Meantone family: Difference between revisions

Line 907:

<span style="display: block; text-align: right;">[[:de:Mohajira|Deutsch]]</span>

[[~~Comma|~~Comma]]s: 81/80, 6144/6125

[[Comma]]s: 81/80, 6144/6125

Mohajira, with wedgie <<2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. [[~~31edo|~~31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.

Mohajira, with wedgie <<2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.

Mohajira can also 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 11-limit). Within this paradigm, mohajira 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, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4.

[[7-limit|7]] and [[9-limit|9-limit]] minimax 1/4 comma

[[7-odd-limit|7]] and [[9-odd-limit|9-limit]] minimax 1/4 comma

[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |6 0 -11/8 0>]

[[~~Eigenmonzo|~~Eigenmonzo]]s: 2, 5

[[Eigenmonzo]]s: 2, 5

[[POTE_tuning|POTE generator]]: ~128/105 = 348.415

Line 927:

Map: [<1 1 0 6|, <0 2 8 -11|]

[[~~generator|~~Generator]]s: 2, 128/105

[[Generator]]s: 2, 128/105

[[~~wedgie|~~Wedgie]]: <<2 8 -11 8 -23 -48||

[[Wedgie]]: <<2 8 -11 8 -23 -48||

EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]]

EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]], [[38edo|38]], [[55edo|55]], [[69edo|69]]

[[~~Badness|~~Badness]]: 0.0557

[[Badness]]: 0.0557

==11-limit==

~~[[Comma|Comma]]s~~: 81/80, 121/120, 176/175

Commas: 81/80, 121/120, 176/175

[[11-limit|11-limit]] minimax 1/4 comma

[[11-odd-limit|11-limit]] minimax 1/4 comma

[|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>,

|6 0 -11/8 0 0>, |2 0 5/8 0 0>]

~~[[Eigenmonzo|Eigenmonzo]]s~~: 2, 5

Eigenmonzos: 2, 5

~~[[POTE_tuning|~~POTE generator]]: ~11/9 = 348.477

POTE generator: ~11/9 = 348.477

Mapping generator: ~11/9

Line 951:

Map: [<1 1 0 6 2|, <0 2 8 -11 5|]

~~[[generator|Generator]]s~~: 2, 11/9

Generators: 2, 11/9

EDOs: ~~[[7edo|~~7]], ~~[[24edo|~~24]], ~~[[31edo|~~31]]

EDOs: 7, 24, 31, 38, 55

[[Badness~~|Badness]]~~: 0.0261

Badness: 0.0261

==13-limit==

Commas: 81/80~~, 121/120~~, 105/104, 66/65

Commas: 66/65, 81/80, 105/104, 121/120

POTE generator: ~11/9 = 348.558

Line 966:

Map: [<1 1 0 6 2 4|, <0 2 8 -11 5 -1|]

EDOs: 7, 24, 31, ~~117ef~~, ~~148bef~~

EDOs: 7, 24, 31, 38, 55

Badness: 0.0234

==17-limit==

Commas: 66/65, 81/80, 105/104, 121/120, 154/153

POTE generator: ~11/9 = 348.736

Mapping generator: ~11/9

Map: [<1 1 0 6 2 4 7|, <0 2 8 -11 5 -1 -10|]

EDOs: 7, 24, 31, 38g, 55

Badness: 0.0206

==19-limit==

Commas: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152

POTE generator: ~11/9 = 348.810

Mapping generator: ~11/9

Map: [<1 1 0 6 2 4 7 6|, <0 2 8 -11 5 -1 -10 -6|]

EDOs: 7, 24, 31, 38gh, 55

Badness: 0.0173

=Ptolemy=

@@ Line 907: / Line 907: @@
 <span style="display: block; text-align: right;">[[:de:Mohajira|Deutsch]]</span>
-[[Comma|Comma]]s: 81/80, 6144/6125
+[[Comma]]s: 81/80, 6144/6125
-Mohajira, with wedgie &lt;&lt;2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. [[31edo|31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.
+Mohajira, with wedgie &lt;&lt;2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.
 Mohajira can also 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 11-limit). Within this paradigm, mohajira 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, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4.
-[[7-limit|7]] and [[9-limit|9-limit]] minimax 1/4 comma
+[[7-odd-limit|7]] and [[9-odd-limit|9-limit]] minimax 1/4 comma
 [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |6 0 -11/8 0&gt;]
-[[Eigenmonzo|Eigenmonzo]]s: 2, 5
+[[Eigenmonzo]]s: 2, 5
 [[POTE_tuning|POTE generator]]: ~128/105 = 348.415
@@ Line 927: / Line 927: @@
 Map: [&lt;1 1 0 6|, &lt;0 2 8 -11|]
-[[generator|Generator]]s: 2, 128/105
+[[Generator]]s: 2, 128/105
-[[wedgie|Wedgie]]: &lt;&lt;2 8 -11 8 -23 -48||
+[[Wedgie]]: &lt;&lt;2 8 -11 8 -23 -48||
-EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]]
+EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]], [[38edo|38]], [[55edo|55]], [[69edo|69]]
-[[Badness|Badness]]: 0.0557
+[[Badness]]: 0.0557
 ==11-limit==
-[[Comma|Comma]]s: 81/80, 121/120, 176/175
+Commas: 81/80, 121/120, 176/175
-[[11-limit|11-limit]] minimax 1/4 comma
+[[11-odd-limit|11-limit]] minimax 1/4 comma
 [|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
 |6 0 -11/8 0 0&gt;, |2 0 5/8 0 0&gt;]
-[[Eigenmonzo|Eigenmonzo]]s: 2, 5
+Eigenmonzos: 2, 5
-[[POTE_tuning|POTE generator]]: ~11/9 = 348.477
+POTE generator: ~11/9 = 348.477
 Mapping generator: ~11/9
@@ Line 951: / Line 951: @@
 Map: [&lt;1 1 0 6 2|, &lt;0 2 8 -11 5|]
-[[generator|Generator]]s: 2, 11/9
+Generators: 2, 11/9
-EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]]
+EDOs: 7, 24, 31, 38, 55
-[[Badness|Badness]]: 0.0261
+Badness: 0.0261
 ==13-limit==
-Commas: 81/80, 121/120, 105/104, 66/65
+Commas: 66/65, 81/80, 105/104, 121/120
 POTE generator: ~11/9 = 348.558
@@ Line 966: / Line 966: @@
 Map: [&lt;1 1 0 6 2 4|, &lt;0 2 8 -11 5 -1|]
-EDOs: 7, 24, 31, 117ef, 148bef
+EDOs: 7, 24, 31, 38, 55
 Badness: 0.0234
+==17-limit==
+Commas: 66/65, 81/80, 105/104, 121/120, 154/153
+POTE generator: ~11/9 = 348.736
+Mapping generator: ~11/9
+Map: [&lt;1 1 0 6 2 4 7|, &lt;0 2 8 -11 5 -1 -10|]
+EDOs: 7, 24, 31, 38g, 55
+Badness: 0.0206
+==19-limit==
+Commas: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
+POTE generator: ~11/9 = 348.810
+Mapping generator: ~11/9
+Map: [&lt;1 1 0 6 2 4 7 6|, &lt;0 2 8 -11 5 -1 -10 -6|]
+EDOs: 7, 24, 31, 38gh, 55
+Badness: 0.0173
 =Ptolemy=