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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|33}}
{{ED intro}}
 
== Theory ==
== Theory ==
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.
=== 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 {{nowrap|L {{=}} 4|s {{=}} 3}}. The 33c ({{val| 33 52 76 93 }}) and 33cd ({{val| 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{{c}} flat, and two stacked fifths fall only 0.6{{c}} 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. The 33cd val also tempers out [[49/48]], which along with the tempering of 81/80 means it supports [[godzilla]].
 
Besides the 33cd val, one may also consider the patent val. This val maps 5/4 and 7/4 much more accurately (though still somewhat questionable), but 6/5 and 7/6, and especially 10/9 and 9/7 are much more damaged. Notable commas this val tempers out include 128/125, 36/35, and 225/224, supporting [[august]].
 
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 [[3/1|3]], [[5/1|5]], or [[7/1|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. From this reasoning, 33edo's triple, [[99edo]], would be a very strong 7-limit system, and it indeed is.


=== Harmonics ===
Instead of the flat 19-step fifth you may use the 20-step sharp fifth, over 25{{c}} 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 {{nowrap|6\33 {{=}} 2\[[11edo|11]]}} interval of 218{{c}}. Together, these add up to {{nowrap|6\33 + 5\33 {{=}} 11\33 {{=}} 1\3}}, or 400{{c}}, the same major third as 12edo. We also have both a 327{{c}} minor third ({{nowrap|9\33 {{=}} 6\22 {{=}} 3\11}}), the same as that of [[22edo]], and a flatter 8\33 third of 291{{c}}, which if you like could also be called a flat 19th harmonic, but much more accurately a 13/11 sharp by 1.7{{c}} (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 is not especially good at representing all rational intervals in the [[7-limit]], but it does very well on the 7-limit [[k*N subgroups|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 [[Subgroup temperaments#Terrain|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 [[Mint_temperaments#Slurpee|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.
33edo contains an accurate approximation of the [[Bohlen–Pierce]] scale with 4\33 near [[13edt|1\13edt]].
{{Harmonics in equal|33}}


=== Structural properties ===
Other notable 33edo scales are [[diasem]] with {{nowrap|L:m:s {{=}} 5:3:1}} and [[5L 4s]] with {{nowrap|L:s {{=}} 5:2}}. This step ratio for 5L 4s is great for its semitone size of 72.7{{c}}.
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 {{nowrap|L {{=}} 4|s {{=}} 3}}. The 33c ({{val| 33 52 76 93 }}) and 33cd ({{val| 33 52 76 92 }}) mappings temper out [[81/80]] and can be used to represent [[1/2-comma meantone]], a [[Meantone family#Flattertone|"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.


Instead of the flat 19-step fifth you may use the 20-step sharp fifth, 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 {{nowrap|6\33 {{=}} 2\[[11edo|11]]}} interval of 218 cents. Together, these add up to {{nowrap|6\33 + 5\33 {{=}} 11\33 {{=}} 1\3}}, or 400 cents, the same major third as 12edo. We also have both a 327¢ minor third ({{nowrap|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 11, 13 and 19 harmonics (taking the generator as a 19/16) which are relatively well in tune.
=== Odd harmonics ===
{{Harmonics in equal|33}}


33edo contains an accurate approximation of the Bohlen-Pierce scale with 4\33 near 1\[[13edt]].
33edo is not especially good at representing all rational intervals in the [[7-limit]], but it does very well on the 7-limit [[k*N subgroups|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.


