Syntonic–limmic equivalence continuum: Difference between revisions

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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.

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| {{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"

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|-

| 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 }}

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.8213{{c}}

: error map: {{val| 0.000 +7.866 +2.078 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 709.393{{c}}~~

~~: [[error map]]: {{val| 0.000 +7.438 -1.774 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 710.078{{c}}

~~: error map: {{val| 0.000 +8.123 +4.385 }} -->~~

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: ''For extensions, see [[Trienstonic clan #Uncle]].''

The 5-limit version of uncle tempers out [[4096/3645]]. It is generated by a fifth that is supposedly sharper than [[5edo|3\5]], so it leads to an [[5L 3s|oneirotonic]] scale, or otherwise a [[5L 2s|diatonic]] scale with negative small steps. The interval class of 5 is found at -6 fifths, as a major 2-step in oneirotonic, or a diminished fifth (C–Gb) in diatonic. It corresponds to {{nowrap| ''n'' {{=}} -1/2 }} or {{nowrap| ''m'' {{=}} 1/3 }}.

The 5-limit version of uncle tempers out [[4096/3645]]. It is generated by a fifth that is supposedly sharper than [[5edo|3\5]], so it leads to an [[5L 3s|oneirotonic]] scale, or otherwise a [[5L 2s|diatonic]] scale with negative small steps. The interval class of 5 is found at −6 fifths, as a major 2-step in oneirotonic, or a diminished fifth (C–Gb) in diatonic. It corresponds to {{nowrap| ''n'' {{=}} −1/2 }} or {{nowrap| ''m'' {{=}} 1/3 }}.

[[Subgroup]]: 2.3.5

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[[Optimal tuning]]s:

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 733.721{{c}}~~

~~: [[error map]]: {{val| 0.000 +31.766 +11.362 }} -->~~

* [[WE]]: ~2 = 1189.7544{{c}}, ~3/2 = 724.6670{{c}}

: [[error map]]: {{val| -10.246 +12.466 +4.210 }}

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5968{{c}}

: error map: {{val| 0.000 +11.642 +4.041 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 713.185{{c}}~~

~~: [[error map]]: {{val| 0.000 +11.230 -1.722 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 713.829{{c}}

~~: error map: {{val| 0.000 +11.874 +7.289 }} -->~~

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Dirt tempers out the [[dirt comma]], 1342177280/1162261467. It is generated by a perfect fifth. The interval class of 5 is found at +19 fifths, as a double-augmented seventh (C–Bx). It corresponds to {{nowrap| ''n'' {{=}} 1/3 }} and {{nowrap| ''m'' {{=}} -1/2 }}.

Dirt tempers out the [[dirt comma]], 1342177280/1162261467. It is generated by a perfect fifth. The interval class of 5 is found at +19 fifths, as a double-augmented seventh (C–Bx). It corresponds to {{nowrap| ''n'' {{=}} 1/3 }} and {{nowrap| ''m'' {{=}} −1/2 }}.

[[Subgroup]]: 2.3.5

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[[Optimal tuning]]s:

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 714.992{{c}}~~

~~: [[error map]]: {{val| 0.000 +13.037 -1.473 }} -->~~

* [[WE]]: ~2 = 1195.8566{{c}}, ~3/2 = 713.0611{{c}}

: [[error map]]: {{val| -4.143 +6.963 -0.863 }}

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: ''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

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* [[CWE]]: ~2 = 1200.000{{c}}, ~729/640 = 234.4999{{c}}

: error map: {{val| 0.000 +1.545 +0.185 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~729/640 = 234.457{{c}}~~

~~: [[error map]]: {{val| 0.000 +1.417 -0.537 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~729/640 = 234.528{{c}}

~~: error map: {{val| 0.000 +1.629 +0.663 }} -->~~

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: ''For extensions, see [[Gamelismic clan #Gorgo]].''

Laconic tempers out [[2187/2000]], which is the difference between a stack of three [[10/9|ptolemaic whole tones (10/9)]]'s and a perfect fifth (3/2). Although a higher-error temperament, it does pop up enough in the low-numbered edos to be useful, most notably in [[16edo]] and [[21edo]]. 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 }}.

