Xenharmonic Wiki

Meantone family: Difference between revisions

← Older editNewer edit →
VisualWikitext
KingHyperio (talk | contribs)
autopatrolled, emailconfirmed
216 edits
m “Tritone” is too ambiguous
Tags: Visual edit Mobile edit Mobile web edit
Place mohaha right under meantone as a subgroup extension
Line 7: Line 7:
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.


== Meantone (12&19, 2.3.5) ==
The [[7-limit]] [[extension]]s of meantone include:
{{main| Meantone }}
* 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 }}]]],
* Squares, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -3 9 0 -4 }}],
* Liese, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -9 11 0 -3 }}],
 
all considered below.  
 
Notable subgroup extensions include mohaha.  
 
== Meantone ==
{{Main| Meantone }}


Subgroup: 2.3.5
Subgroup: 2.3.5
Line 37: Line 53:
Scales: [[meantone5]], [[meantone7]], [[meantone12]]
Scales: [[meantone5]], [[meantone7]], [[meantone12]]


=== Seven-limit extensions ===
=== Mohaha ===
The [[7-limit]] [[extension]]s of meantone are:
{{See also| No-sevens subgroup temperaments #Mohaha }}
* 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 }}],
Mohaha is the 2.3.5.11 subgroup temperament with a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 11/9. Mohaha can be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 2.3.5.11 subgroup). Within this paradigm, mohaha is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, and that maps four 3/2's to 5/1. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs. Taking septimal meantone mapping of 7 leads to [[#Migration]], flattone mapping of 7 leads to [[#Ptolemy]], and dominant mapping of 7 leads to [[#Neutrominant]].
* 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 }}]]],
Subgroup: 2.3.5.11
* 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 }}],
[[Comma list]]: 81/80, 121/120
* 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 }}]]],
[[Sval]] [[mapping]]: [{{val| 1 1 0 2 }}, {{val| 0 2 8 5 }}]
* Squares, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -3 9 0 -4 }}], and
 
* Liese, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -9 11 0 -3 }}].
Sval mapping generators: ~2, ~11/9
 
Gencom mapping: [{{val| 1 1 0 0 2 }}, {{val| 0 2 8 0 5 }}]
 
[[Gencom]]: [2 11/9; 81/80 121/120]
 
[[POTE generator]]: ~11/9 = 348.0938
 
{{Val list|legend=1| 7, 17c, 24, 31, 100e, 131bee }}
 
Scales: [[mohaha7]], [[mohaha10]]
 
==== Mohoho ====
Subgroup: 2.3.5.11.13
 
Comma list: 66/65, 81/80, 121/120
 
Sval mapping: [{{val| 1 1 0 2 4 }}, {{val| 0 2 8 5 -1 }}]
 
Sval mapping generators: ~2, ~11/9
 
Gencom mapping: [{{val| 1 1 0 0 2 4 }}, {{val| 0 2 8 0 5 -1 }}]
 
Gencom: [2 11/9; 66/65 81/80 121/120]
 
POTE generator: ~11/9 = 348.9155
 
Vals: {{Val list| 7, 17c, 24, 31, 55, 86ef, 141ceff }}
 
Scales: [[mohaha7]], [[mohaha10]]


== Septimal meantone ==
== Septimal meantone ==
Line 1,384: Line 1,429:


Badness: 0.016548
Badness: 0.016548
== Mohaha ==
{{see also| Subgroup temperaments #Mohaha }}
Mohaha is the 2.3.5.11 subgroup temperament with a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 11/9. Mohaha can be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 2.3.5.11 subgroup). Within this paradigm, mohaha is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, and that maps four 3/2's to 5/1. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs. Taking septimal meantone mapping of 7 leads migration, flattone mapping of 7 leads ptolemy, and dominant mapping of 7 leads maqamic.
Subgroup: 2.3.5.11
[[Comma list]]: 81/80, 121/120
[[Sval]] [[mapping]]: [{{val|1 1 0 2}}, {{val|0 2 8 5}}]
Sval mapping generators: ~2, ~11/9
Gencom mapping: [{{val|1 1 0 0 2}}, {{val|0 2 8 0 5}}]
[[Gencom]]: [2 11/9; 81/80 121/120]
[[POTE generator]]: ~11/9 = 348.0938
{{Val list|legend=1| 7, 17c, 24, 31, 100e, 131bee }}
Scales: [[mohaha7]], [[mohaha10]]
=== Mohoho ===
Subgroup: 2.3.5.11.13
Comma list: 66/65, 81/80, 121/120
Sval mapping: [{{val|1 1 0 2 4}}, {{val|0 2 8 5 -1}}]
Sval mapping generators: ~2, ~11/9
Gencom mapping: [{{val|1 1 0 0 2 4}}, {{val|0 2 8 0 5 -1}}]
Gencom: [2 11/9; 66/65 81/80 121/120]
POTE generator: ~11/9 = 348.9155
Vals: {{Val list| 7, 17c, 24, 31, 55, 86ef, 141ceff }}
Scales: [[mohaha7]], [[mohaha10]]


== Mohajira ==
== Mohajira ==
{{main|Mohajira}}
{{Main| Mohajira }}


Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. [[31edo|31EDO]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31.  
Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. [[31edo|31EDO]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31.  
Retrieved from "https://en.xen.wiki/w/Meantone_family"