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Miscellaneous 7-limit temperaments: Difference between revisions

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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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[[Projection pair]]: <code>5 2401/486</code> to 2.3.7
[[Projection pair]]: <code>5 2401/486</code> to 2.3.7
== Schismean ==
Schismean tempers out the [[3645/3584|schismean comma]] in the 7-limit. It equates [[7/5]] with a stack of three [[9/8]]'s.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 3645/3584
{{Mapping|legend=1| 1 0 0 -9 | 0 1 0 6 | 0 0 1 1 }}
: Mapping generators: ~2, ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.420{{c}}, ~3/2 = 698.145{{c}}, ~5/4 = 382.612{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 697.459{{c}}, ~5/4 = 383.104{{c}}
{{Optimal ET sequence|legend=1| 5c, 7d, 12, 19, 31, 81 }}
[[Badness]] (Sintel): 2.96


== Keegic ==
== Keegic ==
Keegic tempers out the [[keega]] in the 7-limit, and finds the [[3/1|3rd]] [[harmonic]] by a stack of three [[10/7]]'s.
[[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 ==
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[[Badness]] (Sintel): 6.18
[[Badness]] (Sintel): 6.18
== Tolerant ==
: ''For extensions, see [[Pentacircle clan #Tolerant]].''
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 179200/177147
{{Mapping|legend=1| 1 0 0 -10 | 0 1 0 11 | 0 0 1 -2 }}
: Mapping generators: ~2, ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.539{{c}}, ~3 = 1903.226{{c}}, ~5 = 2785.816{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3 = 1903.805{{c}}, ~5 = 2786.356{{c}}
{{Optimal ET sequence|legend=1| 34d, 39d, 41, 80, 87, 121, 167, 208, 329b, 375b, 496bd }}
[[Badness]] (Sintel): 7.26
== History ==
: ''For extensions, see [[Werckismic temperaments #History]].''
History tempers out the [[historisma]] in the 7-limit, and splits the fourth in six.
[[Comma list]]: 257298363/256000000
{{Mapping|legend=1| 1 2 0 0 | 0 -6 0 7 | 0 0 1 1 }}
: Mapping generators: ~2, ~21/20, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.154{{c}}, ~21/20 = 83.091{{c}}, ~5 = 2786.669{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~21/20 = 83.067{{c}}, ~5 = 2786.515{{c}}
{{Optimal ET sequence|legend=1| 14c, 15, 29, 43, 58, 72, 130, 202}}
[[Badness]] (Sintel): 7.95


== Sensibeta ==
== Sensibeta ==
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== Septimagic ==
== Septimagic ==
Septimagic tempers out the [[septimagic comma]] in the 7-limit and gives the [[2.3.7 subgroup|2.3.7-]][[subgroup]] [[magic]] [[restriction]] an independent generator for the [[5/1|5th]] [[harmonic]].
[[Subgroup]]: 2.3.5.7  
[[Subgroup]]: 2.3.5.7  


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[[Badness]] (Sintel): 15.7
[[Badness]] (Sintel): 15.7
== Naiad ==
: ''For extensions, see [[Wizardharry clan #Naiad]].''
Naiad tempers out the [[naiadisma]] in the 7-limit. An obvious 13-limit interpretation of one generator (~98/75) is [[13/10]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 161414428/158203125
{{Mapping|legend=1| 1 5 0 2 | 0 -9 0 -4 | 0 0 1 1 }}
: Mapping generators: ~2, ~98/75, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.937{{c}}, ~98/75 = 455.269{{c}}, ~5/4 = 387.946{{c}}
: [[error map]]: {{val| -0.063, +0.311, +1.506, -2.207 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~98/75 = 455.298{{c}}, ~5/4 = 387.896{{c}}
: error map: {{val| 0.000, +0.362, +1.582, -2.123 }}
{{Optimal ET sequence|legend=1| 8d, 21, 29, 37, 50, 58, 87, 145 }}
[[Badness]] (Sintel): 52.5


[[Category:Temperament collections]]
[[Category:Temperament collections]]
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