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Würschmidt family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The [[5-limit]] parent comma for the '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt. The [[generator]] is a [[5/4|classical major third]], and to get to the interval class of 3 requires eight of these. In fact, (5/4)<sup>8</sup> × 393216/390625 = 6.  
The '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') of [[regular temperament|temperaments]] [[tempering out|tempers out]] [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt.  


Similar to [[meantone]], würschmidt implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore [[6/5]] will be tempered flat. Unlike meantone, it is far more accurate. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo|11\34]] and especially [[65edo|21\65]], which notably is the point where it is combined with [[schismic]] and [[gravity]]. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]]. Another tuning solution is to sharpen the major third by 1/8 of a würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure perfect fifths; this is the [[minimax tuning]] for the [[5-odd-limit]].  
== Würschmidt ==
{{Main| Würschmidt }}
 
The [[generator]] of würschmidt is a [[5/4|classical major third]], and to get to the interval class of the [[3/1|3rd harmonic]] requires eight of these. In fact, (5/4)<sup>8</sup> × 393216/390625 = 6.
 
Similar to [[meantone]], würschmidt implies that [[3/2]] will be tempered flat and/or 5/4 will be tempered sharp, and therefore [[6/5]] will be tempered flat. Unlike meantone, it is far more accurate. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo|11\34]] and especially [[65edo|21\65]], which notably is the point where it is combined with [[schismic]] and [[gravity]]. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]]. Another tuning solution is to sharpen the major third by 1/8 of a würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure perfect fifths; this is the [[minimax tuning]] for the [[5-odd-limit]].  


[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Rothenberg propriety|Rothenberg-proper]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning.  
[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Rothenberg propriety|Rothenberg-proper]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning.  
== Würschmidt ==
{{Main| Würschmidt }}


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.734{{c}}
* [[WE]]: ~2 = 1199.6942{{c}}, ~5/4 = 387.7005{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.799{{c}}
: [[error map]]: {{val| -0.306 -0.045 +0.775 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 387.7762{{c}}
: error map: {{val| 0.000 +0.255 +1.463 }}


{{Optimal ET sequence|legend=1| 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc }}
{{Optimal ET sequence|legend=1| 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc }}


[[Badness]] (Smith): 0.040603
[[Badness]] (Sintel): 0.951


=== Overview to extensions ===
=== Overview to extensions ===
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The "free" higher prime for würschmidt outside the 5-limit is in fact [[23/1|23]], via tempering out [[576/575]] ({{S|24}}) and [[12167/12150]] ([[S-expression|S46<sup>2</sup>⋅S47]]). This is considered immediately below, and the no-11 restriction thereof is considered in [[#Other subgroup extensions]].
The "free" higher prime for würschmidt outside the 5-limit is in fact [[23/1|23]], via tempering out [[576/575]] ({{S|24}}) and [[12167/12150]] ([[S-expression|S46<sup>2</sup>⋅S47]]). This is considered immediately below, and the no-11 restriction thereof is considered in [[#Other subgroup extensions]].
=== 2.3.5.11 subgroup ===
Subgroup: 2.3.5.11
Comma list: 243/242, 5632/5625
Subgroup-val mapping: {{mapping| 1 -1 2 -3 | 0 8 1 20 }}
Optimal tuning:
* WE: ~2 = 1199.7508{{c}}, ~5/4 = 387.6058{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.6697{{c}}
{{Optimal ET sequence|legend=0| 31, 34, 65 }}
Badness (Sintel): 0.477


==== 2.3.5.11.23 subgroup ====
==== 2.3.5.11.23 subgroup ====
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Optimal tuning:  
Optimal tuning:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.652{{c}}
* WE: ~2 = 1199.8205{{c}}, ~5/4 = 387.6316{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.6770{{c}}


{{Optimal ET sequence|legend=0| 31, 34, 65 }}
{{Optimal ET sequence|legend=0| 31, 34, 65 }}


