Semaphoresmic clan

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Revision as of 05:50, 25 August 2024 by Lériendil (talk | contribs) (Lériendil moved page Slendro clan to Semaphore clan: The current name of 49/48 as the "slendro diesis" is needlessly overloaded with that of the "slendric" temperament (which belongs to the gamelismic family instead of the slendric).)
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The slendro clan tempers out the slendro diesis, 49/48, a triprime comma with factors of 2, 3 and 7.

Semaphore

Main article: Semaphore and godzilla

Subgroup: 2.3.7

Comma list: 49/48

Subgroup-val mapping: [⟨1 0 2], ⟨0 2 1]]

sval mapping generators: ~2, ~7/4

Gencom mapping: [⟨1 2 0 3], ⟨0 -2 0 -1]]

gencom: [2 7/6; 49/48]

Optimal tunings:

  • CTE: ~2 = 1\1, ~7/4 = 952.2948
  • POTE: ~2 = 1\1, ~7/4 = 949.615

Optimal ET sequence: 5, 14, 19, 24, 67dd, 91dd, 115ddd

Scales: semaphore5, semaphore9, semaphore14

Overview to extensions

The second comma of the comma list defines which 7-limit family member we are looking at.

Godzilla adds 81/80. Immunity adds 2240/2187. Superpelog adds 135/128. Beep adds 21/20. Baba adds 16/15. These all use the same nominal generator as semaphore, though some of them are of very low accuracy.

Decimal adds 25/24, splitting the octave in two. Negri adds 225/224, splitting the hemifourth in two. Triforce adds 128/125, splitting the octave in three. Keemun adds 126/125, splitting the hemitwelfth in three. Nautilus adds 250/243, splitting the hemifourth in three. Nuke is like nautilus, but adds 3584/3375 instead. Hemidim adds 648/625 with a 1/4-octave period. Blacksmith adds 28/27, splitting the octave in five. Spell adds 3125/3072, splitting the hemitwelfth in five. Hemiripple adds 6561/6250, splitting the hemifourth in five. Finally, mabila adds 28672/28125 and splits an interval of two octaves plus a hemifourth in five.

Discussed elsewhere are

Considered below are godzilla, superpelog, negri, nuke, mabila, and hemiripple.

Godzilla

Deutsch

Main article: Semaphore and godzilla

Godzilla tempers out 81/80, equating 9/8 and 10/9, so it finds the prime 5 at a stack of four fifths, as does any temperament in the meantone family. 19edo is close to being the optimal generator tuning; hence it can be more or less equated with taking 4\19 as a generator. Mos scales are of 5, 9, or 14 notes.

Subgroup: 2.3.5.7

Comma list: 49/48, 81/80

Mapping: [⟨1 0 -4 2], ⟨0 2 8 1]]

mapping generators: ~2, ~7/4

Wedgie: ⟨⟨ 2 8 1 8 -4 -20 ]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~7/4 = 948.7959
  • POTE: ~2 = 1\1, ~7/4 = 947.365

Tuning ranges:

Optimal ET sequence: 5, 14c, 19

Badness: 0.026747

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 49/48, 81/80

Mapping: [⟨1 0 -4 2 -6], ⟨0 2 8 1 12]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~7/4 = 947.4563
  • POTE: ~2 = 1\1, ~7/4 = 945.973

Tuning ranges:

  • 11-odd-limit diamond monotone: ~7/4 = [942.857, 947.368] (11\14 to 15\19)
  • 11-odd-limit diamond tradeoff: ~7/4 = [933.129, 968.826]

Optimal ET sequence: 14c, 19, 33cd

Badness: 0.028947

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 49/48, 78/77, 81/80

Mapping: [⟨1 0 -4 2 -6 -5], ⟨0 2 8 1 12 11]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~7/4 = 947.8877
  • POTE: ~2 = 1\1, ~7/4 = 946.397

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~7/4 = 947.368 (15\19)
  • 13- and 15-odd-limit diamond tradeoff: ~7/4 = [910.890, 968.826]

