Semaphoresmic clan: Difference between revisions

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{{Technical data page}}<br><br>
{{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
Line 10: Line 14:


{{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:  
* [[CTE]]: ~2 = 1200.000, ~7/4 = 952.295
* [[WE]]: ~2 = 1202.8324{{c}}, ~7/4 = 951.8567{{c}}
: [[error map]]: {{val| 0.000 +2.635 -16.531 }}
: [[error map]]: {{val| +2.832 +1.758 -11.304 }}
* [[POTE]]: ~2 = 1200.000, ~7/4 = 949.615
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 950.6890{{c}}
: error map: {{val| 0.000 -2.724 -19.211 }}
: error map: {{val| 0.000 -0.577 -18.137 }}


{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 67dd, 91dd, 115ddd }}
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 67dd, 91dd, 115ddd }}
[[Badness]] (Sintel): 0.193


Scales: [[semaphore5]], [[semaphore9]], [[semaphore14]]
Scales: [[semaphore5]], [[semaphore9]], [[semaphore14]]


=== 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:
* 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.  


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.  
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  
Line 44: Line 54:
* [[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]]
* [[Blacksmith]] (+28/27) → [[Limmic temperaments #Blacksmith|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]]
* ''[[Hemiripple]]'' (+6561/6250) → [[Ripple family #Hemiripple|Ripple family]]
* ''[[Hemiripple]]'' (+6561/6250) → [[Ripple family #Hemiripple|Ripple family]]
Line 56: Line 66:
{{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


Line 65: Line 76:


: mapping generators: ~2, ~7/4
: mapping generators: ~2, ~7/4
{{Multival|legend=1| 2 8 1 8 -4 -20 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~7/4 = 948.796
* [[WE]]: ~2 = 1203.8275{{c}}, ~7/4 = 950.3867{{c}}
: [[error map]]: {{val| 0.000 -4.363 +4.054 -20.030 }}
: [[error map]]: {{val| +3.827 -1.182 +1.470 -10.784 }}
* [[POTE]]: ~2 = 1200.000, ~7/4 = 947.365
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 947.8216{{c}}
: error map: {{val| 0.000 -7.225 -7.394 -21.461 }}
: error map: {{val| 0.000 -6.312 -3.741 -21.004 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 80: Line 89:
{{Optimal ET sequence|legend=1| 5, 14c, 19 }}
{{Optimal ET sequence|legend=1| 5, 14c, 19 }}


[[Badness]] (Smith): 0.026747
[[Badness]] (Sintel): 0.677
 
==== 2.3.5.7.13 subgroup ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 81/80, 91/90


=== 11-limit ===
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


Line 89: Line 113:
Mapping: {{mapping| 1 0 -4 2 -6 | 0 2 8 1 12 }}
Mapping: {{mapping| 1 0 -4 2 -6 | 0 2 8 1 12 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 947.456
* WE: ~2 = 1204.4129{{c}}, ~7/4 = 949.4513{{c}}
* POTE: ~2 = 1200.000, ~7/4 = 945.973
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 946.4361{{c}}


Tuning ranges:  
Tuning ranges:  
Line 99: Line 123:
{{Optimal ET sequence|legend=0| 14c, 19, 33cd }}
{{Optimal ET sequence|legend=0| 14c, 19, 33cd }}


Badness (Smith): 0.028947
Badness (Sintel): 0.957


==== 13-limit ====
==== 13-limit ====
Line 108: Line 132:
Mapping: {{mapping| 1 0 -4 2 -6 -5 | 0 2 8 1 12 11 }}
Mapping: {{mapping| 1 0 -4 2 -6 -5 | 0 2 8 1 12 11 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 947.888
* WE: ~2 = 1203.7164{{c}}, ~7/4 = 949.2061{{c}}
* POTE: ~2 = 1200.000, ~7/4 = 946.397
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 946.4131{{c}}


