Xenharmonic Wiki

Meantone family: Difference between revisions

← Older editNewer edit →
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
fixed nowiki markup showing up in HTML; set monzos via template, space between heading and metachars, removed redundant TOC forcing
m Some cleanup
Line 1: Line 1:
<span style="display: block; text-align: right;">
{{interwiki
[[de:mitteltönig]]</span>
| de = Mitteltönig
 
| en = Meantone family
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 [[monzo]] for 81/80 goes {{monzo| -4 4 -1 }}, and that can be flipped around to the corresponding [[Wedgies and Multivals|wedgie]], &lt;&lt;1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.
| es =  
| ja =
}}
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 [[monzo]] for 81/80 goes {{monzo| -4 4 -1 }}, and that can be flipped around to the corresponding [[wedgie]], &lt;&lt;1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.


= 5-limit meantone =
= 5-limit meantone =
Line 11: Line 14:
Mapping generator: ~3
Mapping generator: ~3


[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [685.714, 720.000] (7 to 5)
[[Tuning Ranges of Regular Temperaments|valid range]]: [685.714, 720.000] (7 to 5)


nice range: [694.786, 701.955] (1/3 comma to Pythagorean)
nice range: [694.786, 701.955] (1/3 comma to Pythagorean)
Line 19: Line 22:
[[Map]]: [&lt;1 0 -4|, &lt;0 1 4|]
[[Map]]: [&lt;1 0 -4|, &lt;0 1 4|]


EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[26edo|26]], [[31edo|31]], [[43edo|43]], [[45edo|45]], [[50edo|50]], [[55edo|55]], [[67edo|67]], [[69edo|69]], [[74edo|74]], [[81edo|81]], [[88edo|88]], [[98edo|98]], [[105edo|105]], [[117edo|117]], [[131edo|131b]], 212bb, 293bb
{{EDOs|legend=1| 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb }}


[[Badness]]: 0.00736
[[Badness]]: 0.00736
Line 29: Line 32:
<span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>
<span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>


[https://en.wikipedia.org/wiki/Septimal_meantone_temperament Wikipedia article]
* [[Wikipedia: Septimal meantone temperament]]


The comma {{Monzo| -13 10 0 -1 }} for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the [[7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and [[7/5]], C-F#, the tritone. The [[Wedgies_and_Multivals|wedgie]] for septimal meantone is &lt;&lt;1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and [[31edo]] is a good tuning for it.
The comma [[Harrison's comma|{{Monzo| -13 10 0 -1 }}]] for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the [[7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are [[7/6]], C-D#, the augmented second, and [[7/5]], C-F#, the tritone. The [[wedgie]] for septimal meantone is &lt;&lt;1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out [[126/125]] and [[225/224]], and [[31edo]] is a good tuning for it.


[[Comma]]s: 81/80, 126/125
[[Comma]]s: 81/80, 126/125
Line 41: Line 44:
[[Eigenmonzo]]s: 2, 5
[[Eigenmonzo]]s: 2, 5


[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [694.737, 700.000] (19 to 12)
[[Tuning Ranges of Regular Temperaments|valid range]]: [694.737, 700.000] (19 to 12)


nice range: [694.786, 701.955]
nice range: [694.786, 701.955]
Line 51: Line 54:
Mapping generator: ~3
Mapping generator: ~3


Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.
Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.


[[Map]]: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]
[[Map]]: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]
Line 59: Line 62:
[[Wedgie]]: &lt;&lt;1 4 10 4 13 12||
[[Wedgie]]: &lt;&lt;1 4 10 4 13 12||


EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[43edo|43]], [[50edo|50]], [[74edo|74]], [[81edo|81]], [[105edo|105]], [[143edo|143b]]
{{EDOs|legend=1| 12, 19, 31, 43, 50, 74, 81, 105, 143b }}


[[Badness]]: 0.0137
[[Badness]]: 0.0137


== 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
Commas: 81/80, 126/125, 245/242
Line 72: Line 75:
Map: [&lt;2 0 -8 -26 -31|, &lt;0 1 4 10 12|]
Map: [&lt;2 0 -8 -26 -31|, &lt;0 1 4 10 12|]


