Xenharmonic Wiki

Passion family: Difference between revisions

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== 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 16: Line 18:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 98.7335{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 98.7335{{c}}
: error map: {{val| 0.000 +4.378 +8.620 }}
: error map: {{val| 0.000 +4.378 +8.620 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~16/15 = 98.670{{c}} -->


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


[[Badness]] (Sintel): 3.95
[[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 34: Line 47:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 98.1019{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 98.1019{{c}}
: error map: {{val| 0.000 +7.535 +6.094 +12.193 }}
: error map: {{val| 0.000 +7.535 +6.094 +12.193 }}
<!-- [[POTE]]: ~2 = 1200.000{{c}}, ~16/15 = 98.153{{c}} -->


{{Optimal ET sequence|legend=1| 12, 37, 49, 110bcd }}
{{Optimal ET sequence|legend=1| 12, 37, 49, 110bcd }}
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* WE: ~2 = 1196.9304{{c}}, ~21/20 = 97.7686{{c}}
* WE: ~2 = 1196.9304{{c}}, ~21/20 = 97.7686{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 97.9333{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 97.9333{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~16/15 = 98.019{{c}} -->


{{Optimal ET sequence|legend=0| 12, 37, 49 }}
{{Optimal ET sequence|legend=0| 12, 37, 49 }}
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* WE: ~2 = 1196.8712{{c}}, ~21/20 = 97.6548{{c}}
* WE: ~2 = 1196.8712{{c}}, ~21/20 = 97.6548{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 97.8141{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 97.8141{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~16/15 = 97.910{{c}} -->


{{Optimal ET sequence|legend=0| 12f, 25f, 37, 49f }}
{{Optimal ET sequence|legend=0| 12f, 25f, 37, 49f }}
Line 73: Line 83:


== 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 84: Line 96:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 98.7770{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 98.7770{{c}}
: error map: {{val| 0.000 +4.160 +8.794 +4.269 }}
: error map: {{val| 0.000 +4.160 +8.794 +4.269 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~16/15 = 98.817{{c}} -->


{{Optimal ET sequence|legend=1| 12, 49d, 61, 73, 85c }}
{{Optimal ET sequence|legend=1| 12, 49d, 61, 73, 85c }}
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* WE: ~2 = 1197.8835{{c}}, ~16/15 = 98.5469{{c}}
* WE: ~2 = 1197.8835{{c}}, ~16/15 = 98.5469{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/15 = 98.6733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/15 = 98.6733{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~16/15 = 98.721{{c}} -->


{{Optimal ET sequence|legend=0| 12, 49de, 61, 73, 158bccee }}
{{Optimal ET sequence|legend=0| 12, 49de, 61, 73, 158bccee }}
Line 116: Line 126:
* WE: ~2 = 1197.8309{{c}}, ~16/15 = 98.4485{{c}}
* WE: ~2 = 1197.8309{{c}}, ~16/15 = 98.4485{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/15 = 98.5760{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/15 = 98.5760{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~16/15 = 98.627{{c}} -->


{{Optimal ET sequence|legend=0| 12f, 49de, 61, 73f }}
{{Optimal ET sequence|legend=0| 12f, 49de, 61, 73f }}
Line 132: Line 141:
* WE: ~2 = 1197.8306{{c}}, ~16/15 = 98.6124{{c}}
* WE: ~2 = 1197.8306{{c}}, ~16/15 = 98.6124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/15 = 98.7468{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/15 = 98.7468{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~16/15 = 98.791{{c}} -->


{{Optimal ET sequence|legend=0| 12f, 73, 85ce }}
{{Optimal ET sequence|legend=0| 12f, 73, 85ce }}
Line 139: Line 147:


== 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 150: Line 160:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 98.9690{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 98.9690{{c}}
: error map: {{val| 0.000 +3.200 +9.562 +33.236 }}
: error map: {{val| 0.000 +3.200 +9.562 +33.236 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~16/15 = 98.809{{c}} -->


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


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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.4021{{c}}, ~48/35 = 550.2038{{c}}
* [[WE]]: ~2 = 1198.4021{{c}}, ~48/35 = 550.2038{{c}}
: [[error map]]: {{val| -4.645 -3.378 -1.897 +20.386 }}
: [[error map]]: {{val| -1.598 +4.877 +2.468 -7.371 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 550.8980{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 550.8980{{c}}
: error map: {{val| 0.000 +3.200 +9.562 +33.236 }}
: error map: {{val| 0.000 +7.025 +6.502 -3.559 }}
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~48/35 = 550.9374{{c}} -->


{{Optimal ET sequence|legend=1| 13d, 24, 37, 61, 98b }}
{{Optimal ET sequence|legend=1| 13d, 24, 37, 61, 98b }}
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* WE: ~2 = 1198.8053{{c}}, ~11/8 = 550.3522{{c}}
* WE: ~2 = 1198.8053{{c}}, ~11/8 = 550.3522{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 550.8760{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 550.8760{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~11/8 = 550.9007{{c}} -->


{{Optimal ET sequence|legend=0| 13d, 24, 37, 61 }}
{{Optimal ET sequence|legend=0| 13d, 24, 37, 61 }}
Line 202: Line 211:
* WE: ~2 = 1198.7706{{c}}, ~11/8 = 550.3418{{c}}
* WE: ~2 = 1198.7706{{c}}, ~11/8 = 550.3418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 550.8892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 550.8892{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~11/8 = 550.9062{{c}} -->


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