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| Line 1,297: |
Line 1,297: |
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| ==== 34nejis ==== | | ==== 34nejis ==== |
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| == Rank-2 temperaments ==
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| Oneirotonic temperaments have a sort of analogy to diatonic temperaments superpyth and meantone in how they treat the large step. In diatonic the large step approximates 9/8 (a very good 9/8 in 12edo), but superpyth has 9/8 ~ 8/7, and meantone has 9/8 ~ 10/9. In oneirotonic the large step tends to approximate 10/9 (and is a very good 10/9 in 13edo which is the oneirotonic analogue to 12edo), but different oneiro temperaments do different things with it. In A-Team (13&18), 10/9 is equated with 9/8, making the major oneirothird a 5/4 (thus is "meantone" in that sense). In both Petrtri (13&21) and Tridec (21&29), 10/9 is equated with 11/10, making the major oneirothird a 11/9; and the perfect oneirofourth is equated to 13/10. So the compressed major triad add2 (R-M2-M3-M5, M5 = major oneirofifth = minor fifth in 13edo) is interpreted as 9:10:11:13 in petrtri, analogous to meantone's 8:9:10:12. Thus Petrtri and Tridec are the same temperament when you only care about the 9:10:11:13, or equivalently the 2.9/5.11/5.13/5 subgroup. This is one reason why Tridec can be viewed as the oneirotonic analogue of [[flattone]]: it's a flatter variant of the flat-of-13edo oneiro temperament on the 2.9/5.11/5.13/5 subgroup.
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|
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| Vulture/[[Hemifamity_temperaments|Buzzard]], in which four generators make a 3/1 (and three generators approximate an octave plus 8/7), is the only [[harmonic entropy]] minimum in the oneirotonic range. However, the rest of this region is still rich in notable subgroup temperaments.
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| === Tridec ===
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| Subgroup: 2.3.7/5.11/5.13/5
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|
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| Period: 1\1
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|
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| Optimal ([[POTE]]) generator: 455.2178
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|
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| EDO generators: [[21edo|8\21]], [[29edo|11\29]], [[37edo|14\37]]
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|
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">Technical data</div>
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| <div class="mw-collapsible-content">
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|
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| [[Comma]] list: 196/195, 847/845, 1001/1000
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|
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| [[Mapping]] (for 2, 3, 7/5, 11/5, 13/5): [{{val|1 5 2 0 1}}, {{val|0 -9 -4 3 1}}]
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|
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| Mapping generators: ~2, ~13/10
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|
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| {{Vals|legend=1| 21, 29, 37 }}
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|
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| </div></div>
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|
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| ==== Intervals ====
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| Sortable table of intervals in the Dylathian mode and their Tridec interpretations:
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|
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| {| class="wikitable right-2 right-3 right-4 sortable"
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| |-
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| ! Degree
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| ! Size in 21edo
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| ! Size in 29edo
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| ! Size in 37edo
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| ! Size in POTE tuning
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| ! Note name on Q
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| ! class="unsortable"| Approximate ratios
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| ! #Gens up
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| |-
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| | 1
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| | 0\21, 0.00
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| | 0\29, 0.00
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| | 0\37, 0.00
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| | 0.00
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| | Q
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| | 1/1
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| | 0
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| |-
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| | 2
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| | 3\21, 171.43
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| | 4\29, 165.52
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| | 5\37, 163.16
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| | 165.65
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| | J
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| | 11/10, 10/9
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| | +3
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| |-
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| | 3
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| | 6\21, 342.86
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| | 8\29, 331.03
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| | 10\37, 324.32
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| | 331.31
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| | K
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| | 11/9, 6/5
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| | +6
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| |-
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| | 4
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| | 8\21, 457.14
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| | 11\29, 455.17
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| | 14\37, 454.05
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| | 455.17
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| | L
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| | 13/10, 9/7
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| | +1
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| |-
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| | 5
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| | 11\21, 628.57
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| | 15\29, 620.69
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| | 19\37, 616.22
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| | 620.87
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| | M
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| | 13/9, 10/7
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| | +4
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| |-
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| | 6
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| | 14\21, 800.00
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| | 19\29, 786.21
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| | 23\37, 778.38
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| | 786.52
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| | N
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| | 11/7
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| | +7
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| |-
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| | 7
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| | 16\21, 914.29
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| | 22\29, 910.34
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| | 28\37, 908.11
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| | 910.44
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| | O
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| | 22/13
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| | +2
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| |-
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| | 8
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| | 19\21, 1085.71
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| | 26\29, 1075.86
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| | 33\37, 1070.27
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| | 1076.09
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| | P
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| | 13/7, 28/15
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| | +5
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| |}
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|
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| === Petrtri ===
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| Subgroup: 2.5.9.11.13.17
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|
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| Period: 1\1
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| Optimal ([[POTE]]) generator: 459.1502
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|
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| EDO generators: [[13edo|5\13]], [[21edo|8\21]], [[34edo|13\34]]
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|
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">Technical data</div>
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| <div class="mw-collapsible-content">
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|
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| [[Comma]] list: 100/99, 144/143, 170/169, 221/220
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|
