# 43edo

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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 pions 7mus Approximate 13-limit Ratios ups and downs notation 0 1/1 P1 perfect unison D 1 27.907 29.581 35.721 (23.B8916) ^1, d2 up unison, dim 2nd D^, Ebb 2 55.814 59.163 71.442 (47.61216) vA1, ^d2 downaug unison, updim 2nd D#v, Ebb^ 3 83.721 88.744 107.163 (6B.29B16) vm2 downminor 2nd Ebv 4 111.628 118.326 142.884 (8E.E2416) 17/16, 16/15, 15/14 m2 minor 2nd Eb 5 139.535 147.907 178.605 (B2.9AD16) 12/11, 13/12, 14/13 ^m2 upminor 2nd Eb^ 6 167.442 177.488 214.326 (D6.53616) 11/10 vM2 downmajor 2nd Ev 7 195.349 207.07 250.0465 (FA.0BE816) 9/8, 10/9 M2 major 2nd E 8 223.256 236.651 285.767 (11D.C4716) 8/7 ^M2 upmajor 2nd E^ 9 251.163 266.233 321.488 (141.7D0816) 15/13 vA2, ^d3 downaug 2nd, updim 3rd E#v, Fb^ 10 279.07 295.814 357.209 (165.35916) 7/6, 13/11 vm3 downminor 3rd Fv 11 306.977 325.395 392.93 (188.EE216) 6/5 m3 minor 3rd F 12 334.884 354.977 428.651 (1AC.A6B16) 17/14, 39/32 ^m3 upminor 3rd F^ 13 362.791 384.558 464.372 (1D0.6F416) 11/9, 16/13 vM3 downmajor 3rd F#v 14 390.698 414.1395 500.093 (1F4.2FD16) 5/4 M3 major 3rd F# 15 418.605 443.721 535.814 (217.D0616) 9/7, 14/11 ^M3 upmajor 3rd F#^ 16 446.512 473.302 571.535 (23B.88F16) 13/10 vA3, ^d4 downaug 3rd, updim 4th Fxv, Gb^ 17 474.419 502.884 607.256 (25F.41816) 21/16 v4 down 4th Gv 18 502.326 532.465 642.977 (282.FA116) 4/3 P4 perfect 4th G 19 530.233 562.0465 678.698 (2A5.B2A16) 15/11 ^4 up 4th G^ 20 558.1395 591.628 714.419 (2C8.6FD16) 11/8, 18/13 vA4 downaug 4th G#v 21 586.0465 621.209 750.1395 (2EE.531816) 7/5 A4, vd5 aug 4th, downdim 5th G#, Abv 22 613.9535 650.791 785.8605 (311.ACE816) 10/7 ^A4, d5 upaug 4th, dim 5th G#^, Ab 23 641.8605 680.372 821.581 (335,A0316) 16/11, 13/9 ^d5 updim 5th Ab^ 24 669.767 709.9535 857.302 (359.4D616) 22/15 v5 down 5th Av 25 697.674 739.535 893.023 (37D.05F16) 3/2 P5 perfect 5th A 26 725.581 769.116 928.744 (3A0.BE816) 32/21 ^5 up 5th A^ 27 753.488 798.698 964.465 (3D4.77116) 20/13 vA5, ^d6 downaug 5th, updim 6th A#v, Bbb^ 28 781.395 828.279 1000.186 (3E8.2FA16) 14/9, 11/7 vm6 downminor 6th Bbv 29 809.302 857.8605 1035.917 (40B.D0216) 8/5 m6 minor 6th Bb 30 837.209 887.442 1071.628 (42F.90C16) 18/11, 13/8 ^m6 upminor 6th Bb^ 31 865.116 917.023 1107.349 (453.59516) 28\17, 64\39 vM6 downmajor 6th Bv 32 893.023 946.605 1143.07 (477.11E16) 5/3 M6 major 6th B 33 920.93 976.186 1178.791 (49A.DB716) 12/7, 22/13 ^M6 upmajor 6th B^ 34 948.837 1005.767 1214.512 (4BF.82F816) 26/15 vA6, ^d7 downaug 6th, updim 7th B#v, Cb^ 35 976.744 1035.349 1250.233 (4E2.3B916) 7/4 vm7 downminor 7th Cv 36 1004.651 1064.93 1285.9535 (505.F41816) 16/9, 9/5 m7 minor 7th C 37 1032.558 1094.512 1321.674 (529.ACA16) 20/11 ^m7 upminor 7th C^ 38 1060.465 1124.093 1357.395 (54D.65216) 11/6, 24/13, 13/7 vM7 downmajor 7th C#v 39 1088.372 1153.674 1393.116 (571.1DC16) 15/8, 28/15 M7 major 7th C# 40 1116.279 1183.256 1428.847 (594.D6516) ^M7 upmajor 7th C#^ 41 1144.186 1212.837 1464.488 (5B8.BEE16) vA7, ^d8 downaug 7th, updim 8ve Cxv, Db^ 42 1172.093 1242.419 1500.279 (5DC.47616) A7, v8 aug 7th, down 8ve Cx, Dv 43 1200 1272 1536 (60016) 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.

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 43edo (ordered by absolute error).

Interval, complement Error (abs., in cents)
16/15, 15/8 0.103
13/12, 24/13 0.962
14/11, 11/7 1.097
11/10, 20/11 2.438
16/13, 13/8 3.318
15/13, 26/15 3.422
7/5, 10/7 3.534
4/3, 3/2 4.281
5/4, 8/5 4.384
18/13, 13/9 5.243
15/11, 22/15 6.718
11/8, 16/11 6.822
13/10, 20/13 7.702
15/14, 28/15 7.815
8/7, 7/4 7.918
9/8, 16/9 8.561
6/5, 5/3 8.665
13/11, 22/13 10.140
12/11, 11/6 11.102
14/13, 13/7 11.237
9/7, 14/9 11.4275
7/6, 12/7 12.199
11/9, 18/11 12.524
10/9, 9/5 12.945

## 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)