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Syntonic–diatonic equivalence continuum: Difference between revisions

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| −2
| −2
| −3
| −3
| [[Gamelismic clan #Gorgo|Laconic]]
| [[Laconic]]
| [[2187/2000]]
| [[2187/2000]]
| {{Monzo| -4 7 -3 }}
| {{Monzo| -4 7 -3 }}
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| {{Monzo| -32 29 -6 }}
| {{Monzo| -32 29 -6 }}
|-
|-
| …
| …
| …
| …
| …
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|-
|-
| 21/5 = 4.2 || 21/16 = 1.3125 || 559 & 2513 || {{Monzo| -124 109 -21 }}
| 21/5 = 4.2 || 21/16 = 1.3125 || 559 & 2513 || {{Monzo| -124 109 -21 }}
|-
| 13/3 = 4.{{overline|3}} || 13/10 = 1.3 || [[Tokko]] || {{Monzo| -76 67 -13 }}
|-
|-
| 9/2 = 4.5 || 9/7 = 1.{{overline|285714}} || 5 & 118 || {{Monzo| -52 46 -9 }}
| 9/2 = 4.5 || 9/7 = 1.{{overline|285714}} || 5 & 118 || {{Monzo| -52 46 -9 }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 709.393
* [[WE]]: ~2 = 1197.6520{{c}}, ~3/2 = 708.6882{{c}}
: [[error map]]: {{val| 0.000 +7.438 -1.774 }}
: [[error map]]: {{val| -2.348 +4.385 -1.076 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 710.078
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.8213{{c}}
: error map: {{val| 0.000 +8.123 +4.385 }}
: error map: {{val| 0.000 +7.866 +2.078 }}


{{Optimal ET sequence|legend=1| 5, 17, 22, 49, 120b, 169bbc }}
{{Optimal ET sequence|legend=1| 5, 17, 22, 49, 120b, 169bbc }}


[[Badness]] (Smith): 0.135141
[[Badness]] (Sintel): 3.17


== Uncle (5-limit) ==
== Uncle (5-limit) ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 733.721
* [[WE]]: ~2 = 1189.7544{{c}}, ~3/2 = 724.6670{{c}}
: [[error map]]: {{val| 0.000 +31.766 +11.362 }}
: [[error map]]: {{val| -10.246 +12.466 +4.210 }}
* [[CWE]]: ~2 = 1200.000, ~3/2 = 731.732
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 731.7318{{c}}
: error map: {{val| 0.000 +29.777 +23.296 }}
: error map: {{val| 0.000 +29.777 +23.296 }}


{{Optimal ET sequence|legend=1| 5, 13, 18, 23bc }}
{{Optimal ET sequence|legend=1| 5, 13, 18, 23bc }}


[[Badness]]:
[[Badness]] (Sintel): 6.33
* Smith: 0.270
* Dirichlet: 6.33


== Ultrapyth (5-limit) ==
== Ultrapyth (5-limit) ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 713.185
* [[WE]]: ~2 = 1196.4357{{c}}, ~3/2 = 711.7085{{c}}
: [[error map]]: {{val| 0.000 +11.230 -1.722 }}
: [[error map]]: {{val| -3.564 +6.189 -1.009 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 713.829
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5968{{c}}
: error map: {{val| 0.000 +11.874 +7.289 }}
: error map: {{val| 0.000 +11.642 +4.041 }}


{{Optimal ET sequence|legend=1| 5, 27c, 32, 37, 79bc, 116bbc }}
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37, 79bc, 116bbc }}


[[Badness]] (Smith): 0.795243
[[Badness]] (Sintel): 18.7


== Dirt ==
== Dirt ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 714.992
* [[WE]]: ~2 = 1195.8566{{c}}, ~3/2 = 713.0611{{c}}
: [[error map]]: {{val| 0.000 +13.037 -1.473 }}
: [[error map]]: {{val| -4.143 +6.963 -0.863 }}
* [[CWE]]: ~2 = 1200.000, ~3/2 = 715.341
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 715.3406{{c}}
: error map: {{val| 0.000 +13.386 +5.157 }}
: error map: {{val| 0.000 +13.386 +5.157 }}


{{Optimal ET sequence|legend=1| 5, 42c, 47b, 52b, 109bbc }}
{{Optimal ET sequence|legend=1| 5, 42c, 47b, 52b, 109bbc }}


[[Badness]] (Smith): 2.36
[[Badness]] (Sintel): 55.3


== Rodan (5-limit) ==
== Rodan (5-limit) ==
: ''For extensions, see [[Gamelismic clan #Rodan]].''
: ''For extensions, see [[Gamelismic clan #Rodan]].''