Tuning ranges:  
Tuning ranges:  
Line 118: Line 142:
{{Optimal ET sequence|legend=0| 14cf, 19, 33cdff }}
{{Optimal ET sequence|legend=0| 14cf, 19, 33cdff }}


Badness (Smith): 0.022503
Badness (Sintel): 0.930


=== Semafour ===
=== Semafour ===
Line 128: Line 152:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 948.209
* WE: ~2 = 1206.9595{{c}}, ~7/4 = 951.4440{{c}}
* POTE: ~2 = 1200.000, ~7/4 = 945.958
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 946.4472{{c}}


{{Optimal ET sequence|legend=0| 14c, 19e, 33cdee, 52cdeee }}
{{Optimal ET sequence|legend=0| 14c, 19e, 33cdee, 52cdeee }}


Badness (Smith): 0.028510
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 ===
Line 143: Line 182:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 949.616
* WE: ~2 = 1202.5842{{c}}, ~7/4 = 950.9647{{c}}
* POTE: ~2 = 1200.000, ~7/4 = 948.921
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 949.1239{{c}}


{{Optimal ET sequence|legend=0| 19e, 24, 43de }}
{{Optimal ET sequence|legend=0| 19e, 24, 43de }}


Badness (Smith): 0.039647
Badness (Sintel): 1.31


==== 13-limit ====
==== 13-limit ====
Line 158: Line 197:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 949.525
* WE: ~2 = 1202.4367{{c}}, ~7/4 = 950.7615{{c}}
* POTE: ~2 = 1200.000, ~7/4 = 948.835
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 949.0338{{c}}


{{Optimal ET sequence|legend=0| 19e, 24, 43de }}
{{Optimal ET sequence|legend=0| 19e, 24, 43de }}


Badness (Smith): 0.025676
Badness (Sintel): 1.06


=== Baragon ===
=== Baragon ===
Line 172: Line 211:
Mapping: {{mapping| 1 0 -4 2 9 | 0 2 8 1 -7 }}
Mapping: {{mapping| 1 0 -4 2 9 | 0 2 8 1 -7 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 949.031
* WE: ~2 = 1201.1412{{c}}, ~7/4 = 949.7291{{c}}
* POTE: ~2 = 1200.000, ~7/4 = 948.827
* 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 (Smith): 0.035673
Badness (Sintel): 1.18


==== 13-limit ====
==== 13-limit ====
Line 187: Line 226:
Mapping: {{mapping| 1 0 -4 2 9 -5 | 0 2 8 1 -7 11 }}
Mapping: {{mapping| 1 0 -4 2 9 -5 | 0 2 8 1 -7 11 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 949.067
* WE: ~2 = 1201.1228{{c}}, ~7/4 = 949.6894{{c}}
* POTE: ~2 = 1200.000, ~7/4 = 948.802
* 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 (Smith): 0.026703
Badness (Sintel): 1.10


== Helayo ==
== Helayo ==
: ''For the 5-limit version of this temperament see [[High badness temperaments #Hogzilla]].''
: ''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
Line 203: Line 244:


{{Mapping|legend=1| 1 0 11 2 | 0 2 -11 1 }}
{{Mapping|legend=1| 1 0 11 2 | 0 2 -11 1 }}
{{Multival|legend=1| 2 -6 1 -14 -4 19 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~7/4 = 947.097
* [[WE]]: ~2 = 1204.0199{{c}}, ~7/4 = 950.7917{{c}}
: [[error map]]: {{val| 0.000 -7.761 -4.380 -21.729 }}
: [[error map]]: {{val| +4.020 -0.372 -0.804 -9.995 }}
* [[CWE]]: ~2 = 1200.000, ~7/4 = 947.505
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 947.5047{{c}}
: error map: {{val| 0.000 -6.946 -8.866 -21.321 }}
: error map: {{val| 0.000 -6.946 -8.866 -21.321 }}


