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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
| en = Würschmidt
| de = Würschmidt
}}
{{Infobox regtemp
| Title = Würschmidt
| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.11.23
| 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
| Mapping = 1; 8 1 20 14
| Generators = 5/4 | Generators tuning = 387.7 | 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 = 10
| 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.


{{Tdlink|Würschmidt family #Würschmidt}}
For technical data, see [[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 ([[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 S46<sup>2</sup> × 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]].
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 [[2401/2400]] and [[6144/6125]].  


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 S161 = [[25921/25920]]).  
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.


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.
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]])⋅([[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]] ({{S|24}}) and [[12167/12150]] ([[S-expression|S46<sup>2</sup>⋅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]] ([[2401/2400|S49]]/([[25921/25920|S161]]<sup>2</sup>)).


== Interval chain ==
== Interval chain ==
In the below, octave-reduced harmonics 1–23 are indicated in '''bold'''.
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 17: 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 || 0.00 || '''1/1''' ||
| 0 || 0.0 || '''1/1''' ||
|-
|-
| 1 || 387.78 || '''5/4''', 144/115 || 69/55
| 1 || 387.8 || '''5/4''' ||  
|-
|-
| 2 || 775.55 || 25/16, 36/23 || 69/44, 352/225
| 2 || 775.6 || 25/16 || 36/23, 69/44
|-
|-
| 3 || 1163.33 || 45/23, 125/64, 736/375 || 88/45, 108/55
| 3 || 1163.3 || 125/64 || 45/23, 88/45, 108/55
|-
|-
| 4 || 351.10 || 92/75, 225/184, 625/512 || 11/9, 27/22
| 4 || 351.1 || 625/512, 768/625 || 11/9, 27/22
|-
|-
| 5 || 738.88 || 23/15, 192/125 || 55/36, 135/88
| 5 || 738.9 || 192/125 || 23/15, 55/36
|-
|-
| 6 || 1126.66 || 23/12, 48/25 || 44/23, 275/144
| 6 || 1126.7 || 48/25 || 23/12, 44/23
|-
|-
| 7 || 314.43 || 6/5, 115/96 || 55/46
| 7 || 314.4 || 6/5 ||  
|-
|-
| 8 || 702.20 || '''3/2''' || 121/81
| 8 || 702.2 || '''3/2''' ||  
|-
|-
| 9 || 1089.98 || '''15/8''', 216/115 || 207/110, 253/135
| 9 || 1090.0 || '''15/8''' ||  
|-
|-
| 10 || 277.76 || 27/23, 75/64 || 88/75, 207/176
| 10 || 277.8 || 75/64 || 27/23, 88/75
|-
|-
| 11 || 665.54 || 184/125, 135/92, 375/256 || 22/15, 81/55
| 11 || 665.5 || 375/256 || 22/15, 81/55
|-
|-
| 12 || 1053.31 || 46/25, 675/368 || 11/6, 81/44
| 12 || 1053.3 || 1152/625, 1875/1024 || 11/6, 46/25, 81/44
|-
|-
| 13 || 241.09 || 23/20, 144/125 || 55/48, 132/115
| 13 || 241.1 || 144/125 || 23/20, 55/48
|-
|-
| 14 || 628.86 || '''23/16''', 36/25 || 33/23, 275/192
| 14 || 628.9 || 36/25 || '''23/16''', 33/23
|-
|-
| 15 || 1016.64 || 9/5, 115/64 || 165/92, 242/135
| 15 || 1016.6 || 9/5 ||  
|-
|-
| 16 || 204.42 || '''9/8''' || 121/108
| 16 || 204.4 || '''9/8''' ||  
|-
|-
| 17 || 592.19 || 45/32, 162/115 || 253/180
| 17 || 592.2 || 45/32 ||  
|-
|-
| 18 || 979.97 || 81/46, 225/128 || 44/25
| 18 || 980.0 || 225/128 || 44/25, 81/46
|-
|-
| 19 || 167.74 || 138/125, 405/368 || 11/10, 243/220
| 19 || 167.7 || 1125/1024 || 11/10
|-
|-
| 20 || 555.52 || 69/50, 864/625 || '''11/8''', 243/176
| 20 || 555.5 || 864/625 || '''11/8'''
|-
|-
| 21 || 943.30 || 69/40 || 55/32
| 21 || 943.3 || 216/125 || 55/32
|-
|-
| 22 || 131.07 || 69/64 ||
| 22 || 131.1 || 27/25 || 69/64, 99/92
|-
|-
| 23 || 518.84 || 27/20 ||  
| 23 || 518.9 || 27/20 ||  
|-
|-
| 24 || 906.62 || 27/16 ||  
| 24 || 906.6 || 27/16 ||  
|-
|-
| 25 || 94.40 || 135/128 || 132/125
| 25 || 94.4 || 135/128 || 132/125
|-
|-
| 26 || 482.18 || 243/184 || 33/25
| 26 || 482.2 || 675/512 || 33/25
|-
|-
| 27 || 869.95 || 207/125 || 33/20
| 27 || 870.0 || 3375/2048, 5184/3125 || 33/20
|-
|-
| 28 || 57.73 || 648/625 || 33/32
| 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
|}
|}
<nowiki>*</nowiki> In 5-limit CWE tuning
<nowiki/>* In 5-limit [[CWE]] tuning


