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{{EDO intro|33}}
{{EDO intro|33}}
== 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, making notation unwieldy in distant keys.
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.


=== Harmonics ===
=== Harmonics ===
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.
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.


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.
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.
{{Harmonics in equal|33}}
{{Harmonics in equal|33}}


=== 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|3L 7s]] with {{nowrap|L {{=}} 4|s {{=}} 3}}. The 33c mapping (which has val {{val| 33 52 76 }}) tempers 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 cents flat, and two stacked fifths fall only 0.6 cents flat of 10/9. Leaving the scale be would result in the standard diatonic scale ([[5L 2s]]) having minor seconds of four steps and whole tones of five steps. This also results in common practice minor and major chords becoming more supraminor and submajor in character, making everything sound almost neutral in quality.
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\33 fifth you may use the sharp fifth of 20\33, over 25 cents sharp. Two of these lead to a 9/8 of 7\33, which is about 22/19 in size and may be counted as a small third. Between the flat 5\33 version of 9/8 and the sharp 7\33 version there is, of course, a 6\33 = 2\11 [[11edo]] interval of 218 cents. Now 6\33 + 5\33 = 11\33 = 1\3 of an octave, or 400 cents, the same major third as 12edo. Also, we have both a 327 minor third from 9\33 = 3\11, the same as the [[22edo]] minor third, and a flatter 8\33 third of 291 cents, which if you like could also be called a flat 19th harmonic, but much more accurately a 13/11 sharp by 1.7 cents (if you use the patent val it is an extremely inaccurate 6/5). Another talent it has is that 7/5 is tuned quite accurately by 16\33, and we may put two 8\33 versions of 13/11 together to produce the [[cuthbert triad]]. The 8\33 generator, with MOS of size 5, 9 and 13, gives plenty of scope for these, as well as the 11, 13 and 19 harmonics (taking the generator as a 19/16) which are relatively well in tune.
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.


33edo contains an accurate approximation of the Bohlen-Pierce scale with 4\33 near 1\[[13edt]].
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¢.
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¢.


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.


== Intervals ==
== Intervals ==
Line 27: Line 27:
! ET
! ET
! colspan="2" | Just
! colspan="2" | Just
! rowspan="2" | Difference <br> (ET minus Just)
! rowspan="2" | Difference<br>(ET minus Just)
! rowspan="2" colspan="3" |Extended Pythagorean Notation
! rowspan="2" colspan="3" | Extended Pythagorean notation
|-
|-
! Cents
! Cents
Line 342: Line 342:


== Notation ==
== Notation ==
 
=== 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 479: Line 478:


== Scales ==
== Scales ==
* {{main|List of MOS scales in {{ROOTPAGENAME}}}}
Brightest mode is listed except where noted.
Brightest mode is listed except where noted.
* Deeptone[7], 5 5 5 4 5 5 4 (diatonic)
* Deeptone[7], 5 5 5 4 5 5 4 (diatonic)
Line 494: Line 494:
== Delta-rational harmony ==
== Delta-rational harmony ==
The tables below show chords that approximate 3-integer-limit [[delta-rational]] chords with least-squares error less than 0.001.
