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'''33ed4''' is the [[ed4|Equal Divisions of the Double Octave]] into 33 narrow chromatic semitones each of 72.727 [[cent|cent]]s. It takes out every second step of [[33edo|33edo]] and falls between [[16edo|16edo]] and [[17edo|17edo]]. So even degree 16 or degree 17 can play the role of the [[Octave|octave]], depending on the actual melodic or harmonic situation in a given composition. So it can be seen as a kind of '''<span style="color: #00cc00;">E</span><span style="color: #00cc00;">quivocal Tuning</span>'''.
'''33ed4''' is the [[ed4|Equal Divisions of the Double Octave]] into 33 narrow chromatic semitones each of 72.727 [[cent]]s. It takes out every second step of [[33edo]] and falls between [[16edo]] and [[17edo]]. So even degree 16 or degree 17 can play the role of the [[octave]], depending on the actual melodic or harmonic situation in a given composition. So it can be seen as a kind of '''<span style="color: #00cc00;">E</span><span style="color: #00cc00;">quivocal Tuning</span>'''.


It has a [[9/5|9/5]] which is 0.6 cents sharp, a [[7/5|7/5]] which is 0.7 cents flat, and a [[9/7|9/7]] which is 1.3 cents sharp. Therefore it is closely related to [[13edt|13edt]], the [[Bohlen-Pierce|Bohlen-Pierce]] scale, although it has no pure [[3/1|3/1]], which is 11.1 cents flat. The lack of a [[3/2|pure fifth]] makes it also interesting.
It has a [[9/5]] which is 0.6 cents sharp, a [[7/5]] which is 0.7 cents flat, and a [[9/7]] which is 1.3 cents sharp. Therefore it is closely related to [[13edt]], the [[Bohlen-Pierce]] scale, although it has no pure [[3/1]], which is 11.1 cents flat. The lack of a [[3/2|pure fifth]] makes it also interesting.


Furthermore it has some [[11-limit|11-limit]], [[13-limit|13-limit]], [[17-limit|17-limit]] and even [[23-limit|23-limit]] which are very close (most of them under or nearby 1 cent).
Furthermore it has some [[11-limit]], [[13-limit]], [[17-limit]] and even [[23-limit]] which are very close (most of them under or nearby 1 cent).


===Intervals===
== Intervals ==


{| class="wikitable"
{| class="wikitable right-all"
|-
|-
! | degree
! degree
! | in cents
! in cents
! | nearest JI
! nearest JI <br> interval
 
