Semaphoresmic clan

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This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The semaphoresmic clan (or semaphore family) of temperaments tempers out the large septimal diesis, or semaphoresma, 49/48, a triprime comma with factors of 2, 3 and 7.

This article focuses on rank-2 temperaments. See Semaphoresmic family for the rank-3 temperament resulting from tempering out 49/48 alone in the full 7-limit.

Semaphore

Main article: Semaphore and godzilla

Semaphore tempers out 49/48, and splits a perfect twelfth into two halfs of 7/4~12/7, and a perfect fourth into two halfs of 7/6~8/7, hence the name semaphore, which sounds like semifourth; its ploidacot is alpha-dicot. 19edo and 24edo are among the possible edo tunings.

Subgroup: 2.3.7

Comma list: 49/48

Sval mapping: [⟨1 0 2], ⟨0 2 1]]

sval mapping generators: ~2, ~7/4

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

gencom: [2 7/4; 49/48]

Optimal tunings:

  • WE: ~2 = 1202.8324 ¢, ~7/4 = 951.8567 ¢
error map: ⟨+2.832 +1.758 -11.304]
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 950.6890 ¢
error map: ⟨0.000 -0.577 -18.137]

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

Badness (Sintel): 0.193

Scales: semaphore5, semaphore9, semaphore14

Overview to extensions

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

  • Beep adds 21/20, for a tuning flat of 9edo;
  • Superpelog adds 135/128, for a tuning between 9edo and 14c-edo;
  • Godzilla adds 81/80, for a tuning between 14c-edo and 24edo;
  • Helayo adds 3645/3584, for a tuning between 14edo and 24c-edo;
  • Immunity adds 2240/2187, for a tuning sharp of 29edo;
  • Baba adds 16/15, for a niche exotemperament well tuned around 11b-edo.

These all use the same nominal generator as semaphore, though some of them are of very low accuracy.

Decimal adds 25/24. Anguirus adds 2048/2025. Those split 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, semabila adds 28672/28125 and splits an interval of two octaves plus a hemifourth in five.

Discussed elsewhere are

Considered below are godzilla, helayo, superpelog, baba, negri, and nuke.

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

Optimal tunings:

  • WE: ~2 = 1203.8275 ¢, ~7/4 = 950.3867 ¢
error map: ⟨+3.827 -1.182 +1.470 -10.784]
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 947.8216 ¢
error map: ⟨0.000 -6.312 -3.741 -21.004]

Tuning ranges:

Optimal ET sequence: 5, 14c, 19

Badness (Sintel): 0.677

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:

  • WE: ~2 = 1204.4129 ¢, ~7/4 = 949.4513 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 946.4361 ¢

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 (Sintel): 0.957

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:

  • WE: ~2 = 1203.7164 ¢, ~7/4 = 949.2061 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 946.4131 ¢

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 (Sintel): 0.930

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:

  • WE: ~2 = 1206.9595 ¢, ~7/4 = 951.4440 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 946.4472 ¢

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

Badness (Sintel): 0.943

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:

  • WE: ~2 = 1202.5842 ¢, ~7/4 = 950.9647 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 949.1239 ¢

Optimal ET sequence: 19e, 24, 43de

Badness (Sintel): 1.31

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:

  • WE: ~2 = 1202.4367 ¢, ~7/4 = 950.7615 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 949.0338 ¢

Optimal ET sequence: 19e, 24, 43de

Badness (Sintel): 1.06

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:

  • WE: ~2 = 1201.1412 ¢, ~7/4 = 949.7291 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 948.8625 ¢

Optimal ET sequence: 5, 19, 24

Badness (Sintel): 1.18

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:

  • WE: ~2 = 1201.1228 ¢, ~7/4 = 949.6894 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 948.8468 ¢

Optimal ET sequence: 5, 19, 24

Badness (Sintel): 1.10

Helayo

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

Helayo tempers out 3645/3584 and may be thought of as the opposite of godzilla with respect to 19edo. Like godzilla, 19edo's generator is close to the optimum.

Subgroup: 2.3.5.7

Comma list: 49/48, 3645/3584

Mapping: [⟨1 0 11 2], ⟨0 2 -11 1]]

Optimal tunings:

  • WE: ~2 = 1204.0199 ¢, ~7/4 = 950.7917 ¢
error map: ⟨+4.020 -0.372 -0.804 -9.995]
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 947.5047 ¢
error map: ⟨0.000 -6.946 -8.866 -21.321]

Optimal ET sequence: 5c, 14, 19

Badness (Sintel): 2.00

Music

Superpelog

Superpelog tempers out 135/128 and finds the prime 5 at a stack of three fourths, as does any temperament in the mavila family. It may be described as 9 & 14c, with 23edo (23d val) giving a tuning close to the optimum.

Subgroup: 2.3.5.7

Comma list: 49/48, 135/128

Mapping: [⟨1 0 7 2], ⟨0 2 -6 1]]

Optimal tunings:

  • WE: ~2 = 1208.8222 ¢, ~7/4 = 946.9590 ¢
error map: ⟨+8.822 -8.037 -6.313 -4.223]
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 939.8419 ¢
error map: ⟨0.000 -22.271 -25.365 -28.984]

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

Badness (Sintel): 1.47

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:

  • WE: ~2 = 1208.8663 ¢, ~7/4 = 946.9861 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 939.7687 ¢

Optimal ET sequence: 9, 14c, 23de, 37bcde, 60bbccddeee

Badness (Sintel): 0.943

Music
Mindaugas Rex Lithuaniae (2012) by Chris Vaisvil – listen | blog – in Superpelog[9], 23edo tuning

Baba

This low-accuracy extension tempers out 16/15, so the perfect fifth stands in for ~8/5 as in father.