Laconic tempers out [[2187/2000]], which is the difference between a stack of three [[10/9|ptolemaic whole tones (10/9)]]'s and a perfect fifth (3/2). Although a higher-error temperament, it does pop up enough in the low-numbered edos to be useful, most notably in [[16edo]] and [[21edo]]. 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 }}.

[[Subgroup]]: 2.3.5

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* [[CWE]]: ~2 = 1200.000{{c}}, ~10/9 = 228.0128{{c}}

: error map: {{val| 0.000 -17.917 +9.776 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 228.700{{c}}~~

~~: [[error map]]: {{val| 0.000 -15.856 +14.584 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 227.426{{c}}

~~: error map: {{val| 0.000 -19.679 +5.664 }} -->~~

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== University ==

~~{{Main| University }}~~

: ''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]].

University widens the classical major and minor chords to [[Extraclassical tonality|tendo and arto chords.]]

[[Subgroup]]: 2.3.5

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* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 231.4822{{c}}

: error map: {{val| 0.000 -7.509 +76.651 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 226.980{{c}}~~

~~: [[error map]]: {{val| 0.000 -21.014 +67.647 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 235.442{{c}}

~~: error map: {{val| 0.000 +4.370 +84.569 }} -->~~

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[[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) ==

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[http://x31eq.com/cgi-bin/rt.cgi?ets=5_56&limit=5 The temperament finder - 5-limit 5 & 56]

-->

== Hemiseven (5-limit) ==

: ''For extensions, see [[Gamelismic clan #Hemiseven]].''

