78edo: Difference between revisions

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This tuning [[tempering out|tempers out]] [[2048/2025]] in the [[5-limit]]; [[875/864]] and [[2401/2400]] in the [[7-limit]]; and [[100/99]], [[385/384]] and [[1375/1372]] in the [[11-limit]]. It provides the [[optimal patent val]] for 11-limit [[keen]] temperament.

Much like [[100edo|100bddd]], the 78dd val can be used to construct an alternative to [[22edo]] for [[pajara]]. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7{{cent}}. The major third is 384.6~~{{cent}}~~; less than two cents flat of just. The harmonic seventh is 984.6~~{{cent}}~~, or about 15.8~~{{cent}}~~ sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16~~{{cent}~~} off. The 22-note ~~2MOS~~ generated in this way could be used to build straight-fretted guitars that would be {{w|Augmented-fourths tuning|tuned in tritones}}. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6~~{{cent}}~~ "flat minor whole tone" of 22edo is now 169.2~~{{cent}}~~, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.

Much like [[100edo|100bddd]], the 78dd val can be used to construct an alternative to [[22edo]] for [[pajara]]. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7 [[cent]]s. The major third is 384.6 cents; less than two cents flat of just. The harmonic seventh is 984.6 cents, or about 15.8 cents sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16 cents} off. The 22-note [[2mos]] generated in this way could be used to build straight-fretted guitars that would be {{w|Augmented-fourths tuning|tuned in tritones}}. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6 cents "flat minor whole tone" of 22edo is now 169.2 cents, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.

=== Odd harmonics ===

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=== Subsets and supersets ===

Since 78 factors into {{factorization|78}}, 78edo has subset edos {{EDOs| 2, 3, 6, 13, 26, and 39 }}. [[156edo]], which doubles it, is a notable tuning.

== Intervals ==

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 This tuning [[tempering out|tempers out]] [[2048/2025]] in the [[5-limit]]; [[875/864]] and [[2401/2400]] in the [[7-limit]]; and [[100/99]], [[385/384]] and [[1375/1372]] in the [[11-limit]]. It provides the [[optimal patent val]] for 11-limit [[keen]] temperament.
-Much like [[100edo|100bddd]], the 78dd val can be used to construct an alternative to [[22edo]] for [[pajara]]. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7{{cent}}. The major third is 384.6{{cent}}; less than two cents flat of just. The harmonic seventh is 984.6{{cent}}, or about 15.8{{cent}} sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16{{cent}} off. The 22-note 2MOS generated in this way could be used to build straight-fretted guitars that would be {{w|Augmented-fourths tuning|tuned in tritones}}. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6{{cent}} "flat minor whole tone" of 22edo is now 169.2{{cent}}, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.
+Much like [[100edo|100bddd]], the 78dd val can be used to construct an alternative to [[22edo]] for [[pajara]]. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7 [[cent]]s. The major third is 384.6 cents; less than two cents flat of just. The harmonic seventh is 984.6 cents, or about 15.8 cents sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16 cents} off. The 22-note [[2mos]] generated in this way could be used to build straight-fretted guitars that would be {{w|Augmented-fourths tuning|tuned in tritones}}. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6 cents "flat minor whole tone" of 22edo is now 169.2 cents, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.
 === Odd harmonics ===
@@ Line 13: / Line 13: @@
 === Subsets and supersets ===
 Since 78 factors into {{factorization|78}}, 78edo has subset edos {{EDOs| 2, 3, 6, 13, 26, and 39 }}. [[156edo]], which doubles it, is a notable tuning.
 == Intervals ==