Decic: Difference between revisions

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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.
+There are three mappings for 11, 13 and 17-limit that are comparable in complexity and error: ''decic'' (10&amp;50), ''splendic'' (10e&amp;50) and ''prodigic'' (10&amp;50e).
 == Temperament data ==
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 [[POTE generator]]s:
-* 7-limit: ~3/2 = 698.69596
+* 7-limit: ~15/14 = 120.00000, ~3/2 = 698.69596
-* 11-limit: ~3/2 = 696.79119
+* 11-limit: ~15/14 = 120.00000, ~3/2 = 696.79119
-* 13-limit: ~3/2 = 696.99342
+* 13-limit: ~14/13 = 120.00000, ~3/2 = 696.99342
-* 17-limit: ~3/2 = 697.08527
+* 17-limit: ~14/13 = 120.00000, ~3/2 = 697.08527
 [[TOP tuning|TOP generator]]s:
@@ Line 48: / Line 50: @@
 * 13-limit: 0.036880
 * 17-limit: 0.025064
+</div></div>
+<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+<div style="line-height:1.6;">'''Splendic temperament (10e&amp;50)'''</div>
+<div class="mw-collapsible-content">
+Subgroup: 2.3.5.7.11.13.17
+[[Comma list]]: 105/104, 170/169, 196/195, 289/288, 375/374
+[[Mapping]]: [{{val| 10 16 23 28 34 37 41 }}, {{val| 0 -1 1 0 3 0 -1 }}]
+[[POTE generator]]s:
+* 11-limit: ~15/14 = 120.00000, ~3/2 = 698.51792
+* 13-limit: ~14/13 = 120.00000, ~3/2 = 698.36508
+* 17-limit: ~14/13 = 120.00000, ~3/2 = 698.37473
+[[TOP tuning|TOP generator]]s:
+* 11-limit: ~15/14 = 120.18780, ~3/2 = 699.61110
+* 13-limit: ~14/13 = 120.15710, ~3/2 = 699.27937
+* 17-limit: ~14/13 = 120.15778, ~3/2 = 699.29299
+[[Diamond monotone]] ranges:
+* 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000] (29\50 to 35\60)
+[[Diamond tradeoff]] ranges:
+* 11, 13, and 15-odd-limit: ~3/2 = [693.12909, 702.51219]
+* 17-odd-limit: ~3/2 = [693.12909, 704.95541]
+Diamond monotone and tradeoff ranges:
+* 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000]
+[[Optimal GPV sequence]]s:
+* 11 and 13-limit: {{Vals| 10e, 30bee, 40e, 50, 60e, 110de }}
+* 17-limit: {{Vals| 10e, 30beeg, 40e, 50, 60e, 110deg }}
+[[Badness]]:
+* 11-limit: 0.059802
+* 13-limit: 0.037989
+* 17-limit: 0.026050
+</div></div>
+<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+<div style="line-height:1.6;">'''Prodigic temperament (10&amp;50e)'''</div>
+<div class="mw-collapsible-content">
+Subgroup: 2.3.5.7.11.13.17
+[[Comma list]]: 105/104, 154/153, 170/169, 196/195, 289/288
+[[Mapping]]: [{{val| 10 16 23 28 35 37 41 }}, {{val| 0 -1 1 0 -3 0 -1 }}]
+[[POTE generator]]s:
+* 11-limit: ~15/14 = 120.00000, ~3/2 = 700.20964
+* 13-limit: ~14/13 = 120.00000, ~3/2 = 700.50272
+* 17-limit: ~14/13 = 120.00000, ~3/2 = 700.47431
+[[TOP tuning|TOP generator]]s:
+* 11-limit: ~15/14 = 120.20310, ~3/2 = 701.39473
+* 13-limit: ~14/13 = 120.16631, ~3/2 = 701.47358
+* 17-limit: ~14/13 = 120.15866, ~3/2 = 701.40045
+[[Diamond monotone]] ranges:
+* 11, 13, 15, and 17-odd-limit: ~3/2 = [700.00000, 702.85714] (35\60 to 41\70)
+[[Diamond tradeoff]] ranges:
+* 11, 13, 15, and 17-odd-limit: ~3/2 = [693.12909, 707.40794]
+Diamond monotone and tradeoff ranges:
+* 11, 13, 15, and 17-odd-limit: ~3/2 = [700.00000, 702.85714]
+[[Optimal GPV sequence]]s:
+* 11, 13, and 17-limit: {{Vals| 10, 40ee, 50e, 60e }}
+[[Badness]]:
+* 11-limit: 0.066600
+* 13-limit: 0.041788
+* 17-limit: 0.027590
 </div></div>
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 == Scales ==
-* [[Decic50]] - proper [[10L 40s]] scale
+* [[Decic50]] - [[10L 40s]] scale
 [[Category:Cloudy clan]]