Würschmidt: Difference between revisions

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Rework the infobox and interval table to reflect that 11 is natural
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== Tunings ==
== Tunings ==
=== Optimized tunings ===
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Norm-based tunings
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
|-
! rowspan="2" | Weight-skew\Order !! colspan="2" | Euclidean
! rowspan="2" | !! colspan="3" | Euclidean
|-
|-
! Constrained !! Destretched
! Constrained !! Constrained & skewed !! Destretched
|-
|-
! Tenney
! Tenney
| (2.3.5) CTE: ~5/4 = 387.734¢ || (2.3.5) POTE: ~5/4 = 387.7993¢
| CTE: ~5/4 = 387.734{{c}} || CWE: ~5/4 = 387.776{{c}} || POTE: ~5/4 = 387.7993{{c}}
|-
|-
! Weil
! Equilateral
| (2.3.5) CWE: ~5/4 = 387.776¢ ||
| CEE: ~5/4 = 387.7224{{c}}<br>(8/65-comma) ||  ||
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit add-23 norm-based tunings
|-
! rowspan="2" |  !! colspan="3" | Euclidean
|-
|-
! Equilateral
! Constrained !! Constrained & skewed !! Destretched
| (2.3.5) CEE: ~5/4 = 387.7224¢
(8/65-comma)
|-
|-
! Tenney
! Tenney
| (2.3.5.23) CTE: ~5/4 = 387.734¢ || (2.3.5.23) POTE: ~5/4 = 387.8051¢
| CTE: ~5/4 = 387.734{{c}} || CWE: ~5/4 = 387.781{{c}} ||POTE: ~5/4 = 387.8051{{c}}
|-
! Weil
| (2.3.5.23) CWE: ~5/4 = 387.781¢ ||
|}
|}


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=== Tuning spectrum ===
=== Tuning spectrum ===
The below assumes the 2.3.5.11.23 subgroup extension. Note that "e" and "i" are the warts for primes 11 and 23, respectively.
The below assumes the 2.3.5.11.23-subgroup extension. Note that "e" and "i" are the warts for primes 11 and 23, respectively.


{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
! Edo<br />generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br />(unchanged interval)]]*
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
! Generator (¢)
! Generator&nbsp;(¢)
! Comments
! Comments
|-
|-
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|  
|  
| 385.7143
| 385.7143
| 28ei val, major thirds slightly flatter than this fall under 25&28 or [[magic]]
| 28ei val, major thirds slightly flatter than this fall under 25 & 28 or [[magic]]
|-
|-
|  
|  

Revision as of 09:00, 9 September 2026

Würschmidt
Subgroups 2.3.5, 2.3.5.11, 2.3.5.11.23
Comma basis 393216/390625 (2.3.5);
243/242, 5632/5625 (2.3.5.11);
243/242, 276/275, 529/528 (2.3.5.11.23)
Reduced mapping ⟨1; 8 1 20 14]
ET join 31 & 34
Generators (CWE) ~5/4 = 387.7 ¢
MOS scales 3L 1s, 3L 4s, …, 3L 28s, 31L 3s
Ploidacot beta-octacot
Pergen (P8, ccP5/8)
Color name Saquadbiguti
Minimax error 5-odd-limit: 1.43 ¢;
2.3.5.11.23 23-odd-limit: 3.12 ¢
Target scale size 5-odd-limit: 10 notes;
2.3.5.11.23 23-odd-limit: 22 notes

Würschmidt is a rank-2 temperament and parent of the würschmidt family, characterized by tempering out the würschmidt comma (ratio: 393216/390625, monzo: [17 1 -8⟩). It can be treated as analogous to schismic with the roles of the primes 3 and 5 reversed, since würschmidt is generated by a classical major third (5/4), very slightly sharpened so that eight of them make the sixth harmonic (6/1), giving 3/2 the same complexity 5/4 has in schismic, but with comparable accuracy on the part of the generator. Four generators, therefore, reach the interval 625/512, which is equated to 768/625 and functions as a neutral third.

For technical data, see Würschmidt family #Würschmidt.

Extensions

Strong extensions to the 7-limit include septimal würschmidt (tempering out 225/224, finding 7 at +18 generator steps), worschmidt (tempering out 126/125, finding 7 at -13 generator steps), and whirrschmidt (tempering out 4375/4374, finding 7 at +52 generator steps), but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be the weak extension hemiwürschmidt, which splits the ~5/4 generator into two ~28/25's by tempering out 3136/3125 alongside 2401/2400 and 6144/6125.

Therefore, it may be advisable to consider würschmidt a no-7's system, specifically in the 2.3.5.11 subgroup, where an extension that equates 128/125 with 45/44 and therefore 625/512 with 11/9 (by tempering out 243/242 and 5632/5625), finding the 11th harmonic at 20 generators up, is highly natural.

Another useful interpretation of the würschmidt comma is that it makes the interval of 25/24 equal to two-thirds the size of 16/15. This can be exploited, as 16/15 factorizes into near-2:1 parts as (24/23)⋅(46/45), and therefore, if one is interested in adding another prime to this temperament, it is illogical not to set 25/24 equal to 24/23 (and 128/125 equal to 46/45) as well and set the remainder, 46/45, equal to a third of 16/15, by tempering out 576/575 (S24) and 12167/12150 (S462⋅S47). A perhaps more direct way of seeing why equating 25/24 with 24/23 is natural is that würschmidt's generator is a slightly sharpened 5/4 with a slightly flat 3/2 in an optimised tuning, so that 25/24 is sharpened and equating it with 24/23 takes advantage of the natural tempering tendency. 14 generators turn out to stack to 23/1. Notably, 6/1 stacked 7 times and 23/1 stacked four times (at 56 generators) differ only by the 0.59-cent comma 279936/279841 (S49/(S1612)).