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 ===
=== Fully delta-rational triads ===
{| class="mw-collapsible mw-collapsed class="wikitable sortable"
{| class="mw-collapsible mw-collapsed class="wikitable sortable"
!|Steps
!|Delta signature
!|Least-squares error
|-
|-
|0,1,2
! Steps
! Delta signature
! Least-squares error
|-
| 0,1,2
| +1+1
| +1+1
|0.00021
| 0.00021
|-
|-
|0,1,3
| 0,1,3
| +1+2
| +1+2
|0.00048
| 0.00048
|-
|-
|0,1,4
| 0,1,4
| +1+3
| +1+3
|0.00078
| 0.00078
|-
|-
|0,2,3
| 0,2,3
| +2+1
| +2+1
|0.00039
| 0.00039
|-
|-
|0,2,4
| 0,2,4
| +1+1
| +1+1
|0.00087
| 0.00087
|-
|-
|0,3,4
| 0,3,4
| +3+1
| +3+1
|0.00056
| 0.00056
|-
|-
|0,3,11
| 0,3,11
| +1+3
| +1+3
|0.00007
| 0.00007
|-
|-
|0,5,8
| 0,5,8
| +3+2
| +3+2
|0.00084
| 0.00084
|-
|-
|0,8,18
| 0,8,18
| +2+3
| +2+3
|0.00082
| 0.00082
|-
|-
|0,9,20
| 0,9,20
| +2+3
| +2+3
|0.00076
| 0.00076
|-
|-
|0,12,17
| 0,12,17
| +2+1
| +2+1
|0.00048
| 0.00048
|-
|-
|0,13,20
| 0,13,20
| +3+2
| +3+2
|0.00063
| 0.00063
|-
|-
|0,15,21
| 0,15,21
| +2+1
| +2+1
|0.00063
| 0.00063
|-
|-
|0,16,28
| 0,16,28
| +1+1
| +1+1
|0.00082
| 0.00082
|-
|-
|0,18,25
| 0,18,25
| +2+1
| +2+1
|0.00081
| 0.00081
|-
|-
|0,18,31
| 0,18,31
| +1+1
| +1+1
|0.00058
| 0.00058
|-
|-
|0,19,24
| 0,19,24
| +3+1
| +3+1
|0.00095
| 0.00095
|}
|}


=== Partially delta-rational tetrads ===
=== Partially delta-rational tetrads ===
{| class="mw-collapsible mw-collapsed class="wikitable sortable"
{| class="mw-collapsible mw-collapsed class="wikitable sortable"
!|Steps
!|Delta signature
!|Least-squares error
|-
|-
|0,1,2,3
! Steps
! Delta signature
! Least-squares error
|-
| 0,1,2,3
| +1+?+1
| +1+?+1
|0.00053
| 0.00053
|-
|-
|0,1,2,4
| 0,1,2,4
| +1+?+2
| +1+?+2
|0.00094
| 0.00094
|-
|-
|0,1,3,4
| 0,1,3,4
| +1+?+1
| +1+?+1
|0.00080
| 0.00080
|-
|-
|0,1,17,18
| 0,1,17,18
| +2+?+3
| +2+?+3
|0.00073
| 0.00073
|-
|-
|0,1,17,19
| 0,1,17,19
| +1+?+3
| +1+?+3
|0.00071
| 0.00071
|-
|-
|0,1,18,19
| 0,1,18,19
| +2+?+3
| +2+?+3
|0.00042
| 0.00042
|-
|-
|0,1,18,20
| 0,1,18,20
| +1+?+3
| +1+?+3
|0.00032
| 0.00032
|-
|-
|0,1,19,20
| 0,1,19,20
| +2+?+3
| +2+?+3
|0.00010
| 0.00010
|-
|-
|0,1,19,21
| 0,1,19,21
| +1+?+3
| +1+?+3
|0.00008