== 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;" | Constrained Prime-Optimized Tunings
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
! Weight-skew\Order !! Euclidean
|-
|-
| Tenney || (2.3.5) CTE: ~5/4 = 387.734¢
! rowspan="2" | !! colspan="3" | Euclidean
|-
|-
| Weil || (2.3.5) CWE: ~5/4 = 387.776¢
! Constrained !! Constrained & skewed !! Destretched
|-
|-
| Tenney || (2.3.5.23) CTE: ~6/5 = 387.734¢
! Tenney
| CTE: ~5/4 = 387.734{{c}} || CWE: ~5/4 = 387.776{{c}} || POTE: ~5/4 = 387.7993{{c}}
|-
|-
| Weil || (2.3.5.23) CWE: ~6/5 = 387.781¢
! Equilateral
| CEE: ~5/4 = 387.7224{{c}}<br>(8/65-comma) ||  ||
|}
|}
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Destretched Prime-Optimized Tunings
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit add-23 norm-based tunings
|-
|-
! Weight-skew\Order !! Euclidean
! rowspan="2" |  !! colspan="3" | Euclidean
|-
|-
| Tenney || (2.3.5) POTE: ~5/4 = 387.7993¢
! Constrained !! Constrained & skewed !! Destretched
|-
|-
| Tenney || (2.3.5.23) POTE: ~6/5 = 387.8051¢
! Tenney
| CTE: ~5/4 = 387.734{{c}} || CWE: ~5/4 = 387.781{{c}} ||POTE: ~5/4 = 387.8051{{c}}
|}
|}


=== Target tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings
|+ style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings
|-
|-
! Optimized chord !! Generator value !! Polynomial !! Further notes
! Optimized chord !! Generator value !! Polynomial !! Further notes
|-
|-
| 3:4:5 (+1 +1) || ~5/4 = 387.4975 || ''g''<sup>8</sup> + 8''g'' - 16 = 0 || 1-3-5 equal-beating tuning, close to 3/29-comma
| 3:4:5 (+1 +1) || ~5/4 = 387.4975 || ''g''<sup>8</sup> + 8''g'' &minus; 16 = 0 || {{dash|1, 3, 5|med}} equal-beating tuning, close to 3/29-comma
|-
|-
| 4:5:6 (+1 +1) || ~5/4 = 388.1207 || ''g''<sup>8</sup> - 8''g'' + 8 = 0 || 1-3-5 equal-beating tuning, close to 3/19-comma
| 4:5:6 (+1 +1) || ~5/4 = 388.1207 || ''g''<sup>8</sup> &minus; 8''g'' + 8 = 0 || {{dash|1, 3, 5|med}} equal-beating tuning, close to 3/19-comma
|-
|-
| 10:12:15 (+2 +3) || ~5/4 = 388.2216 || ''g''<sup>8</sup> - 2''g''<sup>7</sup> + 4 = 0 || Close to 1/6-comma
| 10:12:15 (+2 +3) || ~5/4 = 388.2216 || ''g''<sup>8</sup> &minus; 2''g''<sup>7</sup> + 4 = 0 || Close to 1/6-comma
|-
|-
| 15:18:23 (+3 +5) || ~5/4 = 387.9215 || 4''g''<sup>7</sup> - 3''g''<sup>5</sup> - 10 = 0 ||
| 15:18:23 (+3 +5) || ~5/4 = 387.9215 || 4''g''<sup>7</sup> &minus; 3''g''<sup>5</sup> &minus; 10 = 0 ||
|}
 
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~3/2 = 387.7444{{c}}
| 3/2
|}
|}


=== 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
|-
|-
Line 131: Line 168:
|  
|  
| 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 223: Line 255:
|
|
|-
|-
| [[294edo|95\294]]
| [[359edo|116\359]]
|  
|  
| 387.7551
| 387.7437
| 294e val
| 359ee val
|-
|-
|  
|  
| [[3/2]]
| [[3/2]]
| 387.7444
| 387.7444
| 1/8-comma
| 1/8-comma, 5-odd-limit minimax
|-
| [[294edo|95\294]]
|
| 387.7551
| 294e val
|-
|-
| [[229edo|74\229]]
| [[229edo|74\229]]
Line 331: Line 368:
|  
|  
| '''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>*</nowiki> besides the octave
<nowiki />* Besides the octave


=== Other tunings ===
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5): ~2 = 1\1, ~5/4 = 387.8015
* 5-limit [[DKW theory|DKW]]: ~5/4 = 387.8015{{c}}
* [[5-odd-limit]] minimax: ~2 = 1\1, ~5/4 = 387.7444 ([[eigenmonzo]] 3/2 aka 1/8-comma, generator = 6<sup>1/8</sup>)


== Music ==
== Music ==
Line 346: Line 382:
* ''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]]
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