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== Theory ==
== Theory ==
=== Structural properties ===
=== 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 [[Meantone family#Flattertone|"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.
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.


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.
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.
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| 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 ==
== 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.
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]]
[[File:33edo.png|alt=33edo.png|966x199px|33edo.png]]
What follows is a comparison of stretched- and compressed-octave 33edo tunings.
; [[ed5|76ed5]]
* 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 ==
* {{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
** 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 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 ==
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* [https://www.youtube.com/watch?v=GypR6x_Ih1I ''33edo jam''] (2025)
* [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/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 1,481: Line 1,436:


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


; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
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