! in cents
interval
! difference <br> in cents
! | in cents
! | difference
 
in cents
|-
| style="text-align:right;" | 1
| style="text-align:right;" | 72,7
| style="text-align:right;" | 24/23
| style="text-align:right;" | 73,7
| style="text-align:right;" | -1,0
|-
|-
| style="text-align:right;" | 2
| 1
| style="text-align:right;" | 145,5
| 72.7
| style="text-align:right;" | 25/23
| 24/23
| style="text-align:right;" | 144,4
| 73.7
| style="text-align:right;" | 1,1
| -1.0
|-
|-
| style="text-align:right;" | 3
| 2
| style="text-align:right;" | 218,2
| 145.5
| style="text-align:right;" | 17/15
| 25/23
| style="text-align:right;" | 216,6
| 144.4
| style="text-align:right;" | 1,6
| 1.1
|-
|-
| style="text-align:right;" | 4
| 3
| style="text-align:right;" | 290,9
| 218.2
| style="text-align:right;" | 13/11
| 17/15
| style="text-align:right;" | 289,2
| 216.6
| style="text-align:right;" | 1,7
| 1.6
|-
|-
| style="text-align:right;" | 5
| 4
| style="text-align:right;" | 363,6
| 290.9
| style="text-align:right;" | 16/13
| 13/11
| style="text-align:right;" | 359,5
| 289.2
| style="text-align:right;" | 4,1
| 1.7
|-
|-
| style="text-align:right;" | '''6'''
| 5
| style="text-align:right;" | '''436,4'''
| 363.6
| style="text-align:right;" | '''9/7'''
| 16/13
| style="text-align:right;" | '''435,1'''
| 359.5
| style="text-align:right;" | '''1,3'''
| 4.1
|- style="font-weight: bold"
| 6
| 436.4
| 9/7
| 435.1
| 1.3
|-
|-
| style="text-align:right;" | 7
| 7
| style="text-align:right;" | 509,1
| 509.1
| style="text-align:right;" | 51/38
| 51/38
| style="text-align:right;" | 509,4
| 509.4
| style="text-align:right;" | -0,3
| -0.3
|- style="font-weight: bold"
| 8
| 581.8
| 7/5
| 582.5
| -0.7
|-
|-
| style="text-align:right;" | '''8'''
| 9
| style="text-align:right;" | '''581,8'''
| 654.5
| style="text-align:right;" | '''7/5'''
| 19/13
| style="text-align:right;" | '''582,5'''
| 657.0
| style="text-align:right;" | '''-0,7'''
| -2.5
|-
|-
| style="text-align:right;" | 9
| 10
| style="text-align:right;" | 654,5
| 727.3
| style="text-align:right;" | 19/13
| 35/23
| style="text-align:right;" | 657,0
| 726.9
| style="text-align:right;" | -2,5
| 0.4
|-
|-
| style="text-align:right;" | 10
| 11
| style="text-align:right;" | 727,3
| 800.0
| style="text-align:right;" | 35/23
| 27/17
| style="text-align:right;" | 726,9
| 800.9
| style="text-align:right;" | 0,4
| -0.9
|-
|-
| style="text-align:right;" | 11
| 12
| style="text-align:right;" | 800,0
| 872.7
| style="text-align:right;" | 27/17
| 53/32
| style="text-align:right;" | 800,9
| 873.5
| style="text-align:right;" | -0,9
| -0.8
|-
|-
| style="text-align:right;" | 12
| 13
| style="text-align:right;" | 872,7
| 945.5
| style="text-align:right;" | 53/32
| 19/11
| style="text-align:right;" | 873,5
| 946.2
| style="text-align:right;" | -0,8
| -0.7
|- style="font-weight: bold"
| 14
| 1018.2
| 9/5
| 1017.6
| 0.6
|-
|-
| style="text-align:right;" | 13
| 15
| style="text-align:right;" | 945,5
| 1090.9
| style="text-align:right;" | 19/11
| 15/8
| style="text-align:right;" | 946,2
| 1088.3
| style="text-align:right;" | -0,7
| 2.6
|- style="font-weight: bold; color: #00cc00;"
| 16
| 1163.6
| 45/23
| 1161.9
| 1.7
|- style="font-weight: bold; color: #00cc00;"
| 17
| 1236.4
| 49/24
| 1235.7
| 0.7
|-
|-
| style="text-align:right;" | '''14'''
| 18
| style="text-align:right;" | '''1018,2'''
| 1309.1
| style="text-align:right;" | '''9/5'''
| 32/15
| style="text-align:right;" | '''1017,6'''
| 1311.7
| style="text-align:right;" | '''0,6'''
| -2.6
|- style="font-weight: bold"
| 19
| 1381.8
| 20/9
| 1382.4
| -0.6
|-
|-
| style="text-align:right;" | 15
| 20
| style="text-align:right;" | 1090,9
| 1454.5
| style="text-align:right;" | 15/8
| 44/19
| style="text-align:right;" | 1088,3
| 1453.8
| style="text-align:right;" | 2,6
| 0.7
|-
|-
| style="text-align:right;" | '''<span style="color: #00cc00;">16</span>'''
| 21
| style="text-align:right;" | '''<span style="color: #00cc00;">1163,6</span>'''
| 1527.3
| style="text-align:right;" | '''<span style="color: #00cc00;">45/23</span>'''
| 29/12
