3159811edo: Difference between revisions
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3159811edo is [[consistent]] in the 65-odd-limit with a lower [[relative error]] than any previous equal temperaments in the 61-limit. It is the smallest | == Theory == | ||
3159811edo is [[consistent]] in the 65-odd-limit with a lower [[relative error]] than any previous equal temperaments in the 61-limit. It is the smallest edo which is purely consistent{{idio}} in the 63-odd-limit (i.e. does not exceed 25% relative error on the first 63 harmonics of the [[harmonic series]]). | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|3159811 | {{Harmonics in equal|3159811|columns=9}} | ||
{{Harmonics in equal|3159811 | {{Harmonics in equal|3159811|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 3159811edo (continued)}} | ||
== Scales == | == Scales == | ||
Revision as of 08:32, 8 May 2025
| ← 3159810edo | 3159811edo | 3159812edo → |
3159811 equal divisions of the octave (abbreviated 3159811edo or 3159811ed2), also called 3159811-tone equal temperament (3159811tet) or 3159811 equal temperament (3159811et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3159811 equal parts of about 0.00038 ¢ each. Each step represents a frequency ratio of 21/3159811, or the 3159811th root of 2.
Theory
3159811edo is consistent in the 65-odd-limit with a lower relative error than any previous equal temperaments in the 61-limit. It is the smallest edo which is purely consistent[idiosyncratic term] in the 63-odd-limit (i.e. does not exceed 25% relative error on the first 63 harmonics of the harmonic series).
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000000 | +0.000021 | +0.000025 | +0.000014 | -0.000031 | -0.000048 | -0.000018 | -0.000032 | +0.000065 |
| Relative (%) | +0.0 | +5.6 | +6.5 | +3.6 | -8.2 | -12.6 | -4.8 | -8.4 | +17.2 | |
| Steps (reduced) |
3159811 (0) |
5008182 (1848371) |
7336854 (1017232) |
8870711 (2551089) |
10931150 (1451717) |
11692690 (2213257) |
12915610 (276366) |
13422648 (783404) |
14293601 (1654357) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000081 | +0.000001 | -0.000018 | +0.000092 | -0.000023 | +0.000017 | -0.000023 | -0.000082 | -0.000029 |
| Relative (%) | +21.4 | +0.2 | -4.9 | +24.3 | -6.1 | +4.6 | -5.9 | -21.6 | -7.7 | |
| Steps (reduced) |
15350302 (2711058) |
15654324 (3015080) |
16460888 (661833) |
16928852 (1129797) |
17145971 (1346916) |
17551451 (1752396) |
18099146 (2300091) |
18588040 (2788985) |
18740009 (2940954) | |
Scales
Harmonic scales
3159811edo accurately approximates the mode 32 of harmonic series. All interval pairs are distinguished.
| Overtones | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
|---|---|---|---|---|---|---|---|---|---|
| JI Ratios | 1/1 | 33/32 | 17/16 | 35/32 | 9/8 | 37/32 | 19/16 | 39/32 | 5/4 |
| … in cents | 0 | 53.273 | 104.955 | 155.14 | 203.91 | 251.344 | 297.513 | 342.483 | 386.314 |
| Degrees in 3159811edo | 0 | 140277 | 276366 | 408510 | 536931 | 661833 | 783404 | 901817 | 1017232 |
| Overtones | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 |
|---|---|---|---|---|---|---|---|---|
| JI Ratios | 41/32 | 21/16 | 43/32 | 11/8 | 45/32 | 23/16 | 47/32 | 3/2 |
| … in cents | 429.062 | 470.781 | 511.518 | 551.318 | 590.224 | 628.274 | 665.507 | 701.955 |
| Degrees in 3159811edo | 1129797 | 1239649 | 1346916 | 1451717 | 1554163 | 1654357 | 1752396 | 1848371 |
| Overtones | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 |
|---|---|---|---|---|---|---|---|---|
| JI Ratios | 49/32 | 25/16 | 51/32 | 13/8 | 53/32 | 27/16 | 55/32 | 7/4 |
| … in cents | 737.652 | 772.627 | 806.91 | 840.528 | 873.505 | 905.865 | 937.632 | 968.826 |
| Degrees in 3159811edo | 1942367 | 2034464 | 2124737 | 2213257 | 2300091 | 2385302 | 2468949 | 2551089 |
| Overtones | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 |
|---|---|---|---|---|---|---|---|---|
| JI Ratios | 57/32 | 29/16 | 59/32 | 15/8 | 61/32 | 31/16 | 63/32 | 2/1 |
| … in cents | 999.468 | 1029.577 | 1059.172 | 1088.269 | 1116.885 | 1145.036 | 1172.736 | 1200 |
| Degrees in 3159811edo | 2631775 | 2711058 | 2788985 | 2865603 | 2940954 | 3015080 | 3088020 | 3159811 |
- The scale in adjacent steps is 140277, 136089, 132144, 128421, 124902, 121571, 118413, 115415, 112565, 109852, 107267, 104801, 102446, 100194, 98039, 95975, 93996, 92097, 90273, 88520, 86834, 85211, 83647, 82140, 80686, 79283, 77927, 76618, 75351, 74126, 72940, 71791.