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 ==

[[~~Category:Equal divisions of the octave~~]]

78edo is [[consistent]] to the [[7-odd-limit]], but the error of [[harmonic]] [[3/1|3]], inherited from [[39edo]], is quite large for the size of the system.

[[Category:~~keen~~]]

[[~~Category~~:~~theory]~~]

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]]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 ===

=== 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 ==

== Scales ==

* [[Maeve Gutierrez|Gutierrez Moonglade scale]]

== Instruments ==

[[Lumatone mapping for 78edo|Lumatone mappings for 78edo]] are available.

[[Category:Keen]]

== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/jxYRAo6jHaE ''microtonal improvisation in 78edo''] (2025)

@@ Line 1: / Line 1: @@
-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]].
+{{Infobox ET}}
+{{ED intro}}
-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 ==
-[[Category:Equal divisions of the octave]]
+edo is [[consistent]] to the [[7-odd-limit]], but the error of [[harmonic]] [[3/1|3]], inherited from [[39edo]], is quite large for the size of the system.
-[[Category:keen]]
-[[Category:theory]]
+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]]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 ===
+{{Harmonics in equal|78}}
+=== 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 ==
+{{Interval table}}
+== Scales ==
+* [[Maeve Gutierrez|Gutierrez Moonglade scale]]
+== Instruments ==
+[[Lumatone mapping for 78edo|Lumatone mappings for 78edo]] are available.
+[[Category:Keen]]
+== Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/jxYRAo6jHaE ''microtonal improvisation in 78edo''] (2025)