Interval chain

In the below, octave-reduced harmonics 1–23 are indicated in bold.

# Cents* Approximate ratios
5-limit Add-11 add-23 extension
0 0.0 1/1
1 387.8 5/4
2 775.6 25/16 36/23, 69/44
3 1163.3 125/64 45/23, 88/45, 108/55
4 351.1 625/512, 768/625 11/9, 27/22
5 738.9 192/125 23/15, 55/36
6 1126.7 48/25 23/12, 44/23
7 314.4 6/5
8 702.2 3/2
9 1090.0 15/8
10 277.8 75/64 27/23, 88/75
11 665.5 375/256 22/15, 81/55
12 1053.3 1152/625, 1875/1024 11/6, 46/25, 81/44
13 241.1 144/125 23/20, 55/48
14 628.9 36/25 23/16, 33/23
15 1016.6 9/5
16 204.4 9/8
17 592.2 45/32
18 980.0 225/128 44/25, 81/46
19 167.7 1125/1024 11/10
20 555.5 864/625 11/8
21 943.3 216/125 55/32
22 131.1 27/25 69/64, 99/92
23 518.9 27/20
24 906.6 27/16
25 94.4 135/128 132/125
26 482.2 675/512 33/25
27 870.0 3375/2048, 5184/3125 33/20
28 57.7 648/625 33/32
29 445.5 162/125 165/128
30 833.3 81/50 121/75
31 21.1 81/80 121/120

* In 5-limit CWE tuning

Tunings

Norm-based tunings

5-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~5/4 = 387.734 ¢ CWE: ~5/4 = 387.776 ¢ POTE: ~5/4 = 387.7993 ¢
Equilateral CEE: ~5/4 = 387.7224 ¢
(8/65-comma)
5-limit add-23 norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~5/4 = 387.734 ¢ CWE: ~5/4 = 387.781 ¢ POTE: ~5/4 = 387.8051 ¢
DR and equal-beating tunings
Optimized chord Generator value Polynomial Further notes
3:4:5 (+1 +1) ~5/4 = 387.4975 g8 + 8g − 16 = 0 1 – 3 – 5 equal-beating tuning, close to 3/29-comma
4:5:6 (+1 +1) ~5/4 = 388.1207 g8 − 8g + 8 = 0 1 – 3 – 5 equal-beating tuning, close to 3/19-comma
10:12:15 (+2 +3) ~5/4 = 388.2216 g8 − 2g7 + 4 = 0 Close to 1/6-comma
15:18:23 (+3 +5) ~5/4 = 387.9215 4g7 − 3g5 − 10 = 0

Tuning spectrum

The below assumes the 2.3.5.11.23-subgroup extension. Note that "e" and "i" are the warts for primes 11 and 23, respectively.

Edo
generator 		Eigenmonzo
(unchanged interval)* 		Generator (¢) Comments
9\28 385.7143 28ei val, major thirds slightly flatter than this fall under 25 & 28 or magic
5/4 386.3137 Untempered tuning, lower bound of 5-odd-limit diamond tradeoff
10\31 387.0968 Lower bound of 2.3.5.23-subgroup 25-odd-limit diamond monotone
23/22 387.1739
375/256 387.3542 1/11-comma
41\127 387.4016 127e val
11/6 387.4469
75/64 387.4582 1/10-comma
31\96 387.5000
11/8 387.5659
52\161 387.5776
15/8 387.5854 1/9-comma
73\226 387.6106
11/10 387.6318
45/32 387.6602 2/17-comma
21\65 387.6923
23/12 387.7199
23/16 387.7338
116\359 387.7437 359ee val
3/2 387.7444 1/8-comma
95\294 387.7551 294e val
74\229 387.7729 229e val
53\164 387.8049 164e val
23/18 387.8178 1/2 S24
85\263 387.8327 263ee val
9/5 387.8393 2/15-comma
23/20 387.8431
32\99 387.8788 99e val
75\232 387.9310 232eei val
5/3 387.9490 1/7-comma, upper bound of 5-odd-limit diamond tradeoff
43\133 387.9699 133e val
25/23 387.9706
23/15 388.0011
54\167 388.0240 167eei val
25/24 388.2213 1/6-comma, upper bound of 2.3.5.23-subgroup 25-odd-limit diamond tradeoff
11\34 388.2353
125/96 388.6028 1/5-comma
23\71 388.7324 71eei val
625/384 389.1750 1/4-comma
12\37 389.1892 37eei val
1\3 400.0000 Upper bound of 2.3.5.23-subgroup 25-odd-limit diamond monotone, major thirds slightly sharper than this fall under smate

* Besides the octave

Other tunings

  • DKW (2.3.5): ~2 = 1\1, ~5/4 = 387.8015
  • 5-odd-limit minimax: ~2 = 1\1, ~5/4 = 387.7444 (eigenmonzo 3/2 aka 1/8-comma, generator = 61/8)

Music

Chris Vaisvil
  • Ancient Stardust (2013) – blog | play – Würschmidt[13] in 5-odd-limit minimax tuning
Jake Freivald
  • Extrospection (2013) – play | SoundCloud – Würschmidt[16] in 31edo tuning.
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