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[[File:33edo.png|alt=33edo.png|966x199px|33edo.png]]
[[File:33edo.png|alt=33edo.png|966x199px|33edo.png]]


What follows is a comparison of stretched- and compressed-octave 33edo tunings.
Useful options include:
 
* Stretched: [[ed5|76ed5]], [[ed7|92ed7]], [[52edt]], [[zpi|138zpi]]
; [[ed5|76ed5]]  
* Compressed: [[ed7|93ed7]], [[ed5|77ed5]], [[equal tuning|115ed11]]
* Octave size: 1209.8{{c}}
Stretching the octave of 33edo by around 10{{c}} results in improved primes 3 and 7, but worse primes 2 and 11. This approximates all harmonics up to 16 within 17.0{{c}}. The tuning 76ed5 does this.
{{Harmonics in equal|76|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 76ed5}}
{{Harmonics in equal|76|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 76ed5 (continued)}}
 
; [[ed7|92ed7]]
* Octave size: 1208.4{{c}}
Stretching the octave of 33edo by around 8.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 17.7{{c}}. The tuning 92ed7 does this. So does the tuning [[zpi|137zpi]] whose octave differs by only 0.3{{c}}.
{{Harmonics in equal|92|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 92ed7}}
{{Harmonics in equal|92|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 92ed7 (continued)}}
 
; [[52edt]]  
* Step size: 36.576, octave size: 1207.0{{c}}
Stretching the octave of 33edo by around 7{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 14. This approximates all harmonics up to 16 within 18.2{{c}}. The tuning 52edt does this.
{{Harmonics in equal|52|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 76ed5}}
{{Harmonics in equal|52|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 52edt (continued)}}
 
; [[equal tuning|114ed11]]
* Octave size: 1201.7{{c}}
Stretching the octave of 33edo by around 2{{c}} results in improved primes 3, 11 and 13, but worse primes 2, 5 and 7. This approximates all harmonics up to 16 within 17.8{{c}}. The tuning 114ed11 does this.
{{Harmonics in equal|114|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 114ed11}}
{{Harmonics in equal|114|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 114ed11 (continued)}}
 
; [[zpi|138zpi]]
* Step size: 36.394{{c}}, octave size: 1201.0{{c}}
Stretching the octave of 33edo by around 1{{c}} results in improved primes 3, 11 and 13, but worse primes 2, 5 and 7. This approximates all harmonics up to 16 within 17.5{{c}}. The tuning 138zpi does this. So does the tuning [[equal tuning|122ed13]] whose octave differs by only 0.1{{c}}.
{{Harmonics in cet|36.394|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 138zpi}}
{{Harmonics in cet|36.394|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 138zpi (continued)}}
 
; 33edo
* Step size: 36.363{{c}}, octave size: 1200.0{{c}}
Pure-octaves 33edo approximates all harmonics up to 16 within 14.3{{c}}.
{{Harmonics in equal|33|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 33edo}}
{{Harmonics in equal|33|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 33edo (continued)}}
 
; [[WE|33et, 13-limit WE tuning]]
* Step size: 36.357{{c}}, octave size: 1199.8{{c}}
Compressing the octave of 33edo by a fifth of a cent results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 13.6{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in cet|36.357|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 33et, 13-limit WE tuning}}
{{Harmonics in cet|36.357|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 33et, 13-limit WE tuning (continued)}}
 
; [[ed7|93ed7]]
* Octave size: 1196.4{{c}}
Compressing the octave of 33edo by around 4.5{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 17.9{{c}}. If one wishes to use both 33edo's sharp and flat fifths simultaneously (see [[dual-fifth tuning]]), then this amount of stretch is ideal, because it evenly shares error between the two fifths. The tuning 93ed7 does this. So does the tuning [[equal tuning|52ed13]] whose octave differs by only 0.1{{c}}.
{{Harmonics in equal|93|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 93ed7}}
{{Harmonics in equal|93|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 93ed7 (continued)}}
 
; [[ed5|77ed5]]
* Octave size: 1194.1{{c}}
Compressing the octave of 33edo by around 6{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 17.6{{c}}. The tuning 77ed5 does this. So does the tuning [[zpi|139zpi]] whose octave differs by only 0.2{{c}}.
{{Harmonics in equal|77|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 77ed5}}
{{Harmonics in equal|77|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 77ed5 (continued)}}
 
