78edo: Difference between revisions

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~~The 78 equal division divides the octave into~~ 78 equal parts of size 15.385 [[cent|cent]]s each. It tempers out 2048/2025 in the [[5-limit|5-limit]]; 875/864 and 2401/2400 in the [[7-limit|7-limit]]; and 100/99, 385/384 and 1375/1372 in the [[11-limit|11-limit]]. It provides the [[Optimal_patent_val|optimal patent val]] for 11-limit [[Diaschismic_family|keen temperament]].

Much like [[100edo|100bddd]], the 78ddd 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 ~~cents~~. 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 [https://en.wikipedia.org/wiki/Augmented-fourths_tuning tuned in tritones]. The appeal of this scale is that it is less xenharmonic than [[~~22edo|~~22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6¢ "flat minor whole tone" of 22edo is now 169.2¢, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.

== Theory ==

~~{{harmonics in equal|78}}~~

This tuning 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 [[Diaschismic_family|keen temperament]].

Much like [[100edo|100bddd]], the 78ddd 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 [https://en.wikipedia.org/wiki/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.

[[Category:Equal divisions of the octave|##]]

[[Category:~~keen~~]]

[[Category:Keen]]

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-The 78 equal division divides the octave into 78 equal parts of size 15.385 [[cent|cent]]s each. It tempers out 2048/2025 in the [[5-limit|5-limit]]; 875/864 and 2401/2400 in the [[7-limit|7-limit]]; and 100/99, 385/384 and 1375/1372 in the [[11-limit|11-limit]]. It provides the [[Optimal_patent_val|optimal patent val]] for 11-limit [[Diaschismic_family|keen temperament]].
+{{EDO intro|78}}
-Much like [[100edo|100bddd]], the 78ddd 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 cents. 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 [https://en.wikipedia.org/wiki/Augmented-fourths_tuning tuned in tritones]. The appeal of this scale is that it is less xenharmonic than [[22edo|22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6¢ "flat minor whole tone" of 22edo is now 169.2¢, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.
+== Theory ==
-{{harmonics in equal|78}}
+{{Harmonics in equal|78}}
+This tuning 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 [[Diaschismic_family|keen temperament]].
+Much like [[100edo|100bddd]], the 78ddd 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 [https://en.wikipedia.org/wiki/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.
 [[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
-[[Category:keen]]
+[[Category:Keen]]