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Semaphoresmic clan: Difference between revisions

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The '''semaphoresmic clan''' (or '''semaphore family''') [[tempering out|tempers out]] the slendro 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 10: Line 15:
{{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 2 0 3 | 0 -2 0 -1 }}
: mapping generators: ~2, ~7/4
 
: [[gencom]]: [2 7/6; 49/48]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~7/4 = 952.2948
* [[WE]]: ~2 = 1202.8324{{c}}, ~7/4 = 951.8567{{c}}
* [[POTE]]: ~2 = 1\1, ~7/4 = 949.615
: [[error map]]: {{val| +2.832 +1.758 -11.304 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 950.6890{{c}}
: 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]]. 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, mabila 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. Rhinoceros adds [[4375/4374]], splitting the hemifourth in four. Spell adds [[3125/3072]], splitting the hemitwelfth in five. Hemiripple adds 6561/6250, splitting the hemifourth in five. Semabila adds 28672/28125 and splits an interval of two octaves plus a hemifourth in five. Finally, blackwood adds [[28/27]], reducing intervals of 3 and 7 to the same chain of 1/5-octave periods.  


Discussed elsewhere are  
Discussed elsewhere are  
* ''[[Beep]]'' (+21/20) → [[Bug family #Beep|Bug family]]
* ''[[Nessus]]'' (+10/9) → [[Very low accuracy temperaments #Nessus|Very low accuracy temperaments]]
* ''[[Malacoda]]'' (+15/14) → [[Very low accuracy temperaments #Malacoda|Very low accuracy temperaments]]
* [[Beep]] (+21/20) → [[Bug family #Beep|Bug family]]
* ''[[Immunity]]'' (+2240/2187) → [[Immunity family #Septimal immunity|Immunity family]]
* ''[[Immunity]]'' (+2240/2187) → [[Immunity family #Septimal immunity|Immunity family]]
* [[Decimal]] (+25/24) → [[Dicot family #Decimal|Dicot family]]
* [[Decimal]] (+25/24) → [[Dicot family #Decimal|Dicot family]]
Line 39: 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]]
* ''[[Rhinoceros]]'' (+4375/4374) → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Spell]]'' (+3125/3072) → [[Hemimean clan #Spell|Hemimean clan]]
* ''[[Spell]]'' (+3125/3072) → [[Hemimean clan #Spell|Hemimean clan]]
* ''[[Hemiripple]]'' (+6561/6250) → [[Ripple family #Hemiripple|Ripple family]]
* ''[[Semabila]]'' (+28672/28125) → [[Mabila family #Mabila|Mabila family]]
* [[Blackwood]] (+28/27) → [[Blackwood family #Blackwood|Blackwood family]]
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}}


Considered below are godzilla, superpelog, negri, nuke, mabila, and hemiripple.
{{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


Line 58: Line 108:


: 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 = 1\1, ~7/4 = 948.7959
* [[WE]]: ~2 = 1203.8275{{c}}, ~7/4 = 950.3867{{c}}
* [[POTE]]: ~2 = 1\1, ~7/4 = 947.365
: [[error map]]: {{val| +3.827 -1.182 +1.470 -10.784 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 947.8216{{c}}
: error map: {{val| 0.000 -6.312 -3.741 -21.004 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 71: Line 121:
{{Optimal ET sequence|legend=1| 5, 14c, 19 }}
{{Optimal ET sequence|legend=1| 5, 14c, 19 }}


[[Badness]]: 0.026747
[[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: {{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 }}


=== 11-limit ===
Badness (Sintel): 0.591
 
=== Undecimal godzilla ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 80: Line 145:
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 = 1\1, ~7/4 = 947.4563
* WE: ~2 = 1204.4129{{c}}, ~7/4 = 949.4513{{c}}
* POTE: ~2 = 1\1, ~7/4 = 945.973
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 946.4361{{c}}


Tuning ranges:  
Tuning ranges:  
Line 88: Line 153:
* 11-odd-limit diamond tradeoff: ~7/4 = [933.129, 968.826]
* 11-odd-limit diamond tradeoff: ~7/4 = [933.129, 968.826]