Optimal ET sequence: 14cf, 19, 33cdff

Badness: 0.022503

Semafour

Subgroup: 2.3.5.7.11

Comma list: 33/32, 49/48, 55/54

Mapping: [⟨1 0 -4 2 5], ⟨0 2 8 1 -2]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~7/4 = 948.2089
  • POTE: ~2 = 1\1, ~7/4 = 945.958

Optimal ET sequence: 14c, 19e, 33cdee, 52cdeee

Badness: 0.028510

Varan

Subgroup: 2.3.5.7.11

Comma list: 49/48, 77/75, 81/80

Mapping: [⟨1 0 -4 2 -10], ⟨0 2 8 1 17]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~7/4 = 949.6160
  • POTE: ~2 = 1\1, ~7/4 = 948.921

Optimal ET sequence: 19e, 24, 43de

Badness: 0.039647

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 66/65, 77/75, 81/80

Mapping: [⟨1 0 -4 2 -10 -5], ⟨0 2 8 1 17 11]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~7/4 = 949.5255
  • POTE: ~2 = 1\1, ~7/4 = 948.835

Optimal ET sequence: 19e, 24, 43de

Badness: 0.025676

Baragon

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 81/80

Mapping: [⟨1 0 -4 2 9], ⟨0 2 8 1 -7]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~7/4 = 949.0311
  • POTE: ~2 = 1\1, ~7/4 = 948.827

Optimal ET sequence: 5, 19, 24

Badness: 0.035673

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 81/80, 91/90

Mapping: [⟨1 0 -4 2 9 -5], ⟨0 2 8 1 -7 11]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~7/4 = 949.0670
  • POTE: ~2 = 1\1, ~7/4 = 948.802

Optimal ET sequence: 5, 19, 24

Badness: 0.026703

Helayo

For the 5-limit version of this temperament see High badness temperaments #Hogzilla.

Subgroup: 2.3.5.7

Comma list: 49/48, 3645/3584

Mapping: [⟨1 0 11 2], ⟨0 2 -11 1]]

Wedgie: ⟨⟨ 2 -6 1 -14 -4 19 ]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~7/4 = 947.0969

Optimal ET sequence: 5c, 14, 19

Badness: 0.0791

Music

Superpelog

Subgroup: 2.3.5.7

Comma list: 49/48, 135/128

Mapping: [⟨1 0 7 2], ⟨0 2 -6 1]]

Wedgie: ⟨⟨ 2 -6 1 -14 -4 19 ]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~7/4 = 939.0297
  • POTE: ~2 = 1\1, ~7/4 = 940.048

Optimal ET sequence: 9, 14c, 23d, 37bcd, 60bbccdd

Badness: 0.058216

11-limit

Subgroup: 2.3.5.7.11

Comma list: 33/32, 45/44, 49/48

Mapping: [⟨1 0 7 2 5], ⟨0 2 -6 1 -2]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~7/4 = 938.4673
  • POTE: ~2 = 1\1, ~7/4 = 940.041

Optimal ET sequence: 9, 14c, 23de, 37bcde

Badness: 0.028535

Music
Mindaugas Rex Lithuaniae by Chris Vaisvil (blog) (superpelog[9] in 23edo tuning)

Baba

Subgroup: 2.3.5.7

Comma list: 16/15, 49/45

Mapping: [⟨1 0 4 2], ⟨0 2 -2 1]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~7/4 = 973.296

Wedgie: ⟨⟨ 2 -2 1 -8 -4 8 ]]

Optimal ET sequence: 5, 11b, 16bc

Badness: 0.044321

11-limit

Subgroup: 2.3.5.7.11

Comma list: 16/15, 22/21, 49/45

Mapping: [⟨1 0 4 2 1], ⟨0 2 -2 1 3]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~7/4 = 978.164

Optimal ET sequence: 5, 11b, 16bc, 27bbcc

Badness: 0.036538

Negri

Main article: Negri

Negri tempers out the negri comma in the 5-limit, 49/48 and 225/224 in the 7-limit. It can be extended naturally to the 2.3.5.7.13 subgroup by adding 91/90 to the comma list; this will be discussed below under the title of negra.