{{Optimal ET sequence|legend=1| 5c, 14, 19 }}
{{Optimal ET sequence|legend=1| 5c, 14, 19 }}


[[Badness]] (Smith): 0.0791
[[Badness]] (Sintel): 2.00


; Music
; Music
Line 220: Line 259:


== 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


Line 226: Line 267:
{{Mapping|legend=1| 1 0 7 2 | 0 2 -6 1 }}
{{Mapping|legend=1| 1 0 7 2 | 0 2 -6 1 }}


{{Multival|legend=1| 2 -6 1 -14 -4 19 }}
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1208.8222{{c}}, ~7/4 = 946.9590{{c}}
[[Optimal tuning]]s:
: [[error map]]: {{val| +8.822 -8.037 -6.313 -4.223 }}
* [[CTE]]: ~2 = 1200.000, ~7/4 = 939.030
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 939.8419{{c}}
: [[error map]]: {{val| 0.000 -23.896 -20.492 -29.796 }}
: error map: {{val| 0.000 -22.271 -25.365 -28.984 }}
* [[POTE]]: ~2 = 1200.000, ~7/4 = 940.048
: error map: {{val| 0.000 -21.859 -26.602 -28.778 }}


{{Optimal ET sequence|legend=1| 9, 14c, 23d, 37bcd, 60bbccdd }}
{{Optimal ET sequence|legend=1| 9, 14c, 23d, 37bcd, 60bbccdd }}


[[Badness]] (Smith): 0.058216
[[Badness]] (Sintel): 1.47


=== 11-limit ===
=== 11-limit ===
Line 246: Line 285:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 938.467
* WE: ~2 = 1208.8663{{c}}, ~7/4 = 946.9861{{c}}
* POTE: ~2 = 1200.000, ~7/4 = 940.041
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 939.7687{{c}}


{{Optimal ET sequence|legend=0| 9, 14c, 23de, 37bcde }}
{{Optimal ET sequence|legend=0| 9, 14c, 23de, 37bcde, 60bbccddeee }}


Badness (Smith): 0.028535
Badness (Sintel): 0.943


; 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] in 23edo tuning
: ''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


Line 264: Line 305:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~7/4 = 968.739
* [[WE]]: ~2 = 1184.7407{{c}}, ~7/4 = 960.9196{{c}}
: [[error map]]: {{val| 0.000 +35.523 +76.208 -0.087 }}
: [[error map]]: {{val| -15.259 +19.884 +30.810 -38.425 }}
* [[POTE]]: ~2 = 1200.000, ~7/4 = 973.296
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 972.2994{{c}}
: error map: {{val| 0.000 +42.644 +69.088 +3.473 }}
: error map: {{val| 0.000 +42.644 +69.088 +3.473 }}
{{Multival|legend=1| 2 -2 1 -8 -4 8 }}


{{Optimal ET sequence|legend=1| 5, 11b, 16bc }}
{{Optimal ET sequence|legend=1| 5, 11b, 16bc }}


[[Badness]] (Smith): 0.044321
[[Badness]] (Sintel): 1.12


=== 11-limit ===
=== 11-limit ===
Line 283: Line 322:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 972.258
* WE: ~2 = 1187.4876{{c}}, ~7/4 = 967.9643{{c}}
* POTE: ~2 = 1200.000, ~7/4 = 978.164
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9298{{c}}


{{Optimal ET sequence|legend=0| 5, 11b }}
{{Optimal ET sequence|legend=0| 5, 11b }}


Badness (Smith): 0.036538
Badness (Sintel): 1.21


== 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 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.  
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 16875/16384
 
{{Mapping|legend=1| 1 2 2 | 0 -4 3 }}
 
: mapping generators: ~2, ~16/15
 
{{Multival|legend=1| 4 -3 -14 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~16/15 = 125.396
: [[error map]]: {{val| 0.000 -3.539 -10.126 }}
* [[POTE]]: ~2 = 1200.000, ~16/15 = 125.755
: error map: {{val| 0.000 -4.975 -9.049 }}
 