Badness (Smith): 0.00660
Badness (Sintel): 0.300


== Septimal würschmidt ==
== Septimal würschmidt ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.379{{c}}
* [[WE]]: ~2 = 1199.9741{{c}}, ~5/4 = 387.3742{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.383{{c}}
: [[error map]]: {{val| -0.026 -2.936 +1.009 +3.987 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 387.3809{{c}}
: error map: {{val| 0.000 -2.907 +1.067 +4.031 }}


{{Optimal ET sequence|legend=1| 31, 96, 127 }}
{{Optimal ET sequence|legend=1| 31, 96, 127 }}


[[Badness]] (Smith): 0.050776
[[Badness]] (Sintel): 1.28


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.441{{c}}
* WE: ~2 = 1199.9618{{c}}, ~5/4 = 387.4347{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.447{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.4446{{c}}


{{Optimal ET sequence|legend=0| 31, 65d, 96, 127 }}
{{Optimal ET sequence|legend=0| 31, 65d, 96, 127 }}


Badness (Smith): 0.024413
Badness (Sintel): 0.807


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.469{{c}}
* WE: ~2 = 1199.0325{{c}}, ~5/4 = 387.3137{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.626{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.5893{{c}}


{{Optimal ET sequence|legend=0| 31, 65d }}
{{Optimal ET sequence|legend=0| 31, 65d }}


Badness (Smith): 0.023593
Badness (Sintel): 0.975


==== Worseschmidt ====
==== Worseschmidt ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.179{{c}}
* WE: ~2 = 1200.4712{{c}}, ~5/4 = 387.2511{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.099{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.1252{{c}}


{{Optimal ET sequence|legend=0| 3def, 28def, 31 }}
{{Optimal ET sequence|legend=0| 3def, 28def, 31 }}


Badness (Smith): 0.034382
Badness (Sintel): 1.42


== Worschmidt ==
== Worschmidt ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.406{{c}}
* [[WE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.406{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.392{{c}}
: [[error map]]: {{val| +0.763 -1.610 +2.851 -2.786 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 387.3950{{c}}
: error map: {{val| 0.000 -2.795 +1.081 -4.960 }}


{{Optimal ET sequence|legend=1| 31, 96d, 127d }}
{{Optimal ET sequence|legend=1| 31, 96d, 127d }}


[[Badness]] (Smith): 0.064614
[[Badness]] (Sintel): 1.64


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.472{{c}}
* WE: ~2 = 1200.7911{{c}}, ~5/4 = 387.6624{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.407{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.4189{{c}}


{{Optimal ET sequence|legend=0| 31, 65, 96d, 127d }}
{{Optimal ET sequence|legend=0| 31, 65, 96d, 127d }}


Badness (Smith): 0.033436
Badness (Sintel): 1.11


== Whirrschmidt ==
== Whirrschmidt ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.853
* [[WE]]: ~2 = 1199.5703{{c}}, ~5/4 = 387.7422{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.881
: [[error map]]: {{val| -0.430 +0.412 +0.569 -0.216 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 387.8729{{c}}
: error map: {{val| 0.000 +1.029 +1.559 +0.567 }}


{{Optimal ET sequence|legend=1| 34d, 65, 99 }}
{{Optimal ET sequence|legend=1| 34d, 65, 99 }}


[[Badness]] (Smith): 0.086334
[[Badness]] (Sintel): 2.18


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.829
* WE: ~2 = 1199.2839{{c}}, ~5/4 = 387.6507{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.882
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.8682{{c}}


{{Optimal ET sequence|legend=0| 34d, 65, 99e }}
{{Optimal ET sequence|legend=0| 34d, 65, 99e }}


Badness (Smith): 0.058325
Badness (Sintel): 1.93


== Other subgroup extensions ==
== Other subgroup extensions ==
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.734{{c}}
* WE: ~2 = 1199.7075{{c}}, ~5/4 = 387.7106{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.805{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.7807{{c}}


{{Optimal ET sequence|legend=0| 3, …, 28i, 31, 34, 65, 99, 164 }}
{{Optimal ET sequence|legend=0| 31, 34, 65, 99, 164 }}


Badness (Smith): 0.00530
Badness (Sintel): 0.216


[[Category:Temperament families]]
[[Category:Temperament families]]
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