Miscellaneous 7-limit temperaments: Difference between revisions
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Move mirkwai here as we're dissolving the mirkwai family |
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[[Badness]] (Sintel): 0.661 | [[Badness]] (Sintel): 0.661 | ||
== Canopic a.k.a. mirkwai == | |||
: ''For extensions, see [[Swetismic temperaments #Indra]].'' | |||
Canopic, a.k.a. mirkwai, tempers out the [[canopic comma]] in the 7-limit. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 16875/16807 | |||
{{Mapping|legend=1| 1 0 -5 -4 | 0 1 3 3 | 0 0 5 4 }} | |||
: Mapping generators: ~2, ~3, ~10/7 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.9999{{c}}, ~3/2 = 701.7827{{c}}, ~10/7 = 616.0944{{c}} | |||
: [[error map]]: {{val| -0.000 -0.172 -0.493 +0.900 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7827{{c}}, ~10/7 = 616.0945{{c}} | |||
: error map: {{val| 0.000 -0.172 -0.493 +0.900 }} | |||
[[Minimax tuning]]: | |||
* [[7-odd-limit]] | |||
: {{monzo list| 1 0 0 0 | 0 4/7 -4/7 5/7 | 0 -3/7 3/7 5/7 | 0 0 0 1 }} | |||
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3.7 | |||
* [[9-odd-limit]] | |||
: {{monzo list| 1 0 0 0 | 0 8/11 -4/11 5/11 | 0 -6/11 3/11 10/11 | 0 0 0 1 }} | |||
: eigenmonzo (unchanged-interval) basis: 2.9/5.7 | |||
{{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224 }} | |||
[[Badness]] (Sintel): 1.51 | |||
[[Projection pair]]s: <code>5 84375/16807 7 16875/2401</code> to 2.3.7/5 | |||
== Greenwoodmic == | == Greenwoodmic == | ||
Greenwoodmic tempers out the [[greenwoodma]] in the 7-limit. It equates [[5/2]] with a stack of two [[14/9]]'s. This implies [[prime interval|primes]] [[3/1|3]] and [[5/1|5]] should be tuned flat, and [[7/1|7]] should be tuned sharp. A rank-2 temperament that does that is [[injera]], which introduces little extra [[damage]] over greenwoodmic. | Greenwoodmic tempers out the [[greenwoodma]] in the 7-limit. It equates [[5/2]] with a stack of two [[14/9]]'s. This implies [[prime interval|primes]] [[3/1|3]] and [[5/1|5]] should be tuned flat, and [[7/1|7]] should be tuned sharp. A rank-2 temperament that does that is [[injera]], which introduces little extra [[damage]] over greenwoodmic. | ||
In contrast to [[sensamagic]], where two [[9/7]]'s stack to [[5/3]], here two 9/7's stack to [[8/5]]. As such, greenwoodmic induces [[essentially tempered chord]]s in the [[9-odd-limit]]. An obvious 11-limit extension then equates [[5/4]] with [[11/9]] and equates 9/7 with [[14/11]], tempering out [[45/44]] as well as [[99/98]] using the identity 405/392 = (45/44)⋅(99/98). | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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== Avicennmic == | == Avicennmic == | ||
Avicennmic tempers out the [[avicennma]] in the 7-limit. It equates [[32/21]] with a stack of two [[5/4]]'s, and [[12/7]] with a stack of two [[15/8]]'s octave reduced. This implies [[prime interval|primes]] [[3/1|3]], [[5/1|5]] and [[7/1|7]] should all be tuned flat. A rank-2 temperament that does that is [[flattone]], which introduces little extra [[damage]] over avicennmic. | Avicennmic tempers out the [[avicennma]] in the 7-limit. It equates [[32/21]] with a stack of two [[5/4]]'s, and [[12/7]] with a stack of two [[15/8]]'s octave reduced. This implies [[prime interval|primes]] [[3/1|3]], [[5/1|5]] and [[7/1|7]] should all be tuned flat. A rank-2 temperament that does that is [[flattone]], which introduces little extra [[damage]] over avicennmic. | ||
One possible extension of avicennmic to the 11-limit is via [[45/44]] and [[385/384]], using the identity 525/512 = (45/44)⋅(385/384), but the result is somewhat less accurate. Instead, it is more natural to extend it to the [[2.3.5.7.13 subgroup]] by tempering out [[65/64]] and [[105/104]], using the identity 525/512 = (65/64)⋅(105/104). | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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[[Badness]] (Sintel): 3.40 | [[Badness]] (Sintel): 3.40 | ||
== Decovulture == | |||
: ''For extensions, see [[Olympic clan #Baffin]].'' | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 67108864/66976875 | |||
{{Mapping|legend=1| 1 0 0 13 | 0 2 0 -7 | 0 0 1 -2 }} | |||
: mapping generators: ~2, ~8192/4725, ~5 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.9033{{c}}, ~8192/4725 = 951.0102{{c}}, ~5/4 = 386.5872{{c}} | |||
: [[error map]]: {{val| -0.097 +0.065 +0.080 +0.059 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8192/4725 = 951.0899{{c}}, ~5/4 = 386.6184{{c}} | |||
: error map: {{val| 0.000 +0.225 +0.305 +0.308 }} | |||
{{Optimal ET sequence|legend=1| 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd }} | |||
[[Badness]] (Sintel): 3.82 | |||
== Trimyna == | == Trimyna == | ||