; [[equal tuning|115ed11]]  
* Octave size: 1191.2{{c}}
Compressing the octave of 33edo by around 9{{c}} results in improved primes 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 17.5{{c}}. The tuning 115ed11 does this. So do the tunings [[equal tuning|123ed13]] and [[AS|1ed47/46]] whose octaves are within 0.3{{c}} of 115ed11.
{{Harmonics in equal|115|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 115ed11}}
{{Harmonics in equal|115|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 115ed11 (continued)}}


== Scales ==
== Scales ==

Revision as of 03:09, 23 September 2025

← 32edo 33edo 34edo →
Prime factorization 3 × 11
Step size 36.3636 ¢ 
Fifth 19\33 (690.909 ¢)
Semitones (A1:m2) 1:4 (36.36 ¢ : 145.5 ¢)
Consistency limit 3
Distinct consistency limit 3

33 equal divisions of the octave (abbreviated 33edo or 33ed2), also called 33-tone equal temperament (33tet) or 33 equal temperament (33et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 33 equal parts of about 36.4 ¢ each. Each step represents a frequency ratio of 21/33, or the 33rd root of 2.

Theory

Structural properties

While relatively uncommon, 33edo is actually quite an interesting system. As a multiple of 11edo, it approximates the 7th and 11th harmonics via orgone temperament (see 26edo). 33edo also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having a 3L 7s with L = 4, s = 3. The 33c (⟨33 52 76 93]) and 33cd (⟨33 52 76 92]) mappings temper out 81/80 and can be used to represent 1/2-comma meantone, a "flattertone" tuning where the whole tone is 10/9 in size. Indeed, the perfect fifth is tuned about 11 ¢ flat, and two stacked fifths fall only 0.6 ¢ flat of 10/9. Leaving the scale be would result in the standard diatonic scale (5L 2s) having minor seconds of four steps and whole tones of five steps. This also results in common practice minor and major chords becoming more supraminor and submajor in character, making everything sound almost neutral in quality.

33edo maps both the 4:5:6 and 6:7:8 chords inconsistently, with the third harmonic being about a third of a step flat and the 5th and 7th harmonics being about a third of a step sharp. It is thus reasonable to use the second-best approximation of 3, 5, or 7 in either chord, but in any case, the worst of the three intervals in the chord is detuned by over 22 cents, meaning 33edo is near-maximally bad for its size for tonal harmony.

Instead of the flat 19-step fifth you may use the 20-step sharp fifth, over 25 ¢ sharp. Two of these lead to a 9/8 of 7\33, which is about 22/19 in size and may be counted as a small third. Between the flat 5\33 version of 9/8 and the sharp 7\33 version there is, of course, a 6\33 = 2\11 interval of 218 ¢. Together, these add up to 6\33 + 5\33 = 11\33 = 1\3, or 400 ¢, the same major third as 12edo. We also have both a 327 ¢ minor third (9\33 = 6\22 = 3\11), the same as that of 22edo, and a flatter 8\33 third of 291 ¢, which if you like could also be called a flat 19th harmonic, but much more accurately a 13/11 sharp by 1.7 ¢ (if you use the patent val it is an extremely inaccurate 6/5). Another talent it has is that 7/5 is tuned quite accurately by 16\33, and we may put two 8\33 versions of 13/11 together to produce the cuthbert triad. The 8\33 generator, with MOS of size 5, 9, and 13, gives plenty of scope for these, as well as the 11th, 13th, and 19th harmonics (taking the generator as a 19/16) which are relatively well in tune.

33edo contains an accurate approximation of the Bohlen–Pierce scale with 4\33 near 1\13edt.

Other notable 33edo scales are diasem with L:m:s = 5:3:1 and 5L 4s with L:s = 5:2. This step ratio for 5L 4s is great for its semitone size of 72.7 ¢.

Odd harmonics

Approximation of odd harmonics in 33edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -11.0 +13.7 +13.0 +14.3 -5.9 -4.2 +2.6 +4.1 -6.6 +1.9 -10.1
Relative (%) -30.4 +37.6 +35.7 +39.2 -16.1 -11.5 +7.3 +11.4 -18.2 +5.4 -27.8
Steps
(reduced) 		52
(19) 		77
(11) 		93
(27) 		105
(6) 		114
(15) 		122
(23) 		129
(30) 		135
(3) 		140
(8) 		145
(13) 		149
(17)

33edo is not especially good at representing all rational intervals in the 7-limit, but it does very well on the 7-limit 3*33 subgroup 2.27.15.21.11.13. On this subgroup it tunes things to the same tuning as 99edo, and as a subgroup patent val it tempers out the same commas. The 99 equal temperaments hemififths, amity, parakleismic, hemiwuerschmidt, ennealimmal and hendecatonic can be reduced to this subgroup and give various possibilities for MOS scales, etc. In particular, the terrain 2.7/5.9/5 subgroup temperament can be tuned via the 5\33 generator. The full system of harmony provides the optimal patent val for slurpee temperament in the 5-, 7-, 11-, and 13-limits.