{{Optimal ET sequence|legend=1| 14c, 19, 33cd }}
{{Optimal ET sequence|legend=0| 14c, 19, 33cd }}


Badness: 0.028947
Badness (Sintel): 0.957


==== 13-limit ====
==== 13-limit ====
Line 99: Line 164:
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 = 1\1, ~7/4 = 947.8877
* WE: ~2 = 1203.7164{{c}}, ~7/4 = 949.2061{{c}}
* POTE: ~2 = 1\1, ~7/4 = 946.397
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 946.4131{{c}}


Tuning ranges:  
Tuning ranges:  
Line 107: Line 172:
* 13- and 15-odd-limit diamond tradeoff: ~7/4 = [910.890, 968.826]
* 13- and 15-odd-limit diamond tradeoff: ~7/4 = [910.890, 968.826]


{{Optimal ET sequence|legend=1| 14cf, 19, 33cdff }}
{{Optimal ET sequence|legend=0| 14cf, 19, 33cdff }}


Badness: 0.022503
Badness (Sintel): 0.930


=== Semafour ===
=== Semafour ===
Line 119: Line 184:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~7/4 = 948.2089
* WE: ~2 = 1206.9595{{c}}, ~7/4 = 951.4440{{c}}
* POTE: ~2 = 1\1, ~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=1| 14c, 19e, 33cdee, 52cdeee }}
Badness (Sintel): 0.943


Badness: 0.028510
==== 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 134: Line 214:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~7/4 = 949.6160
* WE: ~2 = 1202.5842{{c}}, ~7/4 = 950.9647{{c}}
* POTE: ~2 = 1\1, ~7/4 = 948.921
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 949.1239{{c}}


{{Optimal ET sequence|legend=1| 19e, 24, 43de }}
{{Optimal ET sequence|legend=0| 19e, 24, 43de }}


Badness: 0.039647
Badness (Sintel): 1.31


==== 13-limit ====
==== 13-limit ====
Line 149: Line 229:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~7/4 = 949.5255
* WE: ~2 = 1202.4367{{c}}, ~7/4 = 950.7615{{c}}
* POTE: ~2 = 1\1, ~7/4 = 948.835
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 949.0338{{c}}


{{Optimal ET sequence|legend=1| 19e, 24, 43de }}
{{Optimal ET sequence|legend=0| 19e, 24, 43de }}


Badness: 0.025676
Badness (Sintel): 1.06


=== Baragon ===
=== Baragon ===
Line 163: Line 243:
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 = 1\1, ~7/4 = 949.0311
* WE: ~2 = 1201.1412{{c}}, ~7/4 = 949.7291{{c}}
* POTE: ~2 = 1\1, ~7/4 = 948.827
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 948.8625{{c}}


{{Optimal ET sequence|legend=1| 5, 19, 24 }}
{{Optimal ET sequence|legend=0| 19, 24 }}


Badness: 0.035673
Badness (Sintel): 1.18


==== 13-limit ====
==== 13-limit ====
Line 178: Line 258:
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 = 1\1, ~7/4 = 949.0670
* WE: ~2 = 1201.1228{{c}}, ~7/4 = 949.6894{{c}}
* POTE: ~2 = 1\1, ~7/4 = 948.802
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 948.8468{{c}}


{{Optimal ET sequence|legend=1| 5, 19, 24 }}
{{Optimal ET sequence|legend=0| 19, 24 }}


Badness: 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 [[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
[[Subgroup]]: 2.3.5.7
Line 194: Line 276:


{{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 = 1\1, ~7/4 = 947.0969
* [[WE]]: ~2 = 1204.0199{{c}}, ~7/4 = 950.7917{{c}}
: [[error map]]: {{val| +4.020 -0.372 -0.804 -9.995 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 947.5047{{c}}
: 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]]: 0.0791
[[Badness]] (Sintel): 2.00
 
Scales: [[Helayo14]], [[Helayo19]], [[Helayo24]]


; Music
; Music
Line 208: Line 293:


