8th-octave temperaments: Difference between revisions

Line 3:

An 8th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 8.

~~Discussed~~ elsewhere ~~are~~:

Temperaments discussed elsewhere include:

* ''Octant'', → [[Schismatic family#Octant|Schismatic family]]

* ''Octant'' → [[Schismatic family#Octant|Schismatic family]]

* ''Octoid'', → [[Ragismic microtemperaments#Octoid|Ragismic microtemperaments]]

== Octatonic ==

[[12/11]] is very close to 1 step of 8edo, and hence this temperament tempers out the octatonic comma, the difference between a stack of 8 12/11's and the octave. The octatonic temperament makes a [[consistent circle]].

Subgroup: 2.3.11

[[Subgroup]]: 2.3.11

Comma list: {{monzo|15 8 0 0 -8}}

[[Comma list]]: {{monzo|15 8 0 0 -8}}

: Mapping generators: ~12/11 ~~= 1\8~~, ~3

: Mapping generators: ~12/11, ~3

[[Support]]ing [[ET]]s: {{EDOs|16, 24, 32, 48, 56, 80~~, 72~~, 120, 128, 152}}, ...

[[Support]]ing [[ET]]s: {{EDOs|16, 24, 32, 48, 56, 72, 80, 120, 128, 152}}, ...

== Octium ==

Line 29:

Line 28:

: Mapping generators: ~10051/9216 ~~= 1\8~~, ~24/23 ~~= 76.284~~

: Mapping generators: ~10051/9216, ~24/23

[[Optimal tuning]] ([[CTE]]): ~24/23 = 76.284

Line 35:

Line 34:

[[Support]]ing [[ET]]s: {{EDOs|472, 1400, 1872, 3272, 5144}}

== Octoid ~~(5-limit)~~ ==

== Octoid ==

''For the 7-limit temperament, see [[Ragismic microtemperaments#Octoid]].''

: ''For the 7-limit temperament, see [[Ragismic microtemperaments #Octoid]].''

Subgroup: 2.3.5

[[Subgroup]]: 2.3.5

Comma list: 59604644775390625/59296646043258912

[[Comma list]]: 59604644775390625/59296646043258912

~~[[Optimal tuning]] ([[CWE]])~~: ~2125764/1953125 ~~= 1\8~~, ~4374/3125 ~~= 584.025~~

: Mapping generators: ~2125764/1953125, ~4374/3125

[[~~Support~~]]~~ing~~ [[ET]]s: {{~~EDOs~~|8, ~~152~~, ~~224~~, 72, ~~376~~, 80, ~~600, 528, 296, 64c, 232, 88bc, 824~~, ~~680~~}}

[[Optimal tuning]]s:

* [[WE]]: ~2125764/1953125 = 150.001{{c}}, ~4374/3125 = 584.027{{c}}

* [[CWE]]: ~2125764/1953125 = 150.000{{c}}, ~4374/3125 = 584.025{{c}}

{{Optimal ET sequence|legend=1| 8, 56bcc, 64c, 72, 152, 224, 376, 976 }}

[[Badness]] (Sintel): 6.687

@@ Line 3: / Line 3: @@
 An 8th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 8.
-Discussed elsewhere are:
+Temperaments discussed elsewhere include:
-* ''Octant'', → [[Schismatic family#Octant|Schismatic family]]
+* ''Octant'' → [[Schismatic family#Octant|Schismatic family]]
-* ''Octoid'', → [[Ragismic microtemperaments#Octoid|Ragismic microtemperaments]]
 == Octatonic ==
 [[12/11]] is very close to 1 step of 8edo, and hence this temperament tempers out the octatonic comma, the difference between a stack of 8 12/11's and the octave. The octatonic temperament makes a [[consistent circle]].
-Subgroup: 2.3.11
+[[Subgroup]]: 2.3.11
-Comma list: {{monzo|15 8 0 0 -8}}
+[[Comma list]]: {{monzo|15 8 0 0 -8}}
 {{Mapping|legend=1|8 0 15|0 1 1}}
-: Mapping generators: ~12/11 = 1\8, ~3
+: Mapping generators: ~12/11, ~3
-[[Support]]ing [[ET]]s: {{EDOs|16, 24, 32, 48, 56, 80, 72, 120, 128, 152}}, ...
+[[Support]]ing [[ET]]s: {{EDOs|16, 24, 32, 48, 56, 72, 80, 120, 128, 152}}, ...
 == Octium ==
@@ Line 29: / Line 28: @@
 {{Mapping|legend=2| 8 2 35 25 47 2 | 0 21 -2 22 -16 74 }}
-: Mapping generators: ~10051/9216 = 1\8, ~24/23 = 76.284
+: Mapping generators: ~10051/9216, ~24/23
 [[Optimal tuning]] ([[CTE]]): ~24/23 = 76.284
@@ Line 35: / Line 34: @@
 [[Support]]ing [[ET]]s: {{EDOs|472, 1400, 1872, 3272, 5144}}
-== Octoid (5-limit) ==
+== Octoid ==
-''For the 7-limit temperament, see [[Ragismic microtemperaments#Octoid]].''
+: ''For the 7-limit temperament, see [[Ragismic microtemperaments #Octoid]].''
-Subgroup: 2.3.5
+[[Subgroup]]: 2.3.5
-Comma list: 59604644775390625/59296646043258912
+[[Comma list]]: 59604644775390625/59296646043258912
-{{Mapping|legend=2| 8 1 3 | 0 3 4 }}
+{{Mapping|legend=1| 8 1 3 | 0 3 4 }}
-[[Optimal tuning]] ([[CWE]]): ~2125764/1953125 = 1\8, ~4374/3125 = 584.025
+: Mapping generators: ~2125764/1953125, ~4374/3125
-[[Support]]ing [[ET]]s: {{EDOs|8, 152, 224, 72, 376, 80, 600, 528, 296, 64c, 232, 88bc, 824, 680}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2125764/1953125 = 150.001{{c}}, ~4374/3125 = 584.027{{c}}
+* [[CWE]]: ~2125764/1953125 = 150.000{{c}}, ~4374/3125 = 584.025{{c}}
+{{Optimal ET sequence|legend=1| 8, 56bcc, 64c, 72, 152, 224, 376, 976 }}
+[[Badness]] (Sintel): 6.687
 {{Navbox fractional-octave}}