8th-octave temperaments: Difference between revisions

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

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

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

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

Temperaments discussed elsewhere include:

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

== 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 ==

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

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

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

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[[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

[[Comma list]]: 59604644775390625/59296646043258912

~~Subgroup: 2.~~3.5

~~Comma list~~: ~~59604644775390625~~/~~59296646043258912~~

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

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

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

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

[[Badness]] (Sintel): 6.687

@@ Line 1: / Line 1: @@
-{{Fractional-octave navigation|8}}
+{{Technical data page}}
-Discussed elsewhere are:
+{{Infobox fractional-octave|8}}
-* ''Octant'', → [[Schismatic family#Octant|Schismatic family]]
+An 8th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 8.
-* ''Octoid'', → [[Ragismic microtemperaments#Octoid|Ragismic microtemperaments]]
+Temperaments discussed elsewhere include:
+* ''Octant'' → [[Schismatic family#Octant|Schismatic family]]
 == 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 25: / 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 31: / 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
+[[Comma list]]: 59604644775390625/59296646043258912
-Subgroup: 2.3.5
+{{Mapping|legend=1| 8 1 3 | 0 3 4 }}
-Comma list: 59604644775390625/59296646043258912
+: Mapping generators: ~2125764/1953125, ~4374/3125
-{{Mapping|legend=2| 8 1 3 | 0 3 4 }}
+[[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 tuning]] ([[CWE]]): ~2125764/1953125 = 1\8, ~4374/3125 = 584.025
+{{Optimal ET sequence|legend=1| 8, 56bcc, 64c, 72, 152, 224, 376, 976 }}
-[[Support]]ing [[ET]]s: {{EDOs|8, 152, 224, 72, 376, 80, 600, 528, 296, 64c, 232, 88bc, 824, 680}}
+[[Badness]] (Sintel): 6.687
 {{Navbox fractional-octave}}