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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 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]]).
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]]).


== Theory ==
=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|3159811|intervals=odd|prec=8|columns=7|title=Approximation of odd harmonics in 3159811edo (3–15)}}
{{Harmonics in equal|3159811|intervals=odd|prec=8|columns=7|title=Approximation of odd harmonics in 3159811edo (3–15)}}
Line 10: Line 11:
{{Harmonics in equal|3159811|intervals=odd|prec=8|columns=8|start=16|collapsed=true|title=Approximation of odd harmonics in 3159811edo (33–47)}}
{{Harmonics in equal|3159811|intervals=odd|prec=8|columns=8|start=16|collapsed=true|title=Approximation of odd harmonics in 3159811edo (33–47)}}
{{Harmonics in equal|3159811|intervals=odd|prec=8|columns=8|start=24|collapsed=true|title=Approximation of odd harmonics in 3159811edo (49–63)}}
{{Harmonics in equal|3159811|intervals=odd|prec=8|columns=8|start=24|collapsed=true|title=Approximation of odd harmonics in 3159811edo (49–63)}}
== Scales ==
=== Harmonic scales ===
3159811edo accurately approximates the mode 32 of [[harmonic series]]. All interval pairs are distinguished.
{| class="wikitable center-all"
|-
! Overtones
! 32
! 33
! 34
! 35
! 36
! 37
! 38
! 39
|-
! JI Ratios
| 1/1
| 33/32
| 17/16
| 35/32
| 9/8
| 37/32
| 19/16
| 39/32
|-
! … in cents
| 0
| 53.273
| 104.955
| 155.14
| 203.91
| 251.344
| 297.513
| 342.483
|-
! Degrees in 3159811edo
| 0
| 140277
| 276366
| 408510
| 536931
| 661833
| 783404
| 901817
|}
{| class="wikitable center-all"
|-
! Overtones
! 40
! 41
! 42
! 43
! 44
! 45
! 46
! 47
|-
! JI Ratios
| 5/4
| 41/32
| 21/16
| 43/32
| 11/8
| 45/32
| 23/16
| 47/32
|-
! … in cents
| 386.314
| 429.062
| 470.781
| 511.518
| 551.318
| 590.224
| 628.274
| 665.507
|-
! Degrees in 3159811edo
| 1017232
| 1129797
| 1239649
| 1346916
| 1451717
| 1554163
| 1654357
| 1752396
|}
{| class="wikitable center-all"
|-
! Overtones
! 48
! 49
! 50
! 51
! 52
! 53
! 54
! 55
|-
! JI Ratios
| 3/2
| 49/32
| 25/16
| 51/32
| 13/8
| 53/32
| 27/16
| 55/32
|-
! … in cents
| 701.955
| 737.652
| 772.627
| 806.91
| 840.528
| 873.505
| 905.865
| 937.632
|-
! Degrees in 3159811edo
| 1848371
| 1942367
| 2034464
| 2124737
| 2213257
| 2300091
| 2385302
| 2468949
|}
{| class="wikitable center-all"
|-
! Overtones
! 56
! 57
! 58
! 59
! 60
! 61
! 62
! 63
! 64
|-
! JI Ratios
| 7/4
| 57/32
| 29/16
| 59/32
| 15/8
| 61/32
| 31/16
| 63/32
| 2/1
|-
! … in cents
| 968.826
| 999.468
| 1029.577
| 1059.172
| 1088.269
| 1116.885
| 1145.036
| 1172.736
| 1200
|-
! Degrees in 3159811edo
| 2551089
| 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.
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