| 0.00008
|-
|-
|0,1,20,21
| 0,1,20,21
| +2+?+3
| +2+?+3
|0.00023
| 0.00023
|-
|-
|0,1,20,22
| 0,1,20,22
| +1+?+3
| +1+?+3
|0.00049
| 0.00049
|-
|-
|0,1,21,22
| 0,1,21,22
| +2+?+3
| +2+?+3
|0.00056
| 0.00056
|-
|-
|0,1,21,23
| 0,1,21,23
| +1+?+3
| +1+?+3
|0.00091
| 0.00091
|-
|-
|0,1,22,23
| 0,1,22,23
| +2+?+3
| +2+?+3
|0.00090
| 0.00090
|-
|-
|0,1,31,32
| 0,1,31,32
| +1+?+2
| +1+?+2
|0.00071
| 0.00071
|-
|-
|0,2,3,4
| 0,2,3,4
| +2+?+1
| +2+?+1
|0.00077
| 0.00077
|-
|-
|0,2,6,11
| 0,2,6,11
| +1+?+3
| +1+?+3
|0.00094
| 0.00094
|-
|-
|0,2,7,12
| 0,2,7,12
| +1+?+3
| +1+?+3
|0.00013
| 0.00013
|-
|-
|0,2,8,13
| 0,2,8,13
| +1+?+3
| +1+?+3
|0.00069
| 0.00069
|-
|-
|0,2,12,13
| 0,2,12,13
| +3+?+2
| +3+?+2
|0.00083
| 0.00083
|-
|-
|0,2,12,15
| 0,2,12,15
| +1+?+2
| +1+?+2
|0.00087
| 0.00087
|-
|-
|0,2,13,14
| 0,2,13,14
| +3+?+2
| +3+?+2
|0.00045
| 0.00045
|-
|-
|0,2,13,16
| 0,2,13,16
| +1+?+2
| +1+?+2
|0.00014
| 0.00014
|-
|-
|0,2,14,15
| 0,2,14,15
| +3+?+2
| +3+?+2
|0.00008
| 0.00008
|-
|-
|0,2,14,17
| 0,2,14,17
| +1+?+2
| +1+?+2
|0.00060
| 0.00060
|-
|-
|0,2,15,16
| 0,2,15,16
| +3+?+2
| +3+?+2
|0.00031
| 0.00031
|-
|-
|0,2,16,17
| 0,2,16,17
| +3+?+2
| +3+?+2
|0.00071
| 0.00071
|-
|-
|0,2,18,20
| 0,2,18,20
| +2+?+3
| +2+?+3
|0.00084
| 0.00084
|-
|-
|0,2,18,22
| 0,2,18,22
| +1+?+3
| +1+?+3
|0.00024
| 0.00024
|-
|-
|0,2,19,21
| 0,2,19,21
| +2+?+3
| +2+?+3
|0.00020
| 0.00020
|-
|-
|0,2,19,23
| 0,2,19,23
| +1+?+3
| +1+?+3
|0.00058
| 0.00058
|-
|-
|0,2,20,22
| 0,2,20,22
| +2+?+3
| +2+?+3
|0.00046
| 0.00046
|-
|-
|0,3,4,5
| 0,3,4,5
| +3+?+1
| +3+?+1
|0.00097
| 0.00097
|-
|-
|0,3,5,9
| 0,3,5,9
| +2+?+3
| +2+?+3
|0.00010
| 0.00010
|-
|-
|0,3,6,10
| 0,3,6,10
| +2+?+3
| +2+?+3
|0.00090
| 0.00090
|-
|-
|0,3,7,12
| 0,3,7,12
| +1+?+2
| +1+?+2
|0.00074
| 0.00074
|-
|-
|0,3,8,13
| 0,3,8,13
| +1+?+2
| +1+?+2
|0.00037
| 0.00037
|-
|-
|0,3,10,17
| 0,3,10,17
| +1+?+3
| +1+?+3
|0.00009
| 0.00009
|-
|-
|0,3,17,23
| 0,3,17,23
| +1+?+3
| +1+?+3
|0.00096
| 0.00096
|-
|-
|0,3,18,22
| 0,3,18,22
| +1+?+2
| +1+?+2
|0.00088
| 0.00088
|-
|-
|0,3,18,24
| 0,3,18,24
| +1+?+3
| +1+?+3
|0.00027
| 0.00027
|-
|-
|0,3,19,20
| 0,3,19,20
| +2+?+1
| +2+?+1
|0.00059
| 0.00059