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/160 = 716.7671{{c}}

: error map: {{val| 0.000 -1.352 -0.067 }}

~~~~

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== Counterpental ==

: ''For extensions, see [[Orwellismic temperaments #~~Pentorwell~~]].''

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

[[Subgroup]]: 2.3.5

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* [[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 }}

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* [[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, …, 94, 99, 193, 292, 391, 1074b, 1465bb }}

[[Badness]] (Sintel): 22.8

== Tokko (5-limit) ==

: ''For extensions, see [[Wizmic microtemperaments #Tokko]].''

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| -76 67 -13 }}

: 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

<!--

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: mapping generators: ~2, ~320/243

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{{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) ==

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[[Comma list]]: {{Monzo| -60 54 -11 }}

: mapping generators: ~2, ~243/160

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

[[Optimal tuning]]s:

* [[POTE]]: ~2 = 1200.000{{c}}, ~~~320~~/~~243~~ = ~~481~~.~~742~~{{c}}

* [[POTE]]: ~2 = 1200.000{{c}}, ~243/160 = 718.258{{c}}

{{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 ==

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[[Comma list]]: {{monzo| -124 109 -21 }}

: mapping generators: ~2, ~{{monzo| -29 26 -5 }}

: mapping generators: ~2, ~~~3355443200000/2541865828329~~

[[Optimal tuning]]s:

* [[POTE]]: ~2 = 1200.0000{{c}}, ~~~3355443200000/2541865828329~~ = ~~480~~.~~8595~~{{c}}

* [[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 }}

{{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]] (~~Smith~~): 0.~~134523~~

[[Badness]] (Sintel): 3.16

~~[http://x31eq.com/cgi-bin/rt.cgi?ets=2513_559&limit=5 The temperament finder - 5-limit 2513 & 559]~~

-->

[[Category:5edo]]

[[Category:Equivalence continua]]

@@ Line 1: / Line 1: @@
 {{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.
@@ Line 94: / Line 94: @@
 | {{Monzo| -32 29 -6 }}
 |-
+| …
 | …
 | …
@@ Line 106: / Line 107: @@
 |}
-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"
@@ Line 170: / Line 171: @@
 |-
 | 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 }}
@@ Line 194: / Line 197: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.8213{{c}}
 : error map: {{val| 0.000 +7.866 +2.078 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 709.393{{c}}
-: [[error map]]: {{val| 0.000 +7.438 -1.774 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 710.078{{c}}
-: error map: {{val| 0.000 +8.123 +4.385 }} -->
 {{Optimal ET sequence|legend=1| 5, 17, 22, 49, 120b, 169bbc }}
@@ Line 206: / Line 205: @@
 : ''For extensions, see [[Trienstonic clan #Uncle]].''
-The 5-limit version of uncle tempers out [[4096/3645]]. It is generated by a fifth that is supposedly sharper than [[5edo|3\5]], so it leads to an [[5L 3s|oneirotonic]] scale, or otherwise a [[5L 2s|diatonic]] scale with negative small steps. The interval class of 5 is found at -6 fifths, as a major 2-step in oneirotonic, or a diminished fifth (C–Gb) in diatonic. It corresponds to {{nowrap| ''n'' {{=}} -1/2 }} or {{nowrap| ''m'' {{=}} 1/3 }}.
+The 5-limit version of uncle tempers out [[4096/3645]]. It is generated by a fifth that is supposedly sharper than [[5edo|3\5]], so it leads to an [[5L 3s|oneirotonic]] scale, or otherwise a [[5L 2s|diatonic]] scale with negative small steps. The interval class of 5 is found at −6 fifths, as a major 2-step in oneirotonic, or a diminished fifth (C–Gb) in diatonic. It corresponds to {{nowrap| ''n'' {{=}} −1/2 }} or {{nowrap| ''m'' {{=}} 1/3 }}.
 [[Subgroup]]: 2.3.5
@@ Line 217: / Line 216: @@
 [[Optimal tuning]]s:
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 733.721{{c}}
-: [[error map]]: {{val| 0.000 +31.766 +11.362 }} -->
 * [[WE]]: ~2 = 1189.7544{{c}}, ~3/2 = 724.6670{{c}}
 : [[error map]]: {{val| -10.246 +12.466 +4.210 }}
@@ Line 246: / Line 243: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5968{{c}}
 : error map: {{val| 0.000 +11.642 +4.041 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 713.185{{c}}
-: [[error map]]: {{val| 0.000 +11.230 -1.722 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 713.829{{c}}
-: error map: {{val| 0.000 +11.874 +7.289 }} -->
 {{Optimal ET sequence|legend=1| 5, 27c, 32, 37, 79bc, 116bbc }}
@@ Line 258: / Line 251: @@
 {{Main| Dirt }}
-Dirt tempers out the [[dirt comma]], 1342177280/1162261467. It is generated by a perfect fifth. The interval class of 5 is found at +19 fifths, as a double-augmented seventh (C–Bx). It corresponds to {{nowrap| ''n'' {{=}} 1/3 }} and {{nowrap| ''m'' {{=}} -1/2 }}.
+Dirt tempers out the [[dirt comma]], 1342177280/1162261467. It is generated by a perfect fifth. The interval class of 5 is found at +19 fifths, as a double-augmented seventh (C–Bx). It corresponds to {{nowrap| ''n'' {{=}} 1/3 }} and {{nowrap| ''m'' {{=}} −1/2 }}.
