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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: ''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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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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: ''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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[[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~~ }}

~~[[Badness]] (Smith): 0.134523~~

{{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 }}

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

[[Badness]] (Sintel): 3.16

-->

[[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 202: / 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 248: / 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 292: / 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 420: / 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 428: / Line 451: @@
 {{Mapping|legend=1| 1 -2 -16 | 0 9 46 }}
 : mapping generators: ~2, ~320/243
@@ Line 436: / 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 443: / 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 461: / 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 }}
-[[Badness]] (Smith): 0.134523
+{{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 }}
-[http://x31eq.com/cgi-bin/rt.cgi?ets=2513_559&limit=5 The temperament finder - 5-limit 2513 & 559]
+[[Badness]] (Sintel): 3.16
 -->
 [[Category:5edo]]
 [[Category:Equivalence continua]]