32edo: Difference between revisions
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The patent val also tempers out [[2048/2025]] in the 5-limit, and [[50/49]] with [[64/63]] in the [[7-limit]], which means it [[support]]s [[pajara]], with a very sharp fifth of 712.5 cents which could be experimented with by those with a penchant for fifths even sharper than the fifth of [[27edo]]; this fifth is in fact very close to the [[minimax tuning]] of the pajara extension [[Diaschismic family #Pajaro|pajaro]], using the 32f val. | The patent val also tempers out [[2048/2025]] in the 5-limit, and [[50/49]] with [[64/63]] in the [[7-limit]], which means it [[support]]s [[pajara]], with a very sharp fifth of 712.5 cents which could be experimented with by those with a penchant for fifths even sharper than the fifth of [[27edo]]; this fifth is in fact very close to the [[minimax tuning]] of the pajara extension [[Diaschismic family #Pajaro|pajaro]], using the 32f val. | ||
The patent fifth of 32edo can be used to generate a very unequal [[archy]] (specifically [[oceanfront]]) [[5L 2s|diatonic scale]], with a [[chromatic semitone]] of 5 steps and a [[diatonic semitone]] of only 1. The diatonic [[major third]] (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450{{c}}, approximating [[9/7]] and [[13/10]], and the minor third is 262.5{{c}}, approximating [[7/6]]. In the 11-limit 32edo provides the [[optimal patent val]] for [[kangaroo]] temperament, tempering out [[55/54]], 64/63, and [[245/242]], whereas the 32c val is equally good for [[progress]]. Both temperaments are generated by a flat [[16/11]] such that three of them reduce to [[3/2]] and six of them reduce to [[8/7]]. The 15-note scale could potentially be used as a [[detempering]] of [[15edo]]. | The patent fifth of 32edo can be used to generate a very unequal [[archy]] (specifically [[oceanfront]]) [[5L 2s|diatonic scale]], with a [[chromatic semitone]] of 5 steps and a [[diatonic semitone]] of only 1. The diatonic [[major third]] (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450{{c}}, approximating [[9/7]] and [[13/10]], and the minor third is 262.5{{c}}, approximating [[7/6]] and [[15/13]]. In the 11-limit 32edo provides the [[optimal patent val]] for [[kangaroo]] temperament, tempering out [[55/54]], 64/63, and [[245/242]], whereas the 32c val is equally good for [[progress]]. Both temperaments are generated by a flat [[16/11]] such that three of them reduce to [[3/2]] and six of them reduce to [[8/7]]. The 15-note scale could potentially be used as a [[detempering]] of [[15edo]]. | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
| Line 56: | Line 56: | ||
| upminor 2nd | | upminor 2nd | ||
| ^Eb | | ^Eb | ||
| [[22/21]], [[26/25]], ''[[33/32]]'', ''[[81/80]]'' | | [[22/21]], [[26/25]], ''[[33/32]]'', ''[[36/35]]'', ''[[81/80]]'' | ||
| 24/23, 23/22 | | 24/23, 23/22 | ||
|- | |- | ||
| Line 160: | Line 160: | ||
| dup 4th, updim 5th | | dup 4th, updim 5th | ||
| ^^G, ^Ab | | ^^G, ^Ab | ||
| [[11/8]], ''[[27/20]]'' | | [[11/8]], [[48/35]], ''[[27/20]]'' | ||
| | | | ||
|- | |- | ||
| Line 176: | Line 176: | ||
| downaug 4th, dud 5th | | downaug 4th, dud 5th | ||
| vG#, vvA | | vG#, vvA | ||
| [[16/11]],, ''[[40/27]]'' | | [[16/11]], [[35/24]], ''[[40/27]]'' | ||
| | | | ||
|- | |- | ||
| Line 272: | Line 272: | ||
| dudmajor 7th | | dudmajor 7th | ||
| vvC# | | vvC# | ||
| | | [[15/8]], ''[[24/13]]'', [[28/15]] | ||
| 92/49 | | 92/49 | ||
|- | |- | ||
| Line 280: | Line 280: | ||
| downmajor 7th | | downmajor 7th | ||
| vC# | | vC# | ||
| | | [[21/11]], [[25/13]], ''[[64/33]]'', ''[[35/18]]'', ''[[160/81]]'' | ||
| 23/12, 44/23 | | 23/12, 44/23 | ||
|- | |- | ||
| Line 288: | Line 288: | ||
| major 7th, down 8ve | | major 7th, down 8ve | ||
| C#, vD | | C#, vD | ||
| | | ''[[48/25]]'', [[88/45]], [[96/49]] | ||
| 45/23, 51/26, 100/51 | | 45/23, 51/26, 100/51 | ||
|- | |- | ||
| Line 296: | Line 296: | ||
| 8ve | | 8ve | ||
| D | | D | ||
| | | [[2/1]] | ||
| | | | ||
|} | |} | ||