43edo

Prime factorization

43 (prime)

Step size

27.907¢

Fifth

25\43 (697.674¢)

Semitones (A1:m2)

3:4 (83.72¢ : 111.6¢)

Consistency limit

7

Distinct consistency limit

7

43 equal divisions of the octave (abbreviated 43edo or 43ed2), also called 43-tone equal temperament (43tet) or 43 equal temperament (43et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 43 equal parts of about 27.9 ¢ each. Each step represents a frequency ratio of 2^1/43, or the 43rd root of 2.

Theory

43edo tempers out 81/80 in the 5-limit, and as such it is strongly associated with meantone. It tunes the major third sharp of 5/4 and perfect fifth flat of 3/2 by slightly more than four cents on both of them. Other 5-limit commas tempered out include the hypovishnuzma and the escapade comma, so that in 43edo six chromatic semitones make a perfect fourth and eight minor seconds make a major sixth.

Except for 9/7, 14/9, 11/9, and 18/11, all 15-odd-limit intervals have consistent approximations in 43edo, making it an excellent tuning in the 7-, 11-, and 13-limit. In the 7-limit, it supports septimal meantone, as it tempers out 126/125, 225/224, and 3136/3125. The version of 11-limit meantone is the one tempering out 99/98, 176/175, and 441/440, sometimes called Huygens. In the 13-limit it supports meridetone, which tempers out 78/77, and grosstone, which tempers out 144/143. Meridetone has generator map ⟨0 1 4 10 18 27], for which 43 supplies the optimal patent val for, and grosstone ⟨0 1 4 10 18 -16].

The French Baroque acoustician Joseph Sauveur, who was ironically hearing and speech impaired, based his tuning system on 43 equal tones to the octave, calling them "mérides". Sauveur favoured 43-tone equal temperament because the small intervals are well represented in it. ^[1]

The composer Juhan Puhm uses 43edo in some of his meantone suites for fortepiano and prefers it to 31edo.

43edo's patent val ⟨43 68 100 121 149 159] 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 even 23-limit. 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.

Prime harmonics

Although not consistent, it performs quite well in very high prime limits. It has unambiguous mappings for all prime harmonics up to 113, with the sole exceptions of 23, 71, 89, and 103, making a great Ringer scale. Mappings for ratios between these prime harmonics can then be derived from those for the primes themselves, allowing for an almost-complete version of the first 32 harmonics in the harmonic series, although the limited 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.

Approximation of prime harmonics in 43edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	absolute (¢)	+0.0	-4.3	+4.4	+7.9	+6.8	-3.3	+6.7	+9.5	+13.6	+3.0	-0.8
Error	relative (%)	+0	-15	+16	+28	+24	-12	+24	+34	+49	+11	-3
Steps (reduced)		43 (0)	68 (25)	100 (14)	121 (35)	149 (20)	159 (30)	176 (4)	183 (11)	195 (23)	209 (37)	213 (41)

Divisors

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

Intervals

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

#	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, 65/64, 66/65	^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, 24/17	A4, vd5	aug 4th, downdim 5th	G#, ^Ab
22	613.953	64/45, 10/7, 17/12	^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, 65/33, 128/65	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 29-limit intervals approximated in 43edo

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, ¢)	Error (rel, %)
16/15, 15/8	0.103	0.4
13/12, 24/13	0.962	3.4
14/11, 11/7	1.097	3.9
11/10, 20/11	2.438	8.7
16/13, 13/8	3.318	11.9
15/13, 26/15	3.422	12.3
7/5, 10/7	3.534	12.7
4/3, 3/2	4.281	15.3
5/4, 8/5	4.384	15.7
18/13, 13/9	5.243	18.8
15/11, 22/15	6.718	24.1
11/8, 16/11	6.822	24.4
13/10, 20/13	7.702	27.6
15/14, 28/15	7.815	28.0
8/7, 7/4	7.918	28.4
9/8, 16/9	8.561	30.7
6/5, 5/3	8.665	31.0
13/11, 22/13	10.140	36.3
12/11, 11/6	11.102	39.8
14/13, 13/7	11.237	40.3
9/7, 14/9	11.428	40.9
7/6, 12/7	12.199	43.7
11/9, 18/11	12.524	44.9
10/9, 9/5	12.945	46.4

15-odd-limit intervals by patent val mapping
Interval, complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
15/8, 16/15	0.103	0.4
13/12, 24/13	0.962	3.4
11/7, 14/11	1.097	3.9
11/10, 20/11	2.438	8.7
13/8, 16/13	3.318	11.9
15/13, 26/15	3.422	12.3
7/5, 10/7	3.534	12.7
3/2, 4/3	4.281	15.3
5/4, 8/5	4.384	15.7
13/9, 18/13	5.243	18.8
15/11, 22/15	6.718	24.1
11/8, 16/11	6.822	24.4
13/10, 20/13	7.702	27.6
15/14, 28/15	7.815	28.0
7/4, 8/7	7.918	28.4
9/8, 16/9	8.561	30.7
5/3, 6/5	8.665	31.0
13/11, 22/13	10.140	36.3
11/6, 12/11	11.102	39.8
13/7, 14/13	11.237	40.3
7/6, 12/7	12.199	43.7
9/5, 10/9	12.945	46.4
11/9, 18/11	15.383	55.1
9/7, 14/9	16.479	59.1

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 B♭ 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.