== 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 214: Line 301:
{{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 = 1\1, ~7/4 = 939.0297
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 939.8419{{c}}
* [[POTE]]: ~2 = 1\1, ~7/4 = 940.048
: error map: {{val| 0.000 -22.271 -25.365 -28.984 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 232: Line 319:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~7/4 = 938.4673
* WE: ~2 = 1208.8663{{c}}, ~7/4 = 946.9861{{c}}
* POTE: ~2 = 1\1, ~7/4 = 940.041
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 939.7687{{c}}


{{Optimal ET sequence|legend=1| 9, 14c, 23de, 37bcde }}
{{Optimal ET sequence|legend=0| 9, 14c, 23de, 37bcde, 60bbccddeee }}


Badness: 0.028535
Badness (Sintel): 0.943


; Music
; Music
: ''[http://micro.soonlabel.com/MOS/20120418-9mos-mindaugas.mp3 Mindaugas Rex Lithuaniae]'' by [[Chris Vaisvil]] ([http://chrisvaisvil.com/?p=2267 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 250: Line 339:


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


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


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


=== 11-limit ===
=== 11-limit ===
Line 266: Line 356:


Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1\1, ~7/4 = 978.164
* WE: ~2 = 1187.4876{{c}}, ~7/4 = 967.9643{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9298{{c}}


{{Optimal ET sequence|legend=1| 5, 11b, 16bc, 27bbcc }}
{{Optimal ET sequence|legend=0| 5, 11b }}


Badness: 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 = 1\1, ~16/15 = 125.396
* [[POTE]]: ~2 = 1\1, ~16/15 = 125.755
 
{{Optimal ET sequence|legend=1| 9, 10, 19, 67c, 86c, 105c }}
 
[[Badness]]: 0.086856


=== 7-limit ===
=== 7-limit ===
Line 301: Line 375:


{{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 = 1\1, ~15/14 = 124.813
* [[WE]]: ~2 = 1203.4810{{c}}, ~15/14 = 125.9724{{c}}
* [[POTE]]: ~2 = 1\1, ~15/14 = 125.608
: [[error map]]: {{val| +3.481 +1.118 -1.435 -10.328 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 125.4347{{c}}
: 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]]: 0.026483
[[Badness]] (Sintel): 0.670


==== 2.3.5.7.13 subgroup (negra) ====
==== 2.3.5.7.13 subgroup (negra) ====
Line 317: Line 391:
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:  
* WE: ~2 = 1203.6981{{c}}, ~14/13 = 125.9545{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 125.3543{{c}}


Optimal tunings:
{{Optimal ET sequence|legend=0| 9, 10, 19, 48df, 67cddf, 86cddff }}
* CTE: ~2 = 1\1, ~14/13 = 124.457
* POTE: ~2 = 1\1, ~14/13 = 125.567


Optimal ET sequence: {{Optimal ET sequence| 9, 10, 19, 48df, 67cddf, 86cddff }}
Badness (Sintel): 0.463


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


Line 335: Line 409:


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 = 1\1, ~15/14 = 125.780
* WE: ~2 = 1202.1045{{c}}, ~15/14 = 126.6961{{c}}
* POTE: ~2 = 1\1, ~15/14 = 126.474
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 126.3382{{c}}


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


Badness: 0.026190
Badness (Sintel): 0.866


==== 13-limit ====
==== 13-limit ====
Line 354: Line 426:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~14/13 = 125.434
* WE: ~2 = 1202.6035{{c}}, ~14/13 = 126.7054{{c}}
* POTE: ~2 = 1\1, ~14/13 = 126.431
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.2534{{c}}


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


Badness: 0.017833
Badness (Sintel): 0.737


=== Negril ===
=== Negril ===
Line 367: Line 439:


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 = 1\1, ~15/14 = 124.998
* WE: ~2 = 1202.7081{{c}}, ~15/14 = 125.0491{{c}}
* POTE: ~2 = 1\1, ~15/14 = 124.767
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 124.8066{{c}}


