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User:Overthink/41edo well temperament: Difference between revisions

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Overthink (talk | contribs)
autopatrolled
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+ table (mostly python-generated, ratios are manual)
 
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Prime 5 is closest to the root on the circle of fifths; being only 8 fifths down due to 41edo tempering out the [[schisma]]. We build our well temperament down the chain of fifths to reach a more accurate prime 5, using slightly flat fifths of 701.792[[{{c}}]], or 1/12 schisma flat of just. We continue this chain of schismic fifths down until we reach -12 fifths, or an exact [[81/80]]. We then stack down 12 [[parapyth]] fifths of 704.002{{c}} to obtain accurate approximations of harmonics 7, 11, and 13. Finally, the remaining fifths are all 1/12-schisma flat, giving a total of 29 schismic fifths and 12 parapyth fifths.
Prime 5 is closest to the root on the circle of fifths; being only 8 fifths down due to 41edo tempering out the [[schisma]]. We build our well temperament down the chain of fifths to reach a more accurate prime 5, using slightly flat fifths of 701.792{{Cent}}, or 1/12 schisma flat of just. We continue this chain of schismic fifths down until we reach -12 fifths, or an exact [[160/81]]. We then stack down 12 [[parapyth]] fifths of 704.002{{C}} to obtain accurate approximations of harmonics 7, 11, and 13. Finally, the remaining fifths are all 1/12-schisma flat, giving a total of 29 schismic fifths and 12 parapyth fifths.
 
== Intervals ==
 
{| class="wikitable center-1 right-all"
! Steps
! Cents (ET)
! Cents (WT)
! Nearest Ratio
|-
| 0
| 0.000
| 0.000
| [[1/1]]
|-
| 1
| 29.268
| 21.506
| [[81/80]]
|-
| 2
| 58.537
| 58.483
| [[33/32]]
|-
| 3
| 87.805
| 91.039
| [[135/128]]
|-
| 4
| 117.073
| 112.545
| [[16/15]]
|-
| 5
| 146.341
| 138.472
| [[13/12]]
|-
| 6
| 175.610
| 182.078
| [[10/9]]
|-
| 7
| 204.878
| 203.584
| [[9/8]]
|-
| 8
| 234.146
| 225.091
| [[256/225]]
|-
| 9
| 263.415
| 266.487
| [[7/6]]
|-
| 10
| 292.683
| 294.623
| [[32/27]]
|-
| 11
| 321.951
| 316.130
| [[6/5]]
|-
| 12
| 351.220
| 346.476
| [[11/9]]
|-
| 13
| 380.488
| 385.662
| [[5/4]]
|-
| 14
| 409.756
| 407.169
| [[81/64]]
|-
| 15
| 439.024
| 428.675
| [[32/25]]
|-
| 16
| 468.293
| 474.492
| [[21/16]]
|-
| 17
| 497.561
| 498.208
| [[4/3]]
|-
| 18
| 526.829
| 519.714
| [[27/20]]
|-
| 19
| 556.098
| 554.480
| [[11/8]]
|-
| 20
| 585.366
| 589.247
| [[45/32]]
|-
| 21
| 614.634
| 610.753
| [[64/45]]
|-
| 22
| 643.902
| 634.469
| [[13/9]]
|-
| 23
| 673.171
| 680.286
| [[40/27]]
|-
| 24
| 702.439
| 701.792
| [[3/2]]
|-
| 25
| 731.707
| 723.298
| [[243/160]]
|-
| 26
| 760.976
| 762.485
| [[14/9]]
|-
| 27
| 790.244
| 792.831
| [[128/81]]
|-
| 28
| 819.512
| 814.338
| [[8/5]]
|-
| 29
| 848.780
| 842.474
| [[13/8]]
|-
| 30
| 878.049
| 883.870
| [[5/3]]
|-
| 31
| 907.317
| 905.377
| [[27/16]]
|-
| 32
| 936.585
| 926.883
| [[128/75]]
|-
| 33
| 965.854
| 970.489
| [[7/4]]
|-
| 34
| 995.122
| 996.416
| [[16/9]]
|-
| 35
| 1024.390
| 1017.922
| [[9/5]]
|-
| 36
| 1053.659
| 1050.478
| [[11/6]]
|-
| 37
| 1082.927
| 1087.455
| [[15/8]]
|-
| 38
| 1112.195
| 1108.961
| [[243/128]]
|-
| 39
| 1141.463
| 1130.467
| [[48/25]]
|-
| 40
| 1170.732
| 1178.494
| [[160/81]]
|-
| 41
| 1200.000
| 1200.000
| [[2/1]]
|}
 
As you can see, this well temperament approximates many 5-limit ratios of low-to-medium complexity with accuracy, while also having a good approximation of the higher limits.
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