43edo: Difference between revisions
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Infobox precision, improve intro, make JI approximation table collapsed, cleanup |
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{{Infobox ET | {{Infobox ET | ||
| Step size = 27. | | Step size = 27.9070¢ | ||
| Fifth = 25\43 ( | | Fifth = 25\43 (697.7¢) | ||
| Major 2nd = 7\43 ( | | Major 2nd = 7\43 (195.3¢) | ||
| Semitones = 3:4 ( | | Semitones = 3:4 (83.7¢ : 111.6¢) | ||
| Consistency = 7 | | Consistency = 7 | ||
}} | }} | ||
'''43edo''' divides the [[octave]] into 43 [[equal]] parts. It is strongly associated with [[meantone]] temperament, especially 1/5-meantone. One step of 43edo was named ''[[méride]]'' by Joseph Sauveur (1653-1716) in 1696. The méride and eptaméride were the first logarithmic interval measures proposed. 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 '''43 equal divisions of the octave''' ('''43edo'''), or the '''43(-tone) equal temperament''' ('''43tet''', '''43et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 43 [[equal]] parts. It is strongly associated with [[meantone]] temperament, especially 1/5-meantone. One step of 43edo was named ''[[méride]]'' by Joseph Sauveur (1653-1716) in 1696. The méride and eptaméride were the first logarithmic interval measures proposed. 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> | ||
== Theory == | == Theory == | ||
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 [http://juhanpuhmmusic.ca 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, [[ | 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 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. | ||
43edo is the 14th [[ | 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 ''64'', with the sole exceptions of 23 and, perhaps, 41. 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 ''64'', with the sole exceptions of 23 and, perhaps, 41. 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}} | |||
== Intervals == | == Intervals == | ||
The distance from C to C# is 3 edosteps (or keys, frets). Thus one edostep equals one third of a sharp. | The distance from C to C# is 3 edosteps (or keys, frets). Thus one edostep equals one third of a sharp. | ||
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The following table shows how [[15-odd-limit intervals]] are represented in 43edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. | The following table shows how [[15-odd-limit intervals]] are represented in 43edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. | ||
{| class="wikitable center-all" | {| class="wikitable center-all mw-collapsible mw-collapsed" | ||
|+Direct mapping (even if inconsistent) | |+style=white-space:nowrap| Direct mapping (even if inconsistent) | ||
|- | |- | ||
! Interval, complement | ! Interval, complement | ||