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| '''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.
| | {{Infobox ET}} |
| {{Primes in edo|105}} | | {{ED intro}} |
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| 105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the [[triangular number]] closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number. | | == Theory == |
| | 105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone, such as [[grosstone]] and [[huygens]]. It [[tempering out|tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (a.k.a. huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone. |
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| == 105edo close-up == | | === Odd harmonics === |
| | {{Harmonics in equal|105}} |
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| <pre>C . . Dbb B## . . C# . . Db . . . C## . . D</pre>
| | === Subsets and supersets === |
| | 105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35. |
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| | As such, the val [105 165 245 294], which is contorted in 2.n for each prime n in the subgroup, may be used to extend the concept of 21edo's 5-limit harmony to the 7-limit, producing an independent dimension for each prime. |
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| | == Intervals == |
| | {{Main|Table of 105edo intervals}} |
| | === 15-odd-limit interval mappings === |
| | {{Q-odd-limit intervals|105}} |
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| | == Instruments == |
| | === Lumatone === |
| | The [[lumatone]] can be used to play 105edo. For key mappings, see: [[Lumatone mapping for 105edo]]. |
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| | [[Category:105edo| ]] <!-- main article --> |
| | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> |
| | [[Category:Huygens]] |
| [[Category:Meantone]] | | [[Category:Meantone]] |
| [[Category:Equal divisions of the octave]]
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| [[Category:105edo| ]]
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| Since 105edo has a step of 11.429 cents, it also allows one to use its MOS scales as circulating temperaments, which it is the first triangular edo to do.
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| {| class="wikitable"
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| |+Circulating temperaments in 105edo
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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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| |[[5edo]]
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| |equal
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| |-
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| |6
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| |[[3L 3s]]
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| |18:17
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| |-
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| |7
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| |[[7edo]]
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| |equal
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| |-
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| |8
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| |[[1L 7s]]
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| |14:13
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| |-
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| |9
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| |[[6L 3s]]
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| |12:11
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| |-
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| |10
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| |[[5L 5s]]
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| |11:10
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| |-
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| |11
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| |[[6L 5s]]
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| |10:9
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| |-
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| |12
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| |[[9L 3s]]
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| | rowspan="2" |9:8
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| |-
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| |13
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| |[[1L 12s]]
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| |-
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| |14
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| |[[7L 7s]]
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| |8:7
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| |-
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| |15
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| |[[15edo]]
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| |equal
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| |-
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| |16
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| |[[9L 7s]]
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| | rowspan="2" |7:6
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| |-
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| |17
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| |[[3L 14s]]
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| |-
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| |18
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| |15L 3s
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| | rowspan="3" |6:5
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| |-
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| |19
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| |[[10L 9s]]
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| |-
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| |20
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| |5L 15s
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| |-
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| |21
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| |[[21edo]]
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| |equal
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| |-
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| |22
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| |[[17L 5s]]
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| | rowspan="5" |5:4
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| |-
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| |23
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| |13L 10s
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| |-
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| |24
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| |9L 15s
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| |-
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| |25
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| |5L 20s
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| |-
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| |26
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| |1L 25s
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| |-
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| |27
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| |24L 3s
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| | rowspan="8" |4:3
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| |-
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| |28
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| |21L 7s
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| |-
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| |29
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| |18L 11s
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| |-
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| |30
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| |15L 15s
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| |-
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| |31
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| |12L 19s
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| |-
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| |32
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| |9L 23s
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| |-
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| |33
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| |6L 27s
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| |-
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| |34
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| |3L 31s
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| |-
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| |35
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| |[[35edo]]
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| |equal
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| |-
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| |36
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| |33L 3s
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| | rowspan="17" |3:2
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| |-
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| |37
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| |31L 6s
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| |-
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| |38
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| |29L 9s
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| |-
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| |39
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| |27L 12s
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| |-
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| |40
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| |25L 15s
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| |-
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| |41
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| |23L 18s
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| |-
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| |42
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| |21L 21s
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| |-
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| |43
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| |19L 24s
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| |-
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| |44
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| |17L 27s
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| |-
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| |45
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| |15L 30s
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| |-
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| |46
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| |13L 33s
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| |-
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| |47
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| |11L 36s
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| |-
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| |48
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| |9L 39s
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| |-
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| |49
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| |7L 42s
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| |-
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| |50
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| |5L 45s
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| |-
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| |51
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| |3L 48s
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| |-
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| |52
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| |1L 51s
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| |-
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| |53
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| |52L 1s
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| | rowspan="31" |2:1
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| |-
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| |54
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| |51L 3s
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| |-
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| |55
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| |50L 5s
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| |-
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| |56
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| |49L 7s
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| |-
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| |57
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| |48L 9s
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| |-
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| |58
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| |47L 11s
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| |-
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| |59
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| |46L 13s
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| |-
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| |60
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| |45L 15s
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| |-
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| |61
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| |44L 17s
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| |-
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| |62
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| |43L 19s
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| |-
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| |63
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| |42L 21s
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| |-
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| |64
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| |41L 23s
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| |-
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| |65
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| |40L 25s
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| |-
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| |66
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| |39L 27s
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| |-
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| |67
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| |38L 29s
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| |-
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| |68
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| |37L 31s
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| |-
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| |69
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| |36L 33s
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| |-
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| |70
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| |35L 35s
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| |-
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| |71
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| |34L 37s
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| |-
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| |72
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| |33L 39s
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| |-
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| |73
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| |32L 41s
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| |-
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| |74
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| |31L 43s
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| |-
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| |75
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| |30L 45s
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| |-
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| |76
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| |29L 47s
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| |-
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| |77
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| |28L 49s
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| |-
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| |78
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| |27L 51s
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| |-
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| |79
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| |26L 53s
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| |-
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| |80
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| |25L 55s
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| |-
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| |81
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| |24L 57s
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| |-
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| |82
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| |23L 59s
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| |-
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| |83
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| |22L 61s
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| |}<!-- main article -->
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| [[Category:Huygens]]
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