While it might not be the most harmonically accurate temperament, it is structurally quite interesting, and it approximates the full 19-limit consort in its own way. You could even say it tunes the 23rd and 29th harmonics ten cents flat if you were so inclined; as well as getting within two cents of the 37th.

Miscellany

33 is also the number of years in the Iranian calendar's leap cycle, where leap year is inserted once every 4 or 5 years. This corresponds to the 1L 7s with the step ratio of 5:4.

Intervals

Step # ET Just Difference
(ET minus Just) 		Extended Pythagorean notation
Cents Interval Cents
0 0 1/1 0 0 Perfect Unison P1 D
1 36.364 48/47 36.448 −0.085 Augmented Unison A1 D#
2 72.727 24/23 73.681 −0.953 Double-aug 1sn AA1 Dx
3 109.091 16/15 111.731 −2.640 Diminished 2nd d2 Ebb
4 145.455 12/11 150.637 −5.183 Minor 2nd m2 Eb
5 181.818 10/9 182.404 −0.586 Major 2nd M2 E
6 218.182 17/15 216.687 +1.495 Augmented 2nd A2 E#
7 254.545 15/13 247.741 +6.804 Double-aug 2nd/Double-dim 3rd AA2/dd3 Ex/Fbb
8 290.909 13/11 289.210 +1.699 Diminished 3rd d3 Fb
9 327.273 6/5 315.641 +11.631 Minor 3rd m3 F
10 363.636 16/13 359.472 +4.164 Major 3rd M3 F#
11 400.000 5/4 386.314 +13.686 Augmented 3rd A3 Fx
12 436.364 9/7 435.084 +1.280 Double-dim 4th dd4 Gbb
13 472.727 21/16 470.781 +1.946 Diminished 4th d4 Gb
14 509.091 4/3 498.045 +11.046 Perfect 4th P4 G
15 545.455 11/8 551.318 −5.863 Augmented 4th A4 G#
16 581.818 7/5 582.513 −0.694 Double-aug 4th AA4 Gx
17 618.182 10/7 617.488 +0.694 Double-dim 5th dd5 Abb
18 654.545 16/11 648.682 +5.863 Diminished 5th d5 Ab
19 690.909 3/2 701.955 −11.046 Perfect 5th P5 A
20 727.273 32/21 729.219 -1.946 Augmented 5th A5 A#
21 763.636 14/9 764.916 −1.280 Double-aug 5th AA5 Ax
22 800.000 8/5 813.686 −13.686 Double-dim 6th d6 Bbb
23 836.364 13/8 840.528 −4.164 Minor 6th m6 Bb
24 872.727 5/3 884.359 −11.631 Major 6th M6 B
25 909.091 22/13 910.790 −1.699 Augmented 6th A6 B#
26 945.455 12/7 933.129 +12.325 Double-aug 6th/Double-dim 7th AA6/dd7 Bx/Cbb
27 981.818 30/17 983.313 −1.495 Diminished 7th d7 Cb
28 1018.182 9/5 1017.596 +0.586 Minor 7th m7 C
29 1054.545 11/6 1049.363 +5.183 Major 7th M7 C#
30 1090.909 15/8 1088.268 +2.640 Augmented 7th A7 Cx
31 1127.273 23/12 1126.319 −0.953 Double-dim 8ve dd8 Dbb
32 1163.636 47/24 1163.551 +0.085 Diminished 8ve d8 Db
33 1200 2/1 1200 0 Perfect Octave P8 D

Notation

Standard notation

Because the chromatic semitone in 33edo is 1 step, 33edo can be notated using only naturals, sharps, and flats. However, many key signatures will require double- and triple-sharps and flats, which means that notation in distant keys can be very unwieldy.

Step offset −2 −1 0 +1 +2
Symbol  
 		 
 		 
 		 
 		 

Sagittal notation

This notation uses the same sagittal sequence as EDOs 23 and 28.

