Semaphoresmic clan: Difference between revisions
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{{Technical data page}} | {{Technical data page}} | ||
The '''semaphoresmic clan''' (or '''semaphore family''') [[tempering out|tempers out]] the large septimal diesis, or semaphoresma, [[49/48]], a triprime comma with factors of 2, 3 and 7. | The '''semaphoresmic clan''' (or '''semaphore family''') of [[regular temperament|temperaments]] [[tempering out|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 == | == Semaphore == | ||
{{Main| Semaphore and godzilla }} | {{Main| Semaphore and godzilla }} | ||
Semaphore tempers out 49/48, and splits a [[3/1|perfect twelfth]] into two halfs of [[7/4]][[~]][[12/7]], and a [[4/3|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 | [[Subgroup]]: 2.3.7 | ||
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=== Overview to extensions === | === Overview to extensions === | ||
The second comma of the comma list defines which 7-limit family member we are looking at. | 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]]; | |||
* Helayo adds [[3645/3584]]; | |||
* 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]]. 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. | 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. | ||
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{{Main| Semaphore and godzilla }} | {{Main| 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 scale]]s are of 5, 9, or 14 notes. | 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 scale]]s are of 5, 9, or 14 notes. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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== Helayo == | == Helayo == | ||
: ''For the 5-limit version of this temperament see [[ | : ''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 | [[Subgroup]]: 2.3.5.7 | ||
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== Superpelog == | == 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 {{nowrap| 9 & 14c }}, with [[23edo]] (23d val) giving a tuning close to the optimum. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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; Music | ; Music | ||
: ''Mindaugas Rex Lithuaniae'' (2012) by [[Chris Vaisvil]] – [https://web.archive.org/web/20201127013438/http://micro.soonlabel.com/MOS/20120418-9mos-mindaugas.mp3 listen] | [https://www.chrisvaisvil.com/mindaugas-rex-lithuaniae/ blog] – Superpelog[9] | : ''Mindaugas Rex Lithuaniae'' (2012) by [[Chris Vaisvil]] – [https://web.archive.org/web/20201127013438/http://micro.soonlabel.com/MOS/20120418-9mos-mindaugas.mp3 listen] | [https://www.chrisvaisvil.com/mindaugas-rex-lithuaniae/ blog] – in Superpelog[9], 23edo tuning | ||
== Baba == | == 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 | [[Subgroup]]: 2.3.5.7 | ||
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{{Main| Negri }} | {{Main| Negri }} | ||
Negri tempers out the [[negri comma]] in the 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 {{nowrap| 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. | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
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== Nuke == | == Nuke == | ||
Nuke tempers out 3584/3375 and is the {{nowrap| 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 | [[Subgroup]]: 2.3.5.7 | ||
Revision as of 11:45, 16 June 2025
- 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
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
Subgroup-val 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]
- CTE: ~2 = 1200.000, ~7/4 = 952.295
- error map: ⟨0.000 +2.635 -16.531]
- POTE: ~2 = 1200.000, ~7/4 = 949.615
- error map: ⟨0.000 -2.724 -19.211]
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;
- Helayo adds 3645/3584;
- 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. 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
- Beep (+21/20) → Bug family
- Immunity (+2240/2187) → Immunity family
- Nessus (+10/9) → Very low accuracy temperaments
- Malacoda (+15/14) → Very low accuracy temperaments
- Decimal (+25/24) → Dicot family
- Anguirus (+2048/2025) → Diaschismic family
- Triforce (+128/125) → Augmented family
- Keemun (+126/125) → Kleismic family
- Nautilus (+250/243) → Porcupine family
- Hemidim (+648/625) → Dimipent family
- Blacksmith (+28/27) → Limmic temperaments
- Spell (+3125/3072) → Hemimean clan
- Hemiripple (+6561/6250) → Ripple family
- Semabila (+28672/28125) → Mabila family
Considered below are godzilla, helayo, superpelog, baba, negri, and nuke.
