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Meantone family: Difference between revisions

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Period: 1\1
Period: 1\1


Optimal ([[POTE]]) generator: 696.495
Optimal ([[POTE]]) generator: ~3/2 = 696.495


EDO generators: [[12edo|7\12]], [[19edo|11\19]], [[31edo|18\31]], [[43edo|25\43]], [[50edo|29\50]]
EDO generators: [[12edo|7\12]], [[19edo|11\19]], [[31edo|18\31]], [[43edo|25\43]], [[50edo|29\50]]
Line 198: Line 198:


== Bimeantone ==
== Bimeantone ==
11/8 is mapped to half octave minus the [[meantone diesis]].
11/8 is mapped to half octave minus the [[meantone diesis]].  


Commas: 81/80, 126/125, 245/242
Period: 1\2


[[POTE generator]]: ~3/2 = 696.016
Optimal ([[POTE]]) generator: ~3/2 = 696.016


Map: [{{val|2 0 -8 -26 -31}}, {{val|0 1 4 10 12}}]
EDO generators: [[38edo|22\38]], [[50edo|29\50]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 81/80, 126/125, 245/242
 
Mapping: [{{val| 2 0 -8 -26 -31 }}, {{val| 0 1 4 10 12 }}]
 
Mapping generators: ~63/44, ~3


{{EDOs|legend=1| 12, 26de, 38d, 50 }}
{{EDOs|legend=1| 12, 26de, 38d, 50 }}


Badness: 0.0381
Badness: 0.0381
</div></div>


=== 13-limit ===
=== 13-limit ===
Commas: 81/80, 105/104, 126/125, 245/242


[[POTE generator]]: ~3/2 = 695.836
Period: 1\2


Map: [&lt;2 0 -8 -26 -31 -40|, &lt;0 1 4 10 12 15|]
Optimal ([[POTE]]) generator: ~3/2 = 695.836
 
EDO generators: [[38edo|22\38]], [[50edo|29\50]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 81/80, 105/104, 126/125, 245/242
 
Mapping: [{{val| 2 0 -8 -26 -31 -40 }}, {{val| 0 1 4 10 12 15}}]
 
Mapping generators: ~55/39, ~3


{{EDOs|legend=1| 12f, 26deff, 38df, 50 }}
{{EDOs|legend=1| 12f, 26deff, 38df, 50 }}


Badness: 0.0288
Badness: 0.0288
</div></div>


== Unidecimal meantone aka Huygens ==
== Unidecimal meantone aka Huygens ==
{{see also| Meantone vs meanpop }}
{{see also| Meantone vs meanpop }}


[[Comma]]s: 81/80, 126/125, 99/98
Period: 1\1


[[11-odd-limit|11-limit]] minimax
Optimal ([[POTE]]) generator: ~3/2 = 696.967


[{{Monzo| 1 0 0 0 0 }}, {{Monzo| 25/16 -1/8 0 0 1/16 }}, {{Monzo| 9/4 -1/2 0 0 1/4 }},
EDO generators: [[31edo|19\31]], [[43edo|25\43]]
{{Monzo| 21/8 -5/4 0 0 5/8 }}, {{Monzo| 25/8 -9/4 0 0 9/8 }}<nowiki>]</nowiki>


[[Eigenmonzo]]s: 2, 11/9
Scales (Scala files):  


valid range: [696.774, 700.000] (31 to 12)
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 81/80, 126/125, 99/98


nice range: [691.202, 701.955]
Mapping: [{{val| 1 0 -4 -13 -25 }}, {{val| 0 1 4 10 18 }}]
 
Mapping generators: ~2, ~3


strict range: [696.774, 700.000]
Minimax tuning:  
* [[11-odd-limit]]
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 25/16 -1/8 0 0 1/16 }}, {{Monzo| 9/4 -1/2 0 0 1/4 }}, {{Monzo| 21/8 -5/4 0 0 5/8 }}, {{Monzo| 25/8 -9/4 0 0 9/8 }}]
: [[Eigenmonzo]]s: 2, 11/9


[[POTE generator]]: ~3/2 = 696.967
Tuning ranges:
* valid range: [696.774, 700.000] (31 to 12)
* nice range: [691.202, 701.955]
* strict range: [696.774, 700.000]