|-
|-
|0,3,19,21
| 0,3,19,21
| +1+?+1
| +1+?+1
|0.00063
| 0.00063
|-
|-
|0,3,19,22
| 0,3,19,22
| +2+?+3
| +2+?+3
|0.00030
| 0.00030
|-
|-
|0,3,19,23
| 0,3,19,23
| +1+?+2
| +1+?+2
|0.00023
| 0.00023
|-
|-
|0,3,20,21
| 0,3,20,21
| +2+?+1
| +2+?+1
|0.00014
| 0.00014
|-
|-
|0,3,20,22
| 0,3,20,22
| +1+?+1
| +1+?+1
|0.00015
| 0.00015
|-
|-
|0,3,20,23
| 0,3,20,23
| +2+?+3
| +2+?+3
|0.00070
| 0.00070
|-
|-
|0,3,21,22
| 0,3,21,22
| +2+?+1
| +2+?+1
|0.00032
| 0.00032
|-
|-
|0,3,21,23
| 0,3,21,23
| +1+?+1
| +1+?+1
|0.00095
| 0.00095
|-
|-
|0,3,22,23
| 0,3,22,23
| +2+?+1
| +2+?+1
|0.00078
| 0.00078
|-
|-
|0,3,27,32
| 0,3,27,32
| +1+?+3
| +1+?+3
|0.00004
| 0.00004
|-
|-
|0,4,5,12
| 0,4,5,12
| +1+?+2
| +1+?+2
|0.00026
| 0.00026
|-
|-
|0,4,6,16
| 0,4,6,16
| +1+?+3
| +1+?+3
|0.00066
| 0.00066
|-
|-
|0,4,8,13
| 0,4,8,13
| +2+?+3
| +2+?+3
|0.00023
| 0.00023
|-
|-
|0,4,11,20
| 0,4,11,20
| +1+?+3
| +1+?+3
|0.00023
| 0.00023
|-
|-
|0,4,13,14
| 0,4,13,14
| +3+?+1
| +3+?+1
|0.00091
| 0.00091
|-
|-
|0,4,13,19
| 0,4,13,19
| +1+?+2
| +1+?+2
|0.00048
| 0.00048
|-
|-
|0,4,14,15
| 0,4,14,15
| +3+?+1
| +3+?+1
|0.00050
| 0.00050
|-
|-
|0,4,14,16
| 0,4,14,16
| +3+?+2
| +3+?+2
|0.00055
| 0.00055
|-
|-
|0,4,14,17
| 0,4,14,17
| +1+?+1
| +1+?+1
|0.00021
| 0.00021
|-
|-
|0,4,15,16
| 0,4,15,16
| +3+?+1
| +3+?+1
|0.00009
| 0.00009
|-
|-
|0,4,15,17
| 0,4,15,17
| +3+?+2
| +3+?+2
|0.00023
| 0.00023
|-
|-
|0,4,15,18
| 0,4,15,18
| +1+?+1
| +1+?+1
|0.00085
| 0.00085
|-
|-
|0,4,16,17
| 0,4,16,17
| +3+?+1
| +3+?+1
|0.00034
| 0.00034
|-
|-
|0,4,17,18
| 0,4,17,18
| +3+?+1
| +3+?+1
|0.00077
| 0.00077
|-
|-
|0,4,17,25
| 0,4,17,25
| +1+?+3
| +1+?+3
|0.00043
| 0.00043
|-
|-
|0,4,19,23
| 0,4,19,23
| +2+?+3
| +2+?+3
|0.00041
| 0.00041
|-
|-
|0,4,20,24
| 0,4,20,24
| +2+?+3
| +2+?+3
|0.00094
| 0.00094
|-
|-
|0,4,22,27
| 0,4,22,27
| +1+?+2
| +1+?+2
|0.00020
| 0.00020
|-
|-
|0,4,24,31
| 0,4,24,31
| +1+?+3
| +1+?+3
|0.00022
| 0.00022
|-
|-
|0,5,6,9
| 0,5,6,9
| +3+?+2
| +3+?+2
|0.00003
| 0.00003
|-
|-
|0,5,7,10
| 0,5,7,10
| +3+?+2
| +3+?+2
|0.00097
| 0.00097
|-
|-
|0,5,7,19
| 0,5,7,19
| +1+?+3
| +1+?+3
|0.00004
| 0.00004
|-
|-
|0,5,9,17
| 0,5,9,17
| +1+?+2
| +1+?+2
|0.00017
| 0.00017
|-
