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43 tone equal temperament

43edo divides the octave into 43 equal parts of 27.907 cents 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 Joseph Sauveur based his system on 43 equal tones to the octave, calling them "merides". Further information: http://tonalsoft.com/enc/m/meride.aspx

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

The 43 patent val <43 68 100 121 149 159| maps 5 to 100 steps, allowing the divison of 5 into 20 equal parts, leading to 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 11- and 13-limit amavil temperament, which is not a meantone temperament. Thuja temperament is also a possibility, in which five generators, (~11/8)^5 = ~5/1, with MOS of 15 and 28.

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

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 septimal comma (64:63), while two steps is close to 32:31, and four steps to 16:15.

Intervals


Degrees
Cents value
Approximate 13-limit Ratios
ups and downs notation
0
0
1/1
P1
perfect unison
D
1
27.907

^1, d2
up unison, dim 2nd
D^, Ebb
2
55.814

vA1, ^d2
downaug unison, updim 2nd
D#v, Ebb^
3
83.721

vm2
downminor 2nd
Ebv
4
111.628
17/16, 16/15, 15/14
m2
minor 2nd
Eb
5
139.535
12/11, 13/12, 14/13
^m2
upminor 2nd
Eb^
6
167.442
11/10
vM2
downmajor 2nd
Ev
7
195.349
9/8, 10/9
M2
major 2nd
E
8
223.256
8/7
^M2
upmajor 2nd
E^
9
251.163
15/13
vA2, ^d3
downaug 2nd, updim 3rd
E#v, Fb^
10
279.07
7/6, 13/11
vm3
downminor 3rd
Fv
11
306.977
6/5
m3
minor 3rd
F
12
334.884
17/14, 39/32
^m3
upminor 3rd
F^
13
362.791
11/9, 16/13
vM3
downmajor 3rd
F#v
14
390.698
5/4
M3
major 3rd
F#
15
418.605
9/7, 14/11
^M3
upmajor 3rd
F#^
16
446.512
13/10
vA3, ^d4
downaug 3rd, updim 4th
Fxv, Gb^
17
474.419
21/16
v4
down 4th
Gv
18
502.326
4/3
P4
perfect 4th
G
19
530.233
15/11
^4
up 4th
G^
20
558.139
11/8, 18/13
vA4
downaug 4th
G#v
21
586.046
7/5
A4, vd5
aug 4th, downdim 5th
G#, Abv
22
613.953
10/7
^A4, d5
upaug 4th, dim 5th
G#^, Ab
23
641.86
16/11, 13/9
^d5
updim 5th
Ab^
24
669.767
22/15
v5
down 5th
Av
25
697.674
3/2
P5
perfect 5th
A
26
725.581
32/21
^5
up 5th
A^
27
753.488
20/13
vA5, ^d6
downaug 5th, updim 6th
A#v, Bbb^
28
781.395
14/9, 11/7
vm6
downminor 6th
Bbv
29
809.302
8/5
m6
minor 6th
Bb
30
837.209
18/11, 13/8
^m6
upminor 6th
Bb^
31
865.116

vM6
downmajor 6th
Bv
32
893.023
5/3
M6
major 6th
B
33
920.93
12/7
^M6
upmajor 6th
B^
34
948.837
26/15
vA6, ^d7
downaug 6th, updim 7th
B#v, Cb^
35
976.744
7/4
vm7
downminor 7th
Cv
36
1004.651
16/9, 9/5
m7
minor 7th
C
37
1032.558
20/11
^m7
upminor 7th
C^
38
1060.465
11/6, 24/13, 13/7
vM7
downmajor 7th
C#v
39
1088.372
15/8, 28/15
M7
major 7th
C#
40
1116.279

^M7
upmajor 7th
C#^
41
1144.186

vA7, ^d8
downaug 7th, updim 8ve
Cxv, Db^
42
1172.093

A7, v8
aug 7th, down 8ve
Cx, Dv
43
1200
2/1
P8
perfect 8ve
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.

Notation of 43edo


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.

Alternatively, a red-note/blue-note system (similar to that proposed for sixth-tones/36edo) can be used. (This is a different use of color than Kite's color notation.) Now we have the following sequence of notes, each separated by one meride: A, red A, blue A#, A#, Bb, red Bb, blue B, B. (Note that there are red flats and blue sharps, but no red sharps or blue flats, because the latter are enharmonically equivalent to simpler notes: blue Bb is actually just A#, for instance).

The diatonic semitone is four steps, so for the region between B and C (or, E and F), we can use: B, Cb, red Cb/blue B# (they are enharmonic equivalents), B#, and C. All of the notes in 43edo therefore have unambiguous names except for two: red Cb/blue B#, and red Fb/blue E#. It might also be possible to design special symbols for those two notes (resembling a cross between the letters B and C in the former case, and E and F in the latter).

If red Cb and blue B# (and red Fb/Blue E#) are instead forced to be distinct, but the requirement that all notes be equally spaced is maintained, then we end up with a completely unambiguous red-note/blue-note notation for 45edo, which is another meantone (actually, a flattone) system.

43 edo counterpoint.mid mp3 Peter Kosmorsky (late 2011) (in meantone)