Semaphoresmic clan: Difference between revisions

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== Semaphore ==

~~: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum]].''~~

Semaphore tempers out 49/48, and splits a [[3/1|perfect twelfth]] into two halfs of [[7/4]][[~]][[12/7]], and a [[4/3|perfect fourth]] into two halfs of [[7/6]]~[[8/7]], hence the name ''semaphore'', which sounds like ''semifourth''; its [[ploidacot]] is alpha-dicot. [[19edo]] and [[24edo]] are among the possible edo tunings.

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~~: sval mapping generators: ~2, ~7/4~~

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

: mapping generators: ~2, ~7/4

[[Optimal tuning]]s:

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These all use the same nominal generator as semaphore, though some of them are of very low accuracy.

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

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* [[Keemun]] (+126/125) → [[Kleismic family #Keemun|Kleismic family]]

* ''[[Nautilus]]'' (+250/243) → [[Porcupine family #Nautilus|Porcupine family]]

* ''[[Hemidim]]'' (+648/625) → [[~~Dimipent~~ family #Hemidim|~~Dimipent~~ family]]

* ''[[Hemidim]]'' (+648/625) → [[Diminished family #Hemidim|Diminished family]]

* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]

* ''[[Spell]]'' (+3125/3072) → [[Hemimean clan #Spell|Hemimean clan]]

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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

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[[Badness]] (Sintel): 0.677

=== 11-~~limit~~ ===

==== 2.3.5.7.13 subgroup ====

Subgroup: 2.3.5.7.11.13

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}}

Badness (Sintel): 0.591

=== Undecimal godzilla ===

Subgroup: 2.3.5.7.11

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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 ===

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

Badness (Sintel): 1.18

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* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 948.8468{{c}}

Badness (Sintel): 1.10

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== Negri ==

: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Negri (5-limit)]].''

Negri tempers out the [[negri comma]] in the 5-limit, 49/48 and [[225/224]] in the 7-limit. It may be described as {{nowrap| 9 & 10 }}; its ploidacot is omega-tetracot. It can be extended naturally to the 2.3.5.7.13 subgroup by adding [[91/90]] to the comma list; this will be discussed below under the title of negra.

Negri tempers out the [[negri comma]] in the 5-limit, 49/48 and [[225/224]] in the 7-limit. It may be described as {{nowrap| 9 & 10 }}; its ploidacot is omega-tetracot. It can be extended naturally to the 2.3.5.7.13 subgroup by adding 91/90 and/or 105/104 to the comma list; this will be discussed below under the title of negra.

=== 7-limit ===

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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: [2 14/13; 49/48 65/64 91/90]~~

Optimal tunings:

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Badness (Sintel): 0.463

=== ~~11-limit~~ ===

=== Undecimal negri ===

Subgroup: 2.3.5.7.11

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=== 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: {{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:

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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:

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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:

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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:

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 == Semaphore ==
 {{Main| Semaphore and godzilla }}
-: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum]].''
 Semaphore tempers out 49/48, and splits a [[3/1|perfect twelfth]] into two halfs of [[7/4]][[~]][[12/7]], and a [[4/3|perfect fourth]] into two halfs of [[7/6]]~[[8/7]], hence the name ''semaphore'', which sounds like ''semifourth''; its [[ploidacot]] is alpha-dicot. [[19edo]] and [[24edo]] are among the possible edo tunings.
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 {{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 }}
-: [[gencom]]: [2 7/4; 49/48]
+: mapping generators: ~2, ~7/4
 [[Optimal tuning]]s:
@@ Line 45: / Line 42: @@
 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
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 * [[Keemun]] (+126/125) → [[Kleismic family #Keemun|Kleismic family]]
 * ''[[Nautilus]]'' (+250/243) → [[Porcupine family #Nautilus|Porcupine family]]
-* ''[[Hemidim]]'' (+648/625) → [[Dimipent family #Hemidim|Dimipent family]]
+* ''[[Hemidim]]'' (+648/625) → [[Diminished family #Hemidim|Diminished family]]
 * [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
 * ''[[Spell]]'' (+3125/3072) → [[Hemimean clan #Spell|Hemimean clan]]
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 {{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
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 [[Badness]] (Sintel): 0.677
-=== 11-limit ===
+==== 2.3.5.7.13 subgroup ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 49/48, 81/80, 91/90
+Subgroup-val mapping: {{mapping| 1 0 -4 2 -5 | 0 2 8 1 11 }}
+Optimal tunings:
+* WE: ~2 = 1203.7816{{c}}, ~7/4 = 950.5570{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 948.0037{{c}}
+{{Optimal ET sequence|legend=0| 5, 14cf, 19 }}
+Badness (Sintel): 0.591
+=== Undecimal godzilla ===
 Subgroup: 2.3.5.7.11
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 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 ===
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 * CWE: ~2 = 1200.0000{{c}}, ~7/4 = 948.8625{{c}}
-{{Optimal ET sequence|legend=0| 5, 19, 24 }}
+{{Optimal ET sequence|legend=0| 19, 24 }}
 Badness (Sintel): 1.18
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 * CWE: ~2 = 1200.0000{{c}}, ~7/4 = 948.8468{{c}}
-{{Optimal ET sequence|legend=0| 5, 19, 24 }}
+{{Optimal ET sequence|legend=0| 19, 24 }}
 Badness (Sintel): 1.10
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 == Negri ==
 {{Main| Negri }}
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Negri (5-limit)]].''
-Negri tempers out the [[negri comma]] in the 5-limit, 49/48 and [[225/224]] in the 7-limit. It may be described as {{nowrap| 9 & 10 }}; its ploidacot is omega-tetracot. It can be extended naturally to the 2.3.5.7.13 subgroup by adding [[91/90]] to the comma list; this will be discussed below under the title of negra.
+Negri tempers out the [[negri comma]] in the 5-limit, 49/48 and [[225/224]] in the 7-limit. It may be described as {{nowrap| 9 & 10 }}; its ploidacot is omega-tetracot. It can be extended naturally to the 2.3.5.7.13 subgroup by adding 91/90 and/or 105/104 to the comma list; this will be discussed below under the title of negra.
 === 7-limit ===
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 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: [2 14/13; 49/48 65/64 91/90]
 Optimal tunings:
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 Badness (Sintel): 0.463
-=== 11-limit ===
+=== Undecimal negri ===
 Subgroup: 2.3.5.7.11
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 === 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: {{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:
@@ Line 484: / Line 513: @@
 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:
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 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:
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 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: