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


Mapping generators: ~2, ~16/15
[[Badness]] (Sintel): 3.95


[[POTE generator]]: ~16/15 = 98.670
=== 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]].  


{{Val list|legend=1| 12, 49, 61, 73 }}
Those all use the same nominal generator as passion.


[[Badness]]: 0.168550
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 ==
Subgroup: 2.3.5.7
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


[[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 }}


[[POTE generator]]: ~16/15 = 98.809
[[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 }}


{{Val list|legend=1| 1, 11, 12, 49dd }}
{{Optimal ET sequence|legend=1| 1, 11, 12 }}


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


== Freivald ==
== Freivald ==
Subgroup: 2.3.5.7
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
[[Comma list]]: 6144/6125, 6272/6075


[[Mapping]]: [{{val| 1 7 -2 12 }}, {{val| 0 -10 8 -17 }}]
{{Mapping|legend=1| 1 -3 6 -5 | 0 10 -8 17 }}


Mapping generator: ~2, ~35/24
: mapping generator: ~2, ~48/35


[[POTE generator]]: ~48/35 = 550.9374
[[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 }}


{{Val list|legend=1| 13d, 24, 37, 61, 98b }}
{{Optimal ET sequence|legend=1| 13d, 24, 37, 61, 98b }}


[[Badness]]: 0.209095
[[Badness]] (Sintel): 5.29


=== 11-limit ===
=== 11-limit ===
Line 146: Line 191:
Comma list: 121/120, 176/175, 2744/2673
Comma list: 121/120, 176/175, 2744/2673


Mapping: [{{val| 1 7 -2 12 4 }}, {{val| 0 -10 8 -17 -1 }}]
Mapping: {{mapping| 1 -3 6 -5 3 | 0 10 -8 17 1 }}


POTE generator: ~11/8 = 550.9007
Optimal tunings:  
* WE: ~2 = 1198.8053{{c}}, ~11/8 = 550.3522{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 550.8760{{c}}


Vals: {{val list| 13d, 24, 37, 61 }}
{{Optimal ET sequence|legend=0| 13d, 24, 37, 61 }}


Badness: 0.077504
Badness (Sintel): 2.56


=== 13-limit ===
=== 13-limit ===
Line 159: Line 206:
Comma list: 91/90, 121/120, 176/175, 512/507
Comma list: 91/90, 121/120, 176/175, 512/507


Mapping: [{{val| 1 7 -2 12 4 1 }}, {{val| 0 -10 8 -17 -1 5 }}]
Mapping: {{mapping| 1 7 -2 12 4 1 | 0 -10 8 -17 -1 5 }}


POTE generator: ~11/8 = 550.9062
Optimal tunings:  
* WE: ~2 = 1198.7706{{c}}, ~11/8 = 550.3418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 550.8892{{c}}


Vals: {{val list| 13d, 24, 37, 61 }}
{{Optimal ET sequence|legend=0| 13d, 24, 37, 61 }}


Badness: 0.045864
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]]

Latest revision as of 13:55, 22 August 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The passion family of temperaments tempers out the passion comma, 262144/253125 ([18 -4 -5⟩) in the 5-limit, which equates five instances of 16/15 with 4/3.

Passion

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

Mapping: [⟨1 2 2], ⟨0 -5 4]]

mapping generators: ~2, ~16/15

Optimal tunings:

  • WE: ~2 = 1197.8074 ¢, ~16/15 = 98.4898 ¢
error map: ⟨-2.193 +1.211 +3.260]
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 98.7335 ¢
error map: ⟨0.000 +4.378 +8.620]

Optimal ET sequence: 12, 49, 61, 73, 231bccc

Badness (Sintel): 3.95

Overview to extensions

The second comma of the comma list defines which 7-limit family member we are looking at:

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 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

Mapping: [⟨1 2 2 2], ⟨0 -5 4 10]]

Optimal tunings:

  • WE: ~2 = 1197.0350 ¢, ~21/20 = 97.9106 ¢
error map: ⟨-2.965 +2.562 -0.601 +4.351]
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 98.1019 ¢
error map: ⟨0.000 +7.535 +6.094 +12.193]

