43edo: Difference between revisions

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43edo is strongly associated with [[meantone]], particularly [[1/5-comma meantone]], being a good tuning system in the 5, 7, 11, and 13-limit. 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 baroque, French, ironically hearing and speech impaired acoustician [[wikipedia: Joseph Sauveur|Joseph Sauveur]] based his system on 43 equal tones to the octave, calling them "merides". Further information: [http://tonalsoft.com/enc/m/meride.aspx Tonalsoft encyclopedia entry of meride].

The composer [~~http://juhanpuhmmusic.ca~~ Juhan Puhm] uses 43edo in some of his meantone suites for fortepiano and prefers it to [[31edo]].

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.

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 [[~~Meantone family #Jerome|~~jerome ~~temperament~~]], an interesting higher-limit system for which 43 supplies the optimal patent val in the 7, 11, 13, 17, 19 and 23 ~~limits~~. It also provides the optimal patent val for the 11- and 13-limit [[~~Marvel temperaments #Amavil|~~amavil ~~temperament~~]], which is not a meantone temperament. The [[~~Starling temperaments #Thuja|~~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 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.

~~43edo is the 14th [[prime edo]], following [[41edo]] and coming before [[47edo]]~~.

=== Prime harmonics ===

Although not [[consistent]], it performs quite decently in very high limits. It has unambiguous mappings for all prime harmonics up to ''~~100~~'', with the sole exceptions of 23, 71 and 89, making a great [[#Ringer 43|Ringer scale]]. The mappings for composite harmonics can then be derived from those for the primes, giving an almost-complete version of mode 32 of the harmonic series, although the aforementioned lack of 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 decently in very high limits. It has unambiguous mappings for all prime harmonics up to ''109'', with the sole exceptions of 23, 71 and 89, making a great [[#Ringer 43|Ringer scale]]. The mappings for composite harmonics can then be derived from those for the primes, giving an almost-complete version of mode 32 of the harmonic series, although the aforementioned lack of 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]].

=== Divisors ===

43edo is the 14th [[prime edo]], following [[41edo]] and coming before [[47edo]].

== Intervals ==

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== Well temperaments ==

=== Ringer 43 ===

The metaphorical color palette that the intervals of [[43edo]] present can be quite appealing for various reasons such as being meantone and splitting 4/3 into 6 equal parts and 3/2 into 5 equal parts, but the accuracy leaves one wanting in many cases, which is why an excellent alternative (given the unambiguity of mappings of all primes in the 97-limit except 71 and 89) is Ringer 43, a [[Ringer scale]] with 43 notes per octave period:

The metaphorical color palette that the intervals of 43edo present can be quite appealing for various reasons such as being meantone and splitting 4/3 into 6 equal parts and 3/2 into 5 equal parts, but the accuracy leaves one wanting in many cases, which is why an excellent alternative (given the unambiguity of mappings of all primes in the 109-limit except 71 and 89) is Ringer 43, a [[Ringer scale]] with 43 notes per octave period:

<pre>

55:56:57:58:59:60:61:62:63:64:65:66:67:68:69:70:72:73:74:75:76:78:79:80:82:83:84:86:87:88:90:91:92:94:96:97:98:100:102:104:106:108:109:110

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== Music ==

* [https://www.youtube.com/watch?v=I68hwh45CyQ Juhan Puhm ~ Meantone Suite V in D Minor ~~~ 43Et EMT~~]

; [[Juhan Puhm]] ([http://juhanpuhmmusic.ca site])

* [[:File:43_edo_counterpoint.mid|43 edo counterpoint.mid]]

* [https://www.youtube.com/watch?v=I68hwh45CyQ ''Meantone Suite V in D Minor''] ([http://juhanpuhmmusic.ca/Juhan-Puhm-Meantone-Suite-V-D-Minor.pdf score])

* [http://micro.soonlabel.com/gene_ward_smith/Others/Kosmorsky/43%20edo%20counterpoint.mp3 mp3] ~~Peter Kosmorsky~~ (late 2011~~) (in meantone~~)

; [[Peter Kosmorsky]]

* [[:File:43_edo_counterpoint.mid|43 edo counterpoint.mid]] [http://micro.soonlabel.com/gene_ward_smith/Others/Kosmorsky/43%20edo%20counterpoint.mp3 mp3]{{dead link}} – in meantone (late 2011)

