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→21st century: Add Bryan Deister's ''33edo riff'' (2025)
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-28 17:02:35 UTC</tt>.<br>
: The original revision id was <tt>243299241</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The //33 equal division// divides the [[octave]] into 33 equal parts of 36.3636 [[cent]]s each. It 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. 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.


While relatively uncommon, 33edo is actually quite an interesting system. As a multiple of [[11edo]], it approximates the 7th and 11th harmonics via Andrew Heathwaite's 4L+3s Orgone modes (see [[26edo]]). 33edo also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having an [[3L 7s|3L+7s]] of L=4 s=3. It tunes the perfect fifth about 11 cents flat, to 19\33, so that two fifths down an octave, 5\33, the approximation of 9/8, is actually half a cent flat from 10/9. The &lt;33 52 76| or 33c val tempers out 81/80 and so leads to a very flat meantone tuning where the major tone is approximately 10/9 in size. Leaving the scale be would result in a flat-tone [[5L 2s|5L+2s]] of L=5 s=4
== 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&nbsp;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&nbsp;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 290 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. Another talent it has is that 7/5 is tuned quite accurately by 16\33, and we may put two 8\11 versions of 13/11 together to produce the [[cuthbert triad]]. 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.  
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.


0: 00.000 1/1
33edo contains an accurate approximation of the [[Bohlen–Pierce]] scale with 4\33 near [[13edt|1\13edt]].
1: 36.364 cents
 
2: 72.727 cents
Other notable 33edo scales are [[diasem]] with {{nowrap|L:m:s {{=}} 5:3:1}} and [[5L&nbsp;4s]] with {{nowrap|L:s {{=}} 5:2}}. This step ratio for 5L&nbsp;4s is great for its semitone size of 72.7{{c}}.
3: 109.091 17/16
 
4: 145.455 cents
=== Odd harmonics ===
5: 181.818 10/9
{{Harmonics in equal|33}}
6: 218.182 8/7 9/8
 
7: 254.545 37/32
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.
8: 290.909 19/16
 
9: 327.273 6/5
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.
10: 363.636 16/13
 
11: 400.000 4/3
=== Miscellany ===
12: 436.364 9/7
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&nbsp;7s]] with the step ratio of 5:4.
13: 472.727 21/16
 
14: 509.091 4/3
== Intervals ==
15: 545.455 11/8
{| class="wikitable center-all"
16: 581.818 7/5
|-
17: 618.182 23/16
! rowspan="2" |Step #
18: 654.545 cents
! ET
19: 690.909 3/2
! colspan="2" | Just
20: 727.273 cents
! rowspan="2" | Difference<br>(ET minus Just)
21: 763.636 cents
! rowspan="2" colspan="3" | Extended Pythagorean notation
22: 800.000 cents
|-
23: 836.364 13/8
! Cents
24: 872.727 cents
! Interval
25: 909.091 cents
! Cents
26: 945.455 7/4
|-
27: 981.818 7/4
| 0
28: 1018.182 9/5
| 0
29: 1054.545 cents
| [[1/1]]
30: 1090.909 15/8
| 0
31: 1127.273 cents
| 0
32: 1163.636 cents
| Perfect Unison
33: 1200.000 cents</pre></div>
| P1
<h4>Original HTML content:</h4>
| D
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;33edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;33 equal division&lt;/em&gt; divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 33 equal parts of 36.3636 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. It is not especially good at representing all rational intervals in the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt;, but it does very well on the 7-limit &lt;a class="wiki_link" href="/k%2AN%20subgroups"&gt;3*33 subgroup&lt;/a&gt; 2.27.15.21. On this subgroup it tunes things to the same tuning as &lt;a class="wiki_link" href="/99edo"&gt;99edo&lt;/a&gt;, 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.&lt;br /&gt;
|-
&lt;br /&gt;
| 1
While relatively uncommon, 33edo is actually quite an interesting system. As a multiple of &lt;a class="wiki_link" href="/11edo"&gt;11edo&lt;/a&gt;, it approximates the 7th and 11th harmonics via Andrew Heathwaite's 4L+3s Orgone modes (see &lt;a class="wiki_link" href="/26edo"&gt;26edo&lt;/a&gt;). 33edo also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having an &lt;a class="wiki_link" href="/3L%207s"&gt;3L+7s&lt;/a&gt; of L=4 s=3. It tunes the perfect fifth about 11 cents flat, to 19\33, so that two fifths down an octave, 5\33, the approximation of 9/8, is actually half a cent flat from 10/9. The &amp;lt;33 52 76| or 33c val tempers out 81/80 and so leads to a very flat meantone tuning where the major tone is approximately 10/9 in size. Leaving the scale be would result in a flat-tone &lt;a class="wiki_link" href="/5L%202s"&gt;5L+2s&lt;/a&gt; of L=5 s=4&lt;br /&gt;
| 36.364
&lt;br /&gt;
| [[48/47]]
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 &lt;a class="wiki_link" href="/11edo"&gt;11edo&lt;/a&gt; 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 &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; minor third, and a flatter 8\33 third of 290 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. Another talent it has is that 7/5 is tuned quite accurately by 16\33, and we may put two 8\11 versions of 13/11 together to produce the &lt;a class="wiki_link" href="/cuthbert%20triad"&gt;cuthbert triad&lt;/a&gt;. 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. &lt;br /&gt;
| 36.448
&lt;br /&gt;
| −0.085
0: 00.000 1/1&lt;br /&gt;
| Augmented Unison
1: 36.364 cents&lt;br /&gt;
| A1
2: 72.727 cents&lt;br /&gt;
| D#
3: 109.091 17/16&lt;br /&gt;
|-
4: 145.455 cents&lt;br /&gt;
| 2
5: 181.818 10/9&lt;br /&gt;
| 72.727
6: 218.182 8/7 9/8&lt;br /&gt;
| [[24/23]]
7: 254.545 37/32&lt;br /&gt;
| 73.681
8: 290.909 19/16&lt;br /&gt;
| −0.953
9: 327.273 6/5&lt;br /&gt;
| Double-aug 1sn
10: 363.636 16/13&lt;br /&gt;
| AA1
11: 400.000 4/3&lt;br /&gt;
| Dx
12: 436.364 9/7&lt;br /&gt;
|-
13: 472.727 21/16&lt;br /&gt;
| 3
14: 509.091 4/3&lt;br /&gt;
| 109.091
15: 545.455 11/8&lt;br /&gt;
| [[16/15]]
16: 581.818 7/5&lt;br /&gt;
| 111.731
17: 618.182 23/16&lt;br /&gt;
| −2.640
18: 654.545 cents&lt;br /&gt;
| Diminished 2nd
19: 690.909 3/2&lt;br /&gt;
| d2
20: 727.273 cents&lt;br /&gt;
| Ebb
21: 763.636 cents&lt;br /&gt;
|-
22: 800.000 cents&lt;br /&gt;
| 4
23: 836.364 13/8&lt;br /&gt;
| 145.455
24: 872.727 cents&lt;br /&gt;
| [[12/11]]
25: 909.091 cents&lt;br /&gt;
| 150.637
26: 945.455 7/4&lt;br /&gt;
| −5.183
27: 981.818 7/4&lt;br /&gt;
| Minor 2nd
28: 1018.182 9/5&lt;br /&gt;
| m2
29: 1054.545 cents&lt;br /&gt;
| Eb
30: 1090.909 15/8&lt;br /&gt;
|-
31: 1127.273 cents&lt;br /&gt;
| 5
32: 1163.636 cents&lt;br /&gt;
| 181.818
33: 1200.000 cents&lt;/body&gt;&lt;/html&gt;</pre></div>
| [[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.
 
