Diaschismic–gothmic equivalence continuum: Difference between revisions
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3. A characteristic damage of [[34edo]] which is not trivial; gothic is trivial in that it is just a subgroup restriction, and immunity, though not trivial and comparable in complexity to tetracot, is not characteristic, because it asks for a flat tuning of 5, so that it is arguably more or equally as characteristic of [[63edo]]'s or [[97edo]]'s representation of the 5-limit, but the fact that it has so many 3's in it when 34edo does not do so well in consistency of 3's to long distance should be a hint that we can do better. Compare with tetracot, which admits comparatively more and lower damage patent tunings and is clearly a type of imperfect simplification corresponding to a structural awkwardness in [[5-limit]] [[JI]] – the [[20000/19683|minimal diesis]] – so it has characteristic damage on 10/9 (flatwards) and 9/8 (sharpwards) by exaggerating the difference; this is to say, if you look at 34et's tuning of the 5-limit, its damage is strongly characteristic of tetracot. Exaggerating this difference in this way ''is'' also characteristic of immunity, but its general tunings are at odds with those of tetracot's so that they only merge in 34edo, which is arguably a more unusual tuning for immunity than it is for tetracot, where it is clearly characteristic. | 3. A characteristic damage of [[34edo]] which is not trivial; gothic is trivial in that it is just a subgroup restriction, and immunity, though not trivial and comparable in complexity to tetracot, is not characteristic, because it asks for a flat tuning of 5, so that it is arguably more or equally as characteristic of [[63edo]]'s or [[97edo]]'s representation of the 5-limit, but the fact that it has so many 3's in it when 34edo does not do so well in consistency of 3's to long distance should be a hint that we can do better. Compare with tetracot, which admits comparatively more and lower damage patent tunings and is clearly a type of imperfect simplification corresponding to a structural awkwardness in [[5-limit]] [[JI]] – the [[20000/19683|minimal diesis]] – so it has characteristic damage on 10/9 (flatwards) and 9/8 (sharpwards) by exaggerating the difference; this is to say, if you look at 34et's tuning of the 5-limit, its damage is strongly characteristic of tetracot. Exaggerating this difference in this way ''is'' also characteristic of immunity, but its general tunings are at odds with those of tetracot's so that they only merge in 34edo, which is arguably a more unusual tuning for immunity than it is for tetracot, where it is clearly characteristic. | ||
4. As aforementioned, a convenient point to invert the scale to define the ''' | 4. As aforementioned, a convenient point to invert the scale to define the '''kleismic–tetracot continuum''' nicely, discussed below. | ||
== Kleismic–tetracot continuum == | == Kleismic–tetracot continuum == | ||
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: ''For extensions, see [[Keegic temperaments #Aurora]] and [[Breedsmic temperaments #Hemigoldis]].'' | : ''For extensions, see [[Keegic temperaments #Aurora]] and [[Breedsmic temperaments #Hemigoldis]].'' | ||
Goldis tempers out the [[goldis comma]]. As the generator does not admit a useful interpretation in the [[5-limit]], a number of extensions are possible. One possibility is to notice that the generator is close to [[49/32]], resulting in [[hemigoldis]], which splits the generator in half. | Goldis tempers out the [[goldis comma]]. Despite being a quarter-tone in size, due to its complexity, the damage is spread out, so that simple intervals of the [[5-limit]] tend to be tuned reasonably. Possible edo tunings include [[21edo]], [[34edo]], [[55edo]], and [[89edo]]. [[34edo]] is an especially good and tone-efficient tuning (also evidenced by being the largest "golden edo" appearing in the [[optimal ET sequence]]). | ||
As the generator does not admit a useful interpretation in the [[5-limit]], a number of extensions are possible. One possibility is to notice that the generator is close to [[49/32]], resulting in [[hemigoldis]], which splits the generator in half. | |||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
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[[Comma list]]: 549755813888/533935546875 | [[Comma list]]: 549755813888/533935546875 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 -3 5 | 0 12 -7 }} | ||
: mapping generators: ~2, ~84375/65536 | |||
: mapping generators: ~2, ~ | |||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1199.1846{{c}}, ~84375/65536 = 458.3229{{c}} | |||
: [[error map]]: {{val| -0.815 +0.366 +1.349 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~84375/65536 = 458.6193{{c}} | |||
: error map: {{val| 0.000 +1.477 +3.351 }} | |||
{{Optimal ET sequence|legend=1| 13, 21, 34, 123, 157, 191c }} | {{Optimal ET sequence|legend=1| 13, 21, 34, 123, 157, 191c }} | ||