Subgroup: 2.3.5.7

Comma list: 16/15, 49/45

Mapping: [⟨1 0 4 2], ⟨0 2 -2 1]]

Optimal tunings:

  • WE: ~2 = 1184.7407 ¢, ~7/4 = 960.9196 ¢
error map: ⟨-15.259 +19.884 +30.810 -38.425]
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 972.2994 ¢
error map: ⟨0.000 +42.644 +69.088 +3.473]

Optimal ET sequence: 5, 11b, 16bc

Badness (Sintel): 1.12

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:

  • WE: ~2 = 1187.4876 ¢, ~7/4 = 967.9643 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 976.9298 ¢

Optimal ET sequence: 5, 11b

Badness (Sintel): 1.21

Negri

Main article: Negri
For the 5-limit version, see Syntonic–kleismic equivalence continuum #Negri (5-limit).

Negri tempers out the negri comma in the 5-limit, 49/48 and 225/224 in the 7-limit. It may be described as 9 & 10; its ploidacot is omega-tetracot. 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.

7-limit

Subgroup: 2.3.5.7

Comma list: 49/48, 225/224

Mapping: [⟨1 2 2 3], ⟨0 -4 3 -2]]

Optimal tunings:

  • WE: ~2 = 1203.4810 ¢, ~15/14 = 125.9724 ¢
error map: ⟨+3.481 +1.118 -1.435 -10.328]
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 125.4347 ¢
error map: ⟨0.000 -3.694 -10.009 -19.695]

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

Badness (Sintel): 0.670

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:

  • WE: ~2 = 1203.6981 ¢, ~14/13 = 125.9545 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 125.3543 ¢

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

Badness (Sintel): 0.463

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:

  • WE: ~2 = 1202.1045 ¢, ~15/14 = 126.6961 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 126.3382 ¢

Optimal ET sequence: 9, 10, 19

Badness (Sintel): 0.866

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:

  • WE: ~2 = 1202.6035 ¢, ~14/13 = 126.7054 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 126.2534 ¢

Optimal ET sequence: 9, 10, 19

Badness (Sintel): 0.737

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:

  • WE: ~2 = 1202.7081 ¢, ~15/14 = 125.0491 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 124.8066 ¢

Optimal ET sequence: 10e, 19, 29, 48d, 77cdd

Badness (Sintel): 1.28

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:

  • WE: ~2 = 1202.9319 ¢, ~14/13 = 125.0204 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 124.7374 ¢

Optimal ET sequence: 10e, 19, 29, 48df, 77cddf

Badness (Sintel): 1.01

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:

  • WE: ~2 = 1205.7810 ¢, ~15/14 = 127.6513 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 126.9620 ¢

Optimal ET sequence: 9, 19e

Badness (Sintel): 1.01

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:

  • WE: ~2 = 1205.7833 ¢, ~14/13 = 127.6507 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 126.9093 ¢

Optimal ET sequence: 9, 19e

Badness (Sintel): 0.835

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:

  • WE: ~2 = 1203.4738 ¢, ~15/14 = 124.8992 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 124.3642 ¢

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

Badness (Sintel): 1.17

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:

  • WE: ~2 = 1203.5354 ¢, ~14/13 = 124.9118 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 124.3733 ¢

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

Badness (Sintel): 0.890

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]]

mapping generators: ~2, ~16/11

Optimal tunings:

  • WE: ~2 = 1203.6080 ¢, ~11/8 = 538.8007 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 537.2654 ¢

Optimal ET sequence: 9, 20, 29, 38d, 67cdde, 105cdddee

Badness (Sintel): 1.38

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:

  • WE: ~2 = 1203.7672 ¢, ~11/8 = 538.8948 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 537.3053 ¢

Optimal ET sequence: 9, 20, 29, 38df, 67cddef, 105cdddeefff

Badness (Sintel): 1.04

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:

  • WE: ~2 = 1203.7154 ¢, ~11/8 = 538.8932 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 537.2633 ¢

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

Badness (Sintel): 1.11

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:

  • WE: ~2 = 1203.5906 ¢, ~11/8 = 538.8216 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 537.2534 ¢

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

Badness (Sintel): 1.02

Nuke

Nuke tempers out 3584/3375 and is the 14 & 15 temperament. It splits the hemifourth into three generators of ~16/15. Its ploidacot is omega-hexacot. 15edo is about as accurate as it can be tuned.

Subgroup: 2.3.5.7

Comma list: 49/48, 3584/3375

Mapping: [⟨1 2 2 3], ⟨0 -6 5 -3]]

Optimal tunings:

  • WE: ~2 = 1197.0059 ¢, ~16/15 = 80.7519 ¢
error map: ⟨-2.994 +7.546 +11.457 -20.064]
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 81.0408 ¢
error map: ⟨0.000 +11.800 +18.890 -11.948]

Optimal ET sequence: 14, 15

Badness (Sintel): 3.27

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:

  • WE: ~2 = 1196.6821 ¢, ~16/15 = 80.5936 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 80.8326 ¢

Optimal ET sequence: 14e, 15

Badness (Sintel): 2.29

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:

  • WE: ~2 = 1195.6248 ¢, ~16/15 = 80.7288 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 81.0685 ¢

Optimal ET sequence: 14e, 15

Badness (Sintel): 2.01

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