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


Badness: 0.038679
Badness (Sintel): 1.28


==== 13-limit ====
==== 13-limit ====
Line 386: Line 456:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~14/13 = 124.865
* WE: ~2 = 1202.9319{{c}}, ~14/13 = 125.0204{{c}}
* POTE: ~2 = 1\1, ~14/13 = 124.716
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 124.7374{{c}}


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


Badness: 0.024383
Badness (Sintel): 1.01


=== Negric ===
=== Negric ===
Line 399: Line 469:


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 = 1\1, ~15/14 = 126.574
* WE: ~2 = 1205.7810{{c}}, ~15/14 = 127.6513{{c}}
* POTE: ~2 = 1\1, ~15/14 = 127.039
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 126.9620{{c}}


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


Badness: 0.030617
Badness (Sintel): 1.01


==== 13-limit ====
==== 13-limit ====
Line 418: Line 486:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~14/13 = 126.153
* WE: ~2 = 1205.7833{{c}}, ~14/13 = 127.6507{{c}}
* POTE: ~2 = 1\1, ~14/13 = 127.039
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.9093{{c}}


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


Badness: 0.020205
Badness (Sintel): 0.835


=== Negroni ===
=== Negroni ===
Line 433: Line 501:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~15/14 = 123.735
* WE: ~2 = 1203.4738{{c}}, ~15/14 = 124.8992{{c}}
* POTE: ~2 = 1\1, ~15/14 = 124.539
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 124.3642{{c}}


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


Badness: 0.035296
Badness (Sintel): 1.17


==== 13-limit ====
==== 13-limit ====
Line 448: Line 516:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~15/14 = 123.644
* WE: ~2 = 1203.5354{{c}}, ~14/13 = 124.9118{{c}}
* POTE: ~2 = 1\1, ~14/13 = 124.545
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 124.3733{{c}}


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


Badness: 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 = 1\1, ~11/8 = 537.627
* WE: ~2 = 1203.6080{{c}}, ~11/8 = 538.8007{{c}}
* POTE: ~2 = 1\1, ~11/8 = 537.186
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 537.2654{{c}}


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


Badness: 0.041886
Badness (Sintel): 1.38


==== 13-limit ====
==== 13-limit ====
Line 477: Line 547:
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 = 1\1, ~11/8 = 537.802
* WE: ~2 = 1203.7672{{c}}, ~11/8 = 538.8948{{c}}
* POTE: ~2 = 1\1, ~11/8 = 537.208
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 537.3053{{c}}


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


Badness: 0.025192
Badness (Sintel): 1.04


==== 17-limit ====
==== 17-limit ====
Line 492: Line 562:
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 = 1\1, ~11/8 = 537.492
* WE: ~2 = 1203.7154{{c}}, ~11/8 = 538.8932{{c}}
* POTE: ~2 = 1\1, ~11/8 = 537.230
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 537.2633{{c}}


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


Badness: 0.021778
Badness (Sintel): 1.11


==== 19-limit ====
==== 19-limit ====
Line 507: Line 577:
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 = 1\1, ~11/8 = 537.553
* WE: ~2 = 1203.5906{{c}}, ~11/8 = 538.8216{{c}}
* POTE: ~2 = 1\1, ~11/8 = 537.214
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 537.2534{{c}}


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


Badness: 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 525: Line 597:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1\1, ~16/15 = 80.9538
* [[WE]]: ~2 = 1197.0059{{c}}, ~16/15 = 80.7519{{c}}
: [[error map]]: {{val| -2.994 +7.546 +11.457 -20.064 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 81.0408{{c}}
: error map: {{val| 0.000 +11.800 +18.890 -11.948 }}


{{Optimal ET sequence|legend=1| 14, 15, 44cd }}
{{Optimal ET sequence|legend=1| 14, 15 }}


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


=== 11-limit ===
=== 11-limit ===
Line 539: Line 614:


Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1\1, ~16/15 = 80.8171
* WE: ~2 = 1196.6821{{c}}, ~16/15 = 80.5936{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/15 = 80.8326{{c}}


