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. | 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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We may invert the continuum by setting ''m'' such that {{nowrap| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the '' | 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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[[Comma list]]: {{monzo| -76 67 -13 }} | [[Comma list]]: {{monzo| -76 67 -13 }} | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 -1 -11 | 0 13 67 }} | ||
: mapping generators: ~2, ~{{monzo| -35 31 -6 }} | |||
: mapping generators: ~2, ~{{monzo| | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.0377{{c}}, ~{{monzo| | * [[WE]]: ~2 = 1200.0377{{c}}, ~{{monzo| -35 31 -6 }} = 238.6084{{c}} | ||
: [[error map]]: {{val| +0.038 -0.083 +0.035 }} | : [[error map]]: {{val| +0.038 -0.083 +0.035 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| | * [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -35 31 -6 }} = 238.6015{{c}} | ||
: error map: {{val| 0.000 -0.135 -0.011 }} | : error map: {{val| 0.000 -0.135 -0.011 }} | ||
{{Optimal ET sequence|legend=1| 5, …, 166, 171, 860, 1031, 1202, 1373, 1544 }} | {{Optimal ET sequence|legend=1| 5, …, 166, 171, 860, 1031, 1202, 1373, 1544, 3259, 4803b, 6347b }} | ||
[[Badness]] (Sintel): 20.9 | [[Badness]] (Sintel): 20.9 | ||
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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 | ||
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{{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]] ( | [[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 | {{Mapping|legend=1| 1 -5 -30 | 0 11 54 }} | ||
: mapping generators: ~2, ~243/160 | |||
: mapping generators: ~2, ~ | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[POTE]]: ~2 = 1200.000{{c}}, ~ | * [[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]] ( | [[Badness]] (Sintel): 84.9 | ||
== 559 & 2513 == | == 559 & 2513 == | ||
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[[Comma list]]: {{monzo| -124 109 -21 }} | [[Comma list]]: {{monzo| -124 109 -21 }} | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 -11 -63 | 0 21 109 }} | ||
: mapping generators: ~2, ~{{monzo| -29 26 -5 }} | |||
: mapping generators: ~2, ~ | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[POTE]]: ~2 = 1200.0000{{c}}, ~ | * [[POTE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -29 26 -5 }} = 719.1405{{c}} | ||
{{ | |||
{{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]] | ||