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| [[Mapping]] (for 2, 5, 9, 11, 13, 17): [{{val|1 5 7 5 6 6}}, {{val|0 -7 -10 -4 -6 -5}}]
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|
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| Mapping generators: ~2, ~13/10
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|
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| {{Vals|legend=1| 13, 21, 34 }}
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|
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| </div></div>
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| ==== Intervals ====
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| Sortable table of intervals in the Dylathian mode and their Petrtri interpretations:
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| {| class="wikitable right-2 right-3 right-4 right-5 sortable"
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| |-
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| ! Degree
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| ! Size in 13edo
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| ! Size in 21edo
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| ! Size in 34edo
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| ! Size in POTE tuning
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| ! Note name on Q
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| ! class="unsortable"| Approximate ratios
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| ! #Gens up
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| |-
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| | 1
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| | 0\13, 0.00
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| | 0\21, 0.00
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| | 0\34, 0.00
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| | 0.00
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| | Q
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| | 1/1
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| | 0
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| |-
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| | 2
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| | 2\13, 184.62
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| | 3\21, 171.43
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| | 5\34, 176.47
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| | 177.45
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| | J
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| | 10/9, 11/10
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| | +3
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| |-
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| | 3
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| | 4\13, 369.23
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| | 6\21, 342.86
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| | 10\34, 352.94
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| | 354.90
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| | K
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| | 11/9, 16/13
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| | +6
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| |-
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| | 4
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| | 5\13, 461.54
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| | 8\21, 457.14
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| | 13\34, 458.82
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| | 459.15
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| | L
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| | 13/10, 17/13, 22/17
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| | +1
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| |-
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| | 5
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| | 7\13, 646.15
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| | 11\21, 628.57
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| | 18\34, 635.294
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| | 636.60
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| | M
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| | 13/9, 16/11, 23/16 (esp. 21edo)
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| | +4
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| |-
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| | 6
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| | 9\13, 830.77
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| | 14\21, 800.00
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| | 23\34, 811.77
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| | 814.05
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| | N
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| | 8/5
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| | +7
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| |-
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| | 7
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| | 10\13, 923.08
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| | 16\21, 914.29
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| | 26\34, 917.65
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| | 918.30
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| | O
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| | 17/10
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| | +2
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| |-
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| | 8
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| | 12\13, 1107.69
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| | 19\21, 1085.71
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| | 31\34, 1094.12
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| | 1095.75
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| | P
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| | 17/9, 32/17, 15/8
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| | +5
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| |}
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|
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| === A-Team ===
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| Subgroup: 2.5.9.21
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|
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| Period: 1\1
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|
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| Optimal ([[POTE]]) generator: 464.3865
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|
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| EDO generators: [[13edo|5\13]], [[18edo|7\18]], [[31edo|12\31]], [[44edo|17\44]]
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|
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">Technical data</div>
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| <div class="mw-collapsible-content">
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|
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| [[Comma]] list: 81/80, 1029/1024
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|
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| [[Mapping]] (for 2, 5, 9, 21): [{{val|1 0 2 4}}, {{val|0 6 3 1}}]
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|
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| Mapping generators: ~2, ~21/16
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|
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| {{Vals|legend=1| 13, 18, 31, 44 }}
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|
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| </div></div>
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| ==== Intervals ====
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| Sortable table of intervals in the Dylathian mode and their A-Team interpretations:
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|
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| {| class="wikitable right-2 right-3 right-4 sortable"
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| |-
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| ! Degree
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| ! Size in 13edo
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| ! Size in 18edo
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| ! Size in 31edo
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| ! Note name on Q
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| ! class="unsortable"| Approximate ratios<ref>The ratio interpretations that are not valid for 18edo are italicized.</ref>
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| ! #Gens up
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| |-
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| | 1
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| | 0\13, 0.00
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| | 0\18, 0.00
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| | 0\31, 0.00
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| | Q
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| | 1/1
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| | 0
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| |-
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| | 2
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| | 2\13, 184.62
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| | 3\18, 200.00
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| | 5\31, 193.55
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| | J
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| | 9/8, 10/9
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| | +3
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| |-
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| | 3
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| | 4\13, 369.23
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| | 6\18, 400.00
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| | 10\31, 387.10
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| | K
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| | 5/4
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| | +6
