16edf: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''16EDF''' is the [[EDF|equal division of the just perfect fifth]] into 16 parts of 43.8722 [[cent|cents]] each, corresponding to 27.3522 [[edo]] (similar to every third step of [[82edo]]). 16edf contains good approximations of the 7th and 13th harmonics.
{{ED intro}}


It serves as a good approximation to [[halftone]] temperament, containing the ~[[7/5]] generator at 13 steps.
== Theory ==
16edf corresponds to 27.3522…[[edo]]. It is not quite similar to [[27edo]], but it is similar to every third step of [[82edo]]. It contains good approximations of the [[7/1|7th]] and [[13/1|13th]] [[harmonics]].


Lookalikes: [[27edo]], [[43edt]]
It serves as a good approximation to [[halftone]] temperament, containing the [[~]][[7/5]] generator at 13 steps.


== Harmonics ==
=== Harmonics ===
{{Harmonics in equal|16|3|2}}
{{Harmonics in equal|16|3|2}}
{{Harmonics in equal|16|3|2|start=12|collapsed=1}}
{{Harmonics in equal|16|3|2|start=12|columns=12|collapsed=true|title=Approximation of harmonics in 16edf (continued)}}


== Intervals ==
== Intervals ==
{| class="wikitable right-2 mw-collapsible"
{| class="wikitable center-1 right-2 mw-collapsible"
|+ Intervals of 16edf
|+ Intervals of 16edf
|-
|-
! degree
! #
! cents value
! Cents
! corresponding <br>JI intervals
! Approximate ratios
! Halftone[6] notation (using [[ups and downs notation|ups and downs]])
! Halftone[6] notation<br>(using [[ups and downs notation|ups and downs]])
! comments
! Comments
|-
|-
| 0
| 0
| 0.0000
| 0.0
| [[1/1]]
| [[1/1]]
| C
| C
Line 27: Line 28:
|-
|-
| 1
| 1
| 43.8722
| 43.9
| 40/39, 39/38
| 40/39, 39/38
| ^C
| ^C
Line 33: Line 34:
|-
|-
| 2
| 2
| 87.7444
| 87.7
| [[20/19]]
| [[20/19]]
| Db
| Db
Line 39: Line 40:
|-
|-
| 3
| 3
| 131.6166
| 131.6
| 55/51, ([[27/25]])
| 55/51, ([[27/25]])
| vD
| vD
Line 45: Line 46:
|-
|-
| 4
| 4
| 175.4888
| 175.5
| ([[21/19]])
| ([[21/19]])
| D
| D
Line 51: Line 52:
|-
|-
| 5
| 5
| 219.3609
| 219.4
|  
|  
| vE
| vE
Line 57: Line 58:
|-
|-
| 6
| 6
| 263.2331
| 263.2
| ([[7/6]])
| ([[7/6]])
| E
| E
Line 63: Line 64:
|-
|-
| 7
| 7
| 307.1053
| 307.1
|  
|  
| Fb
| Fb
Line 69: Line 70:
|-
|-
| 8
| 8
| 350.9775
| 351.0
| 60/49, 49/40
| 60/49, 49/40
| vF
| vF
Line 75: Line 76:
|-
|-
| 9
| 9
| 394.8497
| 394.8
| (44/35)
| (44/35)
| F
| F
Line 81: Line 82:
|-
|-
| 10
| 10
| 438.7219
| 438.7
| ([[9/7]])
| ([[9/7]])
| Ab
| Ab
Line 87: Line 88:
|-
|-
| 11
| 11
| 482.5941
| 482.6
|  
|  
| vA
| vA
Line 93: Line 94:
|-
|-
| 12
| 12
| 526.4663
| 526.5
| ([[19/14]])
| ([[19/14]])
| A
| A
Line 99: Line 100:
|-
|-
| 13
| 13
| 570.3384
| 570.3
| ([[25/18]]), 153/110, 112/81
| ([[25/18]]), 153/110, 112/81
| B
| B
Line 105: Line 106:
|-
|-
| 14
| 14
| 614.2106
| 614.2
| ([[10/7]])
| ([[10/7]])
| Cb
| Cb
Line 111: Line 112:
|-
|-
| 15
| 15
| 658.0828
| 658.1
| [[19/13]]
| [[19/13]]
| vC
| vC
Line 117: Line 118:
|-
|-
| 16
| 16
| 701.9550
| 702.0
| [[3/2]] (exact)
| [[3/2]]
| C
| C
| just perfect fifth
| Just perfect fifth
|-
|-
| 17
| 17
| 745.8272
| 745.8
| [[20/13]]
| [[20/13]]
|
|
Line 129: Line 130:
|-
|-
| 18
| 18
| 789.6994
| 789.7
| [[30/19]]
| [[30/19]]
|
|
Line 135: Line 136:
|-
|-
| 19
| 19
| 833.5716
| 833.6
| 55/34
| 55/34
|
|
Line 141: Line 142:
|-
|-
| 20
| 20
| 877.4438
| 877.4
|
|
|  
|  
Line 147: Line 148:
|-
|-
| 21
| 21
| 921.3159
| 921.3
|
|
|  
|  
Line 153: Line 154:
|-
|-
| 22
| 22
| 965.1881
| 965.2
|
|
| [[7/4]]
| [[7/4]]
Line 159: Line 160:
|-
|-
| 23
| 23
| 1009.0603
| 1009.0
|
|
|  
|  
Line 165: Line 166:
|-
|-
| 24
| 24
| 1052.9325
| 1052.9
| 90/49, ([[11/6]])
| 90/49, ([[11/6]])
|
|
Line 171: Line 172:
|-
|-
| 25
| 25
| 1096.8047
| 1096.8
| (66/35)
| (66/35)
|
|
Line 177: Line 178:
|-
|-
| 26
| 26
| 1140.6769
| 1140.7
|  
|  
|
|
Line 183: Line 184:
|-
|-
| 27
| 27
| 1184.5491
| 1184.5
|  
|  
|
|
Line 189: Line 190:
|-
|-
| 28
| 28
| 1228.4213
| 1228.4
| 128/63
| 128/63
|  
|  
Line 195: Line 196:
|-
|-
| 29
| 29
| 1272.2934
| 1272.3
| 25/12
| 25/12
|
|
Line 201: Line 202:
|-
|-
| 30
| 30
| 1316.1656
| 1316.2
| 15/7
| 15/7
|
|
Line 207: Line 208:
|-
|-
| 31
| 31
| 1360.0378
| 1360.0
| 57/26
| 57/26
|
|
Line 213: Line 214:
|-
|-
| 32
| 32
| 1403.9100
| 1403.9
| [[9/4]] (exact)
| [[9/4]]
|
|
| pythagorean ninth
| Pythagorean major ninth
|}
|}


