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A highly notable tuning system is 41edo, which is [[distinctly consistent]] and [[consistent to distance]] 2 in the 9-odd-limit, consistent to the [[15-odd-limit]], and is overall great for its size in all prime limits up to 13. However, the accuracy of some intervals is debateable, so we will construct a well temperament in order to fix this issue.
A highly notable tuning system is [[41edo]], which is [[distinctly consistent]] and [[consistent to distance]] 2 in the 9-odd-limit, consistent to the [[15-odd-limit]], and is overall great for its size in all [[prime limit]]s up to [[13-limit|13]]. However, the accuracy of some intervals is debateable, so we will construct a well temperament in order to fix this issue.


== The chain of fifths ==
== The chain of fifths ==
Our well temperament will be mainly based on a chain of fifths, and we will target harmonics [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]]. In 41edo, these prime harmonics are mapped as follows:
Our well temperament will be based on a chain of fifths, and we will target harmonics [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]]. In 41edo, these prime harmonics are mapped as follows (corresponding to [[andromeda]] temperament if the fifths are all the same):


{| class="wikitable right-all center-1"
{| class="wikitable right-all center-1"
|-
|-
! Prime
! Prime
! Fifths<br>down
! Fifths down
! Fifths<br>up
|-
|-
| 5
| 5
| -8
| -8
| +33
|-
|-
| 7
| 7
| -14
| -14
| +27
|-
|-
| 11
| 11
| -18
| -18
| +23
|-
|-
| 13
| 13
| -21
| -21
| +20
|}
|}


Prime 5 is closest to the root on the circle of fifths; being only 8 fifth 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 701.738[[{{c}}]], or 1/9 schisma flat of just. We continue this chain of schismic fifths until we reach -10 fifths, or [[10/9]].
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.


{| class="wikitable mw-collapsible center-1 right-all"
== Intervals ==
|+ style="font-size: 105% ;" | 41edo&nbsp;well&nbsp;temperament (Table incomplete)
 
{| class="wikitable center-1 right-all"
! Steps
! Cents (ET)
! Cents (WT)
! Nearest Ratio
|-
|-
! Degree
! Fifths
! Cents
! Fifth below (¢)
! Fifth above (¢)
|-
| 0
| +0
| 0
| 0
| 701.738
| 0.000
| ?
| 0.000
| [[1/1]]
|-
|-
| 1
| 1
| +12
| 29.268
| ?
| 21.506
| ?
| [[81/80]]
| ?
|-
|-
| 2
| 2
| -17/+24
| 58.537
| ?
| 58.483
| ?
| [[33/32]]
| ?
|-
|-
| 3
| 3
| -5
| 87.805
| ?
| 91.039
| ?
| [[135/128]]
| ?
|-
|-
| 4
| 4
| +7
| 117.073
| ?
| 112.545
| ?
| [[16/15]]
| ?
|-
|-
| 5
| 5
| -22/+19
| 146.341
| ?
| 138.472
| ?
| [[13/12]]
| ?
|-
|-
| 6
| 6
| -10
| 175.610
| ?
| 182.078
| ?
| [[10/9]]
| ?
|-
|-
| 7
| 7
| +2
| 204.878
| ?
| 203.584
| ?
| [[9/8]]
| ?
|-
|-
| 8
| 8
| +14
| 234.146
| ?
| 225.091
| ?
| [[256/225]]
| ?
|-
|-
| 9
| 9
| -15
| 263.415
| ?
| 266.487
| ?
| [[7/6]]
| ?
|-
|-
| 10
| 10
| -3
| 292.683
| ?
| 294.623
| ?
| [[32/27]]
| ?
|-
|-
| 11
| 11
| +9
| 321.951
| ?
| 316.130
| ?
| [[6/5]]
| ?
|-
|-
| 12
| 12
| -20/+21
| 351.220
| ?
| 346.476
| ?
| [[11/9]]
| ?
|-
|-
| 13
| 13
| -8
| 380.488
| ?
| 385.662
| ?
| [[5/4]]
| ?
|-
|-
| 14
| 14
| +4
| 409.756
| ?
| 407.169
| ?
| [[81/64]]
| ?
|-
|-
| 15
| 15
| +16
| 439.024
| ?
| 428.675
| ?
| [[32/25]]
| ?
|-
|-
| 16
| 16
| -13
| 468.293
| ?
| 474.492
| ?
| [[21/16]]
| ?
|-
|-
| 17
| 17
| -1
| 497.561
| 498.262
| 498.208
| 701.738
| [[4/3]]
| 701.738
|-
|-
| 18
| 18
| ?
| 526.829
| ?
| 519.714
| ?
| [[27/20]]
| ?
|-
|-
| 19
| 19
| ?
| 556.098
| ?
| 554.480
| ?
| [[11/8]]
| ?
|-
|-
| 20
| 20
| ?
| 585.366
| ?
| 589.247
| ?
| [[45/32]]
| ?
|-
|-
| 21
| 21
| ?
| 614.634
| ?
| 610.753
| ?
| [[64/45]]
| ?
|-
|-
| 22
| 22
| ?
| 643.902
| ?
| 634.469
| ?
| [[13/9]]
| ?
|-
|-
| 23
| 23
| ?
| 673.171
| ?
| 680.286
| ?
| [[40/27]]
| ?
|-
|-
| 24
| 24
| +1
| 702.439
| ?
| 701.792
| ?
| [[3/2]]
| ?
|-
|-
| 25
| 25
| +13
| 731.707
| ?
| 723.298
| ?
| [[243/160]]
| ?
|-
|-
| 26
| 26
| ?
| 760.976
| ?
| 762.485
| ?
| [[14/9]]
| ?
|-
|-
| 27
| 27
| ?
| 790.244
| ?
| 792.831
| ?
| [[128/81]]
| ?
|-
|-
| 28
| 28
| ?
| 819.512
| ?
| 814.338
| ?
| [[8/5]]
| ?
|-
|-
| 29
| 29
| ?
| 848.780
| ?
| 842.474
| ?
| [[13/8]]
| ?
|-
|-
| 30
| 30
| ?
| 878.049
| ?
| 883.870
| ?
| [[5/3]]
| ?
|-
|-
| 31
| 31
| ?
| 907.317
| ?
| 905.377
| ?
| [[27/16]]
| ?
|-
|-
| 32
| 32
| ?
| 936.585
| ?
| 926.883
| ?
| [[128/75]]
| ?
|-
|-
| 33
| 33
| -14
| 965.854
| ?
| 970.489
| ?
| [[7/4]]
| ?
|-
|-
| 34
| 34
| -2
| 995.122
| ?
| 996.416
| ?
| [[16/9]]
| ?
|-
|-
| 35
| 35
| +10
| 1024.390
| ?
| 1017.922
| ?
| [[9/5]]
| ?
|-
|-
| 36
| 36
| ?
| 1053.659
| ?
| 1050.478
| ?
| [[11/6]]
| ?
|-
|-
| 37
| 37
| ?
| 1082.927
| ?
| 1087.455
| ?
| [[15/8]]
| ?
|-
|-
| 38
| 38
| ?
| 1112.195
| ?
| 1108.961
| ?
| [[243/128]]
| ?
|-
|-
| 39
| 39
| ?
| 1141.463
| ?
| 1130.467
| ?
| [[48/25]]
| ?
|-
|-
| 40
| 40
| ?
| 1170.732
| ?
| 1178.494
| ?
| [[160/81]]
| ?
|-
|-
| 41
| 41
| +0
| 1200.000
| 1200
| 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.
Retrieved from "https://en.xen.wiki/w/User:Overthink/41edo_well_temperament"