Syntonic–diatonic equivalence continuum: Difference between revisions

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| {{Monzo| -32 29 -6 }}

|-

| …

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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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[[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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[[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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* [[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 94: / Line 94: @@
 | {{Monzo| -32 29 -6 }}
 |-
+| …
 | …
 | …
@@ 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 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 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 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]]