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| '''Decic''' is a temperament for the 7, 11, 13, and 17 [[Harmonic limit|prime limits]]. It is a member of [[marvel temperaments]], [[cloudy clan]], and [[15/14 equal-step tuning|linus temperaments]]. It has a period of 1/10 octave and tempers out 225/224 and 16807/16384. The fifth of decic in size is a [[meantone]] fifth, but four of them are not used to reach the 5th harmonic. Instead, [[14/13]], [[15/14]] and [[16/15]] are equated to 1/10 of an octave, and from this it derives its name. Not only the meantone fifth (flat 3/2) or fourth (sharp 4/3), but also the [[magic]] major third (flat 5/4) can be used as a generator.
| | #redirect [[Marvel temperaments #Decic]] |
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| There are three mappings for 11, 13 and 17-limit that are comparable in complexity and error: ''decic'' (10&50), ''splendic'' (10e&50) and ''prodigic'' (10&50e).
| | [[Category:Rank-2 temperaments]] |
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| == Temperament data ==
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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;">'''Decic temperament (10&50)'''</div>
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| <div class="mw-collapsible-content">
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| Subgroup: 2.3.5.7.11.13.17
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| [[Comma list]]: 105/104, 144/143, 170/169, 196/195, 221/220 | |
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| [[Mapping]]: [{{val| 10 16 23 28 35 37 41 }}, {{val| 0 -1 1 0 -2 0 -1 }}]
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| [[POTE generator]]s:
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| * 7-limit: ~15/14 = 120.00000, ~3/2 = 698.69596
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| * 11-limit: ~15/14 = 120.00000, ~3/2 = 696.79119
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| * 13-limit: ~14/13 = 120.00000, ~3/2 = 696.99342
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| * 17-limit: ~14/13 = 120.00000, ~3/2 = 697.08527
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| [[TOP tuning|TOP generator]]s:
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| * 7-limit: ~15/14 = 120.18411, ~3/2 = 699.76795
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| * 11-limit: ~15/14 = 120.14165, ~3/2 = 697.61366
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| * 13-limit: ~14/13 = 120.11775, ~3/2 = 697.67733
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| * 17-limit: ~14/13 = 120.12744, ~3/2 = 697.82559
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| [[Diamond monotone]] ranges:
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| * 7-odd-limit: ~3/2 = [680.00000, 720.00000] (17\30 to 6\10)
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| * 9-odd-limit: ~3/2 = [696.00000, 720.00000] (29\50 to 6\10)
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| * 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000] (29\50 to 35\60)
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| [[Diamond tradeoff]] ranges:
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| * 7 and 9-odd-limit: ~3/2 = [693.12909, 702.51219]
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| * 11, 13, and 15-odd-limit: ~3/2 = [689.36294, 702.51219]
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| * 17-odd-limit: ~3/2 = [689.36294, 704.95541]
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| Diamond monotone and tradeoff ranges:
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| * 7-odd-limit: ~3/2 = [693.12909, 702.51219]
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| * 9-odd-limit: ~3/2 = [696.00000, 702.51219]
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| * 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000]
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| [[Optimal GPV sequence]]s:
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| * 7-limit: {{Vals| 10, 30b, 40, 50, 60, 110d, 170cdd }}
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| * 11 and 13-limit: {{Vals| 10, 30be, 40, 50 }}
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| * 17-limit: {{Vals| 10, 30beg, 40, 50 }}
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| [[Badness]]:
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| * 7-limit: 0.089135
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| * 11-limit: 0.063900
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| * 13-limit: 0.036880
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| * 17-limit: 0.025064
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| </div></div>
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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;">'''Splendic temperament (10e&50)'''</div>
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| <div class="mw-collapsible-content">
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| Subgroup: 2.3.5.7.11.13.17
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| [[Comma list]]: 105/104, 170/169, 196/195, 289/288, 375/374
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| [[Mapping]]: [{{val| 10 16 23 28 34 37 41 }}, {{val| 0 -1 1 0 3 0 -1 }}]
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| [[POTE generator]]s:
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| * 11-limit: ~15/14 = 120.00000, ~3/2 = 698.51792
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| * 13-limit: ~14/13 = 120.00000, ~3/2 = 698.36508
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| * 17-limit: ~14/13 = 120.00000, ~3/2 = 698.37473
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| [[TOP tuning|TOP generator]]s:
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| * 11-limit: ~15/14 = 120.18780, ~3/2 = 699.61110
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| * 13-limit: ~14/13 = 120.15710, ~3/2 = 699.27937
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| * 17-limit: ~14/13 = 120.15778, ~3/2 = 699.29299
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| [[Diamond monotone]] ranges:
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| * 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000] (29\50 to 35\60)
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| [[Diamond tradeoff]] ranges:
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| * 11, 13, and 15-odd-limit: ~3/2 = [693.12909, 702.51219]
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| * 17-odd-limit: ~3/2 = [693.12909, 704.95541]
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| Diamond monotone and tradeoff ranges:
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| * 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000]
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| [[Optimal GPV sequence]]s:
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| * 11 and 13-limit: {{Vals| 10e, 30bee, 40e, 50, 60e, 110de }}
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| * 17-limit: {{Vals| 10e, 30beeg, 40e, 50, 60e, 110deg }}
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| [[Badness]]:
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| * 11-limit: 0.059802
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| * 13-limit: 0.037989
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| * 17-limit: 0.026050
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| </div></div>
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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;">'''Prodigic temperament (10&50e)'''</div>
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| <div class="mw-collapsible-content">
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| Subgroup: 2.3.5.7.11.13.17
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| [[Comma list]]: 105/104, 154/153, 170/169, 196/195, 289/288
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| [[Mapping]]: [{{val| 10 16 23 28 35 37 41 }}, {{val| 0 -1 1 0 -3 0 -1 }}]
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| [[POTE generator]]s:
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| * 11-limit: ~15/14 = 120.00000, ~3/2 = 700.20964
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| * 13-limit: ~14/13 = 120.00000, ~3/2 = 700.50272
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| * 17-limit: ~14/13 = 120.00000, ~3/2 = 700.47431
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| [[TOP tuning|TOP generator]]s:
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| * 11-limit: ~15/14 = 120.20310, ~3/2 = 701.39473
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| * 13-limit: ~14/13 = 120.16631, ~3/2 = 701.47358
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| * 17-limit: ~14/13 = 120.15866, ~3/2 = 701.40045
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| [[Diamond monotone]] ranges:
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| * 11, 13, 15, and 17-odd-limit: ~3/2 = [700.00000, 702.85714] (35\60 to 41\70)
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| [[Diamond tradeoff]] ranges:
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| * 11, 13, 15, and 17-odd-limit: ~3/2 = [693.12909, 707.40794]
