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| '''109edo''' is the [[equal division of the octave]] into 109 parts of 11.009 [[cent]]s each. It [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.
| | {{Infobox ET}} |
| | {{ED intro}} |
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| 109edo is the 29th [[prime EDO]]. | | == Theory == |
| | 109edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot. |
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| | 109edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log<sub>2</sub>7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2254, 2300/2299, 6860/6859, 10241/10240. |
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| Since 109edo has a step of 11.009 cents, it also allows one to use its MOS scales as circulating temperaments.
| | === Prime harmonics === |
| {| class="wikitable" | | {{Harmonics in equal|109|columns=16}} |
| |+Circulating temperaments in 109edo
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| !Tones
| | === Subsets and supersets === |
| !Pattern
| | 109edo is the 29th [[prime edo]], following [[107edo]] and before [[113edo]]. [[436edo]], which slices each step of 109edo in four, provides correction for the approximation to harmonic 3. |
| !L:s
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| | === Nonoctave temperaments === |
| |5
| | Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]]. |
| |[[4L 1s]]
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| |22:21
| | == Intervals == |
| |-
| | {{Interval table}} |
| |6
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| |[[1L 5s]]
| | == Music == |
| |19:18
| | ; [[Francium]] |
| |-
| | * "Teenagerges" from ''TOTMC September to December 2024'' (2024) – [https://open.spotify.com/track/4oQglJSEyp6CsL5RNWuiBy Spotify] | [https://francium223.bandcamp.com/track/teenagerges Bandcamp] | [https://www.youtube.com/watch?v=v_J71U392_k YouTube] – in Tetracot[13], 109edo tuning |
| |7
| | * "Catbabel" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/0T7nW3ziEFvjV8c7v1EaMB Spotify] | [https://francium223.bandcamp.com/track/catbabel Bandcamp] | [https://www.youtube.com/watch?v=gtnTdPqiTDQ YouTube] |
| |[[4L 3s]]
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| |16:15
| | == See also == |
| |-
| | * [[109-7-comma]] |
| |8
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| |[[5L 3s]]
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| |14:13
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| |-
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| |9
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| |[[1L 8s]]
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| |13:12
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| |-
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| |10
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| |[[9L 1s]]
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| |11:10
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| |-
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| |11
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| |[[10L 1s]]
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| | rowspan="2" |10:9
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| |-
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| |12
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| |[[1L 11s]] | |
| |-
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| |13
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| |[[4L 9s]] | |
| |9:8
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| |-
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| |14
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| |[[11L 3s]]
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| | rowspan="2" |8:7
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| |-
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| |15
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| |[[4L 11s]]
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| |- | |
| |16
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| |13L 3s
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| | rowspan="3" |7:6
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| |-
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| |17
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| |[[7L 10s]] | |
| |-
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| |18
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| |1L 17s
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| |-
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| |19
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| |14L 5s
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| | rowspan="3" |6:5
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| |-
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| |20
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| |9L 11s
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| |-
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| |21
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| |4L 17s
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| |-
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| |22
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| |21L 1s
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| | rowspan="6" |5:4
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| |-
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| |23
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| |17L 6s
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| |-
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| |24
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| |13L 11s
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| |-
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| |25
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| |9L 16s
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| |-
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| |26
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| |5L 21s
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| |-
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| |27
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| |1L 26s
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| |-
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| |28
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| |25L 3s
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| | rowspan="9" |4:3
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| |-
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| |29
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| |22L 7s
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| |-
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| |30
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| |19L 11s
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| |-
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| |31
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| |16L 15s
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| |-
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| |32
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| |13L 19s
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| |-
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| |33
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| |10L 23s
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| |-
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| |34
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| |7L 27s
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| |-
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| |35
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| |4L 31s
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| |-
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| |36
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| |1L 35s
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| |-
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| |37
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| |35L 2s
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| | rowspan="18" |3:2
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| |-
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| |38
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| |33L 5s
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| |-
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| |39
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| |31L 8s
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| |-
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| |40
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| |29L 11s
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| |-
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| |41
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| |27L 14s
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| |-
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| |42
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| |25L 17s
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| |-
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| |43
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| |23L 20s
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| |-
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| |44
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| |21L 23s
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| |-
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| |45
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| |19L 26s
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| |-
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| |46
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| |17L 29s
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| |-
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| |47
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| |15L 32s
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| |-
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| |48
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| |13L 35s
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| |-
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| |49
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| |11L 38L
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| |-
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| |50
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| |9L 41s
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| |-
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| |51
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| |7L 44s
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| |-
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| |52
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| |5L 47s
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| |-
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| |53
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| |3L 50s
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| |-
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| |54
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| |1L 53s
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| |-
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| |55
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| |54L 1s
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| | rowspan="34" |2:1
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| |-
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| |56
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| |53L 3s
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| |-
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| |57
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| |52L 5s
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| |-
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| |58
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| |51L 7s
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| |-
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| |59
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| |50L 9s
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| |-
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| |60
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| |49L 11s
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| |-
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| |61
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| |48L 13s
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| |-
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| |62
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| |47L 15s
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| |-
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| |63
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| |46L 17s
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| |-
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| |64
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| |45L 19s
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| |-
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| |65
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| |44L 21s
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| |-
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| |66
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| |43L 23s
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| |-
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| |67
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| |42L 25s
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| |-
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| |68
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| |41L 27s
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| |-
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| |69
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| |40L 29s
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| |-
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| |70
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| |39L 31s
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| |-
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| |71
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| |38L 33s
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| |-
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| |72
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| |37L 35s
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| |-
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| |73
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| |36L 37s
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| |-
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| |74
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| |35L 39s
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| |-
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| |75
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| |34L 41s
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| |-
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| |76
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| |33L 43s
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| |-
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| |77
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| |32L 45s
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| |-
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| |78
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| |31L 47s
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| |-
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| |79
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| |30L 49s
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| |-
| |
| |80
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| |29L 51s
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| |-
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| |81
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| |28L 53s
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| |-
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| |82
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| |27L 55s
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| |-
| |
| |83
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| |26L 57s
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| |-
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| |84
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| |25L 59s
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| |-
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| |85
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| |24L 61s
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| |-
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| |86
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| |23L 63s
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| |-
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| |87
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| |22L 65s
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| |-
| |
| |88
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| |21L 67s
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| |}
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| [[Category:Equal divisions of the octave]]
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| [[Category:Prime EDO]]
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| [[Category:Theory]]
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