|-
|0,5,10,16
| 0,5,10,16
| +2+?+3
| +2+?+3
|0.00019
| 0.00019
|-
|-
|0,5,11,13
| 0,5,11,13
| +2+?+1
| +2+?+1
|0.00087
| 0.00087
|-
|-
|0,5,11,15
| 0,5,11,15
| +1+?+1
| +1+?+1
|0.00018
| 0.00018
|-
|-
|0,5,12,14
| 0,5,12,14
| +2+?+1
| +2+?+1
|0.00011
| 0.00011
|-
|-
|0,5,12,23
| 0,5,12,23
| +1+?+3
| +1+?+3
|0.00067
| 0.00067
|-
|-
|0,5,13,15
| 0,5,13,15
| +2+?+1
| +2+?+1
|0.00067
| 0.00067
|-
|-
|0,5,16,23
| 0,5,16,23
| +1+?+2
| +1+?+2
|0.00008
| 0.00008
|-
|-
|0,5,17,27
| 0,5,17,27
| +1+?+3
| +1+?+3
|0.00055
| 0.00055
|-
|-
|0,5,19,24
| 0,5,19,24
| +2+?+3
| +2+?+3
|0.00051
| 0.00051
|-
|-
|0,5,22,31
| 0,5,22,31
| +1+?+3
| +1+?+3
|0.00057
| 0.00057
|-
|-
|0,5,24,30
| 0,5,24,30
| +1+?+2
| +1+?+2
|0.00036
| 0.00036
|-
|-
|0,5,25,26
| 0,5,25,26
| +3+?+1
| +3+?+1
|0.00071
| 0.00071
|-
|-
|0,5,25,27
| 0,5,25,27
| +3+?+2
| +3+?+2
|0.00082
| 0.00082
|-
|-
|0,5,25,28
| 0,5,25,28
| +1+?+1
| +1+?+1
|0.00045
| 0.00045
|-
|-
|0,5,26,27
| 0,5,26,27
| +3+?+1
| +3+?+1
|0.00018
| 0.00018
|-
|-
|0,5,26,28
| 0,5,26,28
| +3+?+2
| +3+?+2
|0.00016
| 0.00016
|-
|-
|0,5,26,29
| 0,5,26,29
| +1+?+1
| +1+?+1
|0.00090
| 0.00090
|-
|-
|0,5,27,28
| 0,5,27,28
| +3+?+1
| +3+?+1
|0.00035
| 0.00035
|-
|-
|0,5,28,29
| 0,5,28,29
| +3+?+1
| +3+?+1
|0.00090
| 0.00090
|-
|-
|0,6,7,17
| 0,6,7,17
| +1+?+2
| +1+?+2
|0.00087
| 0.00087
|-
|-
|0,6,8,22
| 0,6,8,22
| +1+?+3
| +1+?+3
|0.00045
| 0.00045
|-
|-
|0,6,9,14
| 0,6,9,14
| +1+?+1
| +1+?+1
|0.00031
| 0.00031
|-
|-
|0,6,11,18
| 0,6,11,18
| +2+?+3
| +2+?+3
|0.00093
| 0.00093
|-
|-
|0,6,12,21
| 0,6,12,21
| +1+?+2
| +1+?+2
|0.00036
| 0.00036
|-
|-
|0,6,12,25
| 0,6,12,25
| +1+?+3
| +1+?+3
|0.00032
| 0.00032
|-
|-
|0,6,15,18
| 0,6,15,18
| +3+?+2
| +3+?+2
|0.00026
| 0.00026
|-
|-
|0,6,16,19
| 0,6,16,19
| +3+?+2
| +3+?+2
|0.00095
| 0.00095
|-
|-
|0,6,16,28
| 0,6,16,28
| +1+?+3
| +1+?+3
|0.00053
| 0.00053
|-
|-
|0,6,18,26
| 0,6,18,26
| +1+?+2
| +1+?+2
|0.00064
| 0.00064
|-
|-
|0,6,19,25
| 0,6,19,25
| +2+?+3
| +2+?+3
|0.00062
| 0.00062
|-
|-
|0,6,20,24
| 0,6,20,24
| +1+?+1
| +1+?+1
|0.00052
| 0.00052
|-
|-
|0,6,21,23
| 0,6,21,23
| +2+?+1
| +2+?+1
|0.00031
| 0.00031
|-
|-
|0,6,21,32
| 0,6,21,32
| +1+?+3
| +1+?+3
|0.00033
| 0.00033