[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
[[Map]]: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]
Mapping generator: ~3


{{EDOs|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}
{{EDOs|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}


[[Badness]]: 0.0170
Badness: 0.0170
</div></div>


* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 Twinkle canon – 74 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 Twinkle canon – 74 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]


=== Tridecimal meantone ===
=== Tridecimal meantone ===
Commas: 66/65, 81/80, 99/98, 105/104


[[POTE generator]]: ~3/2 = 696.642
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 696.642
 
EDO generators: [[31edo|19\31]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 66/65, 81/80, 99/98, 105/104


Mapping generator: ~3
Mapping: [{{val| 1 0 -4 -13 -25 -20 }}, {{val| 0 1 4 10 18 15 }}]


Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]
Mapping generators: ~2, ~3


{{EDOs|legend=1| 12f, 19e, 31 }}
{{EDOs|legend=1| 12f, 19e, 31 }}


Badness: 0.0180
Badness: 0.0180
</div></div>


=== Grosstone ===
=== Grosstone ===
Commas: 81/80, 99/98, 126/125, 144/143


valid range: [696.774, 700.000] (31 to 12)
Period: 1\1


nice range: [691.202, 701.955]
Optimal ([[POTE]]) generator: ~3/2 = 697.264


strict range: [696.774, 700.000]
EDO generators: [[31edo|19\31]], [[43edo|25\43]]


[[POTE generator]]: ~3/2 = 697.264
Scales (Scala files):  


Mapping generator: ~3
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 81/80, 99/98, 126/125, 144/143


Map: [&lt;1 0 -4 -13 -25 29|, &lt;0 1 4 10 18 -16|]
Mapping: [{{val| 1 0 -4 -13 -25 29 }}, {{val| 0 1 4 10 18 -16 }}]
 
Mapping generators: ~2, ~3
 
Tuning ranges:
* valid range: [696.774, 700.000] (31 to 12)
* nice range: [691.202, 701.955]
* strict range: [696.774, 700.000]


{{EDOs|legend=1| 12, 19ef, 31, 43, 74 }}
{{EDOs|legend=1| 12, 19ef, 31, 43, 74 }}


Badness: 0.0259
Badness: 0.0259
</div></div>


=== Meridetone ===
=== Meridetone ===
Commas: 78/77, 81/80, 99/98, 126/125


[[POTE generator]]: ~3/2 = 697.529
Period: 1\1


Mapping generator: ~3
Optimal ([[POTE]]) generator: ~3/2 = 697.529


Map: [&lt;1 0 -4 -13 -25 -39|, &lt;0 1 4 10 18 27|]
EDO generators: [[43edo|25\43]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 78/77, 81/80, 99/98, 126/125
 
Mapping: [{{val| 1 0 -4 -13 -25 -39 }}, {{val| 0 1 4 10 18 27 }}]
 
Mapping generators: ~2, ~3


{{EDOs|legend=1| 12f, 31f, 43 }}
{{EDOs|legend=1| 12f, 31f, 43 }}


Badness: 0.0264
Badness: 0.0264
</div></div>


=== Hemimeantone ===
=== Hemimeantone ===
Commas: 81/80, 99/98, 126/125, 169/168


[[POTE generator]]: ~52/45 = 250.304
Period: 1\1
 
Optimal ([[POTE]]) generator: ~15/13 = 250.304
 
EDO generators: [[43edo|9\43]], [[62edo|13\62]]


Mapping generator: ~26/15
Scales (Scala files):  


Map: [&lt;1 0 -4 -13 -25 -5|, &lt;0 2 8 20 36 11|]
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 81/80, 99/98, 126/125, 169/168
 
Mapping: [{{val| 1 0 -4 -13 -25 -5 }}, {{val| 0 2 8 20 36 11 }}]
 
Mapping generators: ~2, ~26/15


{{EDOs|legend=1| 19e, 43, 62, 167bef }}
{{EDOs|legend=1| 19e, 43, 62, 167bef }}


Badness: 0.0314
Badness: 0.0314
</div></div>


== Meanpop ==
== Meanpop ==
Retrieved from "https://en.xen.wiki/w/Meantone_family"