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Ragismic microtemperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (4/)
Switch to Sintel's badness, WE & CWE tunings (5/)
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Optimal tunings:  
Optimal tunings:  
* CWE: ~99/70 = 599.9323{{c}}, ~99/80 = 369.6212{{c}}
* WE: ~99/70 = 599.9323{{c}}, ~99/80 = 369.6212{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~99/80 = 369.6610{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~99/80 = 369.6610{{c}}


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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~332150625/234881024 = 600.0019{{c}}, ~1125/1024 = 162.7479{{c}}
* [[WE]]: ~332150625/234881024 = 600.0019{{c}}, ~1125/1024 = 162.7479{{c}}
* [[error map]]: {{val| +0.004 +0.030 -0.042 -0.014 }}
: [[error map]]: {{val| +0.004 +0.030 -0.042 -0.014 }}
* [[CWE]]: ~332150625/234881024 = 600.0000{{c}}, ~1125/1024 = 162.7474{{c}}
* [[CWE]]: ~332150625/234881024 = 600.0000{{c}}, ~1125/1024 = 162.7474{{c}}
* error map: {{val| 0.000 +0.024 -0.051 -0.022 }}
: error map: {{val| 0.000 +0.024 -0.051 -0.022 }}


{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}
{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~7411887/5242880 = 599.9927{{c}}, ~8/7 = 231.1012{{c}}
* [[WE]]: ~7411887/5242880 = 599.9927{{c}}, ~8/7 = 231.1012{{c}}
: [[error map]]: {{val| -0.015 +0.037 -0.045 +0.029 }}
* [[CWE]]: ~7411887/5242880 = 600.0000{{c}}, ~8/7 = 231.1037{{c}}
* [[CWE]]: ~7411887/5242880 = 600.0000{{c}}, ~8/7 = 231.1037{{c}}
: error map: {{val| 0.000 +0.053 -0.023 +0.070 }}


{{Optimal ET sequence|legend=1| 26, …, 244, 270, 836, 1106, 1376, 2482 }}
{{Optimal ET sequence|legend=1| 26, …, 244, 270, 836, 1106, 1376, 2482 }}
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: ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]].  
: ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]].  


Aside from the ragisma, the seniority temperament (26 & 145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ({{monzo| -17 62 -35 }}, quadla-sepquingu) is tempered out.
Aside from the ragisma, the seniority temperament tempers out the [[wadisma]], 201768035/201326592, and may be described as {{nowrap| 26 & 145 }}. It is so named because the senior comma ({{monzo| -17 62 -35 }}, quadla-sepquingu) is tempered out.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 4375/4374, 201768035/201326592
[[Comma list]]: 4375/4374, 201768035/201326592


{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}
{{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }}
: mapping generators: ~2, ~5120/3087


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3087/2560 = 322.804{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0745{{c}}, ~5120/3087 = 877.2500{{c}}
: [[error map]]: {{val| +0.075 +0.008 -0.016 -0.203 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5120/3087 = 877.1965{{c}}
: error map: {{val| 0.000 -0.077 -0.130 -0.415 }}


{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d }}


[[Badness]] (Sintel): 1.136
[[Badness]] (Sintel): 1.14


=== Senator ===
=== Senator ===
The senator temperament (26 & 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It can be extended to the 13- and 17-limit immediately, by adding 364/363 and 595/594 to the comma list in this order.
Senator (26 & 145) extends seniority by tempering out [[441/440]] and [[65536/65219]], and can be extended to the 13- and 17-limit immediately by adding [[364/363]] and [[595/594]] to the comma list in this order.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Comma list: 441/440, 4375/4374, 65536/65219
Comma list: 441/440, 4375/4374, 65536/65219


Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }}
Mapping: {{mapping| 1 -24 -43 5 2 | 0 35 62 -3 2 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~77/64 = 322.793{{c}}
Optimal tunings:
* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}}


{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e, 487ee }}
{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e }}


Badness (Sintel): 3.049
Badness (Sintel): 3.05


==== 13-limit ====
==== 13-limit ====
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Comma list: 364/363, 441/440, 2200/2197, 4375/4374
Comma list: 364/363, 441/440, 2200/2197, 4375/4374


Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }}
Mapping: {{mapping| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~77/64 = 322.793{{c}}
Optimal tunings:
* WE: ~2 = 1199.7136{{c}}, ~108/65 = 877.9974{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2038{{c}}


{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316ef, 487eef }}
{{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }}


Badness (Sintel): 1.845
Badness (Sintel): 1.85


==== 17-limit ====
==== 17-limit ====
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Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197


Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }}
Mapping: {{mapping| 1 -24 -43 5 2 -27 -31 | 0 35 62 -3 2 42 48 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~77/64 = 322.793{{c}}
Optimal tunings:
* WE: ~2 = 1199.7195{{c}}, ~108/65 = 877.0018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}}


{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316ef, 487eef }}
{{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }}


Badness (Sintel): 1.353
Badness (Sintel): 1.35


== Monzismic ==
== Monzismic ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]].  
: ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]].  


The monzismic temperament (53 & 612) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]]. A notable tuning not appearing on the optimal ET sequence is [[665edo]], which is nearly equivalent to the pure-3's tuning.
The monzismic temperament tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]]. It may be described as the {{nowrap| 53 & 612 }} temperament. A notable tuning not appearing on the optimal ET sequence is [[665edo]], which is nearly equivalent to the pure-3's tuning.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}


{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}
{{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }}
: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~{{monzo| -27 11 3 1 }} = 249.0207{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0128{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9895{{c}}
: [[error map]]: {{val| +0.013 +0.024 -0.049 -0.019 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9793{{c}}
: error map: {{val| 0.000 +0.004 -0.080 -0.050 }}


{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}
{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd }}


[[Badness]] (Sintel): 1.179
[[Badness]] (Sintel): 1.18


=== Monzism ===
=== Monzism ===
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Comma list: 4375/4374, 41503/41472, 184549376/184528125
Comma list: 4375/4374, 41503/41472, 184549376/184528125


Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}
Mapping: {{mapping| 1 0 -27 109 -159 | 0 2 37 -134 205 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~231/200 = 249.0193{{c}}
Optimal tunings:
* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}}


{{Optimal ET sequence|legend=0| 53, 559, 612 }}
{{Optimal ET sequence|legend=0| 53, 559, 612, 3619de, 4231de, …, 6067ddee }}


Badness (Sintel): 1.887
Badness (Sintel): 1.89


==== 13-limit ====
==== 13-limit ====
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Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625


Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}
Mapping: {{mapping| 1 0 -27 109 -159 -70 | 0 2 37 -134 205 93 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~231/200 = 249.0199{{c}}
Optimal tunings:
* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}}


{{Optimal ET sequence|legend=0| 53, 559, 612 }}
{{Optimal ET sequence|legend=0| 53, 559, 612 }}


Badness (Sintel): 2.222
Badness (Sintel): 2.22


== Semidimfourth ==
== Semidimfourth ==
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