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3159811edo: Difference between revisions

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{{Niche}}{{clear}}
{{Mathematical interest}}{{clear}}
{{Infobox ET|Consistency=65|Distinct consistency=65}}
{{Infobox ET|Consistency=65|Distinct consistency=65}}
{{ED intro}}
{{ED intro}}


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 ==
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{{idio}} 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 ===
=== 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|columns=9}}
{{Harmonics in equal|3159811|intervals=odd|prec=8|columns=8|start=8|collapsed=true|title=Approximation of odd harmonics in 3159811edo (17–31)}}
{{Harmonics in equal|3159811|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 3159811edo (continued)}}
{{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)}}
== Scales ==
=== Harmonic scales ===
As mentioned, 3159811edo accurately approximates [[32afdo|mode 32]] of the [[harmonic series]]. Additionally, unlike in [[10edo]]'s approximation of [[4afdo|mode 4]], [[87edo]]'s approximation of [[8afdo|mode 8]], or [[311edo]]'s approximation of [[16afdo|mode 16]], all interval pairs are distinguished.
 
{| class="wikitable center-all"
|-
! 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
|}
 
{| class="wikitable center-all"
|-
! 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
|}
 
{| class="wikitable center-all"
|-
! 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
|}
 
{| class="wikitable center-all"
|-
! 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.
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