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| {{Harmonics in equal|108}} | | {{Harmonics in equal|108}} |
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| === Miscellany ===
| | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> |
| Aside from tuning, 108 is the smallest number with a prime factorization of the form <math>p^p \cdot q^q</math>. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.
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| Since 108edo has a step of 11.111 cents, it also allows one to use its MOS scales as circulating temperaments.
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| {| class="wikitable"
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| |+Circulating temperaments in 108edo
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| !Tones
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| !Pattern
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| !L:s
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| |-
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| |5
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| |[[3L 2s]]
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| |22:21
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| |-
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| |6
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| |[[6edo]]
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| |equal
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| |-
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| |7
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| |[[3L 4s]]
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| |16:15
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| |-
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| |8
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| |[[4L 4s]]
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| |14:13
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| |-
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| |9
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| |[[9edo]]
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| |equal
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| |-
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| |10
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| |[[8L 2s]]
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| |11:10
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| |-
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| |11
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| |[[9L 2s]]
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| |10:9
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| |-
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| |12
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| |[[12edo]]
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| |equal
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| |-
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| |13
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| |[[3L 10s]]
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| |9:8
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| |-
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| |14
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| |[[10L 4s]]
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| | rowspan="2" |8:7
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| |-
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| |15
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| |[[3L 12s]]
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| |-
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| |16
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| |12L 4s
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| | rowspan="2" |7:6
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| |-
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| |17
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| |[[6L 11s]]
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| |-
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| |18
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| |[[18edo]]
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| |equal
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| |-
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| |19
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| |[[13L 6s]]
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| | rowspan="3" |6:5
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| |-
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| |20
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| |8L 12s
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| |-
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| |21
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| |3L 18s
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| |-
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| |22
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| |20L 2s
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| | rowspan="5" |5:4
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| |-
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| |23
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| |16L 7s
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| |-
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| |24
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| |12L 12s
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| |-
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| |25
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| |8L 17s
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| |-
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| |26
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| |4L 22s
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| |-
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| |27
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| |[[27edo]]
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| |equal
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| |-
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| |28
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| |24L 4s
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| | rowspan="8" |4:3
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| |-
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| |29
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| |21L 8s
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| |-
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| |30
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| |18L 12s
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| |-
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| |31
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| |15L 16s
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| |-
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| |32
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| |12L 20s
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| |-
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| |33
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| |9L 24s
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| |-
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| |34
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| |6L 28s
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| |-
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| |35
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| |3L 32s
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| |-
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| |36
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| |[[36edo]]
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| |equal
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| |-
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| |37
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| |34L 3s
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| | rowspan="17" |3:2
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| |-
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| |38
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| |32L 6s
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| |-
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| |39
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| |30L 9s
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| |-
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| |40
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| |28L 12s
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| |-
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| |41
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| |26L 15s
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| |-
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| |42
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| |24L 18s
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| |-
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| |43
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| |22L 21s
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| |-
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| |44
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| |20L 24s
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| |-
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| |45
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| |18L 27s
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| |-
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| |46
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| |16L 30s
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| |-
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| |47
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| |14L 33s
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| |-
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| |48
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| |12L 36s
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| |-
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| |49
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| |10L 39s
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| |-
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| |50
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| |8L 42s
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| |-
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| |51
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| |6L 45s
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| |-
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| |52
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| |4L 48s
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| |-
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| |53
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| |2L 51s
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| |-
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| |54
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| |[[54edo]]
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| |equal
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| |-
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| |55
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| |53L 2s
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| | rowspan="32" |2:1
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| |-
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| |56
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| |52L 4s
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| |-
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| |57
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| |51L 6s
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| |-
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| |58
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| |50L 8s
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| |-
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| |59
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| |49L 10s
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| |-
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| |60
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| |48L 12s
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| |-
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| |61
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| |47L 14s
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| |-
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| |62
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| |46L 16s
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| |-
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| |63
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| |45L 18s
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| |-
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| |64
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| |44L 20s
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| |-
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| |65
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| |43L 22s
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| |-
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| |66
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| |42L 24s
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| |-
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| |67
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| |41L 26s
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| |-
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| |68
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| |40L 28s
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| |-
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| |69
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| |39L 30s
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| |-
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| |70
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| |38L 32s
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| |-
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| |71
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| |37L 34s
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| |-
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| |72
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| |36L 36s
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| |-
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| |73
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| |35L 38s
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| |-
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| |74
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| |34L 40s
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| |-
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| |75
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| |33L 42s
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| |-
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| |76
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| |32L 44s
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| |-
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| |77
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| |31L 46s
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| |-
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| |78
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| |30L 48s
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| |-
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| |79
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| |29L 50s
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| |-
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| |80
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| |28L 52s
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| |-
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| |81
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| |27L 54s
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| |-
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| |82
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| |26L 56s
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| |-
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| |83
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| |25L 58s
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| |-
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| |84
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| |24L 60s
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| |-
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| |85
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| |23L 62s
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| |-
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| |86
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| |22L 64s
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| |}
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| [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | |
| [[Category:Valentine]] | | [[Category:Valentine]] |
Revision as of 21:14, 30 May 2023
| Prime factorization
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22 × 33
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| Step size
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11.1111 ¢
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| Fifth
|
63\108 (700 ¢) (→ 7\12)
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| Semitones (A1:m2)
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9:9 (100 ¢ : 100 ¢)
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| Consistency limit
|
7
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| Distinct consistency limit
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7
|
Template:EDO intro
Theory
108edo tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting valentine temperament, and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.
108edo is the smallest 12n-edo which offers a decent alternative fifth to 12edo's fifth, that is 27edo's superpyth fifth.
Prime harmonics
Approximation of prime harmonics in 108edo
| Harmonic
|
2
|
3
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5
|
7
|
11
|
13
|
17
|
19
|
23
|
29
|
31
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| Error
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Absolute (¢)
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+0.00
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-1.96
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+2.58
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-2.16
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+4.24
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+3.92
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-4.96
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+2.49
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+5.06
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+3.76
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-0.59
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| Relative (%)
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+0.0
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-17.6
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+23.2
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-19.4
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+38.1
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+35.3
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-44.6
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+22.4
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+45.5
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+33.8
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-5.3
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Steps
(reduced)
|
108
(0)
|
171
(63)
|
251
(35)
|
303
(87)
|
374
(50)
|
400
(76)
|
441
(9)
|
459
(27)
|
489
(57)
|
525
(93)
|
535
(103)
|