For people who aren't colorblind, 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♯, B♭, red B♭, 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 B♭ 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, C♭, red C♭/blue B♯ (they are enharmonic equivalents), B♯, and C. All of the notes in 43edo therefore have unambiguous names except for two: red C♭/blue B♯, and red F♭/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 C♭ and blue B♯ (and red F♭/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.

Ups and downs notation

The following shows how ups and downs notation can notate the third-sharps and third-flats through extended Helmholtz–Ellis symbols:

Steps	0	1	2	3	4	5	6	7	8
Sharp Symbol
Sharp Symbol
Flat Symbol
Flat Symbol

Sagittal

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

Steps	0	1	2	3
Symbol

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Absolute (¢)	Relative (%)
2.3	[-68 43⟩	[⟨43 68]]	+1.35	1.35	4.84
2.3.5	81/80, 50331648/48828125	[⟨43 68 100]]	+0.27	1.88	6.75
2.3.5.7	81/80, 126/125, 17280/16807	[⟨43 68 100 121]]	-0.51	2.11	7.56
2.3.5.7.11	81/80, 99/98, 126/125, 864/847	[⟨43 68 100 121 149]]	-0.80	1.98	7.08
2.3.5.7.11.13	78/77, 81/80, 99/98, 126/125, 144/143	[⟨43 68 100 121 149 159]]	-0.52	1.91	6.85
2.3.5.7.11.13.17	78/77, 81/80, 99/98, 120/119, 126/125, 144/143	[⟨43 68 100 121 149 159 176]]	-0.52	1.81	6.49

Commas

This is a partial list of the 19-limit commas that 43edo tempers out with its patent val, ⟨43 68 100 121 149 159 176 183].

Prime Limit	Ratio^[2]	Monzo	Cents	Color name	Name(s)
3	(42 digits)	[-68 43⟩	184.07	Tribilawa	43-comma
5	(18 digits)	[20 5 -12⟩	74.01	Saquadtrigu	Hypovishnuzma
5	81/80	[-4 4 -1⟩	21.51	Gu	Syntonic comma, Didymus comma, meantone comma
5	(20 digits)	[32 -7 -9⟩	9.49	Sasa-tritrigu	Escapade comma
5	(42 digits)	[-68 18 17⟩	2.52	Quinla-seyo	Vavoom comma
7	(14 digits)	[9 7 -5 -3⟩	75.64	Triru-aquingu	Superpine comma
7	59049/57344	[-13 10 0 -1⟩	50.72	Laru	Harrison's comma
7	(14 digits)	[5 -1 7 -7⟩	20.46	Sepruyo	Mermisma
7	126/125	[1 2 -3 1⟩	13.80	Zotrigu	Starling comma
7	(14 digits)	[21 -5 -2 -3⟩	11.12	Satriru-agugu	Bronzisma
7	(18 digits)	[-14 7 -6 6⟩	8.76	Latribizogu	Historisma
7	225/224	[-5 2 2 -1⟩	7.71	Ruyoyo	Marvel comma
7	3136/3125	[6 0 -5 2⟩	6.08	Zozoquingu	Hemimean comma
7	(12 digits)	[-11 2 7 -3⟩	1.63	Latriru-asepyo	Meter comma
11	1350/1331	[1 3 2 0 -3⟩	24.54	Trilu-ayoyo	Large Tetracot diesis
11	99/98	[-1 2 0 -2 1⟩	17.58	Loruru	Mothwellsma
11	176/175	[4 0 -2 -1 1⟩	9.86	Lorugugu	Valinorsma
11	441/440	[-3 2 -1 2 -1⟩	3.93	Luzozogu	Werckisma
11	4000/3993	[5 -1 3 0 -3⟩	3.03	Triluyo	Wizardharry comma, pine comma
11	(12 digits)	[17 -5 0 -2 -1⟩	1.26	Salururu	Olympic comma
11	(18 digits)	[24 -6 0 1 -5⟩	0.51	Saquinlu-azo	Quartisma
13	78/77	[1 1 0 -1 -1 1⟩	22.34	Tholuru	Negustma
13	144/143	[4 2 0 0 -1 -1⟩	12.06	Thulu	Grossma
13	169/168	[-3 -1 0 -1 0 2⟩	10.27	Thothoru	Buzurgisma, dhanvantarisma
13	(12 digits)	[9 6 0 0 0 -5⟩	9.09	Quinthu	Glacier comma
13	364/363	[2 -1 0 1 -2 1⟩	4.76	Tholuluzo	Minor minthma
13	1001/1000	[-3 0 -3 1 1 1⟩	1.73	Tholozotrigu	Fairytale comma, sinbadma
13	2080/2079	[5 -3 1 -1 -1 1⟩	0.83	Tholuruyo	Ibnsinma, sinaisma
13	4096/4095	[12 -2 -1 -1 0 -1⟩	0.42	Sathurugu	Schismina
17	120/119	[3 1 1 -1 0 0 -1⟩	14.49	Suruyo	Lynchisma
17	221/220	[-2 0 -1 0 -1 1 1⟩	7.85	Sotholugu	Minor naiadma
17	256/255	[8 -1 -1 0 0 0 -1⟩	6.78	Sugu	Charisma, septendecimal kleisma
17	273/272	[5 1 -1 0 0 0 0 -1⟩	6.35	Suthozo	Tannisma
17	715/714	[-1 -1 1 -1 1 1 -1⟩	2.42	Sutholoruyo	September comma
19	96/95	[5 1 -1 0 0 0 0 -1⟩	18.13	Nugu	19th Partial chroma
19	153/152	[-3 2 0 0 0 0 1 -1⟩	11.35	Nuso	Ganassisma
19	171/170	[-1 2 -1 0 0 0 -1 1⟩	10.15	Nosugu	Malcolmisma
19	209/208	[-4 0 0 0 1 -1 0 1⟩	8.30	Nothulo	Yama comma
19	210/209	[1 1 1 1 -1 0 0 -1⟩	8.26	Nuluzoyo	Spleen comma