{{Optimal ET sequence|legend=1| 9, 10, 19, 67c, 86c, 105c }}
 
[[Badness]] (Smith): 0.086856


=== 7-limit ===
=== 7-limit ===
Line 321: Line 341:


{{Mapping|legend=1| 1 2 2 3 | 0 -4 3 -2 }}
{{Mapping|legend=1| 1 2 2 3 | 0 -4 3 -2 }}
{{Multival|legend=1| 4 -3 2 -14 -8 13 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~15/14 = 124.813
* [[WE]]: ~2 = 1203.4810{{c}}, ~15/14 = 125.9724{{c}}
: [[error map]]: {{val| 0.000 -1.209 -11.874 -18.453 }}
: [[error map]]: {{val| +3.481 +1.118 -1.435 -10.328 }}
* [[POTE]]: ~2 = 1200.000, ~15/14 = 125.608
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 125.4347{{c}}
: error map: {{val| 0.000 -4.387 -9.490 -20.042 }}
: error map: {{val| 0.000 -3.694 -10.009 -19.695 }}


{{Optimal ET sequence|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
{{Optimal ET sequence|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}


[[Badness]] (Smith): 0.026483
[[Badness]] (Sintel): 0.670


==== 2.3.5.7.13 subgroup (negra) ====
==== 2.3.5.7.13 subgroup (negra) ====
Line 339: Line 357:
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:  
* CTE: ~2 = 1200.000, ~14/13 = 124.457
* WE: ~2 = 1203.6981{{c}}, ~14/13 = 125.9545{{c}}
* POTE: ~2 = 1200.000, ~14/13 = 125.567
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 125.3543{{c}}


{{Optimal ET sequence|legend=0| 9, 10, 19, 48df, 67cddf, 86cddff }}
{{Optimal ET sequence|legend=0| 9, 10, 19, 48df, 67cddf, 86cddff }}


=== 11-limit ===
Badness (Sintel): 0.463
 
=== Undecimal negri ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 357: Line 375:


Mapping: {{mapping| 1 2 2 3 4 | 0 -4 3 -2 -5 }}
Mapping: {{mapping| 1 2 2 3 4 | 0 -4 3 -2 -5 }}
Wedgie: {{multival| 4 -3 2 5 -14 -8 -6 13 22 7 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~15/14 = 125.780
* WE: ~2 = 1202.1045{{c}}, ~15/14 = 126.6961{{c}}
* POTE: ~2 = 1200.000, ~15/14 = 126.474
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 126.3382{{c}}


{{Optimal ET sequence|legend=0| 9, 10, 19 }}
{{Optimal ET sequence|legend=0| 9, 10, 19 }}


Badness (Smith): 0.026190
Badness (Sintel): 0.866


==== 13-limit ====
==== 13-limit ====
Line 376: Line 392:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~14/13 = 125.434
* WE: ~2 = 1202.6035{{c}}, ~14/13 = 126.7054{{c}}
* POTE: ~2 = 1200.000, ~14/13 = 126.431
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.2534{{c}}


{{Optimal ET sequence|legend=0| 9, 10, 19 }}
{{Optimal ET sequence|legend=0| 9, 10, 19 }}


Badness (Smith): 0.017833
Badness (Sintel): 0.737


=== Negril ===
=== Negril ===
Line 389: Line 405:


Mapping: {{mapping| 1 2 2 3 2 | 0 -4 3 -2 14 }}
Mapping: {{mapping| 1 2 2 3 2 | 0 -4 3 -2 14 }}
Wedgie: {{multival| 4 -3 2 -14 -14 -8 -36 13 -22 -46 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~15/14 = 124.998
* WE: ~2 = 1202.7081{{c}}, ~15/14 = 125.0491{{c}}
* POTE: ~2 = 1200.000, ~15/14 = 124.767
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 124.8066{{c}}


