33ed4: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''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: #080">Equivocal 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: #080;">Equivocal Tuning</span>'''.


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.
It has a [[9/5]] which is 0.6{{c}} sharp, a [[7/5]] which is 0.7{{c}} flat, and a [[9/7]] which is 1.3{{c}} 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]], [[13-limit]], [[17-limit]] and even [[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{{c}}).


== Intervals ==
== Intervals ==
{| class="wikitable right-all mw-collapsible"
{| class="wikitable right-all mw-collapsible"
|+ Intervals of 33ed4
|+ style="font-size: 105%;" | Intervals of 33ed4
|-
|-
! degree
! Degree
! in cents
! Cents
! nearest JI <br> interval
! Nearest JI<br />interval
! in cents
! Cents
! difference <br> in cents
! Difference<br />in cents
|-
|-
| 1
| 1
Line 21: Line 20:
| 24/23
| 24/23
| 73.7
| 73.7
| -1.0
| −1.0
|-
|-
| 2
| 2
Line 57: Line 56:
| 51/38
| 51/38
| 509.4
| 509.4
| -0.3
| −0.3
|- style="font-weight: bold"
|- style="font-weight: bold"
| 8
| 8
Line 63: Line 62:
| 7/5
| 7/5
| 582.5
| 582.5
| -0.7
| −0.7
|-
|-
| 9
| 9
Line 69: Line 68:
| 19/13
| 19/13
| 657.0
| 657.0
| -2.5
| −2.5
|-
|-
| 10
| 10
Line 81: Line 80:
| 27/17
| 27/17
| 800.9
| 800.9
| -0.9
| −0.9
|-
|-
| 12
| 12
Line 87: Line 86:
| 53/32
| 53/32
| 873.5
| 873.5
| -0.8
| −0.8
|-
|-
| 13
| 13
Line 93: Line 92:
| 19/11
| 19/11
| 946.2
| 946.2
| -0.7
| −0.7
|- style="font-weight: bold"
|- style="font-weight: bold"
| 14
| 14
Line 123: Line 122:
| 32/15
| 32/15
| 1311.7
| 1311.7
| -2.6
| −2.6
|- style="font-weight: bold"
|- style="font-weight: bold"
| 19
| 19
Line 129: Line 128:
| 20/9
| 20/9
| 1382.4
| 1382.4
| -0.6
| −0.6
|-
|-
| 20
| 20
Line 141: Line 140:
| 29/12
| 29/12
| 1527.6
| 1527.6
| -0.3
| −0.3
|-
|-
| 22
| 22
Line 177: Line 176:
| 28/9
| 28/9
| 1964.9
| 1964.9
| -1.3
| −1.3
|-
|-
| 28
| 28
Line 183: Line 182:
| 13/4
| 13/4
| 2040.5
| 2040.5
| -4.1
| −4.1
|-
|-
| 29
| 29
Line 189: Line 188:
| 44/13
| 44/13
| 2110.8
| 2110.8
| -1.7
| −1.7
|-
|-
| 30
| 30
Line 195: Line 194:
| 60/17
| 60/17
| 2183.3
| 2183.3
| -1.5
| −1.5
|-
|-
| 31
| 31
Line 231: Line 230:


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


Revision as of 18:25, 13 March 2025

← 31ed4 33ed4 35ed4 →
Prime factorization 3 × 11
Step size 72.7273 ¢ 
Octave 17\33ed4 (1236.36 ¢)
Twelfth 26\33ed4 (1890.91 ¢)
Consistency limit 1
Distinct consistency limit 1

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 ¢ sharp, a 7/5 which is 0.7 ¢ flat, and a 9/7 which is 1.3 ¢ 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 ¢).

Intervals

Intervals of 33ed4
Degree Cents Nearest JI
interval 		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

Harmonics

Approximation of harmonics in 33ed4
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +36.4 -11.0 +0.0 -22.7 +25.3 -23.4 +36.4 -22.1 +13.7 -5.9 -11.0
Relative (%) +50.0 -15.2 +0.0 -31.2 +34.8 -32.1 +50.0 -30.4 +18.8 -8.1 -15.2
Steps
(reduced) 		17
(17) 		26
(26) 		33
(0) 		38
(5) 		43
(10) 		46
(13) 		50
(17) 		52
(19) 		55
(22) 		57
(24) 		59
(26)
Approximation of harmonics in 33ed4
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -4.2 +13.0 -33.7 +0.0 -32.2 +14.3 -6.6 -22.7 -34.4 +30.5 +26.3
Relative (%) -5.7 +17.9 -46.4 +0.0 -44.3 +19.6 -9.1 -31.2 -47.3 +41.9 +36.1
Steps
(reduced) 		61
(28) 		63
(30) 		64
(31) 		66
(0) 		67
(1) 		69
(3) 		70
(4) 		71
(5) 		72
(6) 		74
(8) 		75
(9)

Music

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