Sagittal notationPeriodic table of EDOs with sagittal notationlimma-fraction notation

Approximation to JI

The following tables show how 15-odd-limit intervals are represented in 33edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 33edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
9/5, 10/9 0.586 1.6
7/5, 10/7 0.694 1.9
9/7, 14/9 1.280 3.5
13/11, 22/13 1.699 4.7
15/8, 16/15 2.640 7.3
13/8, 16/13 4.164 11.5
11/6, 12/11 5.183 14.3
11/8, 16/11 5.863 16.1
15/13, 26/15 6.804 18.7
13/12, 24/13 6.882 18.9
15/11, 22/15 8.504 23.4
15/14, 28/15 10.352 28.5
3/2, 4/3 11.046 30.4
5/3, 6/5 11.631 32.0
7/6, 12/7 12.325 33.9
7/4, 8/7 12.992 35.7
5/4, 8/5 13.686 37.6
9/8, 16/9 14.272 39.2
11/9, 18/11 16.228 44.6
11/10, 20/11 16.814 46.2
13/7, 14/13 17.156 47.2
11/7, 14/11 17.508 48.1
13/10, 20/13 17.850 49.1
13/9, 18/13 17.928 49.3
15-odd-limit intervals in 33edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/5, 10/7 0.694 1.9
13/11, 22/13 1.699 4.7
15/8, 16/15 2.640 7.3
13/8, 16/13 4.164 11.5
11/6, 12/11 5.183 14.3
11/8, 16/11 5.863 16.1
15/13, 26/15 6.804 18.7
13/12, 24/13 6.882 18.9
15/11, 22/15 8.504 23.4
15/14, 28/15 10.352 28.5
3/2, 4/3 11.046 30.4
7/4, 8/7 12.992 35.7
5/4, 8/5 13.686 37.6
11/9, 18/11 16.228 44.6
13/7, 14/13 17.156 47.2
13/10, 20/13 17.850 49.1
13/9, 18/13 17.928 49.3
11/7, 14/11 18.856 51.9
11/10, 20/11 19.550 53.8
9/8, 16/9 22.092 60.8
7/6, 12/7 24.038 66.1
5/3, 6/5 24.732 68.0
9/7, 14/9 35.084 96.5
9/5, 10/9 35.778 98.4
15-odd-limit intervals by 33cd val mapping
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
9/5, 10/9 0.586 1.6
7/5, 10/7 0.694 1.9
9/7, 14/9 1.280 3.5
13/11, 22/13 1.699 4.7
13/8, 16/13 4.164 11.5
11/6, 12/11 5.183 14.3
11/8, 16/11 5.863 16.1
13/12, 24/13 6.882 18.9
15/14, 28/15 10.352 28.5
3/2, 4/3 11.046 30.4
5/3, 6/5 11.631 32.0
7/6, 12/7 12.325 33.9
11/9, 18/11 16.228 44.6
11/10, 20/11 16.814 46.2
11/7, 14/11 17.508 48.1
13/9, 18/13 17.928 49.3
13/10, 20/13 18.513 50.9
13/7, 14/13 19.207 52.8
9/8, 16/9 22.092 60.8
5/4, 8/5 22.677 62.4
7/4, 8/7 23.371 64.3
15/11, 22/15 27.860 76.6
15/13, 26/15 29.559 81.3
15/8, 16/15 33.723 92.7

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-52 33⟩ [⟨33 52]] +3.48 3.49 9.59
2.3.5 81/80, 1171875/1048576 [⟨33 52 76]] (33c) +5.59 4.13 11.29
2.3.5.7 49/48, 81/80, 1875/1792 [⟨33 52 76 92]] (33cd) +6.29 3.77 10.31
2.3.5.7.11 45/44, 49/48, 81/80, 1375/1344 [⟨33 52 76 92 114]] (33cd) +5.36 3.84 10.50
2.3.5.7.11.13 45/44, 49/48, 65/64, 81/80, 275/273 [⟨33 52 76 92 114 122]] (33cd) +4.65 3.84 10.52

