43edo

Template:Infobox ET 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>Stichting Huygens-Fokker: Logarithmic Interval Measures</ref>

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 Joseph Sauveur based his system on 43 equal tones to the octave, calling them "merides". Further information: Tonalsoft encyclopedia entry of meride.

The composer 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, meridetone, tempering out 78/77, the other, grosstone, 144/143. Meridetone has generator map Template:Val, and grosstone Template:Val; 43 supplies the optimal patent val for meridetone.

The 43 patent val Template:Val maps 5 to 100 steps, allowing the divison of 5 into 20 equal parts, leading to the 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 amavil temperament, which is not a meantone temperament. The 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.

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 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 edosteps (or keys, frets). Thus one edostep equals one third of a sharp.

Degrees	Cents	Approximate 17-limit Ratios	Ups and Downs Notation
0	0.000	1/1	P1	perfect unison	D
1	27.907	36/35, 50/49, 64/63	^1, d2	up unison, dim 2nd	^D, Ebb
2	55.814	49/48, 33/32	vA1, ^d2	downaug unison, updim 2nd	vD#, ^Ebb
3	83.721	25/24, 21/20, 28/27, 22/21, 18/17	vm2	downminor 2nd	vEb
4	111.628	16/15, 15/14, 17/16	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	279.070	7/6, 13/11	vm3	downminor 3rd	vF
11	306.977	6/5	m3	minor 3rd	F
12	334.884	39/32, 17/14	^m3	upminor 3rd	^F
13	362.791	16/13, 21/17, 11/9	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	558.140	11/8, 18/13	vA4	downaug 4th	vG#
21	586.047	45/32, 7/5	A4, vd5	aug 4th, downdim 5th	G#, ^Ab
22	613.953	64/45, 10/7	^A4, d5	upaug 4th, dim 5th	^G#, Ab
23	641.860	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	13/8, 34/21, 18/11	^m6	upminor 6th	^Bb
31	865.116	64/39, 28/17	vM6	downmajor 6th	vB
32	893.023	5/3	M6	major 6th	B
33	920.930	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, 32/17	M7	major 7th	C#
40	1116.279	48/25, 40/21, 27/14, 21/11, 17/9	^M7	upmajor 7th	^C#
41	1144.186	96/49, 64/33	vA7, ^d8	downaug 7th, updim 8ve	vCx, ^Db
42	1172.093	35/18, 49/25, 63/32	A7, v8	aug 7th, down 8ve	Cx, vD
43	1200.000	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.

JI approximation

Selected just intervals

15-odd-limit mappings

The following table shows how 15-odd-limit intervals are represented in 43edo. Prime harmonics are in bold; inconsistent intervals are in italic.

Direct mapping (even if inconsistent)

Interval, complement	Error (abs, ¢)
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	11.428
7/6, 12/7	12.199
11/9, 18/11	12.524
10/9, 9/5	12.945

Patent val mapping

Interval, complement	Error (abs, ¢)
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
7/6, 12/7	12.199
10/9, 9/5	12.945
11/9, 18/11	15.383
9/7, 14/9	16.479

Notation

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.

Sagittal

The following table shows sagittal notation accidentals in one apotome for 43do.

Steps	0	1	2	3
Symbol	File:Sagittal natural.png	File:Sagittal tai.png	File:Sagittal sharp tao.png	File:Sagittal sharp.png

Music

• Juhan Puhm ~ Meantone Suite V in D Minor ~ 43Et EMT
• 43 edo counterpoint.mid
• mp3 Peter Kosmorsky (late 2011) (in meantone)

References

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