Optimal ET sequence: 12, 37, 49, 110bcd

Badness (Sintel): 1.58

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 1375/1372

Mapping: [⟨1 2 2 2 2], ⟨0 -5 4 10 18]]

Optimal tunings:

  • WE: ~2 = 1196.9304 ¢, ~21/20 = 97.7686 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 97.9333 ¢

Optimal ET sequence: 12, 37, 49

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 100/99, 196/195, 275/273

Mapping: [⟨1 2 2 2 2 2], ⟨0 -5 4 10 18 21]]

Optimal tunings:

  • WE: ~2 = 1196.8712 ¢, ~21/20 = 97.6548 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 97.8141 ¢

Optimal ET sequence: 12f, 25f, 37, 49f

Badness (Sintel): 1.28

Passionate

Passionate tempers out 126/125 and may be described as 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

Mapping: [⟨1 2 2 1], ⟨0 -5 4 22]]

Optimal tunings:

  • WE: ~2 = 1197.9486 ¢, ~16/15 = 98.6479 ¢
error map: ⟨-2.051 +0.703 +4.175 -0.623]
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 98.7770 ¢
error map: ⟨0.000 +4.160 +8.794 +4.269]

Optimal ET sequence: 12, 49d, 61, 73, 85c

Badness (Sintel): 3.07

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 8192/8019

Mapping: [⟨1 2 2 1 1], ⟨0 -5 4 22 30]]

Optimal tunings:

  • WE: ~2 = 1197.8835 ¢, ~16/15 = 98.5469 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 98.6733 ¢

Optimal ET sequence: 12, 49de, 61, 73, 158bccee

Badness (Sintel): 2.26

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 343/338, 364/363

Mapping: [⟨1 2 2 1 1 1], ⟨0 -5 4 22 30 33]]

Optimal tunings:

  • WE: ~2 = 1197.8309 ¢, ~16/15 = 98.4485 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 98.5760 ¢

Optimal ET sequence: 12f, 49de, 61, 73f

Badness (Sintel): 2.16

Pash

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 1352/1331

Mapping: [⟨1 2 2 1 1 0], ⟨0 -5 4 22 30 45]]

Optimal tunings:

  • WE: ~2 = 1197.8306 ¢, ~16/15 = 98.6124 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 98.7468 ¢

Optimal ET sequence: 12f, 73, 85ce

Badness (Sintel): 2.33

Passive

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

Comma list: 225/224, 256/245

Mapping: [⟨1 2 2 3], ⟨0 -5 4 -2]]

Optimal tunings:

  • WE: ~2 = 1195.3551 ¢, ~16/15 = 98.4266 ¢
error map: ⟨-4.645 -3.378 -1.897 +20.386]
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 98.9690 ¢
error map: ⟨0.000 +3.200 +9.562 +33.236]

Optimal ET sequence: 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 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: [⟨1 -3 6 -5], ⟨0 10 -8 17]]

mapping generator: ~2, ~48/35

Optimal tunings:

  • WE: ~2 = 1198.4021 ¢, ~48/35 = 550.2038 ¢
error map: ⟨-1.598 +4.877 +2.468 -7.371]
  • CWE: ~2 = 1200.0000 ¢, ~48/35 = 550.8980 ¢
error map: ⟨0.000 +7.025 +6.502 -3.559]

Optimal ET sequence: 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: [⟨1 -3 6 -5 3], ⟨0 10 -8 17 1]]

Optimal tunings:

  • WE: ~2 = 1198.8053 ¢, ~11/8 = 550.3522 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 550.8760 ¢

Optimal ET sequence: 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: [⟨1 7 -2 12 4 1], ⟨0 -10 8 -17 -1 5]]

Optimal tunings:

  • WE: ~2 = 1198.7706 ¢, ~11/8 = 550.3418 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 550.8892 ¢

Optimal ET sequence: 13d, 24, 37, 61

Badness (Sintel): 1.90

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