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.
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 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | |
|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00002113 | +0.00002452 | +0.00001382 | +0.00004226 | -0.00003126 | -0.00004793 | +0.00004565 |
| Relative (%) | +5.6 | +6.5 | +3.6 | +11.1 | -8.2 | -12.6 | +12.0 | |
| Steps (reduced) |
5008182 (1848371) |
7336854 (1017232) |
8870711 (2551089) |
10016364 (536931) |
10931150 (1451717) |
11692690 (2213257) |
12345036 (2865603) | |
| Harmonic | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.00001818 | -0.00003197 | +0.00003494 | +0.00006535 | +0.00004904 | +0.00006338 | +0.00008120 | +0.00000087 |
| Relative (%) | -4.8 | -8.4 | +9.2 | +17.2 | +12.9 | +16.7 | +21.4 | +0.2 | |
| Steps (reduced) |
12915610 (276366) |
13422648 (783404) |
13878893 (1239649) |
14293601 (1654357) |
14673708 (2034464) |
15024546 (2385302) |
15350302 (2711058) |
15654324 (3015080) | |
| Harmonic | 33 | 35 | 37 | 39 | 41 | 43 | 45 | 47 | |
|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.00001013 | +0.00003834 | -0.00001850 | -0.00002681 | +0.00009218 | -0.00002310 | +0.00006678 | +0.00001747 |
| Relative (%) | -2.7 | +10.1 | -4.9 | -7.1 | +24.3 | -6.1 | +17.6 | +4.6 | |
| Steps (reduced) |
15939332 (140277) |
16207565 (408510) |
16460888 (661833) |
16700872 (901817) |
16928852 (1129797) |
17145971 (1346916) |
17353218 (1554163) |
17551451 (1752396) | |
| Harmonic | 49 | 51 | 53 | 55 | 57 | 59 | 61 | 63 | |
|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00002763 | +0.00000295 | -0.00002258 | -0.00000674 | -0.00001084 | -0.00008220 | -0.00002937 | +0.00005607 |
| Relative (%) | +7.3 | +0.8 | -5.9 | -1.8 | -2.9 | -21.6 | -7.7 | +14.8 | |
| Steps (reduced) |
17741422 (1942367) |
17923792 (2124737) |
18099146 (2300091) |
18268004 (2468949) |
18430830 (2631775) |
18588040 (2788985) |
18740009 (2940954) |
18887075 (3088020) | |