== References ==

@@ Line 5: / Line 5: @@
 edo is strongly associated with [[meantone]], particularly [[1/5-comma meantone]], being a good tuning system in the 5, 7, 11, and 13-limit. 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 baroque, French, ironically hearing and speech impaired acoustician [[wikipedia: Joseph Sauveur|Joseph Sauveur]] based his system on 43 equal tones to the octave, calling them "merides". Further information: [http://tonalsoft.com/enc/m/meride.aspx Tonalsoft encyclopedia entry of meride].
-The composer [http://juhanpuhmmusic.ca Juhan Puhm] uses 43edo in some of his meantone suites for fortepiano and prefers it to [[31edo]].
+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.
-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 [[Meantone family #Jerome|jerome temperament]], an interesting higher-limit system for which 43 supplies the optimal patent val in the 7, 11, 13, 17, 19 and 23 limits. It also provides the optimal patent val for the 11- and 13-limit [[Marvel temperaments #Amavil|amavil temperament]], which is not a meantone temperament. The [[Starling temperaments #Thuja|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 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.
-edo is the 14th [[prime edo]], following [[41edo]] and coming before [[47edo]].
 === Prime harmonics ===
-Although not [[consistent]], it performs quite decently in very high limits. It has unambiguous mappings for all prime harmonics up to ''100'', with the sole exceptions of 23, 71 and 89, making a great [[#Ringer 43|Ringer scale]]. The mappings for composite harmonics can then be derived from those for the primes, giving an almost-complete version of mode 32 of the harmonic series, although the aforementioned lack of 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 decently in very high limits. It has unambiguous mappings for all prime harmonics up to ''109'', with the sole exceptions of 23, 71 and 89, making a great [[#Ringer 43|Ringer scale]]. The mappings for composite harmonics can then be derived from those for the primes, giving an almost-complete version of mode 32 of the harmonic series, although the aforementioned lack of 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}}
+=== Divisors ===
+edo is the 14th [[prime edo]], following [[41edo]] and coming before [[47edo]].
 == Intervals ==
@@ Line 532: / Line 533: @@
 == Well temperaments ==
 === Ringer 43 ===
-The metaphorical color palette that the intervals of [[43edo]] present can be quite appealing for various reasons such as being meantone and splitting 4/3 into 6 equal parts and 3/2 into 5 equal parts, but the accuracy leaves one wanting in many cases, which is why an excellent alternative (given the unambiguity of mappings of all primes in the 97-limit except 71 and 89) is Ringer 43, a [[Ringer scale]] with 43 notes per octave period:
+The metaphorical color palette that the intervals of 43edo present can be quite appealing for various reasons such as being meantone and splitting 4/3 into 6 equal parts and 3/2 into 5 equal parts, but the accuracy leaves one wanting in many cases, which is why an excellent alternative (given the unambiguity of mappings of all primes in the 109-limit except 71 and 89) is Ringer 43, a [[Ringer scale]] with 43 notes per octave period:
 <pre>
 :56:57:58:59:60:61:62:63:64:65:66:67:68:69:70:72:73:74:75:76:78:79:80:82:83:84:86:87:88:90:91:92:94:96:97:98:100:102:104:106:108:109:110
@@ Line 541: / Line 542: @@
 == Music ==
-* [https://www.youtube.com/watch?v=I68hwh45CyQ Juhan Puhm ~ Meantone Suite V in D Minor ~ 43Et EMT]
+; [[Juhan Puhm]] ([http://juhanpuhmmusic.ca site])
-* [[:File:43_edo_counterpoint.mid|43 edo counterpoint.mid]]
+* [https://www.youtube.com/watch?v=I68hwh45CyQ ''Meantone Suite V in D Minor''] ([http://juhanpuhmmusic.ca/Juhan-Puhm-Meantone-Suite-V-D-Minor.pdf score])
-* [http://micro.soonlabel.com/gene_ward_smith/Others/Kosmorsky/43%20edo%20counterpoint.mp3 mp3] Peter Kosmorsky (late 2011) (in meantone)
+; [[Peter Kosmorsky]]
+* [[:File:43_edo_counterpoint.mid|43 edo counterpoint.mid]] [http://micro.soonlabel.com/gene_ward_smith/Others/Kosmorsky/43%20edo%20counterpoint.mp3 mp3]{{dead link}} – in meantone (late 2011)
 == References ==

43edo: Difference between revisions

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