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Würschmidt: Difference between revisions

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→Extensions: discuss these primes in increasing order, since the 11 is deemed 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 ==
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== 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"
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! 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 ==
=== 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}}
|-
! Equilateral
| 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
|-
|-
! Weil
! rowspan="2" | !! colspan="3" | Euclidean
| (2.3.5) CWE: ~5/4 = 387.776¢ ||
|-
|-
! 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¢ ||
|}
|}


=== 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
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|-
|-
| 15:18:23 (+3 +5) || ~5/4 = 387.9215 || 4''g''<sup>7</sup> &minus; 3''g''<sup>5</sup> &minus; 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
|-
|-
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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]]
|-
|-
|  
|  
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| [[3/2]]
| [[3/2]]
| 387.7444
| 387.7444
| 1/8-comma
| 1/8-comma, 5-odd-limit minimax
|-
|-
| [[294edo|95\294]]
| [[294edo|95\294]]
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=== 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 ==
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