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods per 8ve	Generator (Reduced)	Cents (Reduced)	Associated Ratio (Reduced)	Temperament
1	1\43	27.91	64/63	Arch
1	2\43	55.81	33/32	Escapade
1	4\43	111.63	16/15	Vavoom
1	5\43	139.53	13/12	Jerome
1	6\43	167.44	11/10	Superpine
1	7\43	195.35	28/25	Didacus
1	8\43	223.26	8/7	Kumonga
1	9\43	251.16	15/13	Hemimeantone
1	10\43	279.07	75/64	Decipentic
1	11\43	334.88	17/14	Cohemimabila
1	13\43	362.79	16/13	Submajor (43e) / interpental (43)
1	14\43	390.70	5/4	Amigo
1	16\43	446.51	13/10	Supersensi
1	18\43	502.33	4/3	Meantone
1	19\43	530.23	15/11	Amavil
1	20\43	558.14	11/8	Thuja
1	21\43	586.05	7/5	Merman

Detemperaments

Ringer 43

The metaphorical color palette that the intervals of 43edo present can be quite appealing for various reasons such as being meantone and splitting 4/3 into 6 equal parts and 3/2 into 5 equal parts, but the accuracy leaves one wanting in many cases, which is why an excellent alternative (given the unambiguity of mappings of all primes in the 109-limit except 71 and 89) is Ringer 43, a Ringer scale with 43 notes per octave period:

55:56:57:58:59:60:61:62:63:64:65:66:67:68:69:70:72:73:74:75:76:78:79:80:82:83:84:86:87:88:90:91:92:94:96:97:98:100:102:104:106:108:109:110

Or equivalently in the form of reduced, rooted intervals:

65/64, 33/32, 67/64, 17/16, 69/64, 35/32, 9/8, 73/64, 37/32, 75/64, 19/16, 39/32, 79/64, 5/4, 41/32, 83/64, 21/16, 43/32, 87/64, 11/8, 45/32, 91/64, 23/16, 47/32, 3/2, 97/64, 49/32, 25/16, 51/64, 13/8, 53/32, 27/16, 109/64, 55/32, 7/4, 57/32, 29/16, 59/32, 15/8, 61/32, 31/16, 63/32, 2/1

Scales

Main article: 5- to 10-tone scales in 43edo

Harmonic scales

43edo represents the first 16 overtones of the harmonic series well (written as a ratio of 8:9:10:11:12:13:14:15:16 in just intonation) with degrees 0, 7, 14, 20, 25, 30, 35, 39, and 43, and scale steps of 7, 7, 6, 5, 5, 5, 4, and 4.

7\43 (195.349¢) stands in for frequency ratio 9/8 (203.910¢) and 10/9 (182.404¢).
6\43 (156.522¢) stands in for 11/10 (165.004¢)
5\46 (130.435¢) stands in for 12/11 (150.637¢), 13/12 (138.573¢), and 14/13 (128.298¢).
4\43 (111.628¢) stands in for 15/14 (119.443¢) and 16/15 (111.731¢).