User:Romeolz/Isomorphic layouts/Harmonic Table extensions: Difference between revisions
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I will refer to "pure HT" when talking exclusively about layouts that map 5/4, 6/5 and 3/2 to the same location as the 12edo HT (including reflections and rotations). | I will refer to "pure HT" when talking exclusively about layouts that map 5/4, 6/5 and 3/2 to the same location as the 12edo HT (including reflections and rotations). | ||
=== Legend === | |||
* dim oct = octave derived from diminished temperament, 2/1 ~ (6/5)^4, NOT A LITERAL DIMINISHED OCTAVE (ex. C4-Cb5) | |||
* aug oct = octave derived from augmented temperament, 2/1 ~ (5/4)^3, NOT A LITERAL AUGMENTED OCTAVE (ex. C4-C#5) | |||
* when one of these is crossed out, it means that octave mapping is no longer there and maps to another interval, and the arrow signifies where it would have been | |||
* magic twelfth = twelfth derived from magic temperament (and so on...) | |||
[[File:12edo harmonic table augmented diminished octave and unison.png|none|thumb|900x900px|12edo HT for reference]] | [[File:12edo harmonic table augmented diminished octave and unison.png|none|thumb|900x900px|12edo HT for reference]] | ||
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"Why such few temperaments? Where is meantone? Why do some temperaments not work?" | "Why such few temperaments? Where is meantone? Why do some temperaments not work?" | ||
A prerequisite for a HT-ish layout is that a single octave is reachable using some combination of the two chosen harmonically close intervals. Using meantone's fifths and thirds we can only reach the double-octave. (this probably has something to do with the monzos of the intervals and commas but I don't know how yet) | A prerequisite for a HT-ish layout is that a single octave is reachable using some combination of the two chosen harmonically close intervals (usually 3/2 and 5/4). Using meantone's fifths and thirds we can only reach the double-octave. (this probably has something to do with the monzos of the intervals and commas but I don't know how yet) | ||
It is always possible to reach the octave in any ET where the interval sizes are coprime, so the ET doesn't have to support any of these temperaments. Using a HT-like layout like this is highly impractical. | |||
== Alternate thirds == | == Alternate thirds == | ||