|-
|-
|0,6,22,24
| 0,6,22,24
| +2+?+1
| +2+?+1
|0.00063
| 0.00063
|-
|-
|0,6,25,32
| 0,6,25,32
| +1+?+2
| +1+?+2
|0.00034
| 0.00034
|-
|-
|0,7,8,14
| 0,7,8,14
| +1+?+1
| +1+?+1
|0.00029
| 0.00029
|-
|-
|0,7,8,24
| 0,7,8,24
| +1+?+3
| +1+?+3
|0.00080
| 0.00080
|-
|-
|0,7,9,11
| 0,7,9,11
| +3+?+1
| +3+?+1
|0.00066
| 0.00066
|-
|-
|0,7,9,12
| 0,7,9,12
| +2+?+1
| +2+?+1
|0.00041
| 0.00041
|-
|-
|0,7,9,13
| 0,7,9,13
| +3+?+2
| +3+?+2
|0.00019
| 0.00019
|-
|-
|0,7,10,12
| 0,7,10,12
| +3+?+1
| +3+?+1
|0.00009
| 0.00009
|-
|-
|0,7,10,13
| 0,7,10,13
| +2+?+1
| +2+?+1
|0.00070
| 0.00070
|-
|-
|0,7,11,13
| 0,7,11,13
| +3+?+1
| +3+?+1
|0.00087
| 0.00087
|-
|-
|0,7,12,27
| 0,7,12,27
| +1+?+3
| +1+?+3
|0.00041
| 0.00041
|-
|-
|0,7,16,30
| 0,7,16,30
| +1+?+3
| +1+?+3
|0.00098
| 0.00098
|-
|-
|0,7,17,22
| 0,7,17,22
| +1+?+1
| +1+?+1
|0.00008
| 0.00008
|-
|-
|0,7,19,26
| 0,7,19,26
| +2+?+3
| +2+?+3
|0.00073
| 0.00073
|-
|-
|0,7,20,29
| 0,7,20,29
| +1+?+2
| +1+?+2
|0.00002
| 0.00002
|-
|-
|0,7,23,26
| 0,7,23,26
| +3+?+2
| +3+?+2
|0.00010
| 0.00010
|-
|-
|0,7,28,32
| 0,7,28,32
| +1+?+1
| +1+?+1
|0.00033
| 0.00033
|-
|-
|0,7,29,31
| 0,7,29,31
| +2+?+1
| +2+?+1
|0.00020
| 0.00020
|-
|-
|0,7,30,32
| 0,7,30,32
| +2+?+1
| +2+?+1
|0.00091
| 0.00091
|-
|-
|0,8,12,29
| 0,8,12,29
| +1+?+3
| +1+?+3
|0.00097
| 0.00097
|-
|-
|0,8,13,22
| 0,8,13,22
| +2+?+3
| +2+?+3
|0.00051
| 0.00051
|-
|-
|0,8,15,21
| 0,8,15,21
| +1+?+1
| +1+?+1
|0.00062
| 0.00062
|-
|-
|0,8,15,31
| 0,8,15,31
| +1+?+3
| +1+?+3
|0.00047
| 0.00047
|-
|-
|0,8,16,18
| 0,8,16,18
| +3+?+1
| +3+?+1
|0.00066
| 0.00066
|-
|-
|0,8,16,19
| 0,8,16,19
| +2+?+1
| +2+?+1
|0.00031
| 0.00031
|-
|-
|0,8,16,20
| 0,8,16,20
| +3+?+2
| +3+?+2
|0.00043
| 0.00043
|-
|-
|0,8,16,27
| 0,8,16,27
| +1+?+2
| +1+?+2
|0.00090
| 0.00090
|-
|-
|0,8,17,19
| 0,8,17,19
| +3+?+1
| +3+?+1
|0.00022
| 0.00022
|-
|-
|0,8,17,20
| 0,8,17,20
| +2+?+1
| +2+?+1
|0.00098
| 0.00098
|-
|-
|0,8,19,27
| 0,8,19,27
| +2+?+3
| +2+?+3
|0.00085
| 0.00085
|-
|-
|0,8,24,29
| 0,8,24,29
| +1+?+1
| +1+?+1
|0.00020
| 0.00020
|-
|-
|0,9,11,16
| 0,9,11,16
| +3+?+2
| +3+?+2
|0.00051
| 0.00051
|-
|-
|0,9,13,20
| 0,9,13,20
| +1+?+1
| +1+?+1
|0.00002