{{Optimal ET sequence|legend=0| 10e, 19, 29, 48d, 77cdd }}
{{Optimal ET sequence|legend=0| 10e, 19, 29, 48d, 77cdd }}


Badness (Smith): 0.038679
Badness (Sintel): 1.28


==== 13-limit ====
==== 13-limit ====
Line 408: Line 422:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~14/13 = 124.865
* WE: ~2 = 1202.9319{{c}}, ~14/13 = 125.0204{{c}}
* POTE: ~2 = 1200.000, ~14/13 = 124.716
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 124.7374{{c}}


{{Optimal ET sequence|legend=0| 10e, 19, 29, 48df, 77cddf }}
{{Optimal ET sequence|legend=0| 10e, 19, 29, 48df, 77cddf }}


Badness (Smith): 0.024383
Badness (Sintel): 1.01


=== Negric ===
=== Negric ===
Line 421: Line 435:


Mapping: {{mapping| 1 2 2 3 3 | 0 -4 3 -2 4 }}
Mapping: {{mapping| 1 2 2 3 3 | 0 -4 3 -2 4 }}
Wedgie: {{multival| 4 -3 2 -4 -14 -8 -20 13 1 -18 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~15/14 = 126.574
* WE: ~2 = 1205.7810{{c}}, ~15/14 = 127.6513{{c}}
* POTE: ~2 = 1200.000, ~15/14 = 127.039
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 126.9620{{c}}


{{Optimal ET sequence|legend=0| 9, 19e }}
{{Optimal ET sequence|legend=0| 9, 19e }}


Badness (Smith): 0.030617
Badness (Sintel): 1.01


==== 13-limit ====
==== 13-limit ====
Line 440: Line 452:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~14/13 = 126.153
* WE: ~2 = 1205.7833{{c}}, ~14/13 = 127.6507{{c}}
* POTE: ~2 = 1200.000, ~14/13 = 127.039
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.9093{{c}}


{{Optimal ET sequence|legend=0| 9, 19e }}
{{Optimal ET sequence|legend=0| 9, 19e }}


Badness (Smith): 0.020205
Badness (Sintel): 0.835


=== Negroni ===
=== Negroni ===
Line 455: Line 467:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~15/14 = 123.735
* WE: ~2 = 1203.4738{{c}}, ~15/14 = 124.8992{{c}}
* POTE: ~2 = 1200.000, ~15/14 = 124.539
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 124.3642{{c}}


{{Optimal ET sequence|legend=0| 10, 19e, 29, 77cddee }}
{{Optimal ET sequence|legend=0| 10, 19e, 29, 77cddee }}


Badness (Smith): 0.035296
Badness (Sintel): 1.17


==== 13-limit ====
==== 13-limit ====
Line 470: Line 482:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~15/14 = 123.644
* WE: ~2 = 1203.5354{{c}}, ~14/13 = 124.9118{{c}}
* POTE: ~2 = 1200.000, ~14/13 = 124.545
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 124.3733{{c}}


{{Optimal ET sequence|legend=0| 10, 19e, 29, 77cddeef }}
{{Optimal ET sequence|legend=0| 10, 19e, 29, 77cddeef }}


Badness (Smith): 0.021559
Badness (Sintel): 0.890


=== 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:  
* CTE: ~2 = 1200.000, ~11/8 = 537.627
* WE: ~2 = 1203.6080{{c}}, ~11/8 = 538.8007{{c}}
* POTE: ~2 = 1200.000, ~11/8 = 537.186
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 537.2654{{c}}


{{Optimal ET sequence|legend=0| 9, 20, 29, 38d, 67cdde, 105cdddee }}
{{Optimal ET sequence|legend=0| 9, 20, 29, 38d, 67cdde, 105cdddee }}


