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 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 six chromatic semitones stack to a perfect fourth and eight minor seconds make up 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. 43-equal has the first good 13-limit meantone available as an equal division of the octave. 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 "merides". Further information: [http://tonalsoft.com/enc/m/meride.aspx Tonalsoft encyclopedia's entry of meride].

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]], tempering out [[78/77]], the other, [[Meantone family #Grosstone|grosstone]], [[144/143]]. Meridetone has generator map {{val| 0 1 4 10 18 27 }}, and grosstone {{val| 0 1 4 10 18 -16 }}; 43 supplies the [[optimal patent val]] for meridetone.. 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 "merides". Further information: [http://tonalsoft.com/enc/m/meride.aspx Tonalsoft encyclopedia's entry of meride].

The composer [[Juhan Puhm]] uses 43edo in some of his meantone suites for fortepiano and prefers it to [[31edo]].

In the 13-limit, we get two versions of meantone equivalent in 43et, one, [[Meantone family #Meridetone|meridetone]], tempering out [[78/77]], the other, [[Meantone family #Grosstone|grosstone]], [[144/143]]. Meridetone has generator map {{val| 0 1 4 10 18 27 }}, and grosstone {{val| 0 1 4 10 18 -16 }}; 43 supplies the [[optimal patent val]] for meridetone.

43edo's patent val {{val| 43 68 100 121 149 159 }} maps 5 to 100 steps, allowing the divison of 5 into 20 equal parts, leading to the [[jerome]] temperament, an interesting higher-limit system for which 43 supplies the optimal patent val in the 7-, 11-, 13-, 17-, 19-, and even 23-limit. It also provides the optimal patent val for the 11- and 13-limit [[amavil]] temperament, which is not a meantone temperament. The [[thuja]] temperament is also a possibility, in which five generators, (~11/8)<sup>5</sup> = ~5/1, with [[mos]] of 15 and 28.

~~The 43~~ patent val {{val| 43 68 100 121 149 159 }} maps 5 to 100 steps, allowing the divison of 5 into 20 equal parts, leading to the [[jerome]] temperament, an interesting higher-limit system for which 43 supplies the optimal patent val in the 7-, 11-, 13-, 17-, 19-, and even 23-limit. It also provides the optimal patent val for the 11- and 13-limit [[amavil]] temperament, which is not a meantone temperament. The [[thuja]] temperament is also a possibility, in which five generators, (~11/8)<sup>5</sup> = ~5/1, with [[mos]] of 15 and 28.

=== Prime harmonics ===

Although not [[consistent]], it performs quite well in very high prime limits. It has unambiguous mappings for all prime harmonics up to ''113'', with the sole exceptions of 23, 71, 89, and 103, making a great [[#Ringer 43|Ringer scale]]. Mappings for ~~composite harmonics and~~ ratios between these prime harmonics can then be derived from those for the primes themselves, allowing for an almost-complete version of the first 32 harmonics in the harmonic series, although the limited consistency will give some unusual results. Indeed, the step size of 43edo is very close to the [[64/63|septimal comma (64/63)]], while two steps is close to [[32/31]], and four steps to [[16/15]].

Although not [[consistent]], it performs quite well in very high prime limits. It has unambiguous mappings for all prime harmonics up to ''113'', with the sole exceptions of 23, 71, 89, and 103, making a great [[#Ringer 43|Ringer scale]]. Mappings for ratios between these prime harmonics can then be derived from those for the primes themselves, allowing for an almost-complete version of the first 32 harmonics in the harmonic series, although the limited consistency will give some unusual results. Indeed, the step size of 43edo is very close to the [[64/63|septimal comma (64/63)]], while two steps is close to [[32/31]], and four steps to [[16/15]].

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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 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 six chromatic semitones stack to a perfect fourth and eight minor seconds make up 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. 43-equal has the first good 13-limit meantone available as an equal division of the octave. 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 "merides". Further information: [http://tonalsoft.com/enc/m/meride.aspx Tonalsoft encyclopedia's entry of meride].
+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]], tempering out [[78/77]], the other, [[Meantone family #Grosstone|grosstone]], [[144/143]]. Meridetone has generator map {{val| 0 1 4 10 18 27 }}, and grosstone {{val| 0 1 4 10 18 -16 }}; 43 supplies the [[optimal patent val]] for meridetone.. 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 "merides". Further information: [http://tonalsoft.com/enc/m/meride.aspx Tonalsoft encyclopedia's entry of meride].
 The composer [[Juhan Puhm]] uses 43edo in some of his meantone suites for fortepiano and prefers it to [[31edo]].
-In the 13-limit, we get two versions of meantone equivalent in 43et, one, [[Meantone family #Meridetone|meridetone]], tempering out [[78/77]], the other, [[Meantone family #Grosstone|grosstone]], [[144/143]]. Meridetone has generator map {{val| 0 1 4 10 18 27 }}, and grosstone {{val| 0 1 4 10 18 -16 }}; 43 supplies the [[optimal patent val]] for meridetone.
+edo's patent val {{val| 43 68 100 121 149 159 }} maps 5 to 100 steps, allowing the divison of 5 into 20 equal parts, leading to the [[jerome]] temperament, an interesting higher-limit system for which 43 supplies the optimal patent val in the 7-, 11-, 13-, 17-, 19-, and even 23-limit. It also provides the optimal patent val for the 11- and 13-limit [[amavil]] temperament, which is not a meantone temperament. The [[thuja]] temperament is also a possibility, in which five generators, (~11/8)<sup>5</sup> = ~5/1, with [[mos]] of 15 and 28.
-The 43 patent val {{val| 43 68 100 121 149 159 }} maps 5 to 100 steps, allowing the divison of 5 into 20 equal parts, leading to the [[jerome]] temperament, an interesting higher-limit system for which 43 supplies the optimal patent val in the 7-, 11-, 13-, 17-, 19-, and even 23-limit. It also provides the optimal patent val for the 11- and 13-limit [[amavil]] temperament, which is not a meantone temperament. The [[thuja]] temperament is also a possibility, in which five generators, (~11/8)<sup>5</sup> = ~5/1, with [[mos]] of 15 and 28.
 === Prime harmonics ===
-Although not [[consistent]], it performs quite well in very high prime limits. It has unambiguous mappings for all prime harmonics up to ''113'', with the sole exceptions of 23, 71, 89, and 103, making a great [[#Ringer 43|Ringer scale]]. Mappings for composite harmonics and ratios between these prime harmonics can then be derived from those for the primes themselves, allowing for an almost-complete version of the first 32 harmonics in the harmonic series, although the limited consistency will give some unusual results. Indeed, the step size of 43edo is very close to the [[64/63|septimal comma (64/63)]], while two steps is close to [[32/31]], and four steps to [[16/15]].
+Although not [[consistent]], it performs quite well in very high prime limits. It has unambiguous mappings for all prime harmonics up to ''113'', with the sole exceptions of 23, 71, 89, and 103, making a great [[#Ringer 43|Ringer scale]]. Mappings for ratios between these prime harmonics can then be derived from those for the primes themselves, allowing for an almost-complete version of the first 32 harmonics in the harmonic series, although the limited consistency will give some unusual results. Indeed, the step size of 43edo is very close to the [[64/63|septimal comma (64/63)]], while two steps is close to [[32/31]], and four steps to [[16/15]].
 {{Harmonics in equal|43}}