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| Diamond monotone and tradeoff ranges:
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| * 11, 13, 15, and 17-odd-limit: ~3/2 = [700.00000, 702.85714]
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| [[Optimal GPV sequence]]s:
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| * 11, 13, and 17-limit: {{Vals| 10, 40ee, 50e, 60e }}
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| [[Badness]]:
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| * 11-limit: 0.066600
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| * 13-limit: 0.041788
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| * 17-limit: 0.027590
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| </div></div>
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| == Interval chains ==
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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;">'''Intervals of decic (10&50)'''</div>
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| <div class="mw-collapsible-content">
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| {| class="wikitable"
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| |-
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| ! colspan="2" | Generator
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| ! | -3
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| ! | -2
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| ! | -1
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| ! | 0
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| ! | 1
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| ! | 2
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| ! | 3
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| |-
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| ! rowspan="2" | Period 0
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| ! | Cents*
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| | | 1131.256
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| | | 1154.171
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| | | 1177.085
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| | | 0.000
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| | | 22.915
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| | | 45.829
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| | | 68.744
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| |-
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| ! | Ratios
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| | | 1/1
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| | | 40/39
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| | | 28/27, 25/24
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| |-
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| ! rowspan="2" | Period 1
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| ! | Cents*
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| | | 51.256
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| | | 74.171
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| | | 97.085
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| | | 120.000
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| | | 142.915
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| | | 165.829
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| | | 188.744
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| |-
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| ! | Ratios
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| | | 36/35, 33/32
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| | | 21/20
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| | | 18/17, 17/16
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| | | 16/15, 15/14, 14/13
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| | | 13/12, 12/11
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| | | 10/9
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| |-
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| ! rowspan="2" | Period 2
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| ! | Cents*
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| | | 171.256
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| | | 194.171
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| | | 217.085
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| | | 240.000
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| | | 262.915
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| | | 285.829
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| | | 308.744
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| |-
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| ! | Ratios
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| | | 11/10
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| | | 9/8
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| | | 17/15
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| | | 8/7, 15/13
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| | | 7/6
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| | | 20/17, 13/11
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| |-
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| ! rowspan="2" | Period 3
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| ! | Cents*
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| | | 291.256
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| | | 314.171
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| | | 337.085
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| | | 360.000
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| | | 382.915
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| | | 405.829
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| | | 428.744
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| |-
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| ! | Ratios
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| | | 6/5
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| | | 17/14
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| | | 11/9, 16/13, 21/17
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| | | 26/21, 5/4
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| | | 14/11
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| |-
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| ! rowspan="2" | Period 4
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| ! | Cents*
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| | | 411.256
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| | | 434.171
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| | | 457.085
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| | | 480.000
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| | | 502.915
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| | | 525.829
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| | | 548.744
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| |-
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| ! | Ratios
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| | | 9/7
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| | | 22/17, 13/10, 17/13, 21/16
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| | | 4/3
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| | | 15/11
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| |-
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| ! rowspan="2" | Period 5
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| ! | Cents*
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| | | 531.256
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| | | 554.171
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| | | 577.085
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| | | 600.000
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| | | 622.915
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| | | 645.829
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| | | 668.744
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| |-
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| ! | Ratios
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| | | 11/8, 18/13
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| | | 7/5
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| | | 24/17, 17/12
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| | | 10/7
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| | | 13/9, 16/11
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| |}
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| <nowiki>*</nowiki> in 17-limit POTE tuning
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| </div></div>
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| == Scales ==
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| * [[Decic50]] - [[10L 40s]] scale
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| [[Category:Cloudy clan]]
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| [[Category:Marvel temperaments]] | | [[Category:Marvel temperaments]] |
| [[Category:Decic| ]]<!-- main article -->
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| {{IoT}}
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