Subgroup: 2.3.5

Comma list: 16875/16384

Mapping: [⟨1 2 2], ⟨0 -4 3]]

mapping generators: ~2, ~16/15

Wedgie: ⟨⟨ 4 -3 -14 ]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~16/15 = 125.7549

Optimal ET sequence: 9, 10, 19, 67c, 86c, 105c

Badness: 0.086856

7-limit

Subgroup: 2.3.5.7

Comma list: 49/48, 225/224

Mapping: [⟨1 2 2 3], ⟨0 -4 3 -2]]

Wedgie: ⟨⟨ 4 -3 2 -14 -8 13 ]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~15/14 = 125.608

Optimal ET sequence: 9, 10, 19, 48d, 67cdd, 86cdd

Badness: 0.026483

2.3.5.7.13 subgroup (negra)

Subgroup: 2.3.5.7.13

Comma list: 49/48, 65/64, 91/90

Sval mapping: [⟨1 2 2 3 4], ⟨0 -4 3 -2 -3]]

Gencom mapping: [⟨1 2 2 3 0 4], ⟨0 -4 3 -2 0 -3]]

gencom: [2 14/13; 49/48 65/64 91/90]

Optimal tunings:

  • POTE: ~2 = 1\1, ~14/13 = 125.567

Optimal ET sequence: 9, 10, 19, 48df, 67cddf, 86cddff

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 49/48, 56/55

Mapping: [⟨1 2 2 3 4], ⟨0 -4 3 -2 -5]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~15/14 = 126.474

Optimal ET sequence: 9, 10, 19

Badness: 0.026190

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 49/48, 56/55, 78/77

Mapping: [⟨1 2 2 3 4 4], ⟨0 -4 3 -2 -5 -3]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~14/13 = 126.431

Optimal ET sequence: 9, 10, 19

Badness: 0.017833

Negril

Subgroup: 2.3.5.7.11

Comma list: 49/48, 100/99, 225/224

Mapping: [⟨1 2 2 3 2], ⟨0 -4 3 -2 14]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~15/14 = 124.767

Optimal ET sequence: 19, 29, 48d, 77cdd

Badness: 0.038679

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 65/64, 91/90, 875/858

Mapping: [⟨1 2 2 3 2 4], ⟨0 -4 3 -2 14 -3]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~14/13 = 124.716

Optimal ET sequence: 19, 29, 48df, 77cddf

Badness: 0.024383

Negric

Subgroup: 2.3.5.7.11

Comma list: 33/32, 49/48, 77/75

Mapping: [⟨1 2 2 3 3], ⟨0 -4 3 -2 4]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~15/14 = 127.039

Optimal ET sequence: 9, 19e

Badness: 0.030617

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 33/32, 49/48, 65/64, 91/90

Mapping: [⟨1 2 2 3 3 4], ⟨0 -4 3 -2 4 -3]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~14/13 = 127.039

Optimal ET sequence: 9, 19e

Badness: 0.020205

Negroni

Subgroup: 2.3.5.7.11

Comma list: 49/48, 55/54, 225/224

Mapping: [⟨1 2 2 3 5], ⟨0 -4 3 -2 -15]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~15/14 = 124.539

Optimal ET sequence: 10, 19e, 29, 77cddee

Badness: 0.035296

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 55/54, 65/64, 91/90

Mapping: [⟨1 2 2 3 5 4], ⟨0 -4 3 -2 -15 -3]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~14/13 = 124.545

Optimal ET sequence: 10, 19e, 29, 77cddeef

Badness: 0.021559

Wilsec

Subgroup: 2.3.5.7.11

Comma list: 49/48, 121/120, 225/224

Mapping: [⟨1 6 -1 5 4], ⟨0 -8 6 -4 -1]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~11/8 = 537.186