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 2\33 72.73 21/20 Slurpee (33)
1 4\33 145.45 12/11 Bohpier (33cd)
1 7\33 254.55 8/7 Godzilla (33cd)
1 8\33 290.91 25/21 Quasitemp (33b)
1 10\33 363.64 49/40 Submajor (33ee) / interpental (33e)
1 14\33 509.09 4/3 Flattertone (33cd)
Deeptone a.k.a. tragicomical (33)
1 16\33 581.82 7/5 Tritonic (33)
3 7\33
(4\33) 		254.55
(145.45) 		8/7
(12/11) 		Triforce (33d)
3 13\33
(2\33) 		472.73
(72.73) 		4/3
(25/24) 		Inflated (33bcddd)
3 14\33
(3\33) 		509.09
(98.09) 		4/3
(16/15) 		August (33cd)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Octave stretch or compression

33edo is nearby to many other equal tunings which can act as stretched or compressed versions of 33edo, improving some of its harmonics at the expense of others.

33edo.png

Useful options include:

Scales

Main article: List of MOS scales in 33edo

Brightest mode is listed except where noted.

  • Deeptone[7], 5 5 5 4 5 5 4 (diatonic)
    • Fun 5-tone subset of Deeptone[7] 9 5 5 4 10
  • Deeptone[12], 4 4 1 4 1 4 4 1 4 1 4 1 (chromatic)
  • Deeptone[19], 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 1 (enharmonic)
  • Semiquartal, 5 5 2 5 2 5 2 5 2
  • Semiquartal[14], 3 2 3 2 2 3 2 2 3 2 2
  • Iranian Calendar, 5 4 4 4 4 4 4 4
  • Diasem, 5 3 5 1 5 3 5 1 5 (*right-handed)
  • Diasem, 5 1 5 3 5 1 5 3 5 (*left-handed)
  • Diaslen (4sR), 1 5 1 5 2 5 1 5 1 5 2
  • Diaslen (4sL), 2 5 1 5 1 5 2 5 1 5 1
  • Diaslen (4sC), 1 5 2 5 1 5 1 5 2 5 1

Delta-rational harmony

The tables below show chords that approximate 3-integer-limit delta-rational chords with least-squares error less than 0.001.

Fully delta-rational triads

Steps Delta signature Least-squares error
0,1,2 +1+1 0.00021
0,1,3 +1+2 0.00048
0,1,4 +1+3 0.00078
0,2,3 +2+1 0.00039
0,2,4 +1+1 0.00087
0,3,4 +3+1 0.00056
0,3,11 +1+3 0.00007
0,5,8 +3+2 0.00084
0,8,18 +2+3 0.00082
0,9,20 +2+3 0.00076
0,12,17 +2+1 0.00048
0,13,20 +3+2 0.00063
0,15,21 +2+1 0.00063
0,16,28 +1+1 0.00082
0,18,25 +2+1 0.00081
0,18,31 +1+1 0.00058
0,19,24 +3+1 0.00095

