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43edo: Difference between revisions

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=43 tone equal temperament=
==Theory==


'''43edo''' divides the [[Octave|octave]] into 43 [[Equal|equal]] parts of 27.907 [[cent|cent]]s each. It is strongly associated with meantone temperament, 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 [http://en.wikipedia.org/wiki/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
'''43edo''' divides the [[Octave|octave]] into 43 [[Equal|equal]] parts of 27.907 [[cent|cent]]s each. It is strongly associated with meantone temperament, 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 [http://en.wikipedia.org/wiki/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
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==Intervals==
==Intervals==
 
The distance from C to C# is 3 keys or frets or edosteps. Thus one edostep equals one third of a sharp.
{| class="wikitable"
{| class="wikitable"
|-
|-
| style="text-align:center;" | Degrees
| style="text-align:center;" | Degrees
| style="text-align:center;" | Cents value
| style="text-align:center;" | Cents
| | Approximate 13-limit Ratios
| | Approximate 13-limit Ratios
| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs notation]]
| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs notation]]
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| style="text-align:center;" | ^1, d2
| style="text-align:center;" | ^1, d2
| style="text-align:center;" | up unison, dim 2nd
| style="text-align:center;" | up unison, dim 2nd
| style="text-align:center;" | D^, Ebb
| style="text-align:center;" | ^D, Ebb
|-
|-
| style="text-align:center;" | 2
| style="text-align:center;" | 2
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| style="text-align:center;" | vA1, ^d2
| style="text-align:center;" | vA1, ^d2
| style="text-align:center;" | downaug unison, updim 2nd
| style="text-align:center;" | downaug unison, updim 2nd
| style="text-align:center;" | D#v, Ebb^
| style="text-align:center;" | vD#, ^Ebb
|-
|-
| style="text-align:center;" | 3
| style="text-align:center;" | 3
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| style="text-align:center;" | vm2
| style="text-align:center;" | vm2
| style="text-align:center;" | downminor 2nd
| style="text-align:center;" | downminor 2nd
| style="text-align:center;" | Ebv
| style="text-align:center;" | vEb
|-
|-
| style="text-align:center;" | 4
| style="text-align:center;" | 4
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| style="text-align:center;" | ^m2
| style="text-align:center;" | ^m2
| style="text-align:center;" | upminor 2nd
| style="text-align:center;" | upminor 2nd
| style="text-align:center;" | Eb^
| style="text-align:center;" | ^Eb
|-
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 6
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| style="text-align:center;" | vM2
| style="text-align:center;" | vM2
| style="text-align:center;" | downmajor 2nd
| style="text-align:center;" | downmajor 2nd
| style="text-align:center;" | Ev
| style="text-align:center;" | vE
|-
|-
| style="text-align:center;" | 7
| style="text-align:center;" | 7
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| style="text-align:center;" | ^M2
| style="text-align:center;" | ^M2
| style="text-align:center;" | upmajor 2nd
| style="text-align:center;" | upmajor 2nd
| style="text-align:center;" | E^
| style="text-align:center;" | ^E
|-
|-
| style="text-align:center;" | 9
| style="text-align:center;" | 9
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| style="text-align:center;" | vA2, ^d3
| style="text-align:center;" | vA2, ^d3
| style="text-align:center;" | downaug 2nd, updim 3rd
| style="text-align:center;" | downaug 2nd, updim 3rd
| style="text-align:center;" | E#v, Fb^
| style="text-align:center;" | vE#, ^Fb
|-
|-
| style="text-align:center;" | 10
| style="text-align:center;" | 10
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| style="text-align:center;" | vm3
| style="text-align:center;" | vm3
| style="text-align:center;" | downminor 3rd
| style="text-align:center;" | downminor 3rd
| style="text-align:center;" | Fv
| style="text-align:center;" | vF
|-
|-
| style="text-align:center;" | 11
| style="text-align:center;" | 11
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| style="text-align:center;" | ^m3
| style="text-align:center;" | ^m3
| style="text-align:center;" | upminor 3rd
| style="text-align:center;" | upminor 3rd
| style="text-align:center;" | F^
| style="text-align:center;" | ^F
|-
|-
| style="text-align:center;" | 13
| style="text-align:center;" | 13
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| style="text-align:center;" | vM3
| style="text-align:center;" | vM3
