | 7/6 | 7/4 | 21/16 | 63/32,
160/81 |
40/27 | 10/9 | 5/3 | 5/4 | 15/8 | 45/32 | 135/128 |
| 49/27 | 49/36 | 49/48 | 49/32 | 147/128,
280/243 |
140/81 | 35/27 | 35/18 | 35/24 | 35/32 | 105/64,
400/243 |
315/256,
100/81 |
50/27 | 25/18 | 25/24 |
| 343/192 | 343/256,
980/729 |
1029/1024,
245/243 |
245/162 | 245/216 | 245/144 | 245/192 | 245/128,
1400/729 |
735/512,
350/243 |
175/162 | 175/108 | 175/144 | 175/96 | 175/128 | 250/243,
525/512 |