# 33edo

 ← 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

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 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 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 mapping (which has val 33 52 76]) tempers 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 cents flat, and two stacked fifths fall only 0.6 cents 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.

Instead of the flat 19\33 fifth you may use the sharp fifth of 20\33, over 25 cents 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 11edo interval of 218 cents. Now 6\33 + 5\33 = 11\33 = 1\3 of an octave, or 400 cents, the same major third as 12edo. Also, we have both a 327 minor third from 9\33 = 3\11, the same as the 22edo minor third, and a flatter 8\33 third of 291 cents, which if you like could also be called a flat 19th harmonic, but much more accurately a 13/11 sharp by 1.7 cents (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 11, 13 and 19 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.

So while it might not be the most harmonically accurate temperament, it's structurally quite interesting, and it approximates the full 19-limit consort in it's 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.

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¢.

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.

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, making notation very unwieldy in distant keys.

### 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)

## Intervals

# Cents Approximate ratios Notation
0 1/1 Perfect Unison P1 D
1 36.364 48/47 Augmented Unison A1 D#
2 72.727 24/23 Double-aug 1sn AA1 Dx
3 109.091 16/15 Diminished 2nd d2 Ebb
4 145.455 12/11 Minor 2nd m2 Eb
5 181.818 10/9 Major 2nd M2 E
6 218.182 17/15 Augmented 2nd A2 E#
7 254.545 15/13 Double-aug 2nd/Double-dim 3rd AA2/dd3 Ex/Fbb
8 290.909 13/11 Diminished 3rd d3 Fb
9 327.273 6/5 Minor 3rd m3 F
10 363.636 11/9, 5/4 Major 3rd M3 F#
11 400.000 5/4 Augmented 3rd A3 Fx
12 436.364 9/7 Double-dim 4th dd4 Gbb
13 472.727 21/16 Diminished 4th d4 Gb
14 509.091 4/3 Perfect 4th P4 G
15 545.455 11/8 Augmented 4th A4 G#
16 581.818 7/5 Double-aug 4th AA4 Gx
17 618.182 10/7 Double-dim 5th dd5 Abb
18 654.545 16/11 Diminished 5th d5 Ab
19 690.909 3/2 Perfect 5th P5 A
20 727.273 32/21 Augmented 5th A5 A#
21 763.636 14/9 Double-aug 5th AA5 Ax
22 800.000 8/5 Double-dim 6th d6 Bbb
23 836.364 13/8 Minor 6th m6 Bb
24 872.727 5/3 Major 6th M6 B
25 909.091 22/13 Augmented 6th A6 B#
26 945.455 12/7 Double-aug 6th/Double-dim 7th AA6/dd7 Bx/Cbb
27 981.818 30/17 Diminished 7th d7 Cb
28 1018.182 9/5 Minor 7th m7 C
29 1054.545 11/6 Major 7th M7 C#
30 1090.909 15/8 Augmented 7th A7 Cx
31 1127.273 23/12 Double-dim 8ve dd8 Dbb
32 1163.636 47/24 Diminished 8ve d8 Db
33 1200 2/1 Perfect Octave P8 D

## 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 81/80, 360/343, 525/512 [33 52 76 93]] (33c) +3.01 5.69 15.60
2.3.5.7.11 45/44, 81/80, 360/343, 525/512 [33 52 76 93 114]] (33c) +2.75 5.11 14.03
2.3.5.7.11.13 45/44, 65/64, 81/80, 360/343, 525/512 [33 52 76 93 114 122]] (33c) +2.47 4.71 12.92

### Rank-2 temperaments

Periods
per 8ve
Generator
(Reduced)
Cents
(Reduced)
Associated Ratio
(Reduced)
Temperament
1 2\33 72.727 21/20 Slurpee
1 4\33 145.455 12/11 Bohpier
1 7\33 254.545 8/7 Godzilla
1 8\33 290.909 25/21 Quasitemp
1 10\33 363.636 49/40 Interpental
1 14\33 509.091 4/3 Deeptone a.k.a tragicomical
Flattone (33c)
Dreamtone (33ceeff)
1 16\33 581.818 7/5 Tritonic
3 2\33 72.727 3/2-1\3edo Inflated
3 3\33 98.091 3/2-1\3edo August
3 4\33 145.455 7/4-2\3edo Triforce

## Scales

Brightest mode is listed except where noted.

• Deeptone[7], 5 5 5 4 5 5 4 (diatonic)
• 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[9], 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[9], 5 3 5 1 5 3 5 1 5 (*right-handed)
• Diasem[9], 5 1 5 3 5 1 5 3 5 (*left-handed)

## Music

### Modern renderings

Johann Sebastian Bach

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