| style="text-align:center;" | downmajor 3rd
| style="text-align:center;" | downmajor 3rd
| style="text-align:center;" | F#v
| style="text-align:center;" | vF#
|-
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 14
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| style="text-align:center;" | ^M3
| style="text-align:center;" | ^M3
| style="text-align:center;" | upmajor 3rd
| style="text-align:center;" | upmajor 3rd
| style="text-align:center;" | F#^
| style="text-align:center;" | ^F#
|-
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 16
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| style="text-align:center;" | vA3, ^d4
| style="text-align:center;" | vA3, ^d4
| style="text-align:center;" | downaug 3rd, updim 4th
| style="text-align:center;" | downaug 3rd, updim 4th
| style="text-align:center;" | Fxv, Gb^
| style="text-align:center;" | vFx, ^Gb
|-
|-
| style="text-align:center;" | 17
| style="text-align:center;" | 17
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| style="text-align:center;" | v4
| style="text-align:center;" | v4
| style="text-align:center;" | down 4th
| style="text-align:center;" | down 4th
| style="text-align:center;" | Gv
| style="text-align:center;" | vG
|-
|-
| style="text-align:center;" | 18
| style="text-align:center;" | 18
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| style="text-align:center;" | ^4
| style="text-align:center;" | ^4
| style="text-align:center;" | up 4th
| style="text-align:center;" | up 4th
| style="text-align:center;" | G^
| style="text-align:center;" | ^G
|-
|-
| style="text-align:center;" | 20
| style="text-align:center;" | 20
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| style="text-align:center;" | vA4
| style="text-align:center;" | vA4
| style="text-align:center;" | downaug 4th
| style="text-align:center;" | downaug 4th
| style="text-align:center;" | G#v
| style="text-align:center;" | vG#
|-
|-
| style="text-align:center;" | 21
| style="text-align:center;" | 21
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| style="text-align:center;" | A4, vd5
| style="text-align:center;" | A4, vd5
| style="text-align:center;" | aug 4th, downdim 5th
| style="text-align:center;" | aug 4th, downdim 5th
| style="text-align:center;" | G#, Abv
| style="text-align:center;" | G#, ^Ab
|-
|-
| style="text-align:center;" | 22
| style="text-align:center;" | 22
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| style="text-align:center;" | ^A4, d5
| style="text-align:center;" | ^A4, d5
| style="text-align:center;" | upaug 4th, dim 5th
| style="text-align:center;" | upaug 4th, dim 5th
| style="text-align:center;" | G#^, Ab
| style="text-align:center;" | ^G#, Ab
|-
|-
| style="text-align:center;" | 23
| style="text-align:center;" | 23
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| style="text-align:center;" | ^d5
| style="text-align:center;" | ^d5
| style="text-align:center;" | updim 5th
| style="text-align:center;" | updim 5th
| style="text-align:center;" | Ab^
| style="text-align:center;" | ^Ab
|-
|-
| style="text-align:center;" | 24
| style="text-align:center;" | 24
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| style="text-align:center;" | v5
| style="text-align:center;" | v5
| style="text-align:center;" | down 5th
| style="text-align:center;" | down 5th
| style="text-align:center;" | Av
| style="text-align:center;" | vA
|-
|-
| style="text-align:center;" | 25
| style="text-align:center;" | 25
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| style="text-align:center;" | ^5
| style="text-align:center;" | ^5
| style="text-align:center;" | up 5th
| style="text-align:center;" | up 5th
| style="text-align:center;" | A^
| style="text-align:center;" | ^A
|-
|-
| style="text-align:center;" | 27
| style="text-align:center;" | 27
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| style="text-align:center;" | vA5, ^d6
| style="text-align:center;" | vA5, ^d6
| style="text-align:center;" | downaug 5th, updim 6th
| style="text-align:center;" | downaug 5th, updim 6th
| style="text-align:center;" | A#v, Bbb^
| style="text-align:center;" | vA#, ^Bbb
|-
|-
| style="text-align:center;" | 28
| style="text-align:center;" | 28
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| style="text-align:center;" | vm6
| style="text-align:center;" | vm6
| style="text-align:center;" | downminor 6th
| style="text-align:center;" | downminor 6th
| style="text-align:center;" | Bbv
| style="text-align:center;" | vBb
|-
|-
| style="text-align:center;" | 29
| style="text-align:center;" | 29
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| style="text-align:center;" | ^m6
| style="text-align:center;" | ^m6
| style="text-align:center;" | upminor 6th
| style="text-align:center;" | upminor 6th
| style="text-align:center;" | Bb^
