# 33edo

← 32edo | 33edo | 34edo → |

**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 2^{1/33}, 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

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 | 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 |

## Nearby equal temperaments

## 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

- "Contrapunctus 4" from
*The Art of Fugue*, BWV 1080 (1742–1749) – rendered by Claudi Meneghin (2024) - "Contrapunctus 11" from
*The Art of Fugue*, BWV 1080 (1742–1749) – rendered by Claudi Meneghin (2024)

### 21st century

*groove 33edo*(2023)

*33EDO Track in Dreamtone(7)*(2024)

*Rising Canon on a Ground*, for Baroque Oboe, Bassoon, Violone (2024) – (for Organ)*Lytel Twyelyghte Musicke (Little Twilight Music)*, for Brass and Timpani (2024)

*Mysteries of Thirty-Three*(2024)