43edo

Theory

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

The distance from C to C# is 3 keys or frets or edosteps. Thus one edostep equals one third of a sharp.

Degrees	Cents	Approximate 13-limit Ratios	ups and downs notation
0	0.000	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	vD#, ^Ebb
3	83.721		vm2	downminor 2nd	vEb
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	vE
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	vE#, ^Fb
10	279.07	7/6, 13/11	vm3	downminor 3rd	vF
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	vF#
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	vFx, ^Gb
17	474.419	21/16	v4	down 4th	vG
18	502.326	4/3	P4	perfect 4th	G
19	530.233	15/11	^4	up 4th	^G
20	558.1395	11/8, 18/13	vA4	downaug 4th	vG#
21	586.0465	7/5	A4, vd5	aug 4th, downdim 5th	G#, ^Ab
22	613.9535	10/7	^A4, d5	upaug 4th, dim 5th	^G#, Ab
23	641.8605	16/11, 13/9	^d5	updim 5th	^Ab
24	669.767	22/15	v5	down 5th	vA
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	vA#, ^Bbb
28	781.395	14/9, 11/7	vm6	downminor 6th	vBb
29	809.302	8/5	m6	minor 6th	Bb
30	837.209	18/11, 13/8	^m6	upminor 6th	^Bb
31	865.116	28\17, 64\39	vM6	downmajor 6th	vB
32	893.023	5/3	M6	major 6th	B
33	920.93	12/7, 22/13	^M6	upmajor 6th	^B
34	948.837	26/15	vA6, ^d7	downaug 6th, updim 7th	vB#, ^Cb
35	976.744	7/4	vm7	downminor 7th	vC
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	vC#
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	vCx, ^Db
42	1172.093		A7, v8	aug 7th, down 8ve	Cx, vD
43	1200	2/1	P8	perfect 8ve	D

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See Ups and Downs Notation - Chords and Chord Progressions.

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

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.

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)