Decic

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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), splendecic (10e&50) and prodecic (10&50e). They share the comma list of the no-elevens subgroups, tempering out 105/104, 170/169, 196/195 and 289/288. Decic adds 385/384, splendecic adds 4375/4376, and prodecic adds 441/440 to the 11-limit comma list.

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 ET sequences:

Badness:

  • 7-limit: 0.089135
  • 11-limit: 0.063900
  • 13-limit: 0.036880
  • 17-limit: 0.025064


Splendecic 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 ET sequences:

Badness:

  • 11-limit: 0.059802
  • 13-limit: 0.037989
  • 17-limit: 0.026050


Prodecic 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 ET 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


Intervals of splendecic (10e&50)
Generator -3 -2 -1 0 1 2 3
Period 0 Cents* 1135.124 1156.749 1178.375 0.000 21.625 43.251 64.876
Ratios 1/1 40/39 28/27, 25/24
Period 1 Cents* 55.124 76.749 98.375 120.000 141.625 163.251 184.876
Ratios 36/35 21/20 18/17, 17/16 16/15, 15/14, 14/13 13/12 11/10 10/9
Period 2 Cents* 175.124 196.749 218.375 240.000 261.625 283.251 304.876
Ratios 9/8 17/15 8/7, 15/13 7/6 20/17
Period 3 Cents* 295.124 316.749 338.375 360.000 381.625 403.251 424.876
Ratios 13/11 6/5 17/14 16/13, 21/17 26/21, 5/4
Period 4 Cents* 415.124 436.749 458.375 480.000 501.625 523.251 544.876
Ratios 14/11 9/7 13/10, 17/13, 21/16 4/3 11/8
Period 5 Cents* 535.124 556.749 578.375 600.000 621.625 643.251 664.876
Ratios 15/11 18/13 7/5 24/17, 17/12 10/7 13/9 22/15

* in 17-limit POTE tuning


Intervals of prodecic (10&50e)
Generator -3 -2 -1 0 1 2 3
Period 0 Cents* 1141.423 1160.949 1180.474 0.000 19.526 39.051 58.577
Ratios 1/1 40/39 28/27, 25/24
Period 1 Cents* 61.423 80.949 100.474 120.000 139.526 159.051 178.577
Ratios 36/35 22/21, 21/20 18/17, 17/16 16/15, 15/14, 14/13 13/12 12/11 10/9
Period 2 Cents* 181.423 200.949 220.474 240.000 259.526 279.051 298.577
Ratios 9/8 17/15 8/7, 15/13 7/6 20/17 13/11
Period 3 Cents* 301.423 320.949 340.474 360.000 379.526 399.051 418.577
Ratios 6/5 17/14, 11/9 16/13, 21/17 26/21, 5/4 14/11
Period 4 Cents* 421.423 440.949 460.474 480.000 499.526 519.051 538.577
Ratios 9/7, 22/17 13/10, 17/13, 21/16 4/3 15/11
Period 5 Cents* 541.423 560.949 580.474 600.000 619.526 639.051 658.577
Ratios 11/8 18/13 7/5 24/17, 17/12 10/7 13/9 16/11

* in 17-limit POTE tuning

Scales

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