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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== 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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[[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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== Counterpental ==

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

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

[[Subgroup]]: 2.3.5

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[[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 311: / 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 332: / 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 351: / 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 373: / Line 383: @@
 == Counterpental ==
-: ''For extensions, see [[Orwellismic temperaments #Pentorwell]].''
+: ''For extensions, see [[Orwellismic temperaments #Pentaorwell]].''
 [[Subgroup]]: 2.3.5
@@ Line 413: / Line 423: @@
 [[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 421: / Line 451: @@
 {{Mapping|legend=1| 1 -2 -16 | 0 9 46 }}
 : mapping generators: ~2, ~320/243
@@ Line 429: / 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 436: / 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 454: / 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]]