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'''Würschmidt''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the würschmidt comma, [[393216/390625]]. It can be treated as analogous to [[schismatic]] 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]] does in schismatic, 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.
{{interwiki
| de = Würschmidt
| en = Würschmidt
}}
{{Infobox regtemp
| Title = Würschmidt
| Subgroups = 2.3.5, 2.3.5.23
| Comma basis = [[393216/390625]] (2.3.5); <br>[[576/575]], [[12167/12150]] (2.3.5.23)
| Edo join 1 = 31 | Edo join 2 = 34
| Mapping = 1; 8 1 14
| Generators = 5/4 | Generators tuning = 387.8 | Optimization method = CWE
| MOS scales = [[3L 1s]], [[3L 4s]], …, [[3L 28s]], [[31L 3s]]
| Pergen = (P8, ccP5/8)
| Color name = Saquadbiguti
| Odd limit 1 = 5 | Mistuning 1 = 1.43 | Complexity 1 = 19
| Odd limit 2 = 2.3.5.23 25 | Mistuning 2 = 2.86 | Complexity 2 = 25
}}
'''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]] does 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}}
{{Tdlink|Würschmidt family #Würschmidt}}


== Extensions ==
== Extensions ==
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 {{nowrap|([[24/23]]) &times; ([[46/45]])}}, and therefore 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 {{nowrap|S24 {{=}} [[576/575]]}} and {{nowrap|S46<sup>2</sup> &times; S47 {{=}} [[12167/12150]]}} in the 2.3.5.23 [[subgroup]]. 14 generators turn out to stack to [[23/1]], and 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]] = {{nowrap|[[2401/2400|S49]] / ([[25921/25920|S161]]<sup>2</sup>)}}.
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 {{nowrap|([[24/23]]) &times; ([[46/45]])}}, and therefore 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 {{nowrap|S24 {{=}} [[576/575]]}} and {{nowrap|S46<sup>2</sup> &times; S47 {{=}} [[12167/12150]]}} in the 2.3.5.23 [[subgroup]]. 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 24/n is flattened, so equating it with 24/23 takes advantage of the natural tempering tendency. 14 generators turn out to stack to [[23/1]], and 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]] = {{nowrap|[[2401/2400|S49]] / ([[25921/25920|S161]]<sup>2</sup>)}}.


Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt|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 [[6144/6125]] (notably, in the 2.3.5.7.23 subgroup, this is the extension that tempers out the tiny comma {{nowrap|S161 {{=}} [[25921/25920]]}}).  
Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt|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 [[6144/6125]] and [[2401/2400]] (notably, in the 2.3.5.7.23 subgroup, this is the extension that tempers out the tiny comma {{nowrap|S161 {{=}} [[25921/25920]]}}).  


Therefore, it may be advisable to consider würschmidt a no-sevens 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 in addition to the aforementioned extension to prime 23.
Therefore, it may be advisable to consider würschmidt a no-sevens 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 in addition to the aforementioned extension to prime 23.


== Interval chain ==
== Interval chain ==
In the below, octave-reduced harmonics 1–125 are indicated in '''bold'''. All intervals are in the 625-[[odd limit]].
In the below, octave-reduced harmonics 1–33 are indicated in '''bold'''. All intervals are in the 165-[[odd limit]].


