Meantone family: Difference between revisions
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{{Technical data page}} | {{Technical data page}} | ||
The '''meantone family''' is the family of [[rank-2 temperament]]s that [[tempering out|temper out]] the syntonic comma, [[81/80]], and thus can all be seen as [[extension]]s of [[meantone]]. | The '''meantone family''' is the family of [[rank-2 temperament]]s that [[tempering out|temper out]] the syntonic comma, [[81/80]], and thus can all be seen as [[extension]]s of [[meantone]]. | ||
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{{Main| Meantone }} | {{Main| Meantone }} | ||
Meantone is characterized by an [[ | 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 | [[Subgroup]]: 2.3.5 | ||
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==== 31edo as splitting the fifth into two, three and nine ==== | ==== 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 | Temperaments discussed elsewhere include | ||
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[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
[[Comma list]]: 81/80, 176/175, | [[Comma list]]: 81/80, 176/175, 7056/6875 | ||
{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}} | {{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}} | ||
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[[Subgroup]]: 2.3.5.7.11.13.17 | [[Subgroup]]: 2.3.5.7.11.13.17 | ||
[[Comma list]]: 81/80, 176/175, 189/ | [[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}} | {{Mapping|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}} | ||
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=== 19-limit === | === 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 | [[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825 | ||
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<references/> | <references/> | ||
[[Category:Temperament families | [[Category:Temperament families] | ||
[[Category:Meantone family| ]] <!-- main article --> | [[Category:Meantone family| ]] <!-- main article --> | ||
[[Category:Meantone| ]] <!-- key article --> | [[Category:Meantone| ]] <!-- key article --> | ||
[[Category: | [[Category:Temperament families]] | ||
[[Category:Catalogs of rank-2 temperaments]] | |||
[[Category:Listen]] | [[Category:Listen]] | ||