Semaphoresmic clan: Difference between revisions

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== Semaphore ==
== Semaphore ==
{{Main| Semaphore and godzilla }}
{{Main| Semaphore and godzilla }}
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum]].''


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.  
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.  
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{{Mapping|legend=2| 1 0 2 | 0 2 1 }}
{{Mapping|legend=2| 1 0 2 | 0 2 1 }}
: sval mapping generators: ~2, ~7/4


{{Mapping|legend=3| 1 0 0 2 | 0 2 0 1 }}
{{Mapping|legend=3| 1 0 0 2 | 0 2 0 1 }}


: [[gencom]]: [2 7/4; 49/48]
: mapping generators: ~2, ~7/4


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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These all use the same nominal generator as semaphore, though some of them are of very low accuracy.  
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. Blackwood adds [[28/27]], with a 1/5-octave period. 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  
Discussed elsewhere are  
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* [[Keemun]] (+126/125) → [[Kleismic family #Keemun|Kleismic family]]
* [[Keemun]] (+126/125) → [[Kleismic family #Keemun|Kleismic family]]
* ''[[Nautilus]]'' (+250/243) → [[Porcupine family #Nautilus|Porcupine family]]
* ''[[Nautilus]]'' (+250/243) → [[Porcupine family #Nautilus|Porcupine family]]
* ''[[Hemidim]]'' (+648/625) → [[Dimipent family #Hemidim|Dimipent family]]
* ''[[Hemidim]]'' (+648/625) → [[Diminished family #Hemidim|Diminished family]]
* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* ''[[Spell]]'' (+3125/3072) → [[Hemimean clan #Spell|Hemimean clan]]
* ''[[Spell]]'' (+3125/3072) → [[Hemimean clan #Spell|Hemimean clan]]
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Considered below are godzilla, helayo, superpelog, baba, negri, and nuke.
Considered below are godzilla, helayo, superpelog, baba, negri, and nuke.
=== Semaerophore ===
Named by [[CompactStar]] in 2023, this extension tempers out [[729/704]]. It is the no-5 [[restriction]] of undecimal godzilla.
Subgroup: 2.3.7.11
Comma list: 49/48, 729/704
Subgroup-val mapping: {{mapping| 1 0 2 -6 | 0 2 1 12 }}
Optimal tunings:
* WE: ~2 = 1204.9027{{c}}, ~7/4 = 948.7772{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 945.4959{{c}}
{{Optimal ET sequence|legend=0| 14, 33d, 47de }}
Badness (Sintel): 1.27
==== 2.3.7.11.19 subgroup ====
Subgroup: 2.3.7.11.19
Comma list: 49/48, 77/76, 729/704
Subgroup-val mapping: {{mapping| 1 0 2 -6 -6 | 0 2 1 12 13 }}
Optimal tunings:
* WE: ~2 = 1204.9645{{c}}, ~7/4 = 948.5749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 945.2236{{c}}
{{Optimal ET sequence|legend=0| 14, 33d, 47deh }}
Badness (Sintel): 1.08


== Godzilla ==
== Godzilla ==
<span style="display: block; text-align: right;">[[:de:Semiphor, Semaphor, Godzilla|Deutsch]]</span>
{{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]]. Like many entries of this clan, godzilla can be extended naturally to the 2.3.5.7.13 subgroup by identifying the hemifourth as ~15/13, tempering out [[91/90]] and [[105/104]]. [[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.


=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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[[Badness]] (Sintel): 0.677
[[Badness]] (Sintel): 0.677


=== 11-limit ===
==== 2.3.5.7.13 subgroup ====
Subgroup: 2.3.5.7.13
 
Comma list: 49/48, 81/80, 91/90
 
Subgroup-val mapping: {{mapping| 1 0 -4 2 -5 | 0 2 8 1 11 }}
 
Optimal tunings:
* WE: ~2 = 1203.7816{{c}}, ~7/4 = 950.5570{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 948.0037{{c}}
 
{{Optimal ET sequence|legend=0| 5, 14cf, 19 }}
 
Badness (Sintel): 0.591
 
=== Undecimal godzilla ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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Badness (Sintel): 0.943
Badness (Sintel): 0.943
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 33/32, 49/48, 55/54, 91/90
Mapping: {{mapping| 1 0 -4 2 5 -5 | 0 2 8 1 -2 11 }}
Optimal tunings:
* WE: ~2 = 1206.9737{{c}}, ~7/4 = 951.7738{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 946.7732{{c}}
{{Optimal ET sequence|legend=0| 14cf, 19e, 33cdeeff, 52cdeeeff }}
Badness (Sintel): 0.975


=== Varan ===
=== Varan ===
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* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 948.8625{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 948.8625{{c}}


{{Optimal ET sequence|legend=0| 5, 19, 24 }}
{{Optimal ET sequence|legend=0| 19, 24 }}


Badness (Sintel): 1.18
Badness (Sintel): 1.18
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* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 948.8468{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 948.8468{{c}}


{{Optimal ET sequence|legend=0| 5, 19, 24 }}
{{Optimal ET sequence|legend=0| 19, 24 }}


Badness (Sintel): 1.10
Badness (Sintel): 1.10


== Helayo ==
== Helayo ==
: ''For the 5-limit version of this temperament see [[Miscellaneous 5-limit temperaments #Hogzilla]].''
: ''For the 5-limit version of this temperament see [[Syntonic–kleismic equivalence continuum #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.  
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.  
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[[Badness]] (Sintel): 2.00
[[Badness]] (Sintel): 2.00
Scales: [[Helayo14]], [[Helayo19]], [[Helayo24]]


; Music
; Music
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== Negri ==
== Negri ==
{{Main| Negri }}
{{Main| 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 {{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.  
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 and/or 105/104 to the comma list; this will be discussed below under the title of negra.  


