43edo: Difference between revisions
m →Further reading: Rename to External links, misc. edits |
Split part of Theory section into new History section, bold "méride" lemma (from redirect), reorganize external links, misc. edits |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|43}} | {{EDO intro|43}} | ||
== History == | |||
The French Baroque acoustician {{w|Joseph Sauveur}}, who was ironically hearing and speech impaired, based his tuning system on 43 equal tones to the octave, calling one step a '''méride'''. 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> | |||
The composer [[Juhan Puhm]] uses 43edo in some of his fortepiano suites and prefers it to [[31edo]]. | |||
== Theory == | == Theory == | ||
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Except for 9/7, 11/9, 14/9, and 18/11, all [[15-odd-limit]] intervals have [[consistent]] approximations in 43edo, making it an excellent tuning in the 7-, 11-, and 13-limit. 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 }}. | Except for 9/7, 11/9, 14/9, and 18/11, all [[15-odd-limit]] intervals have [[consistent]] approximations in 43edo, making it an excellent tuning in the 7-, 11-, and 13-limit. 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 }}. | ||
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 meantone. [[Thuja]] is also a possibility, whose 11-limit extension makes five 11/8s stack to a major third (i.e. {{nowrap|(11/8)<sup>5</sup> → 5/1}}), with [[mos]]es of 15 and 28. | 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 meantone. [[Thuja]] is also a possibility, whose 11-limit extension makes five 11/8s stack to a major third (i.e. {{nowrap|(11/8)<sup>5</sup> → 5/1}}), with [[mos]]es of 15 and 28. | ||
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! [[Harmonic limit|Prime<br>Limit]] | ! [[Harmonic limit|Prime<br>Limit]] | ||
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref> | ! [[Ratio]]<ref group="note">Ratios longer than 10 digits are presented by placeholders with informative hints</ref> | ||
! [[Monzo]] | ! [[Monzo]] | ||
! [[Cent]]s | ! [[Cent]]s | ||
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| Spleen comma | | Spleen comma | ||
|} | |} | ||
<references group="note"/> | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
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A possible isomorphic keyboard layout for 43edo: | A possible isomorphic keyboard layout for 43edo: | ||
[[File:Fifth Comma Meantone Keyboard Layout.svg|800px|none|thumb]] | [[File:Fifth Comma Meantone Keyboard Layout.svg|800px|none|thumb]] | ||
==References== | ==References== | ||
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== External links == | == External links == | ||
=== Articles === | |||
* [http://tonalsoft.com/enc/m/meride.aspx méride / 43-ed2 / 43-edo / 43-ET / 43-tone equal-temperament] on [[Tonalsoft Encyclopedia]] | * [http://tonalsoft.com/enc/m/meride.aspx méride / 43-ed2 / 43-edo / 43-ET / 43-tone equal-temperament] on [[Tonalsoft Encyclopedia]] | ||
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Harmonic-Resources-43Et-EMT-43EBMT.pdf ''Harmonic Resources of 43Et EMT and 43EBMT''] by Juhan Puhm (2018) | |||
=== Diagrams === | |||
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Keys-and-Modes-of-43Et.pdf ''Keys and Modes of 43Et''] by Juhan Puhm (2016) | |||
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Keyboard-Mapping-for-43Et.pdf ''Keyboard Mapping for 43Et''] by Juhan Puhm (2017) | |||
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Mapping-Range-for-43Et.pdf ''Mapping Range for 43Et''] by Juhan Puhm (2017) | |||
[[Category:Meantone]] | [[Category:Meantone]] | ||