Würschmidt: Difference between revisions

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→Extensions: discuss these primes in increasing order, since the 11 is deemed natural
Rework the infobox and interval table to reflect that 11 is natural
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{{interwiki
{{Interwiki
| en = Würschmidt
| de = Würschmidt
| de = Würschmidt
| en = Würschmidt
}}
}}
{{Infobox regtemp
{{Infobox regtemp
| Title = Würschmidt
| Title = Würschmidt
| Subgroups = 2.3.5, 2.3.5.23
| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.11.23
| Comma basis = [[393216/390625]] (2.3.5); <br>[[576/575]], [[12167/12150]] (2.3.5.23)
| Comma basis = [[393216/390625]] (2.3.5); <br>[[243/242]], [[5632/5625]] (2.3.5.11);<br>[[243/242]], [[276/275]], [[529/528]] (2.3.5.11.23)
| Edo join 1 = 31 | Edo join 2 = 34
| Edo join 1 = 31 | Edo join 2 = 34
| Mapping = 1; 8 1 14
| Mapping = 1; 8 1 20 14
| Generators = 5/4 | Generators tuning = 387.8 | Optimization method = CWE
| Generators = 5/4 | Generators tuning = 387.7 | Optimization method = CWE
| MOS scales = [[3L 1s]], [[3L 4s]], …, [[3L 28s]], [[31L 3s]]
| MOS scales = [[3L 1s]], [[3L 4s]], …, [[3L 28s]], [[31L 3s]]
| Pergen = (P8, ccP5/8)
| Pergen = (P8, ccP5/8)
| Color name = Saquadbiguti
| Color name = Saquadbiguti
| Odd limit 1 = 5 | Mistuning 1 = 1.43 | Complexity 1 = 19
| Odd limit 1 = 5 | Mistuning 1 = 1.43 | Complexity 1 = 10
| Odd limit 2 = 2.3.5.23 25 | Mistuning 2 = 2.86 | Complexity 2 = 25
| Odd limit 2 = 2.3.5.11.23 23 | Mistuning 2 = 3.12 | Complexity 2 = 22
}}
}}
'''Würschmidt''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the [[würschmidt comma]] ([[ratio]]: 393216/390625, {{monzo|legend=1| 17 1 -8 }}). It can be treated as analogous to [[schismic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator|generated]] by a [[5/4|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.
'''Würschmidt''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the [[würschmidt comma]] ([[ratio]]: 393216/390625, {{monzo|legend=1| 17 1 -8 }}). It can be treated as analogous to [[schismic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator|generated]] by a [[5/4|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.


{{Tdlink|Würschmidt family #Würschmidt}}
For technical data, see [[Würschmidt family #Würschmidt]].


== Extensions ==
== Extensions ==
Line 28: Line 28:


== Interval chain ==
== Interval chain ==
In the below, octave-reduced harmonics 1–33 are indicated in '''bold'''. All intervals are in the 165-[[odd limit]].
In the below, octave-reduced harmonics 1–23 are indicated in '''bold'''.  