{| class="wikitable center-1 right-2"
{| class="wikitable center-all right-2"
|-
|-
! rowspan="2" | &#35; !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
! rowspan="2" | &#35; !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
Line 21: Line 38:
| 0 || 0.00 || '''1/1''' ||
| 0 || 0.00 || '''1/1''' ||
|-
|-
| 1 || 387.78 || '''5/4''', 144/115 || 69/55
| 1 || 387.652 || '''5/4''', 144/115 || 69/55
|-
|-
| 2 || 775.55 || '''25/16''', 36/23 || 69/44, 352/225
| 2 || 775.304 || '''25/16''', 36/23 || 69/44
|-
|-
| 3 || 1163.33 || 45/23, '''125/64''', 736/375 || 88/45, 108/55
| 3 || 1162.956 || 45/23, 125/64 || 88/45, 108/55
|-
|-
| 4 || 351.10 || 92/75, 225/184, 625/512 || 11/9, 27/22
| 4 || 350.608 || 92/75 || 11/9, 27/22
|-
|-
| 5 || 738.88 || 23/15, 192/125 || 55/36, 135/88
| 5 || 738.260 || 23/15, 192/125 || 55/36, 135/88
|-
|-
| 6 || 1126.66 || 23/12, 48/25 || 44/23
| 6 || 1125.912 || 23/12, 48/25 || 44/23
|-
|-
| 7 || 314.43 || 6/5, 115/96 || 55/46
| 7 || 313.564 || 6/5, 115/96 || 55/46
|-
|-
| 8 || 702.20 || '''3/2''' || 121/81
| 8 || 701.216 || '''3/2''' || 121/81
|-
|-
| 9 || 1089.98 || '''15/8''', 216/115 || 207/110, 253/135
| 9 || 1088.868 || '''15/8''', 216/115 ||  
|-
|-
| 10 || 277.76 || 27/23, '''75/64''' || 88/75, 207/176
| 10 || 276.520 || 27/23, 75/64 || 88/75
|-
|-
| 11 || 665.54 || 184/125, 135/92, 375/256 || 22/15, 81/55
| 11 || 664.172 || 184/125, 135/92 || 22/15, 81/55
|-
|-
| 12 || 1053.31 || 46/25, 675/368 || 11/6, 81/44
| 12 || 1051.824 || 46/25 || 11/6, 81/44
|-
|-
| 13 || 241.09 || 23/20, 144/125 || 55/48, 132/115
| 13 || 239.476 || 23/20, 144/125 || 55/48, 132/115
|-
|-
| 14 || 628.86 || '''23/16''', 36/25 || 33/23
| 14 || 627.128 || '''23/16''', 36/25 || 33/23
|-
|-
| 15 || 1016.64 || 9/5, '''115/64''' || 165/92, 242/135
| 15 || 1014.780 || 9/5, 115/64 || 165/92, 242/135
|-
|-
| 16 || 204.42 || '''9/8''' || 121/108
| 16 || 202.432 || '''9/8''' || 121/108
|-
|-
| 17 || 592.19 || '''45/32''', 162/115 || 253/180
| 17 || 590.084 || 45/32, 162/115 ||  
|-
|-
| 18 || 979.97 || 81/46, 225/128 || 44/25
| 18 || 977.736 || 81/46 || 44/25
|-
|-
| 19 || 167.74 || 138/125, 405/368 || 11/10, 243/220
| 19 || 165.388 || 138/125 || 11/10
|-
|-
| 20 || 555.52 || 69/50, 864/625 || '''11/8''', 243/176
| 20 || 553.040 || 69/50 || '''11/8'''
|-
|-
| 21 || 943.30 || 69/40, 216/125 || '''55/32'''
| 21 || 940.692 || 69/40, 216/125 || 55/32
|-
|-
| 22 || 131.07 || 27/25, '''69/64''' || 99/92
| 22 || 128.344 || 27/25, 69/64 || 99/92
|-
|-
| 23 || 518.85 || 27/20, 345/256 ||  
| 23 || 515.996 || 27/20 ||  
|-
|-
| 24 || 906.62 || '''27/16''' || 253/150
| 24 || 903.648 || '''27/16''' ||  
|-
|-
| 25 || 94.40 || 135/128, 243/230 || 132/125, 253/240
| 25 || 91.300 || 135/128 || 132/125
|-
|-
| 26 || 482.18 || 243/184, 828/625 || 33/25, 253/192
| 26 || 478.952 || || 33/25
|-
|-
| 27 || 869.95 || 207/125 || 33/20
| 27 || 866.604 || 207/125 || 33/20
|-
|-
| 28 || 57.73 || 207/200, 648/625 || '''33/32'''
| 28 || 54.256 || || '''33/32'''
|-
|-
| 29 || 445.50 || 162/125, 207/160 || 165/128
| 29 || 441.908 || 162/125 || 165/128
|-
|-
| 30 || 833.28 || 81/50, 207/128 || 121/75
| 30 || 829.560 || 81/50 || 121/75
|-
|-
| 31 || 21.06 || 81/80 || 121/120
| 31 || 17.212 || 81/80 || 121/120
|}
|}
<nowiki />* In 5-limit CWE tuning
<nowiki />* In 5-limit [[CTE]] tuning