=== 7-limit ===
=== 7-limit ===
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Comma list: 49/48, 65/64, 91/90
Comma list: 49/48, 65/64, 91/90


Sval mapping: {{mapping| 1 2 2 3 4 | 0 -4 3 -2 -3 }}
Subgroup-val mapping: {{mapping| 1 2 2 3 4 | 0 -4 3 -2 -3 }}


Gencom mapping: {{mapping| 1 2 2 3 0 4 | 0 -4 3 -2 0 -3 }}
Gencom mapping: {{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:  
Optimal tunings:  
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Badness (Sintel): 0.463
Badness (Sintel): 0.463


=== 11-limit ===
=== Undecimal negri ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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=== Wilsec ===
=== Wilsec ===
Wilsec splits the fifthward generator of negri in half for 11/8~15/11, tempering out [[121/120]]. Its ploidacot is gamma-octacot.
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: {{mapping| 1 6 -1 5 4 | 0 -8 6 -4 -1 }}
Mapping: {{mapping| 1 -2 5 1 3 | 0 8 -6 4 1 }}


: mapping generators: ~2, ~16/11
: mapping generators: ~2, ~11/8


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


Mapping: {{mapping| 1 6 -1 5 4 7 | 0 -8 6 -4 -1 -6 }}
Mapping: {{mapping| 1 -2 5 1 3 1 | 0 8 -6 4 1 6 }}


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


Mapping: {{mapping| 1 6 -1 5 4 7 -2 | 0 -8 6 -4 -1 -6 11 }}
Mapping: {{mapping| 1 -2 5 1 3 1 9 | 0 8 -6 4 1 6 -11 }}


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


Mapping: {{mapping| 1 6 -1 5 4 7 -2 7 | 0 -8 6 -4 -1 -6 11 -5 }}
Mapping: {{mapping| 1 -2 5 1 3 1 9 2 | 0 8 -6 4 1 6 -11 5 }}


Optimal tunings:  
Optimal tunings:  
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[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Semaphoresmic clan| ]] <!-- main article -->
[[Category:Semaphoresmic clan| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 17:45, 8 June 2026

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

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

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

mapping generators: ~2, ~7/4

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. Blackwood adds 28/27, with a 1/5-octave period. 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.

Semaerophore

Named by CompactStar in 2023, this extension tempers out 729/704. It is the no-5 restriction of undecimal godzilla.

Subgroup: 2.3.7.11

Comma list: 49/48, 729/704

Subgroup-val mapping: [⟨1 0 2 -6], ⟨0 2 1 12]]

Optimal tunings:

  • WE: ~2 = 1204.9027 ¢, ~7/4 = 948.7772 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 945.4959 ¢

Optimal ET sequence: 14, 33d, 47de

Badness (Sintel): 1.27

2.3.7.11.19 subgroup

Subgroup: 2.3.7.11.19

Comma list: 49/48, 77/76, 729/704

Subgroup-val mapping: [⟨1 0 2 -6 -6], ⟨0 2 1 12 13]]

Optimal tunings:

  • WE: ~2 = 1204.9645 ¢, ~7/4 = 948.5749 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 945.2236 ¢

Optimal ET sequence: 14, 33d, 47deh

Badness (Sintel): 1.08

Godzilla

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. Like many entries of this clan, godzilla can be extended naturally to the 2.3.5.7.13 subgroup by identifying the hemifourth as ~15/13, tempering out 91/90 and 105/104. 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.

7-limit

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

2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

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

Subgroup-val mapping: [⟨1 0 -4 2 -5], ⟨0 2 8 1 11]]

Optimal tunings:

  • WE: ~2 = 1203.7816 ¢, ~7/4 = 950.5570 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 948.0037 ¢

Optimal ET sequence: 5, 14cf, 19

Badness (Sintel): 0.591

Undecimal godzilla

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

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 33/32, 49/48, 55/54, 91/90

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

Optimal tunings:

  • WE: ~2 = 1206.9737 ¢, ~7/4 = 951.7738 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 946.7732 ¢

Optimal ET sequence: 14cf, 19e, 33cdeeff, 52cdeeeff

Badness (Sintel): 0.975

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: 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: 19, 24

Badness (Sintel): 1.10

Helayo

For the 5-limit version of this temperament see Syntonic–kleismic equivalence continuum #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

Scales: Helayo14, Helayo19, Helayo24

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 and/or 105/104 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

Subgroup-val 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]]

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

Undecimal negri

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

Wilsec splits the fifthward generator of negri in half for 11/8~15/11, tempering out 121/120. Its ploidacot is gamma-octacot.

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 5 1 3], ⟨0 8 -6 4 1]]

mapping generators: ~2, ~11/8

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 -2 5 1 3 1], ⟨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 -2 5 1 3 1 9], ⟨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 -2 5 1 3 1 9 2], ⟨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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