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

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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 well temperament
 
|-
{| class="wikitable center-1 right-all"
! Degree
! Steps
! Fifths
! Cents (ET)
! Cents
! Cents (WT)
! Fifth below (¢)
! Nearest Ratio
! Fifth above (¢)
|-
|-
| 0
| 0
| +0
| 0.000
| 0
| 0.000
| 701.738
| [[1/1]]
| ?
|-
|-
| 1
| 1
| ?
| 29.268
| ?
| 21.506
| ?
| [[81/80]]
| ?
|-
|-
| 2
| 2
| ?
| 58.537
| ?
| 58.483
| ?
| [[33/32]]
| ?
|-
|-
| 3
| 3
| ?
| 87.805
| ?
| 91.039
| ?
| [[135/128]]
| ?
|-
|-
| 4
| 4
| ?
| 117.073
| ?
| 112.545
| ?
| [[16/15]]
| ?
|-
|-
| 5
| 5
| ?
| 146.341
| ?
| 138.472
| ?
| [[13/12]]
| ?
|-
|-
| 6
| 6
| ?
| 175.610
| ?
| 182.078
| ?
| [[10/9]]
| ?
|-
|-
| 7
| 7
| ?
| 204.878
| ?
| 203.584
| ?
| [[9/8]]
| ?
|-
|-
| 8
| 8
| ?
| 234.146
| ?
| 225.091
| ?
| [[256/225]]
| ?
|-
|-
| 9
| 9
| ?
| 263.415
| ?
| 266.487
| ?
| [[7/6]]
| ?
|-
|-
| 10
| 10
| ?
| 292.683
| ?
| 294.623
| ?
| [[32/27]]
| ?
|-
|-
| 11
| 11
| ?
| 321.951
| ?
| 316.130
| ?
| [[6/5]]
| ?
|-
|-
| 12
| 12
| ?
| 351.220
| ?
| 346.476
| ?
| [[11/9]]
| ?
|-
|-
| 13
| 13
| ?
| 380.488
| ?
| 385.662
| ?
| [[5/4]]
| ?
|-
|-
| 14
| 14
| ?
| 409.756
| ?
| 407.169
| ?
| [[81/64]]
| ?
|-
|-
| 15
| 15
| ?
| 439.024
| ?
| 428.675
| ?
| [[32/25]]
| ?
|-
|-
| 16
| 16
| ?
| 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
| ?
| 702.439
| ?
| 701.792
| ?
| [[3/2]]
| ?
|-
|-
| 25
| 25
| ?
| 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
| ?
| 965.854
| ?
| 970.489
| ?
| [[7/4]]
| ?
|-
|-
| 34
| 34
| ?
| 995.122
| ?
| 996.416
| ?
| [[16/9]]
| ?
|-
|-
| 35
| 35
| ?
| 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.

Latest revision as of 05:55, 12 February 2026

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.

The chain of fifths

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

Prime Fifths down
5 -8
7 -14
11 -18
13 -21

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 ¢, 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 ¢ 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

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.

Retrieved from "https://en.xen.wiki/index.php?title=User:Overthink/41edo_well_temperament&oldid=223839"