| style="text-align:right;" | '''<span style="color: #00cc00;">1161,9</span>'''
| 1527.6
| style="text-align:right;" | '''<span style="color: #00cc00;">1,7</span>'''
| -0.3
|-
|-
| style="text-align:right;" | '''<span style="color: #00cc00;">17</span>'''
| 22
| style="text-align:right;" | '''<span style="color: #00cc00;">1236,4</span>'''
| 1600.0
| style="text-align:right;" | '''<span style="color: #00cc00;">49/24</span>'''
| 68/27
| style="text-align:right;" | '''<span style="color: #00cc00;">1235,7</span>'''
| 1599.1
| style="text-align:right;" | '''<span style="color: #00cc00;">0,7</span>'''
| 0.9
|-
|-
| style="text-align:right;" | 18
| 23
| style="text-align:right;" | 1309,1
| 1672.7
| style="text-align:right;" | 32/15
| 21/8
| style="text-align:right;" | 1311,7
| 1670.8
| style="text-align:right;" | -2,6
| 1.9
|-
|-
| style="text-align:right;" | '''19'''
| 24
| style="text-align:right;" | '''1381,8'''
| 1745.5
| style="text-align:right;" | '''20/9'''
| 52/19
| style="text-align:right;" | '''1382,4'''
| 1743.0
| style="text-align:right;" | '''-0,6'''
| 2.5
|- style="font-weight: bold"
| 25
| 1818.2
| 20/7
| 1817.5
| 0.7
|-
|-
| style="text-align:right;" | 20
| 26
| style="text-align:right;" | 1454,5
| 1890.9
| style="text-align:right;" | 44/19
| 116/39
| style="text-align:right;" | 1453,8
| 1887.1
| style="text-align:right;" | 0,7
| 3.8
|- style="font-weight: bold"
| 27
| 1963.6
| 28/9
| 1964.9
| -1.3
|-
|-
| style="text-align:right;" | 21
| 28
| style="text-align:right;" | 1527,3
| 2036.4
| style="text-align:right;" | 29/12
| 13/4
| style="text-align:right;" | 1527,6
| 2040.5
| style="text-align:right;" | -0,3
| -4.1
|-
|-
| style="text-align:right;" | 22
| 29
| style="text-align:right;" | 1600,0
| 2109.1
| style="text-align:right;" | 68/27
| 44/13
| style="text-align:right;" | 1599,1
| 2110.8
| style="text-align:right;" | 0,9
| -1.7
|-
|-
| style="text-align:right;" | 23
| 30
| style="text-align:right;" | 1672,7
| 2181.8
| style="text-align:right;" | 21/8
| 60/17
| style="text-align:right;" | 1670,8
| 2183.3
| style="text-align:right;" | 1,9
| -1.5
|-
|-
| style="text-align:right;" | 24
| 31
| style="text-align:right;" | 1745,5
| 2254.5
| style="text-align:right;" | 52/19
| 114/31
| style="text-align:right;" | 1743,0
| 2254.4
| style="text-align:right;" | 2,5
| 0.1
|-
|-
| style="text-align:right;" | '''25'''
| 32
| style="text-align:right;" | '''1818,2'''
| 2327.3
| style="text-align:right;" | '''20/7'''
| 23/6
| style="text-align:right;" | '''1817,5'''
| 2326.3
| style="text-align:right;" | '''0,7'''
| 1.0
|-
|- style="font-weight: bold"
| style="text-align:right;" | 26
| 33
| style="text-align:right;" | 1890,9
| 2400.0
| style="text-align:right;" | 116/39
| 4/1
| style="text-align:right;" | 1887,1
| 2400.0
| style="text-align:right;" | 3,8
| 0.0
|-
| style="text-align:right;" | '''27'''
| style="text-align:right;" | '''1963,6'''
| style="text-align:right;" | '''28/9'''
| style="text-align:right;" | 1964,9
| style="text-align:right;" | '''-1,3'''
|-
| style="text-align:right;" | 28
| style="text-align:right;" | 2036,4
| style="text-align:right;" | 13/4
| style="text-align:right;" | 2040,5
| style="text-align:right;" | -4,1
|-
| style="text-align:right;" | 29
| style="text-align:right;" | 2109,1
| style="text-align:right;" | 44/13
| style="text-align:right;" | 2110,8
| style="text-align:right;" | -1,7
|-
| style="text-align:right;" | 30
| style="text-align:right;" | 2181,8
| style="text-align:right;" | 60/17
| style="text-align:right;" | 2183,3
| style="text-align:right;" | -1,5
|-
| style="text-align:right;" | 31
| style="text-align:right;" | 2254,5
| style="text-align:right;" | 114/31
| style="text-align:right;" | 2254,4
| style="text-align:right;" | 0,1
|-
| style="text-align:right;" | 32
| style="text-align:right;" | 2327,3
| style="text-align:right;" | 23/6
| style="text-align:right;" | 2326,3
| style="text-align:right;" | 1,0
|-
| style="text-align:right;" | '''33'''
| style="text-align:right;" | '''2400,0'''
| style="text-align:right;" | '''4/1'''
| style="text-align:right;" | '''2400,0'''
| style="text-align:right;" | '''0,0'''
|}
|}


