Hemimean clan: Difference between revisions
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[[Gencom]]: [2 56/25; 3136/3125] | [[Gencom]]: [2 56/25; 3136/3125] | ||
[[Optimal tuning]] ([[POTE]]): | [[Optimal tuning]] ([[POTE]]): 2 = 1\1, ~28/25 = 193.772 | ||
{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }} | {{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }} | ||
Badness (Dirichlet): 0.091 | |||
=== Mediantone === | |||
In the no-3's [[11-limit]], there is a natural extension to prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering [[176/175]], which is the same route that [[undecimal meantone]] uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~[[28/25]] at 2 generators (as a flat ~[[9/8]]) while here it is the generator. Mediantone is named after its whole tone generator serving as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], as well as by the observation that this temperament seems to have been repeatedly rediscovered in parts in a variety of contexts, so that it seems to exist as a "median" of all of these temperaments' logics. It is also an intentional play on "[[meantone]]", as the context one is most likely to first discover this logic is when the tone also represents [[10/9]][[~]][[9/8]]. We will cover related logics as they are relevant to the subgroup shown, such as undecimal meantone being the result of tempering 81/80 to find prime 3 so that the complexity of all mappings except 3 doubles. | |||
Subgroup: 2.5.7.11 (no-3's [[11-limit]]) | |||
Comma list: [[3136/3125]], [[176/175]] | |||
{{Mapping|legend=2| 1 0 -3 -7 | 0 2 5 9 }} | |||
: sval mapping generators: ~2, ~56/25 | |||
Optimal tuning ([[CWE]]): 2 = 1\1, ~19/17 = 194.428 | |||
{{Optimal ET sequence|legend=1| 6, 19e, 25, 31, 37 }} | |||
Badness (Dirichlet): 0.195 | |||
==== 2.5.7.11.13 ==== | |||
In the no-3's [[13-limit]], this temperament is identical to [[roulette]], so its higher-limit extensions may be seen as extensions of roulette if one prefers, but roulette is just one instance of the full logic of this temperament reoccurring across many temperaments with "median tones". | |||
Subgroup: 2.5.7.11.13 (no-3's [[13-limit]]) | |||
Comma list: [[3136/3125]], [[176/175]], [[640/637]] | |||
{{Mapping|legend=2| 1 0 -3 -7 13 | 0 2 5 9 -8 }} | |||
: sval mapping generators: ~2, ~56/25 | |||
Optimal tuning ([[CWE]]): 2 = 1\1, ~19/17 = 194.569 | |||
{{Optimal ET sequence|legend=1| 6, 25, 31, 37 }} | |||
Badness (Dirichlet): 0.324 | |||
==== 2.5.7.11.13.17 ==== | |||
Subgroup: 2.5.7.11.13.17.19 (no-3's [[17-limit]]) | |||
Comma list: [[3136/3125]], [[176/175]], [[640/637]], [[221/220]] | |||
{{Mapping|legend=2| 1 0 -3 -7 13 -18 | 0 2 5 9 -8 19 }} | |||
: sval mapping generators: ~2, ~56/25 | |||
Optimal tuning ([[CWE]]): 2 = 1\1, ~19/17 = 194.887 | |||
{{Optimal ET sequence|legend=1| 6h, 31gh, 37, 80, 117d }} | |||
Badness (Dirichlet): 0.612 | |||
==== 2.5.7.11.13.17.19 ==== | |||
In the full no-3's [[19-limit]], this temperament is a structure common to quite a few temperaments. We might mention: [[hemiwur]], which finds prime 3 through the [[würschmidt]] mapping so that [[6/1]] is found at 16 generators, [[grosstone]], a 19-limit extension of [[undecimal meantone]]. | |||
Subgroup: 2.5.7.11.13.17.19 (no-3's [[19-limit]]) | |||
Comma list: [[3136/3125]], [[176/175]], [[640/637]], [[221/220]], [[476/475]] | |||
{{Mapping|legend=2| 1 0 -3 -7 13 -18 -19 | 0 2 5 9 -8 19 20 }} | |||
: sval mapping generators: ~2, ~56/25 | |||
Optimal tuning ([[CWE]]): 2 = 1\1, ~19/17 = 194.927 | |||
{{Optimal ET sequence|legend=1| 6h, 31gh, 37, 80 }} | |||
Badness (Dirichlet): 0.618 | |||
=== Rectified hebrew === | === Rectified hebrew === | ||