43edo: Difference between revisions

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== Theory ==

43edo is strongly associated with [[meantone]]~~, tuning~~ the major tuned sharp of [[5/4]] and perfect fifth tuned flat of [[3/2]] by ~~approximately 4~~ cents on both intervals, making it an excellent ~~representation~~ of [[1/5-comma meantone]]. It also tempers out the [[hypovishnuzma]] and the [[escapade comma]], so that ~~in 43edo~~ six chromatic semitones ~~stack to~~ a perfect fourth and eight minor seconds make up a major sixth.

43edo tempers out [[81/80]] in the 5-limit, and as such it is strongly associated with [[meantone]]. It tunes the major tuned sharp of [[5/4]] and perfect fifth tuned flat of [[3/2]] by slightly more than four cents on both intervals, making it an excellent approximation of [[1/5-comma meantone]]. It also tempers out the [[hypovishnuzma]] and the [[escapade comma]], so that six chromatic semitones make a perfect fourth and eight minor seconds make a major sixth.

~~It is also an excellent tuning system in the~~ 7, 11, and 13-limit. In the 7-limit, it supports septimal meantone, as it tempers out ~~3136/3125, along with~~ 126/125 ~~and~~ 225/224. The version of 11-limit meantone is the one tempering out [[99/98]], [[176/175]], and [[441/440]], sometimes called [[Huygens temperament|Huygens]]. ~~43-equal has~~ the ~~first good~~ 13-limit ~~meantone available as an equal division of the octave; in the 13-limit we get~~ [[Meantone family #Meridetone|meridetone]], which tempers out [[78/77]], and [[Meantone family #Grosstone|grosstone]], which tempers out [[144/143]]. Meridetone has generator map {{val| 0 1 4 10 18 27 }}, for which 43 supplies the [[optimal patent val]] for, and grosstone {{val| 0 1 4 10 18 -16 }}.

Except for 9/7, 14/9, 11/9, and 18/11, all [[15-odd-limit]] intervals have consistent approximations in 43edo. In the 7-limit, it supports septimal meantone, as it tempers out 126/125, 225/224, and 3136/3125. The version of 11-limit meantone is the one tempering out [[99/98]], [[176/175]], and [[441/440]], sometimes called [[Huygens temperament|Huygens]]. In the 13-limit it supports [[Meantone family #Meridetone|meridetone]], which tempers out [[78/77]], and [[Meantone family #Grosstone|grosstone]], which tempers out [[144/143]]. Meridetone has generator map {{val| 0 1 4 10 18 27 }}, for which 43 supplies the [[optimal patent val]] for, and grosstone {{val| 0 1 4 10 18 -16 }}.

The French Baroque acoustician [[wikipedia: Joseph Sauveur|Joseph Sauveur]], who was ironically hearing and speech impaired, based his tuning system on 43 equal tones to the octave, calling them "mérides". Sauveur favoured 43-tone equal temperament because the small intervals are well represented in it. <ref>[http://www.huygens-fokker.org/docs/measures.html Stichting Huygens-Fokker: Logarithmic Interval Measures]</ref>

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 == Theory ==
-edo is strongly associated with [[meantone]], tuning the major tuned sharp of [[5/4]] and perfect fifth tuned flat of [[3/2]] by approximately 4 cents on both intervals, making it an excellent representation of [[1/5-comma meantone]]. It also tempers out the [[hypovishnuzma]] and the [[escapade comma]], so that in 43edo six chromatic semitones stack to a perfect fourth and eight minor seconds make up a major sixth.
+edo tempers out [[81/80]] in the 5-limit, and as such it is strongly associated with [[meantone]]. It tunes the major tuned sharp of [[5/4]] and perfect fifth tuned flat of [[3/2]] by slightly more than four cents on both intervals, making it an excellent approximation of [[1/5-comma meantone]]. It also tempers out the [[hypovishnuzma]] and the [[escapade comma]], so that six chromatic semitones make a perfect fourth and eight minor seconds make a major sixth.
-It is also an excellent tuning system in the 7, 11, and 13-limit. In the 7-limit, it supports septimal meantone, as it tempers out 3136/3125, along with 126/125 and 225/224. The version of 11-limit meantone is the one tempering out [[99/98]], [[176/175]], and [[441/440]], sometimes called [[Huygens temperament|Huygens]]. 43-equal has the first good 13-limit meantone available as an equal division of the octave; in the 13-limit we get [[Meantone family #Meridetone|meridetone]], which tempers out [[78/77]], and [[Meantone family #Grosstone|grosstone]], which tempers out [[144/143]]. Meridetone has generator map {{val| 0 1 4 10 18 27 }}, for which 43 supplies the [[optimal patent val]] for, and grosstone {{val| 0 1 4 10 18 -16 }}.
+Except for 9/7, 14/9, 11/9, and 18/11, all [[15-odd-limit]] intervals have consistent approximations in 43edo. In the 7-limit, it supports septimal meantone, as it tempers out 126/125, 225/224, and 3136/3125. The version of 11-limit meantone is the one tempering out [[99/98]], [[176/175]], and [[441/440]], sometimes called [[Huygens temperament|Huygens]]. In the 13-limit it supports [[Meantone family #Meridetone|meridetone]], which tempers out [[78/77]], and [[Meantone family #Grosstone|grosstone]], which tempers out [[144/143]]. Meridetone has generator map {{val| 0 1 4 10 18 27 }}, for which 43 supplies the [[optimal patent val]] for, and grosstone {{val| 0 1 4 10 18 -16 }}.
 The French Baroque acoustician [[wikipedia: Joseph Sauveur|Joseph Sauveur]], who was ironically hearing and speech impaired, based his tuning system on 43 equal tones to the octave, calling them "mérides". Sauveur favoured 43-tone equal temperament because the small intervals are well represented in it. <ref>[http://www.huygens-fokker.org/docs/measures.html Stichting Huygens&#45;Fokker&#58; Logarithmic Interval Measures]</ref>