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
- CTE: ~2 = 1200.000, ~7/4 = 948.796
- error map: ⟨0.000 -4.363 +4.054 -20.030]
- POTE: ~2 = 1200.000, ~7/4 = 947.365
- error map: ⟨0.000 -7.225 -7.394 -21.461]
- 7- and 9-odd-limit diamond monotone: ~7/4 = [942.857, 960.000] (11\14 to 4\5)
- 7- and 9-odd-limit diamond tradeoff: ~7/4 = [933.129, 968.826]
Optimal ET sequence: 5, 14c, 19
Badness (Smith): 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 = 1200.000, ~7/4 = 947.456
- POTE: ~2 = 1200.000, ~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 (Smith): 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 = 1200.000, ~7/4 = 947.888
- POTE: ~2 = 1200.000, ~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 (Smith): 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 = 1200.000, ~7/4 = 948.209
- POTE: ~2 = 1200.000, ~7/4 = 945.958
Optimal ET sequence: 14c, 19e, 33cdee, 52cdeee
Badness (Smith): 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 = 1200.000, ~7/4 = 949.616
- POTE: ~2 = 1200.000, ~7/4 = 948.921
Optimal ET sequence: 19e, 24, 43de
Badness (Smith): 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 = 1200.000, ~7/4 = 949.525
- POTE: ~2 = 1200.000, ~7/4 = 948.835
Optimal ET sequence: 19e, 24, 43de
Badness (Smith): 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 = 1200.000, ~7/4 = 949.031
- POTE: ~2 = 1200.000, ~7/4 = 948.827
Optimal ET sequence: 5, 19, 24
Badness (Smith): 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 = 1200.000, ~7/4 = 949.067
- POTE: ~2 = 1200.000, ~7/4 = 948.802
Optimal ET sequence: 5, 19, 24
Badness (Smith): 0.026703
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]]
- CTE: ~2 = 1200.000, ~7/4 = 947.097
- error map: ⟨0.000 -7.761 -4.380 -21.729]
- CWE: ~2 = 1200.000, ~7/4 = 947.505
- error map: ⟨0.000 -6.946 -8.866 -21.321]
Optimal ET sequence: 5c, 14, 19
Badness (Smith): 0.0791
- 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]]
- CTE: ~2 = 1200.000, ~7/4 = 939.030
- error map: ⟨0.000 -23.896 -20.492 -29.796]
- POTE: ~2 = 1200.000, ~7/4 = 940.048
- error map: ⟨0.000 -21.859 -26.602 -28.778]
Optimal ET sequence: 9, 14c, 23d, 37bcd, 60bbccdd
Badness (Smith): 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 = 1200.000, ~7/4 = 938.467
- POTE: ~2 = 1200.000, ~7/4 = 940.041
Optimal ET sequence: 9, 14c, 23de, 37bcde
Badness (Smith): 0.028535
- 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]]
- CTE: ~2 = 1200.000, ~7/4 = 968.739
- error map: ⟨0.000 +35.523 +76.208 -0.087]
- POTE: ~2 = 1200.000, ~7/4 = 973.296
- error map: ⟨0.000 +42.644 +69.088 +3.473]
Optimal ET sequence: 5, 11b, 16bc
Badness (Smith): 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:
- CTE: ~2 = 1200.000, ~7/4 = 972.258
- POTE: ~2 = 1200.000, ~7/4 = 978.164
Badness (Smith): 0.036538
Negri
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.