The 5-limit version of rodan tempers out the [[rodan comma]], which is the difference between a stack of three [[729/640|retroptolemaic whole tones (729/640)]] and a perfect fifth (3/2). The only 7-limit extension that makes any sense to use is to add the gamelisma to the comma list. It corresponds to {{nowrap| ''n'' {{=}} 3 }}.  
The 5-limit version of rodan tempers out the [[rodan comma]], which is the difference between a stack of three [[729/640|retroptolemaic whole tones (729/640)]] and a perfect fifth (3/2). The only 7-limit extension that makes any sense to use is to add the gamelisma to the comma list, whereby the generator represents [[8/7]]. It corresponds to {{nowrap| ''n'' {{=}} 3 }}.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~729/640 = 234.457
* [[WE]]: ~2 = 1199.5618{{c}}, ~729/640 = 234.4424{{c}}
: [[error map]]: {{val| 0.000 +1.417 -0.537 }}
: [[error map]]: {{val| -0.438 +0.934 -0.355 }}
* [[POTE]]: ~2 = 1200.000, ~729/640 = 234.528
* [[CWE]]: ~2 = 1200.000{{c}}, ~729/640 = 234.4999{{c}}
: error map: {{val| 0.000 +1.629 +0.663 }}
: error map: {{val| 0.000 +1.545 +0.185 }}


{{Optimal ET sequence|legend=1| 5, …, 41, 46, 87, 220, 307 }}
{{Optimal ET sequence|legend=1| 5, …, 41, 46, 87, 220, 307 }}


[[Badness]]: 0.168264
[[Badness]] (Sintel): 3.95


== Laconic ==
== Laconic ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~10/9 = 228.700
* [[WE]]: ~2 = 1203.1925{{c}}, ~10/9 = 228.0305{{c}}
: [[error map]]: {{val| 0.000 -15.856 +14.584 }}
: [[error map]]: {{val| +3.193 -14.671 +13.092 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 227.426
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/9 = 228.0128{{c}}
: error map: {{val| 0.000 -19.679 +5.664 }}
: error map: {{val| 0.000 -17.917 +9.776 }}


{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37b }}
{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37b }}


[[Badness]] (Smith): 0.161799
[[Badness]] (Sintel): 3.80


== University ==
== University ==
{{Main| University }}
: ''For extensions, see [[Gamelismic clan #Gidorah]] and [[Mint temperaments #Penta]].''
: ''For extensions, see [[Gamelismic clan #Gidorah]] and [[Mint temperaments #Penta]].''


Named by [[John Moriarty]], university is the {{nowrap| 5 & 6b }} temperament, and tempers out [[144/125]], the triptolemaic diminished third. It corresponds to {{nowrap| ''n'' {{=}} −3/2 }} and {{nowrap| ''m'' {{=}} 3/5 }}. In this temperament, two instances of [[6/5]] make a [[5/4]], and three make a [[3/2]]. Equating 6/5 with [[8/7]] (which makes sense since it is already very flat in the most accurate tunings of this temperament) leads to [[Gamelismic clan #Gidorah|gidorah]], and 6/5 with [[7/6]] leads to [[Mint temperaments #Penta|penta]].
Named by [[John Moriarty]], university is the {{nowrap| 5 & 6b }} temperament, and tempers out [[144/125]], the triptolemaic diminished third. It corresponds to {{nowrap| ''n'' {{=}} −3/2 }} and {{nowrap| ''m'' {{=}} 3/5 }}. In this temperament, two instances of [[6/5]] make a [[5/4]], and three make a [[3/2]]. Equating 6/5 with [[8/7]] (which makes sense since it is already very flat in the most accurate tunings of this temperament) leads to [[Gamelismic clan #Gidorah|gidorah]], and 6/5 with [[7/6]] leads to [[Mint temperaments #Penta|penta]].
University widens the classical major and minor chords to [[Extraclassical tonality|tendo and arto chords.]]


