Meantone family: Difference between revisions

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Meantone is characterized by an [[~~2/1|~~octave]] [[period]], a [[3/2|fifth]] [[generator]], and the relationship that four fifths go to make up a [[5/1|5th harmonic]].

Meantone is characterized by an [[octave]] [[period]], a [[3/2|fifth]] [[generator]], and the relationship that four fifths go to make up a [[5/1|5th harmonic]].

[[Subgroup]]: 2.3.5

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==== 31edo as splitting the fifth into two, three and nine ====

[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering out 225/224, which interestingly, though a rank-2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering out 225/224).

[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering out [[225/224]], which interestingly, though a rank-2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering out 225/224).

Temperaments discussed elsewhere include

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[[Badness]] (Sintel): 0.347

~~=== 2.3.5.7.23 subgroup ===~~

~~There is a reasonable extension to the 2.3.5.7.23 subgroup, which tempers out [[161/160]] and [[162/161]], and maps the [[23/16]] to the diminished fifth (C–G♭).~~

~~Subgroup: 2.3.5.7.23~~

~~Comma list: 81/80, 126/125, 161/160~~

~~Mapping: {{mapping| 1 0 -4 -13 14 | 0 1 4 10 -6 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1201.3627{{c}}, ~3/2 = 697.2263{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.5082{{c}}

~~{{Optimal ET sequence|legend=0| 12, 19, 31, 81, 112bi }}~~

~~Badness (Sintel): 0.451~~

=== Undecimal meantone (huygens) ===

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* [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]

~~==== 2.3.5.7.11.23 subgroup ====~~

~~It is very reasonable to extend meanpop to the 2.3.5.7.11.23 subgroup. In this temperament, [[21/20]], [[23/22]], [[24/23]], [[25/24]] and [[28/27]] are mapped to the augmented unison.~~

~~Subgroup: 2.3.5.7.11.23~~

~~Comma list: 81/80, 126/125, 161/160, 231/230~~

~~Mapping: {{mapping| 1 0 -4 -13 24 14 | 0 1 4 10 -13 -6 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1201.3759{{c}}, ~3/2 = 697.2251{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4318{{c}}

~~{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112bi }}~~

~~Badness (Sintel): 0.615~~

==== Tridecimal meanpop ====

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[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 81/80, 176/175, ~~7058~~/6875

[[Comma list]]: 81/80, 176/175, 7056/6875

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[[Subgroup]]: 2.3.5.7.11.13.17

[[Comma list]]: 81/80, 176/175, 189/~~197~~, 196/195, 832/825

[[Comma list]]: 81/80, 176/175, 189/187, 196/195, 832/825

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=== 19-limit ===

[[Subgroup]]: 2.3.5.7.11.13.19

[[Subgroup]]: 2.3.5.7.11.13.17.19

[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825

@@ Line 11: / Line 11: @@
 {{Main| Meantone }}
-Meantone is characterized by an [[2/1|octave]] [[period]], a [[3/2|fifth]] [[generator]], and the relationship that four fifths go to make up a [[5/1|5th harmonic]].
+Meantone is characterized by an [[octave]] [[period]], a [[3/2|fifth]] [[generator]], and the relationship that four fifths go to make up a [[5/1|5th harmonic]].
 [[Subgroup]]: 2.3.5
@@ Line 71: / Line 71: @@
 ==== 31edo as splitting the fifth into two, three and nine ====
-[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering out 225/224, which interestingly, though a rank-2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering out 225/224).
+[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering out [[225/224]], which interestingly, though a rank-2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering out 225/224).
 Temperaments discussed elsewhere include
@@ Line 115: / Line 115: @@
 [[Badness]] (Sintel): 0.347
-=== 2.3.5.7.23 subgroup ===
-There is a reasonable extension to the 2.3.5.7.23 subgroup, which tempers out [[161/160]] and [[162/161]], and maps the [[23/16]] to the diminished fifth (C–G♭).
-Subgroup: 2.3.5.7.23
-Comma list: 81/80, 126/125, 161/160
-Mapping: {{mapping| 1 0 -4 -13 14 | 0 1 4 10 -6 }}
-Optimal tunings:
-* WE: ~2 = 1201.3627{{c}}, ~3/2 = 697.2263{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.5082{{c}}
-{{Optimal ET sequence|legend=0| 12, 19, 31, 81, 112bi }}
-Badness (Sintel): 0.451
 === Undecimal meantone (huygens) ===
@@ Line 457: / Line 440: @@
 * [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]
-==== 2.3.5.7.11.23 subgroup ====
-It is very reasonable to extend meanpop to the 2.3.5.7.11.23 subgroup. In this temperament, [[21/20]], [[23/22]], [[24/23]], [[25/24]] and [[28/27]] are mapped to the augmented unison.
-Subgroup: 2.3.5.7.11.23
-Comma list: 81/80, 126/125, 161/160, 231/230
-Mapping: {{mapping| 1 0 -4 -13 24 14 | 0 1 4 10 -13 -6 }}
-Optimal tunings:
-* WE: ~2 = 1201.3759{{c}}, ~3/2 = 697.2251{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4318{{c}}
-{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112bi }}
-Badness (Sintel): 0.615
 ==== Tridecimal meanpop ====
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 [[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: 81/80, 176/175, 7058/6875
+[[Comma list]]: 81/80, 176/175, 7056/6875
 {{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}}
@@ Line 1,192: / Line 1,158: @@
 [[Subgroup]]: 2.3.5.7.11.13.17
-[[Comma list]]: 81/80, 176/175, 189/197, 196/195, 832/825
+[[Comma list]]: 81/80, 176/175, 189/187, 196/195, 832/825
 {{Mapping|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}}
@@ Line 1,206: / Line 1,172: @@
 === 19-limit ===
-[[Subgroup]]: 2.3.5.7.11.13.19
+[[Subgroup]]: 2.3.5.7.11.13.17.19
 [[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825