{{Optimal ET sequence|legend=1| 14e, 15 }}
{{Optimal ET sequence|legend=0| 14e, 15 }}


Badness: 0.069398
Badness (Sintel): 2.29


=== 13-limit ===
=== 13-limit ===
Line 553: Line 629:


Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1\1, ~16/15 = 81.0243
* WE: ~2 = 1195.6248{{c}}, ~16/15 = 80.7288{{c}}
 
* CWE: ~2 = 1200.0000{{c}}, ~16/15 = 81.0685{{c}}
{{Optimal ET sequence|legend=1| 14e, 15, 44cdeff }}
 
Badness: 0.048553
 
== Semabila ==
{{See also| Mabila family #Semabila }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 49/48, 28672/28125
 
[[Mapping]]: {{mapping| 1 6 1 5 | 0 -10 3 -5 }}
 
{{Multival|legend=1| 10 -3 5 -28 -20 20 }}
 
[[Optimal tuning]]s:
* POTE: ~2 = 1\1, ~75/56 = 529.667
 
{{Optimal ET sequence|legend=1| 9, 25, 34 }}
 
[[Badness]]: 0.133638
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 49/48, 56/55, 1350/1331
 
Mapping: {{mapping| 1 6 1 5 7 | 0 -10 3 -5 -8 }}
 
Optimal tunings:
* POTE: ~2 = 1\1, ~15/11 = 529.729
 
{{Optimal ET sequence|legend=1| 9, 25e, 34 }}
 
Badness: 0.061501
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 56/55, 91/90, 847/845
 
Mapping: {{mapping| 1 6 1 5 7 9 | 0 -10 3 -5 -8 -12 }}
 
Optimal tunings:
* POTE: ~2 = 1\1, ~15/11 = 529.763
 
{{Optimal ET sequence|legend=1| 9, 25e, 34 }}
 
Badness: 0.037270
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 49/48, 56/55, 91/90, 154/153, 375/374
 
Mapping: {{mapping| 1 6 1 5 7 9 1 | 0 -10 3 -5 -8 -12 7 }}
 
Optimal tunings:
* POTE: ~2 = 1\1, ~15/11 = 529.695
 
{{Optimal ET sequence|legend=1| 9, 25e, 34 }}
 
Badness: 0.031888
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 49/48, 56/55, 76/75, 91/90, 154/153, 190/187
 
Mapping: {{mapping| 1 6 1 5 7 9 1 6 | 0 -10 3 -5 -8 -12 7 -4 }}
 
Optimal tunings:
* POTE: ~2 = 1\1, ~15/11 = 529.736
 
{{Optimal ET sequence|legend=1| 9, 25e, 34 }}
 
Badness: 0.026981
 
== Hemiripple ==
{{See also| Ripple family #Hemiripple }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 49/48, 6561/6250
 
{{Mapping|legend=1| 1 2 3 3 | 0 -10 -16 -5 }}
 
{{Multival|legend=1| 10 16 5 2 -20 -33 }}
 
[[Optimal tuning]]s:
* [[POTE]]: ~2 = 1\1, ~36/35 = 50.826
 
{{Optimal ET sequence|legend=1| 23d, 24, 47d, 71bdd }}
 
[[Badness]]: 0.175113
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 49/48, 121/120, 567/550
 
Mapping: {{mapping| 1 2 3 3 4 | 0 -10 -16 -5 -13 }}
 
Optimal tunings:
* POTE: ~2 = 1\1, ~36/35 = 50.826
 
{{Optimal ET sequence|legend=1| 23de, 24, 47de, 71bdde }}
 
Badness: 0.066834
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 66/65, 121/120, 351/350
 
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -10 -16 -5 -13 -7 }}
 
Optimal tunings:
* POTE: ~2 = 1\1, ~36/35 = 50.635


{{Optimal ET sequence|legend=1| 23de, 24, 47de, 71bdde }}
{{Optimal ET sequence|legend=0| 14e, 15 }}


Badness: 0.046588
Badness (Sintel): 2.01


[[Category:Semaphoresmic clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Slendro clan| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Semaphoresmic_clan"