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| |-
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| | 4
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| | 5\13, 461.54
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| | 7\18, 466.67
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| | 12\31, 464.52
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| | L
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| | 21/16, ''13/10''
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| | +1
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| |-
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| | 5
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| | 7\13, 646.15
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| | 10\18, 666.66
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| | 17\31, 658.06
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| | M
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| | ''13/9'', ''16/11''
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| | +4
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| |-
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| | 6
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| | 9\13, 830.77
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| | 13\18, 866.66
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| | 22\31, 851.61
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| | N
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| | ''13/8'', ''18/11''
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| | +7
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| |-
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| | 7
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| | 10\13, 923.08
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| | 14\18, 933.33
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| | 24\31, 929.03
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| | O
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| | 12/7
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| | +2
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| |-
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| | 8
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| | 12\13, 1107.69
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| | 17\18, 1133.33
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| | 29\31, 1122.58
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| | P
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| | +5
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| |}
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| <references/>
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|
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| === Buzzard ===
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| Subgroup: 2.3.5.7
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|
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| Period: 1\1
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|
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| Optimal ([[POTE]]) generator: ~21/16 = 475.636
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|
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| EDO generators: [[38edo|15\38]], [[43edo|17\43]], [[48edo|19\48]], [[53edo|21\53]], [[58edo|23\58]], [[63edo|25\63]]
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|
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">Technical data</div>
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| <div class="mw-collapsible-content">
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|
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| Commas: 1728/1715, 5120/5103
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|
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| Map: [<1 0 -6 4|, <0 4 21 -3|]
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|
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| Mapping generators: ~2, ~21/16
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|
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| Wedgie: <<4 21 -3 24 -16 -66||
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|
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| {{Vals| 48, 53, 111, 164d, 275d}}
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|
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| Badness: 0.0480
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| </div></div>
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|
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| ==== Intervals ====
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| Sortable table of intervals in the Dylathian mode and their Buzzard interpretations:
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|
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| {| class="wikitable right-2 right-3 right-4 right-5 sortable"
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| |-
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| ! Degree
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| ! Size in 38edo
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| ! Size in 53edo
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| ! Size in 63edo
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| ! Size in POTE tuning
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| ! Note name on Q
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| ! class="unsortable"| Approximate ratios
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| ! #Gens up
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| |-
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| | 1
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| | 0\38, 0.00
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| | 0\53, 0.00
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| | 0\63, 0.00
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| | 0.00
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| | Q
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| | 1/1
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| | 0
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| |-
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| | 2
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| | 7\38, 221.05
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| | 10\53, 226.42
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| | 12\63, 228.57
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| | 227.07
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| | J
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| | 8/7
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| | +3
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| |-
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| | 3
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| | 14\38, 442.10
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| | 20\53, 452.83
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| | 24\63, 457.14
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| | 453.81
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| | K
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| | 13/10, 9/7
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| | +6
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| |-
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| | 4
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| | 15\38, 473.68
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| | 21\53, 475.47
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| | 25\63, 476.19
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| | 475.63
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| | L
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| | 21/16
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| | +1
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| |-
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| | 5
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| | 22\38, 694.73
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| | 31\53, 701.89
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| | 37\63, 704.76
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| | 702.54
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| | M
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| | 3/2
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| | +4
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| |-
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| | 6
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| | 29\38, 915.78
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| | 41\53, 928.30
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| | 49\63, 933.33
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| | 929.45
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| | N
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| | 12/7, 22/13
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| | +7
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| |-
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| | 7
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| | 30\38, 947.36
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| | 42\53, 950.94
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| | 50\63, 952.38
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| | 951.27
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| | O
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| | 26/15
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| | +2
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| |-
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| | 8
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| | 37\38, 1168.42
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| | 52\53, 1177.36
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| | 62\63, 1180.95
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| | 1178.18
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| | P
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| | 108/55, 160/81
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| | +5
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| |}
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| == Samples == | | == Samples == |