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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.
'''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&50), ''splendic'' (10e&50) and ''prodigic'' (10&50e).


== Temperament data ==
== Temperament data ==
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[[POTE generator]]s:  
[[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:  
[[TOP tuning|TOP generator]]s:  
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* 13-limit: 0.036880
* 13-limit: 0.036880
* 17-limit: 0.025064
* 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>
</div></div>


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== Scales ==
== Scales ==
* [[Decic50]] - proper [[10L 40s]] scale
* [[Decic50]] - [[10L 40s]] scale


[[Category:Cloudy clan]]
[[Category:Cloudy clan]]

Revision as of 01:22, 11 February 2022

Decic is a temperament for the 7, 11, 13, and 17 prime limits. It is a member of marvel temperaments, cloudy clan, and 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&50), splendic (10e&50) and prodigic (10&50e).

Temperament data

Decic temperament (10&50)

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 144/143, 170/169, 196/195, 221/220

Mapping: [⟨10 16 23 28 35 37 41], ⟨0 -1 1 0 -2 0 -1]]

POTE generators:

  • 7-limit: ~15/14 = 120.00000, ~3/2 = 698.69596
  • 11-limit: ~15/14 = 120.00000, ~3/2 = 696.79119
  • 13-limit: ~14/13 = 120.00000, ~3/2 = 696.99342
  • 17-limit: ~14/13 = 120.00000, ~3/2 = 697.08527

TOP generators:

  • 7-limit: ~15/14 = 120.18411, ~3/2 = 699.76795
  • 11-limit: ~15/14 = 120.14165, ~3/2 = 697.61366
  • 13-limit: ~14/13 = 120.11775, ~3/2 = 697.67733
  • 17-limit: ~14/13 = 120.12744, ~3/2 = 697.82559

Diamond monotone ranges:

  • 7-odd-limit: ~3/2 = [680.00000, 720.00000] (17\30 to 6\10)
  • 9-odd-limit: ~3/2 = [696.00000, 720.00000] (29\50 to 6\10)
  • 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000] (29\50 to 35\60)

Diamond tradeoff ranges:

  • 7 and 9-odd-limit: ~3/2 = [693.12909, 702.51219]
  • 11, 13, and 15-odd-limit: ~3/2 = [689.36294, 702.51219]
  • 17-odd-limit: ~3/2 = [689.36294, 704.95541]

Diamond monotone and tradeoff ranges:

  • 7-odd-limit: ~3/2 = [693.12909, 702.51219]
  • 9-odd-limit: ~3/2 = [696.00000, 702.51219]
  • 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000]

Optimal GPV sequences:

Badness:

  • 7-limit: 0.089135
  • 11-limit: 0.063900
  • 13-limit: 0.036880
  • 17-limit: 0.025064
Splendic temperament (10e&50)

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 170/169, 196/195, 289/288, 375/374

Mapping: [⟨10 16 23 28 34 37 41], ⟨0 -1 1 0 3 0 -1]]

POTE generators:

  • 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 generators:

  • 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 sequences:

Badness:

  • 11-limit: 0.059802
  • 13-limit: 0.037989
  • 17-limit: 0.026050
Prodigic temperament (10&50e)

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 154/153, 170/169, 196/195, 289/288

Mapping: [⟨10 16 23 28 35 37 41], ⟨0 -1 1 0 -3 0 -1]]

POTE generators:

  • 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 generators:

  • 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 sequences:

Badness:

  • 11-limit: 0.066600
  • 13-limit: 0.041788
  • 17-limit: 0.027590

Interval chains

Intervals of decic (10&50)
Generator -3 -2 -1 0 1 2 3
Period 0 Cents* 1131.256 1154.171 1177.085 0.000 22.915 45.829 68.744
Ratios 1/1 40/39 28/27, 25/24
Period 1 Cents* 51.256 74.171 97.085 120.000 142.915 165.829 188.744
Ratios 36/35, 33/32 21/20 18/17, 17/16 16/15, 15/14, 14/13 13/12, 12/11 10/9
Period 2 Cents* 171.256 194.171 217.085 240.000 262.915 285.829 308.744
Ratios 11/10 9/8 17/15 8/7, 15/13 7/6 20/17, 13/11
Period 3 Cents* 291.256 314.171 337.085 360.000 382.915 405.829 428.744
Ratios 6/5 17/14 11/9, 16/13, 21/17 26/21, 5/4 14/11
Period 4 Cents* 411.256 434.171 457.085 480.000 502.915 525.829 548.744
Ratios 9/7 22/17, 13/10, 17/13, 21/16 4/3 15/11
Period 5 Cents* 531.256 554.171 577.085 600.000 622.915 645.829 668.744
Ratios 11/8, 18/13 7/5 24/17, 17/12 10/7 13/9, 16/11

* in 17-limit POTE tuning

Scales

Template:IoT

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