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


Line 35: Line 40:
|-
|-
| 0
| 0
| 0¢
| 0
| [[1/1]]
| [[1/1]]
| 0
| 0
Line 47: Line 52:
| [[48/47]]
| [[48/47]]
| 36.448
| 36.448
| −0.085
| −0.085
| Augmented Unison
| Augmented Unison
| A1
| A1
Line 56: Line 61:
| [[24/23]]
| [[24/23]]
| 73.681
| 73.681
| −0.953
| −0.953
| Double-aug 1sn
| Double-aug 1sn
| AA1
| AA1
Line 65: Line 70:
| [[16/15]]
| [[16/15]]
| 111.731
| 111.731
| −2.640
| −2.640
| Diminished 2nd
| Diminished 2nd
| d2
| d2
Line 74: Line 79:
| [[12/11]]
| [[12/11]]
| 150.637
| 150.637
| −5.183
| −5.183
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 83: Line 88:
| [[10/9]]
| [[10/9]]
| 182.404
| 182.404
| −0.586
| −0.586
| Major 2nd
| Major 2nd
| M2
| M2
Line 173: Line 178:
| [[11/8]]
| [[11/8]]
| 551.318
| 551.318
| −5.863
| −5.863
| Augmented 4th
| Augmented 4th
| A4
| A4
Line 182: Line 187:
| [[7/5]]
| [[7/5]]
| 582.513
| 582.513
| −0.694
| −0.694
| Double-aug 4th
| Double-aug 4th
| AA4
| AA4
Line 208: Line 213:
| 690.909
| 690.909
| [[3/2]]
| [[3/2]]
| 701.9550
| 701.955
| −11.046
| −11.046
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 226: Line 231:
| 763.636
| 763.636
| [[14/9]]
| [[14/9]]
| 764.9159
| 764.916
| −1.280
| −1.280
| Double-aug 5th
| Double-aug 5th
| AA5
| AA5
Line 236: Line 241:
| [[8/5]]
| [[8/5]]
| 813.686
| 813.686
| −13.686
| −13.686
| Double-dim 6th
| Double-dim 6th
| d6
| d6
Line 244: Line 249:
| 836.364
| 836.364
| [[13/8]]
| [[13/8]]
| 840.5276
| 840.528
| −4.164
| −4.164
| Minor 6th
| Minor 6th
| m6
| m6
Line 254: Line 259:
| [[5/3]]
| [[5/3]]
| 884.359
| 884.359
| −11.631
| −11.631
| Major 6th
| Major 6th
| M6
| M6
Line 262: Line 267:
| 909.091
| 909.091
| [[22/13]]
| [[22/13]]
| 910.7903
| 910.790
| −1.699
| −1.699
| Augmented 6th
| Augmented 6th
| A6
| A6
Line 281: Line 286:
| [[30/17]]
| [[30/17]]
| 983.313
| 983.313
| −1.495
| −1.495
| Diminished 7th
| Diminished 7th
| d7
| d7
Line 317: Line 322:
| [[23/12]]
| [[23/12]]
| 1126.319
| 1126.319
| −0.953
| −0.953
| Double-dim 8ve
| Double-dim 8ve
| dd8
| dd8
Line 342: Line 347:


== Notation ==
== 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.
{{sharpness-sharp1}}
=== Sagittal notation ===
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[23edo#Sagittal notation|23]] and [[28edo#Sagittal notation|28]].
This notation uses the same sagittal sequence as EDOs [[23edo#Sagittal notation|23]] and [[28edo#Sagittal notation|28]].
Line 354: Line 364:
</imagemap>
</imagemap>


== Nearby equal temperaments ==
== Approximation to JI ==
[[File:33edo.png|alt=33edo.png|966x199px|33edo.png]]
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals|32.87|apx=val|header=none|tag=none|title=15-odd-limit intervals by 33cd val mapping}}


== Regular temperament properties ==
== Regular temperament properties ==
Line 378: Line 389:
| 2.3.5
| 2.3.5
| 81/80, 1171875/1048576
| 81/80, 1171875/1048576
| {{mapping| 33 52 76 }} (33cd)
| {{mapping| 33 52 76 }} (33c)
| +5.59
| +5.59
| 4.13
| 4.13
Line 473: Line 484:
| 509.09<br>(98.09)
| 509.09<br>(98.09)
| 4/3<br>(16/15)
| 4/3<br>(16/15)
| [[August]] (33cd)
| [[August]] (33)
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
=== Uniform maps ===
{{Uniform map|min=32.8|max=33.2}}
 
== Octave stretch or compression ==
33edo is nearby to many other [[equal tuning]]s which can act as stretched or compressed versions of 33edo, improving some of its harmonics at the expense of others.
 
Useful options include:
* Stretched: [[ed5|76ed5]], [[ed7|92ed7]], [[52edt]], [[zpi|138zpi]]
* Compressed: [[ed7|93ed7]], [[ed5|77ed5]], [[equal tuning|115ed11]]
 
[[File:33edo.png|alt=33edo.png|966x199px|33edo.png]]


== Scales ==
== Scales ==
* {{main|List of MOS scales in {{ROOTPAGENAME}}}}
* {{main|List of MOS scales in {{ROOTPAGENAME}}}}
Brightest mode is listed except where noted.
* Approximate [[12afdo]], 4 3 4 3 3 2 3 2 3 2 2 2
* Deeptone[7], 5 5 5 4 5 5 4 (diatonic)
* August[12], 3 2 3 3 3 2 3 3 3 2 3 3
* 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 3 5 1 5 3 5 1 5 (*right-handed)
* Diasem, 5 1 5 3 5 1 5 3 5 (*left-handed)
* Diasem, 5 1 5 3 5 1 5 3 5 (*left-handed)
Line 491: Line 509:
* Diaslen (4sL), 2 5 1 5 1 5 2 5 1 5 1
* 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
* Diaslen (4sC), 1 5 2 5 1 5 1 5 2 5 1
* Elevenplus, 3 3 3 3 3 3 1 2 3 3 3 3 (approximated from [[22edo]])
* Flattertone[7], 5 5 4 5 5 5 4 (diatonic)
** Fun 5-tone subset of Flattertone[7], 9 5 5 4 10
* Flattertone[12], 4 1 4 1 4 1 4 4 1 4 1 4 (chromatic)
* Flattertone[19], 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 1 (enharmonic)
* Iranian Calendar, 5 4 4 4 4 4 4 4
* Semiquartal, 5 5 2 5 2 5 2 5 2
* Semiquartal[14], 3 2 3 2 2 3 2 2 3 2 2
* Blended slurpee{{idio}}, 3 1 2 2 3 3 5 3 3 2 2 4 ([[modmos]] of slurpee[12])
{{Todo|expand scales list}}