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~6/5 = 226.980
* [[WE]]: ~2 = 1186.1969{{c}}, ~6/5 = 232.7334{{c}}
: [[error map]]: {{val| 0.000 -21.014 +67.647 }}
: [[error map]]: {{val| -13.803 -17.558 +51.547 }}
* [[POTE]]: ~2 = 1200.000, ~6/5 = 235.442
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 231.4822{{c}}
: error map: {{val| 0.000 +4.370 +84.569 }}
: error map: {{val| 0.000 -7.509 +76.651 }}


{{Optimal ET sequence|legend=1| 1b, …, 4bc, 5 }}
{{Optimal ET sequence|legend=1| 1b, …, 4bc, 5 }}


[[Badness]] (Smith): 0.101806
[[Badness]] (Sintel): 2.39
 
=== Music ===
The purely 5-limit university mapping, using the 21cc [[val]], was in mind when writing this song.  


; [[John Moriarty]]
* [https://soundcloud.com/john-lank1/uni ''Uni''] (2013) – University[6] in approximately 21edo
<!--
== Trisatriyo (5 & 56) ==
== Trisatriyo (5 & 56) ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000, ~2560/2187 = 235.867
* [[POTE]]: ~2 = 1200.000{{c}}, ~2560/2187 = 235.867{{c}}


{{Optimal ET sequence|legend=1| 5, …, 51, 56, 117b, 173b }}
{{Optimal ET sequence|legend=1| 5, …, 51, 56, 117b, 173b }}
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[http://x31eq.com/cgi-bin/rt.cgi?ets=5_56&limit=5 The temperament finder - 5-limit 5 & 56]
[http://x31eq.com/cgi-bin/rt.cgi?ets=5_56&limit=5 The temperament finder - 5-limit 5 & 56]
-->


== Hemiseven (5-limit) ==
== Hemiseven (5-limit) ==
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[[Comma list]]: {{monzo| 32 -29 6 }}
[[Comma list]]: {{monzo| 32 -29 6 }}


{{Mapping|legend=1| 1 4 14 | 0 -6 -29 }}
{{Mapping|legend=1| 1 -2 -15 | 0 6 29 }}


: mapping generators: ~2, ~320/243
: mapping generators: ~2, ~243/160


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000, ~320/243 = 483.247
* [[WE]]: ~2 = 1200.3725{{c}}, ~243/160 = 716.9750{{c}}
: [[error map]]: {{val| +0.373 -0.850 +0.376 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/160 = 716.7671{{c}}
: error map: {{val| 0.000 -1.352 -0.067 }}


{{Optimal ET sequence|legend=1| 5, 62c, 67c, 72, 149, 221, 370, 591b, 961bb }}
{{Optimal ET sequence|legend=1| 5, …, 72, 149, 221, 370, 591b }}


[[Badness]] (Smith): 0.720465
[[Badness]] (Sintel): 16.9


== Counterpental ==
== Counterpental ==
: ''For extensions, see [[Orwellismic temperaments #Pentorwell]].''
: ''For extensions, see [[Orwellismic temperaments #Pentaorwell]].''


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~729/640 = 240.000, ~3/2 = 704.572
* [[WE]]: ~729/640 = 239.8575{{c}}, ~3/2 = 704.1540{{c}}
: [[error map]]: {{val| -0.712 +1.487 -0.535 }}
* [[CWE]]: ~729/640 = 240.0000{{c}}, ~3/2 = 704.4446{{c}}
: error map: {{val| 0.000 +2.490 +0.354 }}


{{Optimal ET sequence|legend=1| 5, …, 75, 80, 155, 390b, 545bbc }}
{{Optimal ET sequence|legend=1| 5, …, 75, 80, 155, 390b, 545bbc }}


[[Badness]] (Smith): 1.500224
[[Badness]] (Sintel): 35.2


== Septiquarter (5-limit) ==
== Septiquarter (5-limit) ==
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[[Comma list]]: {{monzo| 44 -38 7 }}
[[Comma list]]: {{monzo| 44 -38 7 }}


{{Mapping|legend=1| 1 3 10 | 0 -7 -38 }}
{{Mapping|legend=1| 1 -4 -28 | 0 7 38 }}


: mapping generators: ~2, ~204800/177147
: mapping generators: ~2, ~177147/102400


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000, ~204800/177147 = 242.457
* [[WE]]: ~2 = 1199.7741{{c}}, ~177147/102400 = 957.3630{{c}}
: [[error map]]: {{val| -0.226 +0.490 -0.194 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~177147/102400 = 957.5367{{c}}
: error map: {{val| 0.000 +0.802 +0.082 }}


{{Optimal ET sequence|legend=1| 5, 89c, 94, 99, 193, 292, 391 }}
{{Optimal ET sequence|legend=1| 5, …, 94, 99, 193, 292, 391, 1074b, 1465bb }}