| style="text-align:center;" | ^Bb
|-
|-
| style="text-align:center;" | 31
| style="text-align:center;" | 31
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| style="text-align:center;" | vM6
| style="text-align:center;" | vM6
| style="text-align:center;" | downmajor 6th
| style="text-align:center;" | downmajor 6th
| style="text-align:center;" | Bv
| style="text-align:center;" | vB
|-
|-
| style="text-align:center;" | 32
| style="text-align:center;" | 32
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| style="text-align:center;" | ^M6
| style="text-align:center;" | ^M6
| style="text-align:center;" | upmajor 6th
| style="text-align:center;" | upmajor 6th
| style="text-align:center;" | B^
| style="text-align:center;" | ^B
|-
|-
| style="text-align:center;" | 34
| style="text-align:center;" | 34
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| style="text-align:center;" | vA6, ^d7
| style="text-align:center;" | vA6, ^d7
| style="text-align:center;" | downaug 6th, updim 7th
| style="text-align:center;" | downaug 6th, updim 7th
| style="text-align:center;" | B#v, Cb^
| style="text-align:center;" | vB#, ^Cb
|-
|-
| style="text-align:center;" | 35
| style="text-align:center;" | 35
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| style="text-align:center;" | vm7
| style="text-align:center;" | vm7
| style="text-align:center;" | downminor 7th
| style="text-align:center;" | downminor 7th
| style="text-align:center;" | Cv
| style="text-align:center;" | vC
|-
|-
| style="text-align:center;" | 36
| style="text-align:center;" | 36
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| style="text-align:center;" | ^m7
| style="text-align:center;" | ^m7
| style="text-align:center;" | upminor 7th
| style="text-align:center;" | upminor 7th
| style="text-align:center;" | C^
| style="text-align:center;" | ^C
|-
|-
| style="text-align:center;" | 38
| style="text-align:center;" | 38
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| style="text-align:center;" | vM7
| style="text-align:center;" | vM7
| style="text-align:center;" | downmajor 7th
| style="text-align:center;" | downmajor 7th
| style="text-align:center;" | C#v
| style="text-align:center;" | vC#
|-
|-
| style="text-align:center;" | 39
| style="text-align:center;" | 39
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| style="text-align:center;" | ^M7
| style="text-align:center;" | ^M7
| style="text-align:center;" | upmajor 7th
| style="text-align:center;" | upmajor 7th
| style="text-align:center;" | C#^
| style="text-align:center;" | ^C#
|-
|-
| style="text-align:center;" | 41
| style="text-align:center;" | 41
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| style="text-align:center;" | vA7, ^d8
| style="text-align:center;" | vA7, ^d8
| style="text-align:center;" | downaug 7th, updim 8ve
| style="text-align:center;" | downaug 7th, updim 8ve
| style="text-align:center;" | Cxv, Db^
| style="text-align:center;" | vCx, ^Db
|-
|-
| style="text-align:center;" | 42
| style="text-align:center;" | 42
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| style="text-align:center;" | A7, v8
| style="text-align:center;" | A7, v8
| style="text-align:center;" | aug 7th, down 8ve
| style="text-align:center;" | aug 7th, down 8ve
| style="text-align:center;" | Cx, Dv
| style="text-align:center;" | Cx, vD
|-
|-
| style="text-align:center;" | 43
| style="text-align:center;" | 43
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| style="text-align:center;" | D
| style="text-align:center;" | D
|}
|}
The distance from C to C# is 3 keys or frets or EDOsteps. Thus one up equals one third of a sharp. Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].


==Selected just intervals by error==
==Selected just intervals by error==
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|}
|}


==Notation of 43edo==
==Red-Blue Notation==


Because 43edo is a meantone system, this makes it easier to adapt traditional Western notation to it than to some other tunings. A# and Bb are distinct and the distance between them is one meride. The whole tone is divided into seven merides so this means we can use "third-sharps", "two-thirds-sharps", "third-flats", and "two-thirds-flats" to reach the remaining notes between A and B; notes elsewhere on the scale can be notated similarly.
Because 43edo is a meantone system, this makes it easier to adapt traditional Western notation to it than to some other tunings. A# and Bb are distinct and the distance between them is one meride. The whole tone is divided into seven merides so this means we can use "third-sharps", "two-thirds-sharps", "third-flats", and "two-thirds-flats" to reach the remaining notes between A and B; notes elsewhere on the scale can be notated similarly.
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