Optimal ET sequence: 9, 20, 29, 38d, 67cdde

Badness: 0.041886

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 65/64, 91/90, 121/120

Mapping: [⟨1 6 -1 5 4 7], ⟨0 -8 6 -4 -1 -6]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~11/8 = 537.208

Optimal ET sequence: 9, 20, 29, 38df, 67cddef

Badness: 0.025192

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 49/48, 65/64, 91/90, 121/120, 154/153

Mapping: [⟨1 6 -1 5 4 7 -2], ⟨0 -8 6 -4 -1 -6 11]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~11/8 = 537.230

Optimal ET sequence: 9, 20g, 29g, 38df, 67cddefg

Badness: 0.021778

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153

Mapping: [⟨1 6 -1 5 4 7 -2 7], ⟨0 -8 6 -4 -1 -6 11 -5]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~11/8 = 537.214

Optimal ET sequence: 9, 20g, 29g, 38df, 67cddefgh

Badness: 0.016828

Nuke

Subgroup: 2.3.5.7

Comma list: 49/48, 3584/3375

Mapping: [⟨1 2 2 3], ⟨0 -6 5 -3]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~16/15 = 80.9538

Optimal ET sequence: 14, 15, 44cd

Badness: 0.129339

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 77/75, 512/495

Mapping: [⟨1 2 2 3 3], ⟨0 -6 5 -3 7]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~16/15 = 80.8171

Optimal ET sequence: 14e, 15

Badness: 0.069398

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 66/65, 77/75, 448/429

Mapping: [⟨1 2 2 3 3 4], ⟨0 -6 5 -3 7 -4]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~16/15 = 81.0243

Optimal ET sequence: 14e, 15, 44cdeff

Badness: 0.048553

Mabila

See also: Mabila family

Subgroup: 2.3.5.7

Comma list: 49/48, 28672/28125

Mapping: [⟨1 6 1 5], ⟨0 -10 3 -5]]

Wedgie: ⟨⟨ 10 -3 5 -28 -20 20 ]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~75/56 = 529.667

Optimal ET sequence: 9, 25, 34

Badness: 0.133638

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 1350/1331

Mapping: [⟨1 6 1 5 7], ⟨0 -10 3 -5 -8]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~15/11 = 529.729

Optimal ET sequence: 9, 25e, 34

Badness: 0.061501

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 847/845

Mapping: [⟨1 6 1 5 7 9], ⟨0 -10 3 -5 -8 -12]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~15/11 = 529.763

Optimal ET sequence: 9, 25e, 34

Badness: 0.037270

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 49/48, 56/55, 91/90, 154/153, 375/374

Mapping: [⟨1 6 1 5 7 9 1], ⟨0 -10 3 -5 -8 -12 7]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~15/11 = 529.695

Optimal ET sequence: 9, 25e, 34

Badness: 0.031888

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 49/48, 56/55, 76/75, 91/90, 154/153, 190/187

Mapping: [⟨1 6 1 5 7 9 1 6], ⟨0 -10 3 -5 -8 -12 7 -4]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~15/11 = 529.736

Optimal ET sequence: 9, 25e, 34

Badness: 0.026981

Hemiripple

See also: Ripple family § Hemiripple

Subgroup: 2.3.5.7

Comma list: 49/48, 6561/6250

Mapping: [⟨1 2 3 3], ⟨0 -10 -16 -5]]

Wedgie: ⟨⟨ 10 16 5 2 -20 -33 ]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~36/35 = 50.826

Optimal ET sequence: 23d, 24, 47d, 71bdd

Badness: 0.175113

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 121/120, 567/550

Mapping: [⟨1 2 3 3 4], ⟨0 -10 -16 -5 -13]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~36/35 = 50.826

Optimal ET sequence: 23de, 24, 47de, 71bdde

Badness: 0.066834

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 66/65, 121/120, 351/350

Mapping: [⟨1 2 3 3 4 4], ⟨0 -10 -16 -5 -13 -7]]

Optimal tunings:

  • POTE: ~2 = 1\1, ~36/35 = 50.635

Optimal ET sequence: 23de, 24, 47de, 71bdde

Badness: 0.046588

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