1/5-comma meantone: Difference between revisions
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Something to note with septimal meantone |
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'''1/5-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes [[2/1|2]] and [[16/15]] purely. | '''1/5-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes [[2/1|2]] and [[16/15]] purely. | ||
The major third ([[5/4]]) and the fifth are equally off from JI by 1/5 syntonic comma (''c''. 4.3 [[cent]]s), with the major third sharp and the fifth flat. | The major third ([[5/4]]) and the fifth are equally off from JI by 1/5 syntonic comma (''c''. 4.3 [[cent]]s), with the major third sharp and the fifth flat. The augmented sixth is exactly [[225/128]], and for septimal meantone, this means the [[7/4]] is sharp by a [[225/224|septimal kleisma (225/224)]]. | ||
1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.) | 1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.) | ||
Revision as of 15:48, 16 September 2024
1/5-comma meantone is the tuning of meantone temperament in which each perfect fifth is tempered by a fifth of a syntonic comma from its just value of 3/2. This results in major sevenths being exactly 15/8 (and minor seconds being exactly 16/15), and in fact this defines 1/5-comma; it is the regular tuning which tempers out 81/80 and tunes 2 and 16/15 purely.
The major third (5/4) and the fifth are equally off from JI by 1/5 syntonic comma (c. 4.3 cents), with the major third sharp and the fifth flat. The augmented sixth is exactly 225/128, and for septimal meantone, this means the 7/4 is sharp by a septimal kleisma (225/224).
1/5-comma meantone is practically identical to the closed system of 43edo (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)
Tuning profile
| [⟨ | 1 | 4/5 | -4/5 | -5 | ] |
| ⟨ | 0 | 1/5 | 4/5 | 2 | ] |
| ⟨ | 0 | 1/5 | 4/5 | 2 | ] |
| ⟨ | 0 | 0 | 0 | 0 | ]] |
Tuning map: ⟨1200 1897.6537 2790.6150 3376.5374]
Mistuning map: ⟨0 -4.3013 +4.3013 +7.7115]