Badness (Smith): 0.041886
Badness (Sintel): 1.38


==== 13-limit ====
==== 13-limit ====
Line 499: Line 513:
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:  
* CTE: ~2 = 1200.000, ~11/8 = 537.802
* WE: ~2 = 1203.7672{{c}}, ~11/8 = 538.8948{{c}}
* POTE: ~2 = 1200.000, ~11/8 = 537.208
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 537.3053{{c}}


{{Optimal ET sequence|legend=0| 9, 20, 29, 38df, 67cddef, 105cdddeefff }}
{{Optimal ET sequence|legend=0| 9, 20, 29, 38df, 67cddef, 105cdddeefff }}


Badness (Smith): 0.025192
Badness (Sintel): 1.04


==== 17-limit ====
==== 17-limit ====
Line 514: Line 528:
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:  
* CTE: ~2 = 1200.000, ~11/8 = 537.492
* WE: ~2 = 1203.7154{{c}}, ~11/8 = 538.8932{{c}}
* POTE: ~2 = 1200.000, ~11/8 = 537.230
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 537.2633{{c}}


{{Optimal ET sequence|legend=0| 9, 20g, 29g, 38df, 67cddefg }}
{{Optimal ET sequence|legend=0| 9, 20g, 29g, 38df, 67cddefg }}


Badness (Smith): 0.021778
Badness (Sintel): 1.11


==== 19-limit ====
==== 19-limit ====
Line 529: Line 543:
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:  
* CTE: ~2 = 1200.000, ~11/8 = 537.553
* WE: ~2 = 1203.5906{{c}}, ~11/8 = 538.8216{{c}}
* POTE: ~2 = 1200.000, ~11/8 = 537.214
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 537.2534{{c}}


{{Optimal ET sequence|legend=0| 9, 20g, 29g, 38df, 67cddefgh }}
{{Optimal ET sequence|legend=0| 9, 20g, 29g, 38df, 67cddefgh }}


Badness (Smith): 0.016828
Badness (Sintel): 1.02


== 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


Line 547: Line 563:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~16/15 = 81.345
* [[WE]]: ~2 = 1197.0059{{c}}, ~16/15 = 80.7519{{c}}
: [[error map]]: {{val| 0.000 +9.975 +20.411 -12.861 }}
: [[error map]]: {{val| -2.994 +7.546 +11.457 -20.064 }}
* [[POTE]]: ~2 = 1200.000, ~16/15 = 80.954
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 81.0408{{c}}
: error map: {{val| 0.000 +12.322 +18.456 -11.688 }}
: error map: {{val| 0.000 +11.800 +18.890 -11.948 }}


{{Optimal ET sequence|legend=1| 14, 15 }}
{{Optimal ET sequence|legend=1| 14, 15 }}


[[Badness]] (Smith): 0.129339
[[Badness]] (Sintel): 3.27


=== 11-limit ===
=== 11-limit ===
Line 564: Line 580:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~16/15 = 80.908
* WE: ~2 = 1196.6821{{c}}, ~16/15 = 80.5936{{c}}
* POTE: ~2 = 1200.000, ~16/15 = 80.817
* CWE: ~2 = 1200.0000{{c}}, ~16/15 = 80.8326{{c}}


{{Optimal ET sequence|legend=0| 14e, 15 }}
{{Optimal ET sequence|legend=0| 14e, 15 }}


Badness (Smith): 0.069398
Badness (Sintel): 2.29


=== 13-limit ===
=== 13-limit ===
Line 579: Line 595:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~16/15 = 81.320
* WE: ~2 = 1195.6248{{c}}, ~16/15 = 80.7288{{c}}
* POTE: ~2 = 1200.000, ~16/15 = 81.024
* CWE: ~2 = 1200.0000{{c}}, ~16/15 = 81.0685{{c}}


{{Optimal ET sequence|legend=0| 14e, 15 }}
{{Optimal ET sequence|legend=0| 14e, 15 }}


Badness (Smith): 0.048553
Badness (Sintel): 2.01


[[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 11:30, 23 August 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

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.

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. 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: 2.3.5.7.11.13

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 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 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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