Xenharmonic Wiki

Passion family: Difference between revisions

← Older edit
VisualWikitext
m Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}"
Tags: Mobile edit Mobile web edit
+ overview to extensions
 
(7 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{Technical data page}}
{{Technical data page}}
The '''passion family''' [[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]].  
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 ==
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
[[Subgroup]]: 2.3.5


Line 11: Line 13:
: mapping generators: ~2, ~16/15
: mapping generators: ~2, ~16/15


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 }}


{{Optimal ET sequence|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 ==
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
[[Subgroup]]: 2.3.5.7


Line 24: Line 42:
{{Mapping|legend=1| 1 2 2 2 | 0 -5 4 10 }}
{{Mapping|legend=1| 1 2 2 2 | 0 -5 4 10 }}


{{Multival|legend=1| 5 -4 -10 -18 -30 -12 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1197.0350{{c}}, ~21/20 = 97.9106{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/15 = 98.153
: [[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 }}


{{Optimal ET sequence|legend=1| 12, 37, 49, 110bcd }}
{{Optimal ET sequence|legend=1| 12, 37, 49, 110bcd }}


[[Badness]]: 0.062327
[[Badness]] (Sintel): 1.58


=== 11-limit ===
=== 11-limit ===
Line 39: Line 59:
Mapping: {{mapping| 1 2 2 2 2 | 0 -5 4 10 18 }}
Mapping: {{mapping| 1 2 2 2 2 | 0 -5 4 10 18 }}


Optimal tuning (POTE): ~2 = 1\1, ~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}}


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


Badness: 0.040809
Badness (Sintel): 1.35


=== 13-limit ===
=== 13-limit ===
Line 52: Line 74:
Mapping: {{mapping| 1 2 2 2 2 2 | 0 -5 4 10 18 21 }}
Mapping: {{mapping| 1 2 2 2 2 2 | 0 -5 4 10 18 21 }}


Optimal tuning (POTE): ~2 = 1\1, ~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}}


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


Badness: 0.030886
Badness (Sintel): 1.28


== Passionate ==
== Passionate ==
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
[[Subgroup]]: 2.3.5.7


Line 65: Line 91:
{{Mapping|legend=1| 1 2 2 1 | 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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/15 = 98.817
{{Optimal ET sequence|legend=1| 12, 49d, 61, 73, 85c }}


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


=== 11-limit ===
=== 11-limit ===
Line 80: Line 108:
Mapping: {{mapping| 1 2 2 1 1 | 0 -5 4 22 30 }}
Mapping: {{mapping| 1 2 2 1 1 | 0 -5 4 22 30 }}


Optimal tuning (POTE): ~2 = 1\1, ~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}}


Optimal ET sequence: {{Optimal ET sequence| 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 93: Line 123:
Mapping: {{mapping| 1 2 2 1 1 1 | 0 -5 4 22 30 33 }}
Mapping: {{mapping| 1 2 2 1 1 1 | 0 -5 4 22 30 33 }}


Optimal tuning (POTE): ~2 = 1\1, ~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}}


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


Badness: 0.052302
Badness (Sintel): 2.16


==== Pash ====
==== Pash ====
Line 106: Line 138:
Mapping: {{mapping| 1 2 2 1 1 0 | 0 -5 4 22 30 45 }}
Mapping: {{mapping| 1 2 2 1 1 0 | 0 -5 4 22 30 45 }}


Optimal tuning (POTE): ~2 = 1\1, ~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}}


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


Badness: 0.056386
Badness (Sintel): 2.33


== Passive ==
== 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
[[Subgroup]]: 2.3.5.7


Line 119: Line 155:
{{Mapping|legend=1| 1 2 2 3 | 0 -5 4 -2 }}
{{Mapping|legend=1| 1 2 2 3 | 0 -5 4 -2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 }}


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


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


== Freivald ==
== 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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 6272/6075
[[Comma list]]: 6144/6125, 6272/6075


{{Mapping|legend=1| 1 7 -2 12 | 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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 }}


{{Optimal ET sequence|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 145: Line 191:
Comma list: 121/120, 176/175, 2744/2673
Comma list: 121/120, 176/175, 2744/2673


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


Optimal tuning (POTE): ~2 = 1\1, ~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}}


Optimal ET sequence: {{Optimal ET sequence| 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 160: Line 208:
Mapping: {{mapping| 1 7 -2 12 4 1 | 0 -10 8 -17 -1 5 }}
Mapping: {{mapping| 1 7 -2 12 4 1 | 0 -10 8 -17 -1 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~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}}


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


Badness: 0.045864
Badness (Sintel): 1.90


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Passion family| ]] <!-- main article -->
[[Category:Passion family| ]] <!-- main article -->
[[Category:Passion| ]] <!-- key article -->
[[Category:Passion| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Passion_family"