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='''Intervals'''=
{{Infobox ET}}
{{ED intro}}


{| class="wikitable"
== 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.
 
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 ==
{| class="wikitable center-1 right-2 mw-collapsible"
|+ Intervals of 16edf
|-
! #
! Cents
! Approximate ratios
! Halftone[6] notation<br>(using [[ups and downs notation|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
|
|-
|-
| | 1
| 5
| | 43.872 cents
| 219.4
|  
| vE
|  
|-
|-
| | 2
| 6
| | 87.744
| 263.2
| ([[7/6]])
| E
|  
|-
|-
| | 3
| 7
| | 131.61656
| 307.1
|  
| Fb
|  
|-
|-
| | 4
| 8
| | 175.48875
| 351.0
| 60/49, 49/40
| vF
|  
|-
|-
| | 5
| 9
| | 219.3609
| 394.8
| (44/35)
| F
|  
|-
|-
| | 6
| 10
| | 263.233
| 438.7
| ([[9/7]])
| Ab
|  
|-
|-
| | 7
| 11
| | 307.105
| 482.6
|  
| vA
|  
|-
|-
| | 8
| 12
| | 350.9775
| 526.5
| ([[19/14]])
| A
|  
|-
|-
| | 9
| 13
| | 394.84969
| 570.3
| ([[25/18]]), 153/110, 112/81
| B
|  
|-
|-
| | 10
| 14
| | 438.72188
| 614.2
| ([[10/7]])
| Cb
|  
|-
|-
| | 11
| 15
| | 482.594
| 658.1
| [[19/13]]
| vC
|  
|-
|-
| | 12
| 16
| | 526.466
| 702.0
| [[3/2]]
| C
| Just perfect fifth
|-
|-
| | 13
| 17
| | 570.338
| 745.8
| [[20/13]]
|
|  
|-
|-
| | 14
| 18
| | 614.2106
| 789.7
| [[30/19]]
|
|  
|-
|-
| | 15
| 19
| | 658.0828
| 833.6
| 55/34
|
|  
|-
|-
| | 16
| 20
| | 701.955
| 877.4
|
|  
|  
|-
|-
| | 17
| 21
| | 745.827
| 921.3
|
|  
|  
|-
|-
| | 18
| 22
| | 789.699
| 965.2
| [[7/4]]
|  
|  
|-
|-
| | 19
| 23
| | 833.57156
| 1009.0
|
|  
|  
|-
|-
| | 20
| 24
| | 877.44375
| 1052.9
| 90/49, ([[11/6]])
|
|  
|-
|-
| | 21
| 25
| | 921.3159
| 1096.8
| (66/35)
|
|  
|-
|-
| | 22
| 26
| | 965.188
| 1140.7
|  
|
|  
|-
|-
| | 23
| 27
| | 1009.0603
| 1184.5
|  
|
|  
|-
|-
| | 24
| 28
| | 1052.932
| 1228.4
| 128/63
|  
|
|-
|-
| | 25
| 29
| | 1096.80469
| 1272.3
| 25/12
|
|
|-
|-
| | 26
| 30
| | 1140.67688
| 1316.2
| 15/7
|
|
|-
|-
| | 27
| 31
| | 1184.549
| 1360.0
| 57/26
|
|
|-
| 32
| 1403.9
| [[9/4]]
|
| Pythagorean major ninth
|}
|}


=Compositions=
== Music ==
; [[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}}


[http://www.youtube.com/watch?v=-RUeO6hJLBY schizophrenic lullaby fugue]
[[Category:27edo]]

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"