== Delta-rational harmony ==
== Delta-rational harmony ==
Line 1,393: Line 1,421:
=== Modern renderings ===
=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=IhR9oFt5zx4 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742&ndash;1749) &ndash; rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=IhR9oFt5zx4 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=ynPQPm_ekos "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742&ndash;1749) &ndash; rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=ynPQPm_ekos "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)


=== 21st century ===
=== 21st century ===
; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=swyP6tB78k0 ''groove 33edo''] (2023)
* [https://www.youtube.com/watch?v=swyP6tB78k0 ''groove 33edo''] (2023)
* [https://www.youtube.com/watch?v=GypR6x_Ih1I ''33edo jam''] (2025)
* [https://www.youtube.com/shorts/mkaaAJEyGFU ''33edo riff''] (2025)
* [https://www.youtube.com/shorts/Lf0CCX88w_w ''33edo improv''] (2025-10-27)
* [https://www.youtube.com/shorts/IzRhOdnNC64 ''33edo improv''] (2026-04-27)


; [[Peter Kosmorsky]]
; [[Peter Kosmorsky]]
Line 1,404: Line 1,436:


; [[Budjarn Lambeth]]
; [[Budjarn Lambeth]]
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) &ndash; Feb 2024''] (2024)
* [https://youtu.be/scCuGXnj5IY ''Enchanted Shopping Mall''] (2024)


; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=REkrbdesbLo ''Rising Canon on a Ground'', for Baroque Oboe, Bassoon, Violone] (2024) &ndash; ([https://www.youtube.com/watch?v=4fhcNPjFv14 for Organ])
* [https://www.youtube.com/watch?v=REkrbdesbLo ''Rising Canon on a Ground'', for Baroque Oboe, Bassoon, Violone] (2024) – ([https://www.youtube.com/watch?v=4fhcNPjFv14 for Organ])
* [https://www.youtube.com/watch?v=pkYN8SX6luY ''Lytel Twyelyghte Musicke (Little Twilight Music)'', for Brass and Timpani] (2024)
* [https://www.youtube.com/watch?v=pkYN8SX6luY ''Lytel Twyelyghte Musicke (Little Twilight Music)'', for Brass and Timpani] (2024)


Latest revision as of 03:38, 28 April 2026

← 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. The 33cd val also tempers out 49/48, which along with the tempering of 81/80 means it supports godzilla.

Besides the 33cd val, one may also consider the patent val. This val maps 5/4 and 7/4 much more accurately (though still somewhat questionable), but 6/5 and 7/6, and especially 10/9 and 9/7 are much more damaged. Notable commas this val tempers out include 128/125, 36/35, and 225/224, supporting august.

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. From this reasoning, 33edo's triple, 99edo, would be a very strong 7-limit system, and it indeed is.

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 (33)

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

Uniform maps

13-limit uniform maps between 32.8 and 33.2
Min. size Max. size Wart notation Map
32.5930 32.8089 33cdeeff ⟨33 52 76 92 113 121]
32.8089 32.8339 33cdff ⟨33 52 76 92 114 121]
32.8339 32.9468 33cd ⟨33 52 76 92 114 122]
32.9468 32.9492 33d ⟨33 52 77 92 114 122]
32.9492 33.0979 33 ⟨33 52 77 93 114 122]
33.0979 33.1042 33e ⟨33 52 77 93 115 122]
33.1042 33.1238 33ef ⟨33 52 77 93 115 123]
33.1238 33.3054 33bef ⟨33 53 77 93 115 123]

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.

Useful options include:

33edo.png

Scales

Main article: List of MOS scales in 33edo
  • Approximate 12afdo, 4 3 4 3 3 2 3 2 3 2 2 2
  • August[12], 3 2 3 3 3 2 3 3 3 2 3 3
  • 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
  • Elevenplus, 3 3 3 3 3 3 1 2 3 3 3 3 (approximated from 22edo)
  • Flattertone[7], 5 5 4 5 5 5 4 (diatonic)
    • Fun 5-tone subset of Flattertone[7], 9 5 5 4 10
  • Flattertone[12], 4 1 4 1 4 1 4 4 1 4 1 4 (chromatic)
  • Flattertone[19], 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 1 (enharmonic)
  • Iranian Calendar, 5 4 4 4 4 4 4 4
  • Semiquartal, 5 5 2 5 2 5 2 5 2
  • Semiquartal[14], 3 2 3 2 2 3 2 2 3 2 2
  • Blended slurpee[idiosyncratic term], 3 1 2 2 3 3 5 3 3 2 2 4 (modmos of slurpee[12])

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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