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| [[126/125]]
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| {{monzo| 1 2 -3 1 }}
| 13.795
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| Zotrigu
| Zotrigu
| Starling comma
| Starling comma
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| Ruyoyo
| Marvel comma
| Marvel comma

Revision as of 13:35, 9 March 2024

← 42edo 43edo 44edo →
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

Template:EDO intro 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. [1]

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. In the 7-limit, it supports septimal meantone, as it tempers out 3136/3125, along with 126/125 and 225/224. 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 ⟨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 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-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 composite harmonics and 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
Relative (%) +0.0 -15.3 +15.7 +28.4 +24.4 -11.9 +23.9 +33.9 +48.7 +10.7 -3.0
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 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

The following tables show how 15-odd-limit intervals are represented in 43edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 43edo (direct approximation, even if inconsistent)
Interval and 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
9/7, 14/9 11.428 40.9
7/6, 12/7 12.199 43.7
11/9, 18/11 12.524 44.9
9/5, 10/9 12.945 46.4
15-odd-limit intervals in 43edo (patent val mapping)
Interval and 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 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

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
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 commas that 43edo tempers out with its patent val, ⟨43 68 100 121 149 159 176].

Prime
Limit 		Ratio[2] Monzo Cents Color name Name(s)
3 (42 digits) [-68 43⟩ 184.07 43-comma
5 (18 digits) [20 5 -12⟩ 74.01
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 Vavoom comma
7 126/125 [1 2 -3 1⟩ 13.80 Zotrigu Starling comma
7 3136/3125 [6 0 -5 2⟩ 6.08 Zozoquingu Hemimean comma
7 225/224 [-5 2 2 -1⟩ 7.71 Ruyoyo Marvel comma
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
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
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
17 120/119 [3 1 1 -1 0 0 -1⟩ 14.49 Suruyo Lynchisma
19 96/95 [5 1 -1 0 0 0 0 -1⟩ 18.13 Nugu 19th Partial chroma
19 209/208 [-4 0 0 0 1 -1 0 1⟩ 8.30 Nothulo Yama comma
19 273/272 [5 1 -1 0 0 0 0 -1⟩ 18.13 Suthozo Tannisma

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 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

Music

Modern renderings

Johann Sebastian Bach
Nicolaus Bruhns

21st century

Bryan Deister
Peter Kosmorsky
MortisTheneRd
Juhan Puhm (site)
Randy Wells

Articles

Diagrams

Instruments

References

  1. ↑ Stichting Huygens-Fokker: Logarithmic Interval Measures
  2. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
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