3159811edo
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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
Although its step size is far smaller than the human melodic just-noticeable difference, 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).
While not practical to build an acoustic instrument for, one potential use of this system is in electronic music production, where free modulation between higher-limit JI intervals is desired. Instead of keeping track of the intervals directly, the number of steps to the octave for an interval could simply be added or subtracted from one note to get to the next. However, like all other equal temperaments, the consistency of this tuning is finite, and the sequence of intervals may eventually start to deviate from their true JI counterparts.
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 mode 32 of the harmonic series. Additionally, unlike in 87edo's approximation of mode 8 or 311edo's approximation of mode 16, 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 | 155.140 | 203.910 | 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 | 806.910 | 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.