| 0.00002
|-
|-
|0,9,14,24
| 0,9,14,24
| +2+?+3
| +2+?+3
|0.00073
| 0.00073
|-
|-
|0,9,18,30
| 0,9,18,30
| +1+?+2
| +1+?+2
|0.00090
| 0.00090
|-
|-
|0,9,19,28
| 0,9,19,28
| +2+?+3
| +2+?+3
|0.00096
| 0.00096
|-
|-
|0,9,21,27
| 0,9,21,27
| +1+?+1
| +1+?+1
|0.00040
| 0.00040
|-
|-
|0,9,22,24
| 0,9,22,24
| +3+?+1
| +3+?+1
|0.00087
| 0.00087
|-
|-
|0,9,22,25
| 0,9,22,25
| +2+?+1
| +2+?+1
|0.00053
| 0.00053
|-
|-
|0,9,22,26
| 0,9,22,26
| +3+?+2
| +3+?+2
|0.00026
| 0.00026
|-
|-
|0,9,23,25
| 0,9,23,25
| +3+?+1
| +3+?+1
|0.00013
| 0.00013
|-
|-
|0,9,23,26
| 0,9,23,26
| +2+?+1
| +2+?+1
|0.00093
| 0.00093
|-
|-
|0,10,11,26
| 0,10,11,26
| +1+?+2
| +1+?+2
|0.00035
| 0.00035
|-
|-
|0,10,11,32
| 0,10,11,32
| +1+?+3
| +1+?+3
|0.00081
| 0.00081
|-
|-
|0,10,12,20
| 0,10,12,20
| +1+?+1
| +1+?+1
|0.00098
| 0.00098
|-
|-
|0,10,14,18
| 0,10,14,18
| +2+?+1
| +2+?+1
|0.00050
| 0.00050
|-
|-
|0,10,14,25
| 0,10,14,25
| +2+?+3
| +2+?+3
|0.00088
| 0.00088
|-
|-
|0,10,15,29
| 0,10,15,29
| +1+?+2
| +1+?+2
|0.00041
| 0.00041
|-
|-
|0,10,16,21
| 0,10,16,21
| +3+?+2
| +3+?+2
|0.00055
| 0.00055
|-
|-
|0,10,19,32
| 0,10,19,32
| +1+?+2
| +1+?+2
|0.00021
| 0.00021
|-
|-
|0,10,27,31
| 0,10,27,31
| +3+?+2
| +3+?+2
|0.00082
| 0.00082
|-
|-
|0,10,28,30
| 0,10,28,30
| +3+?+1
| +3+?+1
|0.00045
| 0.00045
|-
|-
|0,10,28,31
| 0,10,28,31
| +2+?+1
| +2+?+1
|0.00016
| 0.00016
|-
|-
|0,10,29,31
| 0,10,29,31
| +3+?+1
| +3+?+1
|0.00068
| 0.00068
|-
|-
|0,11,12,18
| 0,11,12,18
| +3+?+2
| +3+?+2
|0.00030
| 0.00030
|-
|-
|0,11,13,16
| 0,11,13,16
| +3+?+1
| +3+?+1
|0.00081
| 0.00081
|-
|-
|0,11,14,17
| 0,11,14,17
| +3+?+1
| +3+?+1
|0.00044
| 0.00044
|-
|-
|0,11,16,31
| 0,11,16,31
| +1+?+2
| +1+?+2
|0.00064
| 0.00064
|-
|-
|0,11,17,25
| 0,11,17,25
| +1+?+1
| +1+?+1
|0.00091
| 0.00091
|-
|-
|0,11,19,23
| 0,11,19,23
| +2+?+1
| +2+?+1
|0.00045
| 0.00045
|-
|-
|0,11,21,26
| 0,11,21,26
| +3+?+2
| +3+?+2
|0.00074
| 0.00074
|-
|-
|0,12,15,24
| 0,12,15,24
| +1+?+1
| +1+?+1
|0.00087
| 0.00087
|-
|-
|0,12,15,28
| 0,12,15,28
| +2+?+3
| +2+?+3
|0.00013
| 0.00013
|-
|-
|0,12,17,23
| 0,12,17,23
| +3+?+2
| +3+?+2
|0.00054
| 0.00054
|-
|-
|0,12,18,21
| 0,12,18,21
| +3+?+1
| +3+?+1
|0.00043
| 0.00043