== Tunings ==
== Tunings ==
=== Optimized tunings ===
=== Optimized tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Prime-optimized tunings
|+ style="font-size: 105%; white-space: nowrap;" | Norm-based tunings
|-
|-
! rowspan="2" | Weight-skew\Order !! colspan="2" | Euclidean
! rowspan="2" | Weight-skew\Order !! colspan="2" | Euclidean
Line 99: Line 116:
! Weil
! Weil
| (2.3.5) CWE: ~5/4 = 387.776¢ ||
| (2.3.5) CWE: ~5/4 = 387.776¢ ||
|-
! Equilateral
| (2.3.5) CEE: ~5/4 = 387.7224¢
(8/65-comma)
|-
|-
! Tenney
! Tenney
Line 126: Line 147:
{| 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 (¢)
! Comments
! Comments
Line 133: Line 154:
|  
|  
| 385.7143
| 385.7143
| 28ei val
| 28ei val, major thirds slightly flatter than this fall under 25&28 or [[magic]]
|-
|-
|  
|  
| [[11/9]]
| [[5/4]]
| 386.3137
| 386.3137
| -1/4 [[5632/5625|vishdel comma]]
|-
|
| [[5/4]]
| 386.8520
| Untempered tuning, lower bound of 5-odd-limit diamond tradeoff
| Untempered tuning, lower bound of 5-odd-limit diamond tradeoff
|-
|-
Line 224: Line 240:
| 387.7338
| 387.7338
|
|
|-
| [[359edo|116\359]]
|
| 387.7437
| 359ee val
|-
|-
|  
|  
Line 333: Line 354:
|  
|  
| '''400.0000'''
| '''400.0000'''
| '''Upper bound of 2.3.5.23-subgroup 25-odd-limit diamond monotone'''
| '''Upper bound of 2.3.5.23-subgroup 25-odd-limit diamond monotone''', major thirds slightly sharper than this fall under [[smate_family|smate]]
|}
|}
<nowiki />* Besides the octave
<nowiki />* Besides the octave
Line 348: Line 369:
* ''Extrospection'' (2013) – [https://web.archive.org/web/20201127013550/http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Wurschmidt{{lbrack}}16{{rbrack}}-out.mp3 play] | [https://soundcloud.com/jdfreivald/extrospection SoundCloud] – Würschmidt[16] in 31edo tuning.
* ''Extrospection'' (2013) – [https://web.archive.org/web/20201127013550/http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Wurschmidt{{lbrack}}16{{rbrack}}-out.mp3 play] | [https://soundcloud.com/jdfreivald/extrospection SoundCloud] – Würschmidt[16] in 31edo tuning.


[[Category:Temperaments]]
[[Category:Würschmidt| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Würschmidt family]]
[[Category:Würschmidt family]]

Latest revision as of 21:27, 8 March 2026

Würschmidt
Subgroups 2.3.5, 2.3.5.23
Comma basis 393216/390625 (2.3.5);
576/575, 12167/12150 (2.3.5.23)
Reduced mapping ⟨1; 8 1 14]
ET join 31 & 34
Generators (CWE) ~5/4 = 387.8 ¢
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 ¢;
2.3.5.23 25-odd-limit: 2.86 ¢
Target scale size 5-odd-limit: 19 notes;
2.3.5.23 25-odd-limit: 25 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 does 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

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 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 S24 = 576/575 and S462 × S47 = 12167/12150 in the 2.3.5.23 subgroup. 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 24/n is flattened, so equating it with 24/23 takes advantage of the natural tempering tendency. 14 generators turn out to stack to 23/1, and 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).

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 6144/6125 and 2401/2400 (notably, in the 2.3.5.7.23 subgroup, this is the extension that tempers out the tiny comma S161 = 25921/25920).

Therefore, it may be advisable to consider würschmidt a no-sevens 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 in addition to the aforementioned extension to prime 23.

Interval chain

In the below, octave-reduced harmonics 1–33 are indicated in bold. All intervals are in the 165-odd limit.

# Cents* Approximate ratios
2.3.5.23 subgroup Add-11 extension
0 0.00 1/1
1 387.652 5/4, 144/115 69/55
2 775.304 25/16, 36/23 69/44
3 1162.956 45/23, 125/64 88/45, 108/55
4 350.608 92/75 11/9, 27/22
5 738.260 23/15, 192/125 55/36, 135/88
6 1125.912 23/12, 48/25 44/23
7 313.564 6/5, 115/96 55/46
8 701.216 3/2 121/81
9 1088.868 15/8, 216/115
10 276.520 27/23, 75/64 88/75
11 664.172 184/125, 135/92 22/15, 81/55
12 1051.824 46/25 11/6, 81/44
13 239.476 23/20, 144/125 55/48, 132/115
14 627.128 23/16, 36/25 33/23
15 1014.780 9/5, 115/64 165/92, 242/135
16 202.432 9/8 121/108
17 590.084 45/32, 162/115
18 977.736 81/46 44/25
19 165.388 138/125 11/10
20 553.040 69/50 11/8
21 940.692 69/40, 216/125 55/32
22 128.344 27/25, 69/64 99/92
23 515.996 27/20
24 903.648 27/16
25 91.300 135/128 132/125
26 478.952 33/25
27 866.604 207/125 33/20
28 54.256 33/32
29 441.908 162/125 165/128
30 829.560 81/50 121/75
31 17.212 81/80 121/120

* In 5-limit CTE 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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