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Würschmidt family: Difference between revisions

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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 ===
Line 38: Line 40:


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
{{Mapping|legend=2| 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 ====
Line 44: Line 61:
Comma list: 243/242, 276/275, 529/528
Comma list: 243/242, 276/275, 529/528


Subgroup-val mapping: {{mapping| 1 -1 2 -3 0 | 0 8 1 20 14 }}
{{Mapping|legend=2| 1 -1 2 -3 0 | 0 8 1 20 14 }}


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 ==
Septimal würschmidt, aside from the commas listed above, also tempers out [[225/224]]. [[31edo]] or [[127edo]] can be used as tunings. It extends naturally to an 11-limit version which also tempers out [[99/98]], [[176/175]] and 243/242. 127edo is again an excellent tuning for 11-limit würschmidt, as well as for [[minerva]], the 11-limit rank-3 temperament tempering out 99/98 and 176/175.
Septimal würschmidt, aside from the commas listed above, also tempers out [[225/224]]. [[31edo]] or [[127edo]] can be used as tunings. It extends naturally to an 11-limit version which also tempers out [[99/98]], [[176/175]] and 243/242. 127edo is again an excellent tuning for 11-limit würschmidt, as well as for [[minerva]], the 11-limit rank-3 temperament tempering out 99/98 and 176/175.


2-würschmidt, the temperament with all the same commas as würschmidt but a generator of twice the size, is equivalent to [[skwares]] as a [[2.3.7.11-subgroup|2.3.7.11 subgroup]] temperament.
2-würschmidt, the temperament with all the same commas as würschmidt but a generator of twice the size, is equivalent to [[skwares]] as a [[2.3.7.11 subgroup|2.3.7.11-subgroup]] temperament.


The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175|S8/S10]], [[243/242|S9/S11]], [[225/224|S15]]}. Tempering out [[81/80|S9]] or [[121/120|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63|S8]] or [[100/99|S10]] results in [[34edo]] in the 34d [[val]], where we accept [[17edo]]'s mapping of prime 7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by tempering them together. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7; even so, it is fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune. However, as 31edo is relatively in-tune already, [[96edo]] (= 65d + 31) is also a reasonable choice, as it has the advantage of being a [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13.
The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175|S8/S10]], [[243/242|S9/S11]], [[225/224|S15]]}. Tempering out [[81/80|S9]] or [[121/120|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63|S8]] or [[100/99|S10]] results in [[34edo]] in the 34d [[val]], where we accept [[17edo]]'s mapping of prime 7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by tempering them together. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7; even so, it is fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune. However, as 31edo is relatively in-tune already, [[96edo]] (= 65d + 31) is also a reasonable choice, as it has the advantage of being a [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13.
Line 68: Line 85:


[[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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Comma list: 99/98, 176/175, 243/242
Comma list: 99/98, 176/175, 243/242


Mapping: {{mapping| 1 -1 2 -3 -3 | 0 8 1 18 20 }}
{{Mapping|legend=0| 1 -1 2 -3 -3 | 0 8 1 18 20 }}


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 ====
Line 95: Line 114:
Comma list: 99/98, 144/143, 176/175, 275/273
Comma list: 99/98, 144/143, 176/175, 275/273


Mapping: {{mapping| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}
{{Mapping|legend=0| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}


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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Commas: 66/65, 99/98, 105/104, 243/242
Commas: 66/65, 99/98, 105/104, 243/242


Mapping: {{mapping| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}
{{Mapping|legend=0| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}


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 ==
Line 130: Line 149:


[[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 ===
Line 142: Line 163:
Comma list: 126/125, 243/242, 385/384
Comma list: 126/125, 243/242, 385/384


Mapping: {{mapping| 1 -1 2 7 -3 | 0 8 1 -13 20 }}
{{Mapping|legend=0| 1 -1 2 7 -3 | 0 8 1 -13 20 }}


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 ==
Line 162: Line 183:


[[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 ===
Line 174: Line 197:
Comma list: 243/242, 896/891, 4375/4356
Comma list: 243/242, 896/891, 4375/4356


Mapping: {{mapping| 1 -1 2 -14 -3 | 0 8 1 52 20 }}
{{Mapping|legend=0| 1 -1 2 -14 -3 | 0 8 1 52 20 }}


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 ==
=== Würschmidt (2.3.5.23) ===
=== Würschmidt (2.3.5.23) ===
Extensions to harmonics [[47/1|47]] and [[49/1|49]] are also available at +11 and +5 generator steps respectively, equalizing the 45::50 [[harmonic series segment|segment]] of the [[harmonic series]]. If we derive a mapping of prime [[7/1|7]] from this mapping of 49, then we get the weak extension [[hemiwürschmidt]].
Extensions for harmonics [[47/1|47]] and [[49/1|49]] are also available at +11 and +5 generator steps respectively, equalizing the 45::50 [[harmonic series segment|segment]] of the [[harmonic series]]. If we derive a mapping of prime [[7/1|7]] from this mapping of 49, then we get the weak extension [[hemiwürschmidt]].


Subgroup: 2.3.5.23
Subgroup: 2.3.5.23
Line 192: Line 215:
Comma list: 576/575, 12167/12150
Comma list: 576/575, 12167/12150


Subgroup-val mapping: {{mapping| 1 -1 2 0 | 0 8 1 14 }}
{{Mapping|legend=2| 1 -1 2 0 | 0 8 1 14 }}


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]]
Retrieved from "https://en.xen.wiki/w/Würschmidt_family"