|-
|-
|0,12,19,22
| 0,12,19,22
| +3+?+1
| +3+?+1
|0.00095
| 0.00095
|-
|-
|0,12,23,27
| 0,12,23,27
| +2+?+1
| +2+?+1
|0.00083
| 0.00083
|-
|-
|0,12,26,31
| 0,12,26,31
| +3+?+2
| +3+?+2
|0.00005
| 0.00005
|-
|-
|0,13,14,24
| 0,13,14,24
| +1+?+1
| +1+?+1
|0.00019
| 0.00019
|-
|-
|0,13,17,22
| 0,13,17,22
| +2+?+1
| +2+?+1
|0.00085
| 0.00085
|-
|-
|0,13,21,27
| 0,13,21,27
| +3+?+2
| +3+?+2
|0.00035
| 0.00035
|-
|-
|0,13,22,25
| 0,13,22,25
| +3+?+1
| +3+?+1
|0.00097
| 0.00097
|-
|-
|0,13,23,26
| 0,13,23,26
| +3+?+1
| +3+?+1
|0.00054
| 0.00054
|-
|-
|0,13,28,32
| 0,13,28,32
| +2+?+1
| +2+?+1
|0.00055
| 0.00055
|-
|-
|0,14,17,24
| 0,14,17,24
| +3+?+2
| +3+?+2
|0.00099
| 0.00099
|-
|-
|0,14,18,28
| 0,14,18,28
| +1+?+1
| +1+?+1
|0.00043
| 0.00043
|-
|-
|0,14,21,26
| 0,14,21,26
| +2+?+1
| +2+?+1
|0.00080
| 0.00080
|-
|-
|0,14,25,31
| 0,14,25,31
| +3+?+2
| +3+?+2
|0.00054
| 0.00054
|-
|-
|0,14,27,30
| 0,14,27,30
| +3+?+1
| +3+?+1
|0.00050
| 0.00050
|-
|-
|0,15,16,20
| 0,15,16,20
| +3+?+1
| +3+?+1
|0.00055
| 0.00055
|-
|-
|0,15,17,28
| 0,15,17,28
| +1+?+1
| +1+?+1
|0.00064
| 0.00064
|-
|-
|0,15,21,28
| 0,15,21,28
| +3+?+2
| +3+?+2
|0.00045
| 0.00045
|-
|-
|0,15,22,32
| 0,15,22,32
| +1+?+1
| +1+?+1
|0.00039
| 0.00039
|-
|-
|0,16,18,26
| 0,16,18,26
| +3+?+2
| +3+?+2
|0.00049
| 0.00049
|-
|-
|0,16,19,25
| 0,16,19,25
| +2+?+1
| +2+?+1
|0.00031
| 0.00031
|-
|-
|0,16,20,24
| 0,16,20,24
| +3+?+1
| +3+?+1
|0.00018
| 0.00018
|-
|-
|0,16,25,32
| 0,16,25,32
| +3+?+2
| +3+?+2
|0.00095
| 0.00095
|-
|-
|0,17,22,28
| 0,17,22,28
| +2+?+1
| +2+?+1
|0.00091
| 0.00091
|-
|-
|0,17,23,27
| 0,17,23,27
| +3+?+1
| +3+?+1
|0.00066
| 0.00066
|-
|-
|0,18,27,31
| 0,18,27,31
| +3+?+1
| +3+?+1
|0.00095
| 0.00095
|-
|-
|0,19,21,28
| 0,19,21,28
| +2+?+1
| +2+?+1
|0.00065
| 0.00065
|-
|-
|0,20,24,31
| 0,20,24,31
| +2+?+1
| +2+?+1
|0.00078
| 0.00078
|-
|-
|0,21,22,32
| 0,21,22,32
| +3+?+2
| +3+?+2
|0.00091
| 0.00091
|-
|-
|0,22,27,32
| 0,22,27,32
| +3+?+1
| +3+?+1
|0.00038
| 0.00038
|}
|}


Line 1,419: Line 1,422:


[[Category:Listen]]
[[Category:Listen]]
[[Category:Meantone]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
Retrieved from "https://en.xen.wiki/w/33edo"