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Semaphoresmic clan: Difference between revisions

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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.  
{{Technical data page}}
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 9: 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 43: 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 55: 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


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: 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]]:  
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{{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 88: 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:  
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{{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 107: 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:  
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{{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 ===
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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 142: 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 157: 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 171: 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 186: 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
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{{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 219: 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 225: 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 245: 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 263: 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 282: 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 320: 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 338: 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 356: 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 375: 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 388: 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 407: 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 420: 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 439: 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 454: 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 469: 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 498: 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 513: 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 528: 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 546: 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 563: 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 578: 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]]
Retrieved from "https://en.xen.wiki/w/Semaphoresmic_clan"