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 | ||
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[[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 | ||
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== 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 | ||
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== 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 | ||
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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. | : [[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 + | : 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 }} | ||
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Badness (Sintel): 1.90 | Badness (Sintel): 1.90 | ||
[[Category:Passion family| ]] <!-- main article --> | [[Category:Passion family| ]] <!-- main article --> | ||
[[Category:Passion| ]] <!-- key article --> | [[Category:Passion| ]] <!-- key article --> | ||
[[Category: | [[Category:Temperament families]] | ||
[[Category:Catalogs of rank-2 temperaments]] | |||