Meantone family: Difference between revisions

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The [[5-limit]] parent [[comma]] of the '''meantone family''' is the Didymus or [[Wikipedia: syntonic comma|syntonic comma]], [[81/80]]. This is the one they all temper out. The period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.
The [[5-limit]] parent [[comma]] of the '''meantone family''' is the Didymus or [[Wikipedia: syntonic comma|syntonic comma]], [[81/80]]. This is the one they all temper out. The period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.
The [[7-limit]] [[extension]]s of meantone include:
* Septimal meantone, with normal comma list [{{Monzo| -4 4 -1 }}, [[Harrison's comma|{{Monzo| -13 10 0 -1 }}]]],
* Flattone, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -17 9 0 1 }}],
* Dominant, with normal list [{{Monzo| -4 4 -1 }}, [[64/63|{{Monzo| 6 -2 0 -1 }}]]],
* Sharptone, with normal list [{{Monzo| -4 4 -1 }}, [[28/27|{{Monzo| 2 -3 0 1 }}]]],
* Injera, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -7 8 0 -2 }}],
* Mohajira, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -23 11 0 2 }}],
* Godzilla, with normal list [{{Monzo| -4 4 -1 }}, [[49/48|{{Monzo| -4 -1 0 2 }}]]],
* Mothra, with normal list [{{Monzo| -4 4 -1 }}, [[1029/1024|{{Monzo| -10 1 0 3 }}]]],
* Liese, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -9 11 0 -3 }}],
* Squares, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -3 9 0 -4 }}],
all considered below.
Notable subgroup extensions include mohaha.


== Meantone ==
== Meantone ==
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Scales: [[meantone5]], [[meantone7]], [[meantone12]]
Scales: [[meantone5]], [[meantone7]], [[meantone12]]
=== Seven-limit extensions ===
The second comma of the normal comma list defines which [[7-limit]] family member we are looking at.
* Septimal meantone adds [[Harrison's comma|{{Monzo| -13 10 0 -1 }}]],
* Flattone adds {{Monzo| -17 9 0 1 }},
* Dominant adds [[64/63|{{Monzo| 6 -2 0 -1 }}]],
* Sharptone adds [[28/27|{{Monzo| 2 -3 0 1 }}]],
Those all have a fifth as generator.
* Injera adds {{Monzo| -7 8 0 -2 }} with a half-octave period.
* Mohajira adds {{Monzo| -23 11 0 2 }} and splits the fifth in two.
* Godzilla adds [[49/48|{{Monzo| -4 -1 0 2 }}]] with an 8/7 generator, two of which give the fourth (4/3, an octave minus a fifth).
* Mothra adds [[1029/1024|{{Monzo| -10 1 0 3 }}]] with an 8/7 generator, three of which give the fifth.
* Liese adds {{Monzo| -9 11 0 -3 }} with a 10/7 generator, three of which give the twelfth (3/1, an octave plus a fifth).
* Squares adds {{Monzo| -3 9 0 -4 }} with a 9/7 generator, four of which give the eleventh (8/3, two octaves minus a fifth).
* Jerome adds {{Monzo| 3 7 0 -5 }} and slices the fifth in five.
Temperaments discussed elsewhere include [[Very low accuracy temperaments #Plutus|plutus]].


=== Mohaha ===
=== Mohaha ===
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Badness: 0.025167
Badness: 0.025167
== Plutus ==
Subgroup: 2.3.5.7
[[Comma list]]: 15/14, 81/80
[[Mapping]]: [{{val| 1 0 -4 -5 }}, {{val| 0 1 4 5 }}]
{{Multival|legend=1| 1 4 5 4 5 0 }}
[[POTE generator]]: ~3/2 = 682.895
{{Val list|legend=1| 2cd, 5d, 7 }}
[[Badness]]: 0.045275
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 15/14, 22/21, 81/80
Mapping: [{{val| 1 0 -4 -5 -6 }}, {{val| 0 1 4 5 6 }}]
POTE generator: ~3/2 = 685.234
Optimal GPV sequence: {{Val list| 2cde, 5de, 7 }}
Badness: 0.032521


== Supermean ==
== Supermean ==