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Passion family: Difference between revisions

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The '''passion family''' tempers out 262144/253125 in the 5-limit, which equates five instances of [[16/15]] with [[4/3]].  
{{Technical data page}}
The '''passion family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[passion comma]], 262144/253125 ({{monzo| 18 -4 -5 }}) in the [[5-limit]], which equates five instances of [[16/15]] with [[4/3]].  


== Passion ==
== Passion ==
Subgroup: 2.3.5
Passion is generated by a ~16/15 semitone, five of which reach a ~4/3 perfect fourth; its [[ploidacot]] is omega-pentacot. It follows that both 3 and 5 should be tuned sharp. [[73edo]] does this and may be recommended as a tuning.
 
[[Subgroup]]: 2.3.5


[[Comma list]]: 262144/253125
[[Comma list]]: 262144/253125


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -5 4 }}]
{{Mapping|legend=1| 1 2 2 | 0 -5 4 }}


Mapping generators: ~2, ~16/15
: mapping generators: ~2, ~16/15


[[POTE generator]]: ~16/15 = 98.670
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1197.8074{{c}}, ~16/15 = 98.4898{{c}}
: [[error map]]: {{val| -2.193 +1.211 +3.260 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 98.7335{{c}}
: error map: {{val| 0.000 +4.378 +8.620 }}


{{Val list|legend=1| 12, 49, 61, 73 }}
{{Optimal ET sequence|legend=1| 12, 49, 61, 73, 231bccc }}


[[Badness]]: 0.168550
[[Badness]] (Sintel): 3.95
 
=== Overview to extensions ===
The second comma of the comma list defines which 7-limit family member we are looking at:
* Septimal passion adds [[64/63]];
* Passionate adds [[126/125]];
* Passive adds [[225/224]].
 
Those all use the same nominal generator as passion.
 
The only weak extension we are considering is freivald, which adds [[6144/6125]] to the comma list and splits the generator in two.  


== Septimal passion ==
== Septimal passion ==
Subgroup: 2.3.5.7
Septimal passion tempers out 64/63, the archytas comma, connecting itself to the [[archytas clan]]. It may be described as the 12 & 49 temperament. The interval class of 7 is found by exactly 10 generator steps, which calls for a flatter tuning of the fourth, and [[49edo]] may be recommended as a tuning.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 3125/3087
[[Comma list]]: 64/63, 3125/3087


[[Mapping]]: [{{val| 1 2 2 2 }}, {{val| 0 -5 4 10 }}]
{{Mapping|legend=1| 1 2 2 2 | 0 -5 4 10 }}


Mapping generators: 2, 16/15
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1197.0350{{c}}, ~21/20 = 97.9106{{c}}
: [[error map]]: {{val| -2.965 +2.562 -0.601 +4.351 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 98.1019{{c}}
: error map: {{val| 0.000 +7.535 +6.094 +12.193 }}


{{Multival|legend=1| 5 -4 -10 -18 -30 -12 }}
{{Optimal ET sequence|legend=1| 12, 37, 49, 110bcd }}


[[POTE generator]]: ~16/15 = 98.153
[[Badness]] (Sintel): 1.58
 
{{Val list|legend=1| 12, 37, 49, 110bcd }}
 
[[Badness]]: 0.062327


=== 11-limit ===
=== 11-limit ===
Line 38: Line 57:
Comma list: 64/63, 100/99, 1375/1372
Comma list: 64/63, 100/99, 1375/1372


Mapping: [{{val| 1 2 2 2 2 }}, {{val| 0 -5 4 10 18 }}]
Mapping: {{mapping| 1 2 2 2 2 | 0 -5 4 10 18 }}


POTE generator: ~16/15 = 98.019
Optimal tunings:  
* WE: ~2 = 1196.9304{{c}}, ~21/20 = 97.7686{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 97.9333{{c}}


Vals: {{Val list| 12, 37, 49 }}
{{Optimal ET sequence|legend=0| 12, 37, 49 }}


Badness: 0.040809
Badness (Sintel): 1.35


=== 13-limit ===
=== 13-limit ===
Line 51: Line 72:
Comma list: 64/63, 100/99, 196/195, 275/273
Comma list: 64/63, 100/99, 196/195, 275/273


Mapping: [{{val| 1 2 2 2 2 2 }}, {{val| 0 -5 4 10 18 21 }}]
Mapping: {{mapping| 1 2 2 2 2 2 | 0 -5 4 10 18 21 }}


POTE generator: ~16/15 = 97.910
Optimal tunings:  
* WE: ~2 = 1196.8712{{c}}, ~21/20 = 97.6548{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 97.8141{{c}}


Vals: {{Val list| 12f, 25f, 37, 49f }}
{{Optimal ET sequence|legend=0| 12f, 25f, 37, 49f }}


Badness: 0.030886
Badness (Sintel): 1.28


== Passionate ==
== Passionate ==
Subgroup: 2.3.5.7
Passionate tempers out 126/125 and may be described as {{nowrap| 12 & 73 }}. The optimum of this temperament is closer to its 5-limit counterpart's, but prime 7 is much more complex, at +22 generator steps away. 73edo and [[85edo]] in the 85c val are among the possible tunings.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 131072/127575
[[Comma list]]: 126/125, 131072/127575


[[Mapping]]: [{{val|1 2 2 1}}, {{val|0 -5 4 22}}]
{{Mapping|legend=1| 1 2 2 1 | 0 -5 4 22 }}


{{Multival|legend=1|5 -4 -22 -18 -49 -40}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.9486{{c}}, ~16/15 = 98.6479{{c}}
: [[error map]]: {{val| -2.051 +0.703 +4.175 -0.623 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 98.7770{{c}}
: error map: {{val| 0.000 +4.160 +8.794 +4.269 }}