 [[Subgroup]]: 2.3.5
@@ Line 269: / Line 262: @@
 [[Optimal tuning]]s:
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 714.992{{c}}
-: [[error map]]: {{val| 0.000 +13.037 -1.473 }} -->
 * [[WE]]: ~2 = 1195.8566{{c}}, ~3/2 = 713.0611{{c}}
 : [[error map]]: {{val| -4.143 +6.963 -0.863 }}
@@ Line 283: / Line 274: @@
 : ''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
@@ Line 296: / Line 287: @@
 * [[CWE]]: ~2 = 1200.000{{c}}, ~729/640 = 234.4999{{c}}
 : error map: {{val| 0.000 +1.545 +0.185 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~729/640 = 234.457{{c}}
-: [[error map]]: {{val| 0.000 +1.417 -0.537 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~729/640 = 234.528{{c}}
-: error map: {{val| 0.000 +1.629 +0.663 }} -->
 {{Optimal ET sequence|legend=1| 5, …, 41, 46, 87, 220, 307 }}
@@ Line 308: / Line 295: @@
 : ''For extensions, see [[Gamelismic clan #Gorgo]].''
-Laconic tempers out [[2187/2000]], which is the difference between a stack of three [[10/9|ptolemaic whole tones (10/9)]]'s and a perfect fifth (3/2). Although a higher-error temperament, it does pop up enough in the low-numbered edos to be useful, most notably in [[16edo]] and [[21edo]]. 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 }}.
+Laconic tempers out [[2187/2000]], which is the difference between a stack of three [[10/9|ptolemaic whole tones (10/9)]]'s and a perfect fifth (3/2). Although a higher-error temperament, it does pop up enough in the low-numbered edos to be useful, most notably in [[16edo]] and [[21edo]]. 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 }}.
 [[Subgroup]]: 2.3.5
@@ Line 321: / Line 308: @@
 * [[CWE]]: ~2 = 1200.000{{c}}, ~10/9 = 228.0128{{c}}
 : error map: {{val| 0.000 -17.917 +9.776 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 228.700{{c}}
-: [[error map]]: {{val| 0.000 -15.856 +14.584 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 227.426{{c}}
-: error map: {{val| 0.000 -19.679 +5.664 }} -->
 {{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37b }}
@@ Line 331: / Line 314: @@
 == University ==
-{{Main| University }}
 : ''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]].
+University widens the classical major and minor chords to [[Extraclassical tonality|tendo and arto chords.]]
 [[Subgroup]]: 2.3.5
@@ Line 347: / Line 331: @@
 * [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 231.4822{{c}}
 : error map: {{val| 0.000 -7.509 +76.651 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 226.980{{c}}
-: [[error map]]: {{val| 0.000 -21.014 +67.647 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 235.442{{c}}
-: error map: {{val| 0.000 +4.370 +84.569 }} -->
 {{Optimal ET sequence|legend=1| 1b, …, 4bc, 5 }}
@@ Line 356: / Line 336: @@
 [[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) ==
@@ Line 375: / Line 360: @@
 [http://x31eq.com/cgi-bin/rt.cgi?ets=5_56&limit=5 The temperament finder - 5-limit 5 & 56]
 -->
 == Hemiseven (5-limit) ==
 : ''For extensions, see [[Gamelismic clan #Hemiseven]].''
@@ Line 391: / Line 377: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~243/160 = 716.7671{{c}}
 : error map: {{val| 0.000 -1.352 -0.067 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~243/160 = 716.753{{c}} -->
 {{Optimal ET sequence|legend=1| 5, …, 72, 149, 221, 370, 591b }}
@@ Line 398: / Line 383: @@
 == Counterpental ==
-: ''For extensions, see [[Orwellismic temperaments #Pentorwell]].''
+: ''For extensions, see [[Orwellismic temperaments #Pentaorwell]].''
 [[Subgroup]]: 2.3.5
@@ Line 413: / Line 398: @@
 * [[CWE]]: ~729/640 = 240.0000{{c}}, ~3/2 = 704.4446{{c}}
 : error map: {{val| 0.000 +2.490 +0.354 }}
-<!-- * [[POTE]]: ~729/640 = 240.000{{c}}, ~3/2 = 704.572{{c}} -->
 {{Optimal ET sequence|legend=1| 5, …, 75, 80, 155, 390b, 545bbc }}
@@ Line 435: / Line 419: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~177147/102400 = 957.5367{{c}}
 : error map: {{val| 0.000 +0.802 +0.082 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~177147/102400 = 957.543{{c}} -->
 {{Optimal ET sequence|legend=1| 5, …, 94, 99, 193, 292, 391, 1074b, 1465bb }}
 [[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
 <!--
@@ Line 448: / Line 451: @@
 {{Mapping|legend=1| 1 -2 -16 | 0 9 46 }}
 : mapping generators: ~2, ~320/243
@@ Line 456: / Line 458: @@
 {{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) ==
@@ Line 463: / Line 465: @@
 [[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:
-* [[POTE]]: ~2 = 1200.000{{c}}, ~320/243 = 481.742{{c}}
+* [[POTE]]: ~2 = 1200.000{{c}}, ~243/160 = 718.258{{c}}
 {{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 ==
@@ Line 481: / Line 480: @@
 [[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:
-* [[POTE]]: ~2 = 1200.0000{{c}}, ~3355443200000/2541865828329 = 480.8595{{c}}
+* [[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 }}
+{{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]] (Smith): 0.134523
+[[Badness]] (Sintel): 3.16
-[http://x31eq.com/cgi-bin/rt.cgi?ets=2513_559&limit=5 The temperament finder - 5-limit 2513 & 559]
 -->
 [[Category:5edo]]
 [[Category:Equivalence continua]]