[[Badness]] (Smith): 0.971284
[[Badness]] (Sintel): 22.8


== Tokko (5-limit) ==
: ''For extensions, see [[Wizmic microtemperaments #Tokko]].''
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| -76 67 -13 }}
{{Mapping|legend=1| 1 -1 -11 | 0 13 67 }}
: mapping generators: ~2, ~{{monzo| -35 31 -6 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0377{{c}}, ~{{monzo| -35 31 -6 }} = 238.6084{{c}}
: [[error map]]: {{val| +0.038 -0.083 +0.035 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -35 31 -6 }} = 238.6015{{c}}
: error map: {{val| 0.000 -0.135 -0.011 }}
{{Optimal ET sequence|legend=1| 5, …, 166, 171, 860, 1031, 1202, 1373, 1544, 3259, 4803b, 6347b }}
[[Badness]] (Sintel): 20.9
<!--
== Quinla-tritrigu (5 & 118) ==
== Quinla-tritrigu (5 & 118) ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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{{Mapping|legend=1| 1 -2 -16 | 0 9 46 }}
{{Mapping|legend=1| 1 -2 -16 | 0 9 46 }}
: mapping generators: ~2, ~320/243
: mapping generators: ~2, ~320/243


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000, ~320/243 = 477.961
* [[POTE]]: ~2 = 1200.000{{c}}, ~320/243 = 477.961{{c}}


{{Optimal ET sequence|legend=1| 5, 108c, 113, 118, 1057, 1175, 1293, 1411, 1529, 1647, 1765, 1883, 2001b, 3884b }}
{{Optimal ET sequence|legend=1| 5, 108c, 113, 118, 1057, 1175, 1293, 1411, 1529, 1647, 1765, 1883, 2001b, 3884b }}


[[Badness]] (Smith): 0.617683
[[Badness]] (Sintel): 14.5


== Tribilalegu (5 & 137) ==
== Tribilalegu (5 & 137) ==
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[[Comma list]]: {{Monzo| -60 54 -11 }}
[[Comma list]]: {{Monzo| -60 54 -11 }}


{{Mapping|legend=1| 1 6 24 | 0 -11 -54 }}
{{Mapping|legend=1| 1 -5 -30 | 0 11 54 }}
 
: mapping generators: ~2, ~243/160
: mapping generators: ~2, ~320/243


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000, ~320/243 = 481.742
* [[POTE]]: ~2 = 1200.000{{c}}, ~243/160 = 718.258{{c}}


{{Optimal ET sequence|legend=1| 5, 127c, 132, 137, 553, 690b, 827b, 964b }}
{{Optimal ET sequence|legend=1| 5, 127c, 132, 137, 553, 690b, 827b, 964b }}


[[Badness]] (Smith): 3.620981
[[Badness]] (Sintel): 84.9
 
[http://x31eq.com/cgi-bin/rt.cgi?ets=5_137&limit=5 The temperament finder - 5-limit 5 & 137]


== 559 & 2513 ==
== 559 & 2513 ==
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[[Comma list]]: {{monzo| -124 109 -21 }}
[[Comma list]]: {{monzo| -124 109 -21 }}


{{Mapping|legend=1| 1 10 46 | 0 -21 -109 }}
{{Mapping|legend=1| 1 -11 -63 | 0 21 109 }}
 
: mapping generators: ~2, ~{{monzo| -29 26 -5 }}
: mapping generators: ~2, ~3355443200000/2541865828329


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.0000, ~3355443200000/2541865828329 = 480.8595
* [[POTE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -29 26 -5 }} = 719.1405{{c}}
 
{{Optimal ET sequence|legend=1| 5, 267c, 272c, 277, 559, 1395, 1954, 2513, 40767, 43280, 45793, 48306, 50819, 53332, 55845, 58358, 60871, 63384, 65897, 68410, 70923, 73436, 75949, 78462 }}
 
[[Badness]] (Smith): 0.134523


[http://x31eq.com/cgi-bin/rt.cgi?ets=2513_559&limit=5 The temperament finder - 5-limit 2513 & 559]
{{Optimal ET sequence|legend=1| 5, …, 277, 559, 1395, 1954, 2513, 40767, 43280, 45793, 48306, 50819, 53332, 55845, 58358, 60871, 63384, 65897, 68410, 70923, 73436, 75949, 78462, 154411b }}


[[Badness]] (Sintel): 3.16
-->
[[Category:5edo]]
[[Category:5edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
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