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| {{Infobox ET}} | | {{Infobox ET}} |
| {{EDO intro|78}} | | {{ED intro}} |
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| == Theory == | | == Theory == |
| | 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. |
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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. |
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| | 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. |
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| | === Odd harmonics === |
| {{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]].
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| 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.
| | === 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. |
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| == Intervals == | | == Intervals == |
| {|class="wikitable" | | {{Interval table}} |
| |-
| | |
| !#
| | == Scales == |
| !Cents
| | * [[Maeve Gutierrez|Gutierrez Moonglade scale]] |
| !Diatonic interval category
| |
| |-
| |
| |0
| |
| |0.0
| |
| |perfect unison
| |
| |-
| |
| |1
| |
| |15.4
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| |superunison
| |
| |-
| |
| |2
| |
| |30.8
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| |superunison
| |
| |-
| |
| |3
| |
| |46.2
| |
| |subminor second
| |
| |-
| |
| |4
| |
| |61.5
| |
| |subminor second
| |
| |-
| |
| |5
| |
| |76.9
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| |subminor second
| |
| |-
| |
| |6
| |
| |92.3
| |
| |minor second
| |
| |-
| |
| |7
| |
| |107.7
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| |minor second
| |
| |-
| |
| |8
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| |123.1
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| |supraminor second
| |
| |-
| |
| |9
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| |138.5
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| |supraminor second
| |
| |-
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| |10
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| |153.8
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| |neutral second
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| |-
| |
| |11
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| |169.2
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| |submajor second
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| |-
| |
| |12
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| |184.6
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| |major second
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| |-
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| |13
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| |200.0
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| |major second
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| |-
| |
| |14
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| |215.4
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| |major second
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| |-
| |
| |15
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| |230.8
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| |supermajor second
| |
| |-
| |
| |16
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| |246.2
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| |ultramajor second
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| |-
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| |17
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| |261.5
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| |subminor third
| |
| |-
| |
| |18
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| |276.9
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| |subminor third
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| |-
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| |19
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| |292.3
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| |minor third
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| |-
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| |20
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| |307.7
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| |minor third
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| |-
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| |21
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| |323.1
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| |supraminor third
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| |-
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| |22
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| |338.5
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| |supraminor third
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| |-
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| |23
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| |353.8
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| |neutral third
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| |-
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| |24
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| |369.2
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| |submajor third
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| |-
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| |25
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| |384.6
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| |major third
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| |-
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| |26
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| |400.0
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| |major third
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| |-
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| |27
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| |415.4
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| |major third
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| |-
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| |28
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| |430.8
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| |supermajor third
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| |-
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| |29
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| |446.2
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| |ultramajor third
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| |-
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| |30
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| |461.5
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| |subfourth
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| |-
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| |31
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| |476.9
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| |subfourth
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| |-
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| |32
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| |492.3
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| |perfect fourth
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| |-
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| |33
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| |507.7
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| |perfect fourth
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| |-
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| |34
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| |523.1
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| |superfourth
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| |-
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| |35
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| |538.5
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| |superfourth
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| |-
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| |36
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| |553.8
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| |superfourth
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| |-
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| |37
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| |569.2
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| |low tritone
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| |-
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| |38
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| |584.6
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| |low tritone
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| |-
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| |39
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| |600.0
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| |high tritone
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| |-
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| |40
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| |615.4
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| |high tritone
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| |-
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| |41
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| |630.8
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| |high tritone
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| |-
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| |42
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| |646.2
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| |subfifth
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| |-
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| |43
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| |661.5
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| |subfifth
| |
| |-
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| |44
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| |676.9
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| |subfifth
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| |-
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| |45
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| |692.3
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| |perfect fifth
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| |-
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| |46
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| |707.7
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| |perfect fifth
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| |-
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| |47
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| |723.1
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| |superfifth
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| |-
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| |48
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| |738.5
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| |superfifth
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| |-
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| |49
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| |753.8
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| |ultrafifth
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| |-
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| |50
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| |769.2
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| |subminor sixth
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| |-
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| |51
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| |784.6
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| |minor sixth
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| |-
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| |52
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| |800.0
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| |minor sixth
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| |-
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| |53
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| |815.4
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| |minor sixth
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| |-
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| |54
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| |830.8
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| |supraminor sixth
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| |-
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| |55
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| |846.2
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| |neutral sixth
| |
| |-
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| |56
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| |861.5
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| |submajor sixth
| |
| |-
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| |57
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| |876.9
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| |submajor sixth
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| |-
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| |58
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| |892.3
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| |major sixth
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| |-
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| |59
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| |907.7 | |
| |major sixth
| |
| |-
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| |60
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| |923.1
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| |supermajor sixth
| |
| |-
| |
| |61
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| |938.5
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| |supermajor sixth
| |
| |-
| |
| |62
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| |953.8
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| |ultramajor sixth
| |
| |-
| |
| |63
| |
| |969.2
| |
| |subminor seventh
| |
| |-
| |
| |64
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| |984.6
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| |minor seventh
| |
| |-
| |
| |65
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| |1000.0
| |
| |minor seventh
| |
| |-
| |
| |66
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| |1015.4
| |
| |minor seventh
| |
| |-
| |
| |67
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| |1030.8
| |
| |supraminor seventh
| |
| |-
| |
| |68
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| |1046.2
| |
| |neutral seventh
| |
| |-
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| |69
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| |1061.5
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| |submajor seventh
| |
| |-
| |
| |70
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| |1076.9
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| |submajor seventh
| |
| |-
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| |71
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| |1092.3
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| |major seventh
| |
| |-
| |
| |72
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| |1107.7
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| |major seventh
| |
| |-
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| |73
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| |1123.1
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| |supermajor seventh
| |
| |-
| |
| |74
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| |1138.5
| |
| |supermajor seventh
| |
| |-
| |
| |75
| |
| |1153.8
| |
| |ultramajor seventh
| |
| |-
| |
| |76
| |
| |1169.2
| |
| |suboctave
| |
| |-
| |
| |77
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| |1184.6
| |
| |suboctave
| |
| |-
| |
| |78
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| |1200.0
| |
| |perfect octave
| |
| |}
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| [[Category:Equal divisions of the octave|##]] <!-- 2-digit number --> | | == Instruments == |
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| | [[Lumatone mapping for 78edo|Lumatone mappings for 78edo]] are available. |
| [[Category:Keen]] | | [[Category:Keen]] |
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| | == Music == |
| | ; [[Bryan Deister]] |
| | * [https://www.youtube.com/shorts/jxYRAo6jHaE ''microtonal improvisation in 78edo''] (2025) |