Subgroup: 2.3.5
Comma list: 16875/16384
Mapping: [⟨1 2 2], ⟨0 -4 3]]
- mapping generators: ~2, ~16/15
- CTE: ~2 = 1200.000, ~16/15 = 125.396
- error map: ⟨0.000 -3.539 -10.126]
- POTE: ~2 = 1200.000, ~16/15 = 125.755
- error map: ⟨0.000 -4.975 -9.049]
Optimal ET sequence: 9, 10, 19, 67c, 86c, 105c
Badness (Smith): 0.086856
7-limit
Subgroup: 2.3.5.7
Comma list: 49/48, 225/224
Mapping: [⟨1 2 2 3], ⟨0 -4 3 -2]]
- CTE: ~2 = 1200.000, ~15/14 = 124.813
- error map: ⟨0.000 -1.209 -11.874 -18.453]
- POTE: ~2 = 1200.000, ~15/14 = 125.608
- error map: ⟨0.000 -4.387 -9.490 -20.042]
Optimal ET sequence: 9, 10, 19, 48d, 67cdd, 86cdd
Badness (Smith): 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:
- CTE: ~2 = 1200.000, ~14/13 = 124.457
- POTE: ~2 = 1200.000, ~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:
- CTE: ~2 = 1200.000, ~15/14 = 125.780
- POTE: ~2 = 1200.000, ~15/14 = 126.474
Optimal ET sequence: 9, 10, 19
Badness (Smith): 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:
- CTE: ~2 = 1200.000, ~14/13 = 125.434
- POTE: ~2 = 1200.000, ~14/13 = 126.431
Optimal ET sequence: 9, 10, 19
Badness (Smith): 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:
- CTE: ~2 = 1200.000, ~15/14 = 124.998
- POTE: ~2 = 1200.000, ~15/14 = 124.767
Optimal ET sequence: 10e, 19, 29, 48d, 77cdd
Badness (Smith): 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:
- CTE: ~2 = 1200.000, ~14/13 = 124.865
- POTE: ~2 = 1200.000, ~14/13 = 124.716
Optimal ET sequence: 10e, 19, 29, 48df, 77cddf
Badness (Smith): 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:
- CTE: ~2 = 1200.000, ~15/14 = 126.574
- POTE: ~2 = 1200.000, ~15/14 = 127.039
Badness (Smith): 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:
- CTE: ~2 = 1200.000, ~14/13 = 126.153
- POTE: ~2 = 1200.000, ~14/13 = 127.039
Badness (Smith): 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:
- CTE: ~2 = 1200.000, ~15/14 = 123.735
- POTE: ~2 = 1200.000, ~15/14 = 124.539
Optimal ET sequence: 10, 19e, 29, 77cddee
Badness (Smith): 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:
- CTE: ~2 = 1200.000, ~15/14 = 123.644
- POTE: ~2 = 1200.000, ~14/13 = 124.545
Optimal ET sequence: 10, 19e, 29, 77cddeef
Badness (Smith): 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]]
- mapping generators: ~2, ~16/11
Optimal tunings:
- CTE: ~2 = 1200.000, ~11/8 = 537.627
- POTE: ~2 = 1200.000, ~11/8 = 537.186
Optimal ET sequence: 9, 20, 29, 38d, 67cdde, 105cdddee
Badness (Smith): 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:
- CTE: ~2 = 1200.000, ~11/8 = 537.802
- POTE: ~2 = 1200.000, ~11/8 = 537.208
Optimal ET sequence: 9, 20, 29, 38df, 67cddef, 105cdddeefff
Badness (Smith): 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:
- CTE: ~2 = 1200.000, ~11/8 = 537.492
- POTE: ~2 = 1200.000, ~11/8 = 537.230
Optimal ET sequence: 9, 20g, 29g, 38df, 67cddefg
Badness (Smith): 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:
- CTE: ~2 = 1200.000, ~11/8 = 537.553
- POTE: ~2 = 1200.000, ~11/8 = 537.214
Optimal ET sequence: 9, 20g, 29g, 38df, 67cddefgh
Badness (Smith): 0.016828
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]]
- CTE: ~2 = 1200.000, ~16/15 = 81.345
- error map: ⟨0.000 +9.975 +20.411 -12.861]
- POTE: ~2 = 1200.000, ~16/15 = 80.954
- error map: ⟨0.000 +12.322 +18.456 -11.688]
Badness (Smith): 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:
- CTE: ~2 = 1200.000, ~16/15 = 80.908
- POTE: ~2 = 1200.000, ~16/15 = 80.817
Badness (Smith): 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:
- CTE: ~2 = 1200.000, ~16/15 = 81.320
- POTE: ~2 = 1200.000, ~16/15 = 81.024
Badness (Smith): 0.048553