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Keemic temperaments: Difference between revisions

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Quasitemp tempers out [[2401/2400]] in addition to 875/864 and may be described as the {{nowrap| 37 & 41 }} temperament. It has a [[strong restriction]] to the 2.5/3.7/3 subgroup, called [[gariberttet]], which is defined by tempering out [[3125/3087]]. In gariberttet, three generators reach [[5/3]] and five reach [[7/3]], so that the generator itself has the interpretation of [[25/21]]. This implies that 3:5:7 and 5:6:7 chords are reached rather quickly. In quasitemp, tempering out 875/864 entails that [[8/7]] is found after 9 generators, from which the mappings of 3 and 5 follow.  
Quasitemp tempers out [[2401/2400]] in addition to 875/864 and may be described as the {{nowrap| 37 & 41 }} temperament. It has a [[strong restriction]] to the 2.5/3.7/3 subgroup, called [[gariberttet]], which is defined by tempering out [[3125/3087]]. In gariberttet, three generators reach [[5/3]] and five reach [[7/3]], so that the generator itself has the interpretation of [[25/21]]. This implies that 3:5:7 and 5:6:7 chords are reached rather quickly. In quasitemp, tempering out 875/864 entails that [[8/7]] is found after 9 generators, from which the mappings of 3 and 5 follow.  


The generator is equated to [[13/11]] for the 13-limit extension, tempering out [[275/273]].  
Note that the generator is given as 25/21's octave complement, 42/25, in the data that follow, since a stack of 14 such generators octave-reduced is the perfect fifth, whence the temperament's [[ploidacot]] is iota-14-cot. This generator is equated to [[22/13]] for the 13-limit extension, tempering out [[275/273]].  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 875/864, 2401/2400
[[Comma list]]: 875/864, 2401/2400


{{Mapping|legend=1| 1 5 5 5 | 0 -14 -11 -9 }}
{{Mapping|legend=1| 1 -9 -6 -4 | 0 14 11 9 }}


: Mapping generators: ~2, ~25/21
: mapping generators: ~2, ~42/25


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25/21 = 292.710{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.9237{{c}}, ~42/25 = 907.9887{{c}}
: [[error map]]: {{val| +0.924 +1.573 -3.981 -0.623 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 907.3471{{c}}
: error map: {{val| 0.000 +0.905 -5.495 -2.702 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~42/25 = 907.290{{c}} -->


{{Optimal ET sequence|legend=1| 4, 37, 41 }}
{{Optimal ET sequence|legend=1| 4, …, 37, 41 }}


[[Badness]] (Smith): 0.060269
[[Badness]] (Sintel): 1.53


=== 11-limit ===
=== 11-limit ===
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Comma list: 100/99, 385/384, 1375/1372
Comma list: 100/99, 385/384, 1375/1372


Mapping: {{mapping| 1 5 5 5 2 | 0 -14 -11 -9 6 }}
Mapping: {{mapping| 1 -9 -6 -4 8 | 0 14 11 9 -6 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~25/21 = 292.547{{c}}
Optimal tunings:
* WE: ~2 = 1199.9585{{c}}, ~42/25 = 907.4221{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.4521{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~42/25 = 907.453{{c}} -->


{{Optimal ET sequence|legend=0| 4, 37, 41, 119 }}
{{Optimal ET sequence|legend=0| 4, 37, 41, 119 }}


Badness (Smith): 0.043209
Badness (Sintel): 1.43


==== 13-limit ====
==== 13-limit ====
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Comma list: 100/99, 196/195, 275/273, 385/384
Comma list: 100/99, 196/195, 275/273, 385/384


Mapping: {{mapping| 1 5 5 5 2 2 | 0 -14 -11 -9 6 7 }}
Mapping: {{mapping| 1 -9 -6 -4 8 9 | 0 14 11 9 -6 -7 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/11 = 292.457{{c}}
Optimal tunings:
* WE: ~2 = 1199.4376{{c}}, ~22/13 = 907.1175{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 907.5314{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~22/13 = 907.543{{c}} -->


{{Optimal ET sequence|legend=0| 4, 37, 41, 78, 119f }}
{{Optimal ET sequence|legend=0| 4, 37, 41, 78, 119f }}


Badness (Smith): 0.032913
Badness (Sintel): 1.36


=== Quato ===
=== Quato ===
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Comma list: 243/242, 441/440, 625/616
Comma list: 243/242, 441/440, 625/616


Mapping: {{mapping| 1 5 5 5 12 | 0 -14 -11 -9 -35 }}
Mapping: {{mapping| 1 -9 -6 -4 -23 | 0 14 11 9 35 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~25/21 = 292.851{{c}}
Optimal tunings:
* WE: ~2 = 1201.2729{{c}}, ~42/25 = 908.1116{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.2109{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~42/25 = 907.149{{c}} -->


{{Optimal ET sequence|legend=0| 41, 127cd, 168cd }}
{{Optimal ET sequence|legend=0| 41, 127cd, 168cd }}


Badness (Smith): 0.041170
Badness (Sintel): 1.36


==== 13-limit ====
==== 13-limit ====
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Comma list: 105/104, 243/242, 275/273, 325/324
Comma list: 105/104, 243/242, 275/273, 325/324


Mapping: {{mapping| 1 5 5 5 12 12 | 0 -14 -11 -9 -35 -34 }}
Mapping: {{mapping| 1 -9 -6 -4 -23 -22 | 0 14 11 9 35 34 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/11 = 292.928{{c}}
Optimal tunings:
* WE: ~2 = 1201.4078{{c}}, ~42/25 = 908.1362{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.1370{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~22/13 = 907.072{{c}} -->


{{Optimal ET sequence|legend=0| 41, 86ce, 127cd }}
{{Optimal ET sequence|legend=0| 41, 86ce }}


Badness (Smith): 0.030081
Badness (Sintel): 1.24


== Chromo ==
== Chromo ==
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