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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|33}}
{{ED intro}}
 
== Theory ==
== Theory ==
=== Harmonics ===
=== Structural properties ===
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 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]].


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 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 in equal|33}}
 
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 contains an accurate approximation of the [[Bohlen–Pierce]] scale with 4\33 near [[13edt|1\13edt]].


=== 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 33: Line 40:
|-
|-
| 0
| 0
| 0¢
| 0
| [[1/1]]
| [[1/1]]
| 0
| 0
Line 45: Line 52:
| [[48/47]]
| [[48/47]]
| 36.448
| 36.448
| −0.085
| −0.085
| Augmented Unison
| Augmented Unison
| A1
| A1
Line 54: Line 61:
| [[24/23]]
| [[24/23]]
| 73.681
| 73.681
| −0.953
| −0.953
| Double-aug 1sn
| Double-aug 1sn
| AA1
| AA1
Line 63: Line 70:
| [[16/15]]
| [[16/15]]
| 111.731
| 111.731
| −2.640
| −2.640
| Diminished 2nd
| Diminished 2nd
| d2
| d2
Line 72: Line 79:
| [[12/11]]
| [[12/11]]
| 150.637
| 150.637
| −5.183
| −5.183
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 81: Line 88:
| [[10/9]]
| [[10/9]]
| 182.404
| 182.404
| −0.586
| −0.586
| Major 2nd
| Major 2nd
| M2
| M2
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| [[11/8]]
| [[11/8]]
| 551.318
| 551.318
| −5.863
| −5.863
| Augmented 4th
| Augmented 4th
| A4
| A4
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| [[7/5]]
| [[7/5]]
| 582.513
| 582.513
| −0.694
| −0.694
| Double-aug 4th
| Double-aug 4th
| AA4
| AA4
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| 690.909
| 690.909
| [[3/2]]
| [[3/2]]
| 701.9550
| 701.955
| −11.046
| −11.046
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 224: 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 234: Line 241:
| [[8/5]]
| [[8/5]]
| 813.686
| 813.686
| −13.686
| −13.686
| Double-dim 6th
| Double-dim 6th
| d6
| d6
Line 242: 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 252: Line 259:
| [[5/3]]
| [[5/3]]
| 884.359
| 884.359
| −11.631
| −11.631
| Major 6th
| Major 6th
| M6
| M6
Line 260: 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 279: Line 286:
| [[30/17]]
| [[30/17]]
| 983.313
| 983.313
| −1.495
| −1.495
| Diminished 7th
| Diminished 7th
| d7
| d7
Line 315: Line 322:
| [[23/12]]
| [[23/12]]
| 1126.319
| 1126.319
| −0.953
| −0.953
| Double-dim 8ve
| Double-dim 8ve
| dd8
| dd8
Line 340: 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]].
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</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 376: 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 471: 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)
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* 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 1,391: 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 1,402: 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)


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