Partially delta-rational tetrads

Steps Delta signature Least-squares error
0,1,2,3 +1+?+1 0.00053
0,1,2,4 +1+?+2 0.00094
0,1,3,4 +1+?+1 0.00080
0,1,17,18 +2+?+3 0.00073
0,1,17,19 +1+?+3 0.00071
0,1,18,19 +2+?+3 0.00042
0,1,18,20 +1+?+3 0.00032
0,1,19,20 +2+?+3 0.00010
0,1,19,21 +1+?+3 0.00008
0,1,20,21 +2+?+3 0.00023
0,1,20,22 +1+?+3 0.00049
0,1,21,22 +2+?+3 0.00056
0,1,21,23 +1+?+3 0.00091
0,1,22,23 +2+?+3 0.00090
0,1,31,32 +1+?+2 0.00071
0,2,3,4 +2+?+1 0.00077
0,2,6,11 +1+?+3 0.00094
0,2,7,12 +1+?+3 0.00013
0,2,8,13 +1+?+3 0.00069
0,2,12,13 +3+?+2 0.00083
0,2,12,15 +1+?+2 0.00087
0,2,13,14 +3+?+2 0.00045
0,2,13,16 +1+?+2 0.00014
0,2,14,15 +3+?+2 0.00008
0,2,14,17 +1+?+2 0.00060
0,2,15,16 +3+?+2 0.00031
0,2,16,17 +3+?+2 0.00071
0,2,18,20 +2+?+3 0.00084
0,2,18,22 +1+?+3 0.00024
0,2,19,21 +2+?+3 0.00020
0,2,19,23 +1+?+3 0.00058
0,2,20,22 +2+?+3 0.00046
0,3,4,5 +3+?+1 0.00097
0,3,5,9 +2+?+3 0.00010
0,3,6,10 +2+?+3 0.00090
0,3,7,12 +1+?+2 0.00074
0,3,8,13 +1+?+2 0.00037
0,3,10,17 +1+?+3 0.00009
0,3,17,23 +1+?+3 0.00096
0,3,18,22 +1+?+2 0.00088
0,3,18,24 +1+?+3 0.00027
0,3,19,20 +2+?+1 0.00059
0,3,19,21 +1+?+1 0.00063
0,3,19,22 +2+?+3 0.00030
0,3,19,23 +1+?+2 0.00023
0,3,20,21 +2+?+1 0.00014
0,3,20,22 +1+?+1 0.00015
0,3,20,23 +2+?+3 0.00070
0,3,21,22 +2+?+1 0.00032
0,3,21,23 +1+?+1 0.00095
0,3,22,23 +2+?+1 0.00078
0,3,27,32 +1+?+3 0.00004
0,4,5,12 +1+?+2 0.00026
0,4,6,16 +1+?+3 0.00066
0,4,8,13 +2+?+3 0.00023
0,4,11,20 +1+?+3 0.00023
0,4,13,14 +3+?+1 0.00091
0,4,13,19 +1+?+2 0.00048
0,4,14,15 +3+?+1 0.00050
0,4,14,16 +3+?+2 0.00055
0,4,14,17 +1+?+1 0.00021
0,4,15,16 +3+?+1 0.00009
0,4,15,17 +3+?+2 0.00023
0,4,15,18 +1+?+1 0.00085
0,4,16,17 +3+?+1 0.00034
0,4,17,18 +3+?+1 0.00077
0,4,17,25 +1+?+3 0.00043
0,4,19,23 +2+?+3 0.00041
0,4,20,24 +2+?+3 0.00094
0,4,22,27 +1+?+2 0.00020
0,4,24,31 +1+?+3 0.00022
0,5,6,9 +3+?+2 0.00003
0,5,7,10 +3+?+2 0.00097
0,5,7,19 +1+?+3 0.00004
0,5,9,17 +1+?+2 0.00017
0,5,10,16 +2+?+3 0.00019
0,5,11,13 +2+?+1 0.00087
0,5,11,15 +1+?+1 0.00018
0,5,12,14 +2+?+1 0.00011
0,5,12,23 +1+?+3 0.00067
0,5,13,15 +2+?+1 0.00067
0,5,16,23 +1+?+2 0.00008
0,5,17,27 +1+?+3 0.00055
0,5,19,24 +2+?+3 0.00051
0,5,22,31 +1+?+3 0.00057
0,5,24,30 +1+?+2 0.00036
0,5,25,26 +3+?+1 0.00071
0,5,25,27 +3+?+2 0.00082
0,5,25,28 +1+?+1 0.00045
0,5,26,27 +3+?+1 0.00018
0,5,26,28 +3+?+2 0.00016
0,5,26,29 +1+?+1 0.00090
0,5,27,28 +3+?+1 0.00035
0,5,28,29 +3+?+1 0.00090
0,6,7,17 +1+?+2 0.00087
0,6,8,22 +1+?+3 0.00045
0,6,9,14 +1+?+1 0.00031
0,6,11,18 +2+?+3 0.00093
0,6,12,21 +1+?+2 0.00036
0,6,12,25 +1+?+3 0.00032
0,6,15,18 +3+?+2 0.00026
0,6,16,19 +3+?+2 0.00095
0,6,16,28 +1+?+3 0.00053
0,6,18,26 +1+?+2 0.00064
0,6,19,25 +2+?+3 0.00062
0,6,20,24 +1+?+1 0.00052