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


Badness: 0.0381
Badness: 0.0381
Line 83: Line 86:
Map: [&lt;2 0 -8 -26 -31 -40|, &lt;0 1 4 10 12 15|]
Map: [&lt;2 0 -8 -26 -31 -40|, &lt;0 1 4 10 12 15|]


EDOs: 12f, 50
{{EDOs|legend=1| 12f, 50 }}


Badness: 0.0288
Badness: 0.0288


== Unidecimal meantone aka Huygens ==
== Unidecimal meantone aka Huygens ==
See also [[Meantone_vs_meanpop|Meantone vs meanpop]]
{{see also| Meantone vs meanpop }}


[[Comma]]s: 81/80, 126/125, 99/98
[[Comma]]s: 81/80, 126/125, 99/98
Line 109: Line 112:
Mapping generator: ~3
Mapping generator: ~3


[[Algebraic_generator|Algebraic generator]]: Traverse, the positive real root of x^4+2x-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|]
[[Map]]: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]
Line 115: Line 118:
[[Generator]]s: 2, 3
[[Generator]]s: 2, 3


EDOs: [[12edo|12]], [[31edo|31]], [[43edo|43]], [[74edo|74]], [[105edo|105]], [[198edo|198be]]
{{EDOs|legend=1| 12, 31, 43, 74, 105, 198be }}


[[Badness]]: 0.0170
[[Badness]]: 0.0170


[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 ===
Line 130: Line 133:
Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]
Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]


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


Badness: 0.0180
Badness: 0.0180
Line 149: Line 152:
Map: [&lt;1 0 -4 -13 -25 29|, &lt;0 1 4 10 18 -16|]
Map: [&lt;1 0 -4 -13 -25 29|, &lt;0 1 4 10 18 -16|]


EDOs: [[12edo|12]], [[31edo|31]], [[43edo|43]], [[74edo|74]], [[105edo|105]]
{{EDOs|legend=1| 12, 31, 43, 74, 105 }}


Badness: 0.0259
Badness: 0.0259
Line 162: Line 165:
Map: [&lt;1 0 -4 -13 -25 -39|, &lt;0 1 4 10 18 27|]
Map: [&lt;1 0 -4 -13 -25 -39|, &lt;0 1 4 10 18 27|]


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


Badness: 0.0264
Badness: 0.0264
Line 175: Line 178:
Map: [&lt;1 0 -4 -13 -25 -5|, &lt;0 2 8 20 36 11|]
Map: [&lt;1 0 -4 -13 -25 -5|, &lt;0 2 8 20 36 11|]


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


Badness: 0.0314
Badness: 0.0314


== Meanpop ==
== Meanpop ==
See also [[Meantone_vs_meanpop|Meantone vs meanpop]]
{{see also| Meantone vs meanpop }}


[[Comma]]s: 81/80, 126/125, 385/384
[[Comma]]s: 81/80, 126/125, 385/384
Line 201: Line 204:
Mapping generator: ~3
Mapping generator: ~3


[[Algebraic_generator|Algebraic generator]]: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
[[Algebraic generator]]: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
 
[http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{Dead link}}


Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]
Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]
Line 209: Line 210:
[[Generator]]s: 2, 3
[[Generator]]s: 2, 3


EDOs: 12e, [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]]
{{EDOs|legend=1| 12, 19, 31, 50, 81 }}


[[Badness]]: 0.0215
[[Badness]]: 0.0215


[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 Twinkle canon – 50 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{Dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 Twinkle canon – 50 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]


=== 13-limit Meanpop ===
=== 13-limit Meanpop ===
Line 230: Line 232:
Map: [&lt;1 0 -4 -13 24 -20|, &lt;0 1 4 10 -13 15|]
Map: [&lt;1 0 -4 -13 24 -20|, &lt;0 1 4 10 -13 15|]


EDOS: 12ef, [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]]
{{EDOs|legend=1| 12ef, 19, 31, 50, 81 }}


Badness: 0.0209
Badness: 0.0209
Line 1,687: Line 1,689:
[[Category:Theory]]
[[Category:Theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Listen]]
[[Category:Meantone]]
[[Category:Meantone]]
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
{{todo|review|improve readability}}
Retrieved from "https://en.xen.wiki/w/Meantone_family"