== Music ==
== Music ==
* [https://www.youtube.com/watch?v=-RUeO6hJLBY schizophrenic lullaby fugue] by [[nationalsolipsism]]
; [[Nae Ayy]]
* [https://www.youtube.com/watch?v=8YegsoiO1Co Neptune] by [[Nae Ayy]]
* [https://www.youtube.com/watch?v=8YegsoiO1Co ''Neptune'']


[[Category:Edf]]
; [[nationalsolipsism]]
[[Category:Edonoi]]
* [https://www.youtube.com/watch?v=-RUeO6hJLBY ''schizophrenic lullaby fugue'']
{{todo|expand}}
 
{{Todo|expand}}

Revision as of 08:27, 4 March 2025

← 15edf 16edf 17edf →
Prime factorization 24
Step size 43.8722 ¢ 
Octave 27\16edf (1184.55 ¢)
Twelfth 43\16edf (1886.5 ¢)
Consistency limit 3
Distinct consistency limit 3

16 equal divisions of the perfect fifth (abbreviated 16edf or 16ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 16 equal parts of about 43.9 ¢ each. Each step represents a frequency ratio of (3/2)1/16, or the 16th root of 3/2.

Theory

16edf corresponds to 27.3522…edo. It is not quite similar to 27edo, but it is similar to every third step of 82edo. It contains good approximations of the 7th and 13th harmonics.

It serves as a good approximation to halftone temperament, containing the ~7/5 generator at 13 steps.

Harmonics

Approximation of harmonics in 16edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -15.5 -15.5 +13.0 +21.5 +13.0 +9.3 -2.5 +13.0 +6.1 +16.5 -2.5
Relative (%) -35.2 -35.2 +29.6 +49.0 +29.6 +21.3 -5.7 +29.6 +13.8 +37.7 -5.7
Steps
(reduced) 		27
(11) 		43
(11) 		55
(7) 		64
(0) 		71
(7) 		77
(13) 		82
(2) 		87
(7) 		91
(11) 		95
(15) 		98
(2)
Approximation of harmonics in 16edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -9.4 -6.1 +6.1 -17.9 +8.7 -2.5 -8.3 -9.4 -6.1 +1.1 +11.9 -17.9
Relative (%) -21.5 -13.9 +13.8 -40.9 +19.9 -5.7 -19.0 -21.4 -13.9 +2.5 +27.1 -40.9
Steps
(reduced) 		101
(5) 		104
(8) 		107
(11) 		109
(13) 		112
(0) 		114
(2) 		116
(4) 		118
(6) 		120
(8) 		122
(10) 		124
(12) 		125
(13)

Intervals

Intervals of 16edf
# Cents Approximate ratios Halftone[6] notation
(using ups and downs) 		Comments
0 0.0 1/1 C
1 43.9 40/39, 39/38 ^C
2 87.7 20/19 Db
3 131.6 55/51, (27/25) vD
4 175.5 (21/19) D
5 219.4 vE
6 263.2 (7/6) E
7 307.1 Fb
8 351.0 60/49, 49/40 vF
9 394.8 (44/35) F
10 438.7 (9/7) Ab
11 482.6 vA
12 526.5 (19/14) A
13 570.3 (25/18), 153/110, 112/81 B
14 614.2 (10/7) Cb
15 658.1 19/13 vC
16 702.0 3/2 C Just perfect fifth
17 745.8 20/13
18 789.7 30/19
19 833.6 55/34
20 877.4
21 921.3
22 965.2 7/4
23 1009.0
24 1052.9 90/49, (11/6)
25 1096.8 (66/35)
26 1140.7
27 1184.5
28 1228.4 128/63
29 1272.3 25/12
30 1316.2 15/7
31 1360.0 57/26
32 1403.9 9/4 Pythagorean major ninth

Music

Nae Ayy
nationalsolipsism
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