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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 similar to every third step of [[82edo]] but not quite similar to [[27edo]]; the octave is compressed by 15.45{{c}}, a small but significant deviation. It contains good approximations of the [[7/1|7th]] and [[13/1|13th]] [[harmonic]]s.


Lookalikes: [[27edo]], [[43edt]]
It serves as a good approximation to [[halftone]] temperament, containing the [[~]][[7/5]] generator at 13 steps.
 
=== Harmonics ===
{{Harmonics in equal|16|3|2}}
{{Harmonics in equal|16|3|2|start=12|columns=12|collapsed=true|title=Approximation of harmonics in 16edf (continued)}}
 
=== Subsets and supersets ===
Since 16 factors into primes as 2<sup>4</sup>, 16edf contains subset edfs {{EDs|equave=f| 2, 4, and 8 }}.


== Intervals ==
== Intervals ==
 
{| class="wikitable center-1 right-2 mw-collapsible"
{| class="wikitable right-2"
|+ 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 23: Line 31:
|-
|-
| 1
| 1
| 43.8722
| 43.9
| 40/39, 39/38
| 40/39, 39/38
| ^C
| ^C
Line 29: Line 37:
|-
|-
| 2
| 2
| 87.7444
| 87.7
| [[20/19]]
| [[20/19]]
| Db
| Db
Line 35: Line 43:
|-
|-
| 3
| 3
| 131.6166
| 131.6
| 55/51, ([[27/25]])
| 55/51, ([[27/25]])
| vD
| vD
Line 41: Line 49:
|-
|-
| 4
| 4
| 175.4888
| 175.5
| ([[21/19]])
| ([[21/19]])
| D
| D
Line 47: Line 55:
|-
|-
| 5
| 5
| 219.3609
| 219.4
|  
|  
| vE
| vE
Line 53: Line 61:
|-
|-
| 6
| 6
| 263.2331
| 263.2
| ([[7/6]])
| ([[7/6]])
| E
| E
Line 59: Line 67:
|-
|-
| 7
| 7
| 307.1053
| 307.1
|  
|  
| Fb
| Fb
Line 65: Line 73:
|-
|-
| 8
| 8
| 350.9775
| 351.0
| 60/49, 49/40
| 60/49, 49/40
| vF
| vF
Line 71: Line 79:
|-
|-
| 9
| 9
| 394.8497
| 394.8
| (44/35)
| (44/35)
| F
| F
Line 77: Line 85:
|-
|-
| 10
| 10
| 438.7219
| 438.7
| ([[9/7]])
| ([[9/7]])
| Ab
| Ab
Line 83: Line 91:
|-
|-
| 11
| 11
| 482.5941
| 482.6
|  
|  
| vA
| vA
Line 89: Line 97:
|-
|-
| 12
| 12
| 526.4663
| 526.5
| ([[19/14]])
| ([[19/14]])
| A
| A
Line 95: Line 103:
|-
|-
| 13
| 13
| 570.3384
| 570.3
| ([[25/18]]), 153/110, 112/81
| ([[25/18]]), 153/110, 112/81
| B
| B
Line 101: Line 109:
|-
|-
| 14
| 14
| 614.2106
| 614.2
| ([[10/7]])
| ([[10/7]])
| Cb
| Cb
Line 107: Line 115:
|-
|-
| 15
| 15
| 658.0828
| 658.1
| [[19/13]]
| [[19/13]]
| vC
| vC
Line 113: Line 121:
|-
|-
| 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 125: Line 133:
|-
|-
| 18
| 18
| 789.6994
| 789.7
| [[30/19]]
| [[30/19]]
|
|
Line 131: Line 139:
|-
|-
| 19
| 19
| 833.5716
| 833.6
| 55/34
| 55/34
|
|
Line 137: Line 145:
|-
|-
| 20
| 20
| 877.4438
| 877.4
|
|
|  
|  
Line 143: Line 151:
|-
|-
| 21
| 21
| 921.3159
| 921.3
|
|
|  
|  
Line 149: Line 157:
|-
|-
| 22
| 22
| 965.1881
| 965.2
|
| [[7/4]]
| [[7/4]]
|
|  
|  
|-
|-
| 23
| 23
| 1009.0603
| 1009.0
|
|
|  
|  
Line 161: Line 169:
|-
|-
| 24
| 24
| 1052.9325
| 1052.9
| 90/49, ([[11/6]])
| 90/49, ([[11/6]])
|
|
Line 167: Line 175:
|-
|-
| 25
| 25
| 1096.8047
| 1096.8
| (66/35)
| (66/35)
|
|
Line 173: Line 181:
|-
|-
| 26
| 26
| 1140.6769
| 1140.7
|  
|  
|
|
Line 179: Line 187:
|-
|-
| 27
| 27
| 1184.5491
| 1184.5
|  
|  
|
|
Line 185: Line 193:
|-
|-
| 28
| 28
| 1228.4213
| 1228.4
| 128/63
| 128/63
|  
|  
Line 191: Line 199:
|-
|-
| 29
| 29
| 1272.2934
| 1272.3
| 25/12
| 25/12
|
|
Line 197: Line 205:
|-
|-
| 30
| 30
| 1316.1656
| 1316.2
| 15/7
| 15/7
|
|
Line 203: Line 211:
|-
|-
| 31
| 31
| 1360.0378
| 1360.0
| 57/26
| 57/26
|
|
Line 209: Line 217:
|-
|-
| 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''] (2021)
 
; [[nationalsolipsism]]
* [https://www.youtube.com/watch?v=-RUeO6hJLBY ''schizophrenic lullaby fugue''] (2011)
 
== See also ==
* [[27edo]] – relative edo
* [[43edt]] – relative edt
* [[70ed6]] – relative ed6
* [[90ed10]] – relative ed10
* [[97ed12]] – relative ed12
 
{{Todo|expand}}


[[Category:Edf]]
[[Category:27edo]]
[[Category:Edonoi]]

Latest revision as of 19:09, 25 June 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 similar to every third step of 82edo but not quite similar to 27edo; the octave is compressed by 15.45 ¢, a small but significant deviation. 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)

Subsets and supersets

Since 16 factors into primes as 24, 16edf contains subset edfs 2, 4, and 8.

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

See also

Retrieved from "https://en.xen.wiki/index.php?title=16edf&oldid=203631"