===Music===
== Music ==
[http://soundcloud.com/ahornberg/sets/equivocal-tuning-33ed4 Equivocal Tuning] by Ahornberg
 
* [http://soundcloud.com/ahornberg/sets/equivocal-tuning-33ed4 Equivocal Tuning] — Set of compositions by Ahornberg
 
[[Category:31edo]]
[[Category:31edo]]
[[Category:what_is]]
[[Category:wiki]]

Revision as of 13:09, 4 December 2021

33ed4 is the Equal Divisions of the Double Octave into 33 narrow chromatic semitones each of 72.727 cents. It takes out every second step of 33edo and falls between 16edo and 17edo. So even degree 16 or degree 17 can play the role of the octave, depending on the actual melodic or harmonic situation in a given composition. So it can be seen as a kind of Equivocal Tuning.

It has a 9/5 which is 0.6 cents sharp, a 7/5 which is 0.7 cents flat, and a 9/7 which is 1.3 cents sharp. Therefore it is closely related to 13edt, the Bohlen-Pierce scale, although it has no pure 3/1, which is 11.1 cents flat. The lack of a pure fifth makes it also interesting.

Furthermore it has some 11-limit, 13-limit, 17-limit and even 23-limit which are very close (most of them under or nearby 1 cent).

Intervals

degree in cents nearest JI
interval 		in cents difference
in cents
1 72.7 24/23 73.7 -1.0
2 145.5 25/23 144.4 1.1
3 218.2 17/15 216.6 1.6
4 290.9 13/11 289.2 1.7
5 363.6 16/13 359.5 4.1
6 436.4 9/7 435.1 1.3
7 509.1 51/38 509.4 -0.3
8 581.8 7/5 582.5 -0.7
9 654.5 19/13 657.0 -2.5
10 727.3 35/23 726.9 0.4
11 800.0 27/17 800.9 -0.9
12 872.7 53/32 873.5 -0.8
13 945.5 19/11 946.2 -0.7
14 1018.2 9/5 1017.6 0.6
15 1090.9 15/8 1088.3 2.6
16 1163.6 45/23 1161.9 1.7
17 1236.4 49/24 1235.7 0.7
18 1309.1 32/15 1311.7 -2.6
19 1381.8 20/9 1382.4 -0.6
20 1454.5 44/19 1453.8 0.7
21 1527.3 29/12 1527.6 -0.3
22 1600.0 68/27 1599.1 0.9
23 1672.7 21/8 1670.8 1.9
24 1745.5 52/19 1743.0 2.5
25 1818.2 20/7 1817.5 0.7
26 1890.9 116/39 1887.1 3.8
27 1963.6 28/9 1964.9 -1.3
28 2036.4 13/4 2040.5 -4.1
29 2109.1 44/13 2110.8 -1.7
30 2181.8 60/17 2183.3 -1.5
31 2254.5 114/31 2254.4 0.1
32 2327.3 23/6 2326.3 1.0
33 2400.0 4/1 2400.0 0.0

Music

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