0,6,21,23 +2+?+1 0.00031
0,6,21,32 +1+?+3 0.00033
0,6,22,24 +2+?+1 0.00063
0,6,25,32 +1+?+2 0.00034
0,7,8,14 +1+?+1 0.00029
0,7,8,24 +1+?+3 0.00080
0,7,9,11 +3+?+1 0.00066
0,7,9,12 +2+?+1 0.00041
0,7,9,13 +3+?+2 0.00019
0,7,10,12 +3+?+1 0.00009
0,7,10,13 +2+?+1 0.00070
0,7,11,13 +3+?+1 0.00087
0,7,12,27 +1+?+3 0.00041
0,7,16,30 +1+?+3 0.00098
0,7,17,22 +1+?+1 0.00008
0,7,19,26 +2+?+3 0.00073
0,7,20,29 +1+?+2 0.00002
0,7,23,26 +3+?+2 0.00010
0,7,28,32 +1+?+1 0.00033
0,7,29,31 +2+?+1 0.00020
0,7,30,32 +2+?+1 0.00091
0,8,12,29 +1+?+3 0.00097
0,8,13,22 +2+?+3 0.00051
0,8,15,21 +1+?+1 0.00062
0,8,15,31 +1+?+3 0.00047
0,8,16,18 +3+?+1 0.00066
0,8,16,19 +2+?+1 0.00031
0,8,16,20 +3+?+2 0.00043
0,8,16,27 +1+?+2 0.00090
0,8,17,19 +3+?+1 0.00022
0,8,17,20 +2+?+1 0.00098
0,8,19,27 +2+?+3 0.00085
0,8,24,29 +1+?+1 0.00020
0,9,11,16 +3+?+2 0.00051
0,9,13,20 +1+?+1 0.00002
0,9,14,24 +2+?+3 0.00073
0,9,18,30 +1+?+2 0.00090
0,9,19,28 +2+?+3 0.00096
0,9,21,27 +1+?+1 0.00040
0,9,22,24 +3+?+1 0.00087
0,9,22,25 +2+?+1 0.00053
0,9,22,26 +3+?+2 0.00026
0,9,23,25 +3+?+1 0.00013
0,9,23,26 +2+?+1 0.00093
0,10,11,26 +1+?+2 0.00035
0,10,11,32 +1+?+3 0.00081
0,10,12,20 +1+?+1 0.00098
0,10,14,18 +2+?+1 0.00050
0,10,14,25 +2+?+3 0.00088
0,10,15,29 +1+?+2 0.00041
0,10,16,21 +3+?+2 0.00055
0,10,19,32 +1+?+2 0.00021
0,10,27,31 +3+?+2 0.00082
0,10,28,30 +3+?+1 0.00045
0,10,28,31 +2+?+1 0.00016
0,10,29,31 +3+?+1 0.00068
0,11,12,18 +3+?+2 0.00030
0,11,13,16 +3+?+1 0.00081
0,11,14,17 +3+?+1 0.00044
0,11,16,31 +1+?+2 0.00064
0,11,17,25 +1+?+1 0.00091
0,11,19,23 +2+?+1 0.00045
0,11,21,26 +3+?+2 0.00074
0,12,15,24 +1+?+1 0.00087
0,12,15,28 +2+?+3 0.00013
0,12,17,23 +3+?+2 0.00054
0,12,18,21 +3+?+1 0.00043
0,12,19,22 +3+?+1 0.00095
0,12,23,27 +2+?+1 0.00083
0,12,26,31 +3+?+2 0.00005
0,13,14,24 +1+?+1 0.00019
0,13,17,22 +2+?+1 0.00085
0,13,21,27 +3+?+2 0.00035
0,13,22,25 +3+?+1 0.00097
0,13,23,26 +3+?+1 0.00054
0,13,28,32 +2+?+1 0.00055
0,14,17,24 +3+?+2 0.00099
0,14,18,28 +1+?+1 0.00043
0,14,21,26 +2+?+1 0.00080
0,14,25,31 +3+?+2 0.00054
0,14,27,30 +3+?+1 0.00050
0,15,16,20 +3+?+1 0.00055
0,15,17,28 +1+?+1 0.00064
0,15,21,28 +3+?+2 0.00045
0,15,22,32 +1+?+1 0.00039
0,16,18,26 +3+?+2 0.00049
0,16,19,25 +2+?+1 0.00031
0,16,20,24 +3+?+1 0.00018
0,16,25,32 +3+?+2 0.00095
0,17,22,28 +2+?+1 0.00091
0,17,23,27 +3+?+1 0.00066
0,18,27,31 +3+?+1 0.00095
0,19,21,28 +2+?+1 0.00065
0,20,24,31 +2+?+1 0.00078
0,21,22,32 +3+?+2 0.00091
0,22,27,32 +3+?+1 0.00038

Instruments

Lumatone mapping for 33edo

Music

Modern renderings

Johann Sebastian Bach

21st century

Bryan Deister
Peter Kosmorsky
Budjarn Lambeth
Claudi Meneghin
Relyt R
Chris Vaisvil
Xeno*n*
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