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

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{| class="wikitable mw-collapsible center-1 right-all"
{| class="wikitable mw-collapsible center-1 right-all"
|+ style="font-size: 105% ;" | 41edo well temperament
|+ style="font-size: 105% ;" | 41edo well temperament (Table incomplete)
|-
|-
! Degree
! Degree
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|-
|-
| 1
| 1
| ?
| +12
| ?
| ?
| ?
| ?
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|-
|-
| 2
| 2
| ?
| -17/+24
| ?
| ?
| ?
| ?
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|-
|-
| 3
| 3
| ?
| -5
| ?
| ?
| ?
| ?
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|-
|-
| 4
| 4
| ?
| +7
| ?
| ?
| ?
| ?
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|-
|-
| 5
| 5
| ?
| -22/+19
| ?
| ?
| ?
| ?
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|-
|-
| 6
| 6
| ?
| -10
| ?
| ?
| ?
| ?
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|-
|-
| 7
| 7
| ?
| +2
| ?
| ?
| ?
| ?
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|-
|-
| 8
| 8
| ?
| +14
| ?
| ?
| ?
| ?
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|-
|-
| 9
| 9
| ?
| -15
| ?
| ?
| ?
| ?
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|-
|-
| 10
| 10
| ?
| -3
| ?
| ?
| ?
| ?
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|-
|-
| 11
| 11
| ?
| +9
| ?
| ?
| ?
| ?
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|-
|-
| 12
| 12
| ?
| -20/+21
| ?
| ?
| ?
| ?
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|-
|-
| 13
| 13
| ?
| -8
| ?
| ?
| ?
| ?
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|-
|-
| 14
| 14
| ?
| +4
| ?
| ?
| ?
| ?
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|-
|-
| 15
| 15
| ?
| +16
| ?
| ?
| ?
| ?
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|-
|-
| 16
| 16
| ?
| -13
| ?
| ?
| ?
| ?
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|-
|-
| 24
| 24
| ?
| +1
| ?
| ?
| ?
| ?
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|-
|-
| 25
| 25
| ?
| +13
| ?
| ?
| ?
| ?
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|-
|-
| 33
| 33
| ?
| -14
| ?
| ?
| ?
| ?
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|-
|-
| 34
| 34
| ?
| -2
| ?
| ?
| ?
| ?
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| 35
| 35
| ?
| +10
| ?
| ?
| ?
| ?

Revision as of 21:07, 14 December 2025

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 mainly 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:

Prime Fifths
down 		Fifths
up
5 -8 +33
7 -14 +27
11 -18 +23
13 -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 ¢, or 1/9 schisma flat of just. We continue this chain of schismic fifths until we reach -10 fifths, or 10/9.

41edo well temperament (Table incomplete)
Degree Fifths Cents Fifth below (¢) Fifth above (¢)
0 +0 0 701.738 ?
1 +12 ? ? ?
2 -17/+24 ? ? ?
3 -5 ? ? ?
4 +7 ? ? ?
5 -22/+19 ? ? ?
6 -10 ? ? ?
7 +2 ? ? ?
8 +14 ? ? ?
9 -15 ? ? ?
10 -3 ? ? ?
11 +9 ? ? ?
12 -20/+21 ? ? ?
13 -8 ? ? ?
14 +4 ? ? ?
15 +16 ? ? ?
16 -13 ? ? ?
17 -1 498.262 701.738 701.738
18 ? ? ? ?
19 ? ? ? ?
20 ? ? ? ?
21 ? ? ? ?
22 ? ? ? ?
23 ? ? ? ?
24 +1 ? ? ?
25 +13 ? ? ?
26 ? ? ? ?
27 ? ? ? ?
28 ? ? ? ?
29 ? ? ? ?
30 ? ? ? ?
31 ? ? ? ?
32 ? ? ? ?
33 -14 ? ? ?
34 -2 ? ? ?
35 +10 ? ? ?
36 ? ? ? ?
37 ? ? ? ?
38 ? ? ? ?
39 ? ? ? ?
40 ? ? ? ?
41 +0 1200 ? ?
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