[[POTE generator]]: ~16/15 = 98.817
{{Optimal ET sequence|legend=1| 12, 49d, 61, 73, 85c }}


{{Val list|legend=1| 12, 61, 73, 85c }}
[[Badness]] (Sintel): 3.07
 
[[Badness]]: 0.121293


=== 11-limit ===
=== 11-limit ===
Line 79: Line 106:
Comma list: 126/125, 176/175, 8192/8019
Comma list: 126/125, 176/175, 8192/8019


Mapping: [{{val|1 2 2 1 1}}, {{val|0 -5 4 22 30}}]
Mapping: {{mapping| 1 2 2 1 1 | 0 -5 4 22 30 }}


POTE generator: ~16/15 = 98.721
Optimal tunings:  
* WE: ~2 = 1197.8835{{c}}, ~16/15 = 98.5469{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/15 = 98.6733{{c}}


Vals: {{Val list| 12, 61, 73, 158bccee }}
{{Optimal ET sequence|legend=0| 12, 49de, 61, 73, 158bccee }}


Badness: 0.068322
Badness (Sintel): 2.26


==== 13-limit ====
==== 13-limit ====
Line 92: Line 121:
Comma list: 126/125, 176/175, 343/338, 364/363
Comma list: 126/125, 176/175, 343/338, 364/363


Mapping: [{{val|1 2 2 1 1 1}}, {{val|0 -5 4 22 30 33}}]
Mapping: {{mapping| 1 2 2 1 1 1 | 0 -5 4 22 30 33 }}


POTE generator: ~16/15 = 98.627
Optimal tunings:  
* WE: ~2 = 1197.8309{{c}}, ~16/15 = 98.4485{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/15 = 98.5760{{c}}


Vals: {{Val list| 12f, 61, 73f }}
{{Optimal ET sequence|legend=0| 12f, 49de, 61, 73f }}


Badness: 0.052302
Badness (Sintel): 2.16


==== Pash ====
==== Pash ====
Line 105: Line 136:
Comma list: 126/125, 176/175, 196/195, 1352/1331
Comma list: 126/125, 176/175, 196/195, 1352/1331


Mapping: [{{val|1 2 2 1 1 0}}, {{val|0 -5 4 22 30 45}}]
Mapping: {{mapping| 1 2 2 1 1 0 | 0 -5 4 22 30 45 }}


POTE generator: ~16/15 = 98.791
Optimal tunings:  
* WE: ~2 = 1197.8306{{c}}, ~16/15 = 98.6124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/15 = 98.7468{{c}}


Vals: {{Val list| 12f, 61f, 73, 85ce }}
{{Optimal ET sequence|legend=0| 12f, 73, 85ce }}


Badness: 0.056386
Badness (Sintel): 2.33


== Passive ==
== Passive ==
Passive (1&11) tempers out the same 5-limit comma as the [[Archytas clan #Passion|passion temperament]] (12&49), but with the marvel (225/224) rather than the archytas (64/63) tempered out.
This low-complexity extension tempers out 225/224 as well as [[256/245]]. Two generator steps represent ~[[8/7]]. As one might expect, 12edo is about as accurate as it can be tuned.  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 256/245
[[Comma list]]: 225/224, 256/245


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
{{Mapping|legend=1| 1 2 2 3 | 0 -5 4 -2 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1195.3551{{c}}, ~16/15 = 98.4266{{c}}
: [[error map]]: {{val| -4.645 -3.378 -1.897 +20.386 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 98.9690{{c}}
: error map: {{val| 0.000 +3.200 +9.562 +33.236 }}
 
{{Optimal ET sequence|legend=1| 1, 11, 12 }}
 
[[Badness]] (Sintel): 1.90
 
== Freivald ==
Freivald splits the [[octave complement]] of the semitone generator of passion in two, each of which is used for ~11/8. It may be described as {{nowrap| 24 & 37 }}; its ploidacot is delta-decacot. [[61edo]] may be recommended as a tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 6272/6075
 
{{Mapping|legend=1| 1 -3 6 -5 | 0 10 -8 17 }}
 
: mapping generator: ~2, ~48/35
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.4021{{c}}, ~48/35 = 550.2038{{c}}
: [[error map]]: {{val| -1.598 +4.877 +2.468 -7.371 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 550.8980{{c}}
: error map: {{val| 0.000 +7.025 +6.502 -3.559 }}
 
{{Optimal ET sequence|legend=1| 13d, 24, 37, 61, 98b }}
 
[[Badness]] (Sintel): 5.29
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 2744/2673
 
Mapping: {{mapping| 1 -3 6 -5 3 | 0 10 -8 17 1 }}
 
Optimal tunings:
* WE: ~2 = 1198.8053{{c}}, ~11/8 = 550.3522{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 550.8760{{c}}
 
{{Optimal ET sequence|legend=0| 13d, 24, 37, 61 }}
 
Badness (Sintel): 2.56
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 121/120, 176/175, 512/507
 
Mapping: {{mapping| 1 7 -2 12 4 1 | 0 -10 8 -17 -1 5 }}


[[POTE generator]]: ~16/15 = 98.809
Optimal tunings:  
* WE: ~2 = 1198.7706{{c}}, ~11/8 = 550.3418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 550.8892{{c}}


{{Val list|legend=1| 1, 11, 12, 49dd }}
{{Optimal ET sequence|legend=0| 13d, 24, 37, 61 }}


[[Badness]]: 0.075055
Badness (Sintel): 1.90


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Passion family]] <!-- main article -->
[[Category:Passion family| ]] <!-- main article -->
[[Category:Passion| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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