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Syntonic–limmic equivalence continuum: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''syntonic–diatonic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|Pythagorean limma (256/243)]]. This continuum is theoretically interesting in that these are all [[5-limit]] temperaments [[support]]ed by [[5edo]].  
The '''syntonic–limmic''' (or '''syntonic–diatonic''') '''equivalence continuum''' is a [[equivalence continuum|continuum]] of [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|Pythagorean limma (256/243)]]. This continuum is theoretically interesting in that these are all [[5-limit]] temperaments [[support]]ed by [[5edo]].  


All temperaments in the continuum satisfy {{nowrap|(81/80)<sup>''n''</sup> ~ 256/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 5edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is 4.1952…, and temperaments near this tend to be the most accurate ones.  
All temperaments in the continuum satisfy {{nowrap|(81/80)<sup>''n''</sup> ~ 256/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 5edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is 4.1952…, and temperaments near this tend to be the most accurate ones.  
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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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|}
|}


We may invert the continuum by setting ''m'' such that {{nowrap| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''superpyth–diatonic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.3130…. The [[superpyth comma]] is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.  
We may invert the continuum by setting ''m'' such that {{nowrap| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''superpyth–limmic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.3130…. The [[superpyth comma]] is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.  


{| class="wikitable center-1"
{| class="wikitable center-1"
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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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