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
Line 34: Line 34:
! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
|-
|-
! 2.3.5.23 subgroup !! Add-11 extension
! 5-limit !! Add-11 add-23 extension
|-
|-
| 0 || 0.0 || '''1/1''' ||
| 0 || 0.0 || '''1/1''' ||
|-
|-
| 1 || 387.7 || '''5/4''', 144/115 || 69/55
| 1 || 387.8 || '''5/4''' ||  
|-
|-
| 2 || 775.3 || '''25/16''', 36/23 || 69/44
| 2 || 775.6 || 25/16 || 36/23, 69/44
|-
|-
| 3 || 1163.0 || 45/23, 125/64 || 88/45, 108/55
| 3 || 1163.3 || 125/64 || 45/23, 88/45, 108/55
|-
|-
| 4 || 350.6 || 92/75 || 11/9, 27/22
| 4 || 351.1 || 625/512, 768/625 || 11/9, 27/22
|-
|-
| 5 || 738.3 || 23/15, 192/125 || 55/36, 135/88
| 5 || 738.9 || 192/125 || 23/15, 55/36
|-
|-
| 6 || 1125.9 || 23/12, 48/25 || 44/23
| 6 || 1126.7 || 48/25 || 23/12, 44/23
|-
|-
| 7 || 313.6 || 6/5, 115/96 || 55/46
| 7 || 314.4 || 6/5 ||  
|-
|-
| 8 || 701.2 || '''3/2''' || 121/81
| 8 || 702.2 || '''3/2''' ||  
|-
|-
| 9 || 1088.9 || '''15/8''', 216/115 ||  
| 9 || 1090.0 || '''15/8''' ||  
|-
|-
| 10 || 276.5 || 27/23, 75/64 || 88/75
| 10 || 277.8 || 75/64 || 27/23, 88/75
|-
|-
| 11 || 664.2 || 184/125, 135/92 || 22/15, 81/55
| 11 || 665.5 || 375/256 || 22/15, 81/55
|-
|-
| 12 || 1051.8 || 46/25 || 11/6, 81/44
| 12 || 1053.3 || 1152/625, 1875/1024 || 11/6, 46/25, 81/44
|-
|-
| 13 || 239.5 || 23/20, 144/125 || 55/48, 132/115
| 13 || 241.1 || 144/125 || 23/20, 55/48
|-
|-
| 14 || 627.1 || '''23/16''', 36/25 || 33/23
| 14 || 628.9 || 36/25 || '''23/16''', 33/23
|-
|-
| 15 || 1014.8 || 9/5, 115/64 || 165/92, 242/135
| 15 || 1016.6 || 9/5 ||  
|-
|-
| 16 || 202.4 || '''9/8''' || 121/108
| 16 || 204.4 || '''9/8''' ||  
|-
|-
| 17 || 590.1 || 45/32, 162/115 ||  
| 17 || 592.2 || 45/32 ||  
|-
|-
| 18 || 977.7 || 81/46 || 44/25
| 18 || 980.0 || 225/128 || 44/25, 81/46
|-
|-
| 19 || 165.4 || 138/125 || 11/10
| 19 || 167.7 || 1125/1024 || 11/10
|-
|-
| 20 || 553.0 || 69/50 || '''11/8'''
| 20 || 555.5 || 864/625 || '''11/8'''
|-
|-
| 21 || 940.7 || 69/40, 216/125 || 55/32
| 21 || 943.3 || 216/125 || 55/32
|-
|-
| 22 || 128.3 || 27/25, 69/64 || 99/92
| 22 || 131.1 || 27/25 || 69/64, 99/92
|-
|-
| 23 || 516.0 || 27/20 ||  
| 23 || 518.9 || 27/20 ||  
|-
|-
| 24 || 903.6 || '''27/16''' ||  
| 24 || 906.6 || 27/16 ||  
|-
|-
| 25 || 91.3 || 135/128 || 132/125
| 25 || 94.4 || 135/128 || 132/125
|-
|-
| 26 || 479.0 || || 33/25
| 26 || 482.2 || 675/512 || 33/25
|-
|-
| 27 || 866.6 || 207/125 || 33/20
| 27 || 870.0 || 3375/2048, 5184/3125 || 33/20
|-
|-
| 28 || 54.3 || || '''33/32'''
| 28 || 57.7 || 648/625 || 33/32
|-
|-
| 29 || 441.9 || 162/125 || 165/128
| 29 || 445.5 || 162/125 || 165/128
|-
|-
| 30 || 829.6 || 81/50 || 121/75
| 30 || 833.3 || 81/50 || 121/75
|-
|-
| 31 || 17.2 || 81/80 || 121/120
| 31 || 21.1 || 81/80 || 121/120
|}
|}
<nowiki/>* In 5-limit [[CTE]] tuning
<nowiki/>* In 5-limit [[CWE]] tuning


== Tunings ==
== Tunings ==

Revision as of 08:51, 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

Optimized tunings

Norm-based tunings
Weight-skew\Order Euclidean
Constrained Destretched
Tenney (2.3.5) CTE: ~5/4 = 387.734¢ (2.3.5) POTE: ~5/4 = 387.7993¢
Weil (2.3.5) CWE: ~5/4 = 387.776¢
Equilateral (2.3.5) CEE: ~5/4 = 387.7224¢

(8/65-comma)

Tenney (2.3.5.23) CTE: ~5/4 = 387.734¢ (2.3.5.23) POTE: ~5/4 = 387.8051¢
Weil (2.3.5.23) CWE: ~5/4 = 387.781¢
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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