{{sharpness-sharp1}}
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[23edo#Sagittal notation|23]] and [[28edo#Sagittal notation|28]].
 
<imagemap>
File:33-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 399 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
default [[File:33-EDO_Sagittal.svg]]
</imagemap>
 
== Approximation to JI ==
{{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}}
 
== Nearby equal temperaments ==
[[File:33edo.png|alt=33edo.png|966x199px|33edo.png]]
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -52 33 }}
| {{mapping| 33 52 }}
| +3.48
| 3.49
| 9.59
|-
| 2.3.5
| 81/80, 1171875/1048576
| {{mapping| 33 52 76 }} (33c)
| +5.59
| 4.13
| 11.29
|-
| 2.3.5.7
| 49/48, 81/80, 1875/1792
| {{mapping| 33 52 76 92 }} (33cd)
| +6.29
| 3.77
| 10.31
|-
| 2.3.5.7.11
| 45/44, 49/48, 81/80, 1375/1344
| {{mapping| 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
| {{mapping| 33 52 76 92 114 122 }} (33cd)
| +4.65
| 3.84
| 10.52
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>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)<br>[[Deeptone]] a.k.a. tragicomical (33)
|-
| 1
| 16\33
| 581.82
| 7/5
| [[Tritonic]] (33)
|-
| 3
| 7\33<br>(4\33)
| 254.55<br>(145.45)
| 8/7<br>(12/11)
| [[Triforce]] (33d)
|-
| 3
| 13\33<br>(2\33)
| 472.73<br>(72.73)
| 4/3<br>(25/24)
| [[Inflated]] (33bcddd)
|-
| 3
| 14\33<br>(3\33)
| 509.09<br>(98.09)
| 4/3<br>(16/15)
| [[August]] (33cd)
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Scales ==
* {{main|List of MOS scales in {{ROOTPAGENAME}}}}
Brightest mode is listed except where noted.
* Deeptone[7], 5 5 5 4 5 5 4 (diatonic)
** 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 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
 
== 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 ===
{| class="mw-collapsible mw-collapsed class="wikitable sortable"
|-
! 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 ===
{| class="mw-collapsible mw-collapsed class="wikitable sortable"
|-
! 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 ===
; {{W|Johann Sebastian Bach}}
* [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] (1742–1749) – rendered by Claudi Meneghin (2024)
 
=== 21st century ===
; [[Bryan Deister]]
* [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)
 
; [[Peter Kosmorsky]]
* [https://www.youtube.com/watch?v=SXgUFxyuLZo ''Deluge'']
 
; [[Budjarn Lambeth]]
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) – Feb 2024''] (2024)
 
; [[Claudi Meneghin]]
* [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)
 
; [[Relyt R]]
* from ''Xuixo'' (2023)
** "Nongenerate" [https://relytr.bandcamp.com/track/nondegenerate-33-edo Bandcamp] | [https://open.spotify.com/track/3e2WbgFlAYC4BccPGOWHMo Spotify]
** "Kolmekymmentäkolme" [https://relytr.bandcamp.com/track/kolme-kymment-kolme-33-edo Bandcamp] | [https://open.spotify.com/track/4fx1yQ1RQtEu8EYhNUtN79 Spotify]
 
; [[Chris Vaisvil]]
* [http://chrisvaisvil.com/5-5-1-mode-of-33-equal-with-video/ 5 5 1 mode of 33 equal (with video)] [http://micro.soonlabel.com/33edo/20130827_551of33.mp3 play]
 
; [[Xeno*n*]]
* [https://www.youtube.com/watch?v=EPB1Rzjwguk ''Mysteries of Thirty-Three''] (2024)
 
[[Category:Listen]]
[[Category:Meantone]]
[[Category:Subgroup temperaments]]
Retrieved from "https://en.xen.wiki/w/33edo"