User:Romeolz/Isomorphic layouts/Harmonic Table extensions: Difference between revisions
fourths, blackwood continuum and table math |
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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 == | ||
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==== Magi ==== | ==== Magi ==== | ||
[[File:"HT" fourths 440cent.png|none|thumb|600x600px|fgdfjhfgds]] | [[File:"HT" fourths 440cent.png|none|thumb|600x600px|fgdfjhfgds]] | ||
Coming up with an equivalence continuum is tough for this one because the generator varies so much. If one were to choose a single JI interval all these should be approximating, the simple temperaments would be unrecognizable. | Coming up with an equivalence continuum is tough for this one because the generator varies so much. If one were to choose a single JI interval all these should be approximating, the simple temperaments would be unrecognizable. | ||
[[File:ProjectiveTuningSpace fourths augmented A.png|thumb|Image A]] | |||
[[File:ProjectiveTuningSpace fourths augmented B.png|thumb|Image B]] | |||
{| class="wikitable" | {| class="wikitable" | ||
|+Continuation of fourths-axial augmented "HT" into various equivalence continua | |+Continuation of fourths-axial augmented "HT" into various equivalence continua | ||
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!5/3 | !5/3 | ||
2 | 2 | ||
|- | |||
!image | |||
! | |||
!A | |||
!B | |||
!todo | |||
|- | |- | ||
! | ! | ||
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==== Orwell/Orson (fourths ver.) ==== | ==== Orwell/Orson (fourths ver.) ==== | ||
[[File:"HT" fourths orwell.png|none|thumb|600x600px|hfhdgfsh]] | [[File:"HT" fourths orwell.png|none|thumb|600x600px|hfhdgfsh]] | ||
Same thing as with the above table. | [[File:ProjectiveTuningSpace fourths diminished A.png|thumb|Image A]] | ||
[[File:ProjectiveTuningSpace fourths diminished B.png|thumb|Image B]] | |||
Same thing as with the above table. | |||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+Continuation of fourths-axial diminished "HT" into various equivalence continua | ||
! | ! | ||
!m | !m | ||
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!5/4 | !5/4 | ||
1 | 1 | ||
|- | |||
!image | |||
! | |||
!A | |||
!B | |||
|- | |- | ||
! | ! | ||
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=== Alternate sixths/seconds === | === Alternate sixths/seconds === | ||
todo | |||
== Hemifourths/fifths ( | == Hemifourths/fifths (might be outside the scope) == | ||