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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
! comments
! Halftone[6] notation<br>(using [[ups and downs notation|ups and downs]])
! Comments
|-
|-
| 0
| 0
| 0.0000
| 0.0
| [[1/1]]
| [[1/1]]
| C
|  
|  
|-
|-
| 1
| 1
| 43.8722
| 43.9
| 40/39, 39/38
| 40/39, 39/38
| ^C
|  
|  
|-
|-
| 2
| 2
| 87.7444
| 87.7
| [[20/19]]
| [[20/19]]
| Db
|  
|  
|-
|-
| 3
| 3
| 131.6166
| 131.6
| 55/51, ([[27/25]])
| 55/51, ([[27/25]])
| vD
|  
|  
|-
|-
| 4
| 4
| 175.4888
| 175.5
| ([[21/19]])
| ([[21/19]])
| D
|  
|  
|-
|-
| 5
| 5
| 219.3609
| 219.4
|  
|  
| vE
|  
|  
|-
|-
| 6
| 6
| 263.2331
| 263.2
| ([[7/6]])
| ([[7/6]])
| E
|  
|  
|-
|-
| 7
| 7
| 307.1053
| 307.1
|  
|  
| Fb
|  
|  
|-
|-
| 8
| 8
| 350.9775
| 351.0
| 60/49, 49/40
| 60/49, 49/40
| vF
|  
|  
|-
|-
| 9
| 9
| 394.8497
| 394.8
| (44/35)
| (44/35)
| F
|  
|  
|-
|-
| 10
| 10
| 438.7219
| 438.7
| ([[9/7]])
| ([[9/7]])
| Ab
|  
|  
|-
|-
| 11
| 11
| 482.5941
| 482.6
|  
|  
| vA
|  
|  
|-
|-
| 12
| 12
| 526.4663
| 526.5
| ([[19/14]])
| ([[19/14]])
| A
|  
|  
|-
|-
| 13
| 13
| 570.3384
| 570.3
| ([[25/18]]), 153/110, 112/81
| ([[25/18]]), 153/110, 112/81
| B
|  
|  
|-
|-
| 14
| 14
| 614.2106
| 614.2
| ([[10/7]])
| ([[10/7]])
| Cb
|  
|  
|-
|-
| 15
| 15
| 658.0828
| 658.1
| [[19/13]]
| [[19/13]]
| vC
|  
|  
|-
|-
| 16
| 16
| 701.9550
| 702.0
| [[3/2]] (exact)
| [[3/2]]
| just perfect fifth
| C
| Just perfect fifth
|-
|-
| 17
| 17
| 745.8272
| 745.8
| [[20/13]]
| [[20/13]]
|
|  
|  
|-
|-
| 18
| 18
| 789.6994
| 789.7
| [[30/19]]
| [[30/19]]
|
|  
|  
|-
|-
| 19
| 19
| 833.5716
| 833.6
| 55/34
| 55/34
|
|  
|  
|-
|-
| 20
| 20
| 877.4438
| 877.4
|
|  
|  
|  
|  
|-
|-
| 21
| 21
| 921.3159
| 921.3
|
|  
|  
|  
|  
|-
|-
| 22
| 22
| 965.1881
| 965.2
| [[7/4]]
| [[7/4]]
|
|  
|  
|-
|-
| 23
| 23
| 1009.0603
| 1009.0
|
|  
|  
|  
|  
|-
|-
| 24
| 24
| 1052.9325
| 1052.9
| 90/49, ([[11/6]])
| 90/49, ([[11/6]])
|
|  
|  
|-
|-
| 25
| 25
| 1096.8047
| 1096.8
| (66/35)
| (66/35)
|
|  
|  
|-
|-
| 26
| 26
| 1140.6769
| 1140.7
|  
|  
|
|  
|  
|-
|-
| 27
| 27
| 1184.5491
| 1184.5
|  
|  
|
|  
|  
|-
|-
| 28
| 28
| 1228.4213
| 1228.4
| 128/63
| 128/63
|  
|  
|
|-
|-
| 29
| 29
| 1272.2934
| 1272.3
| 25/12
| 25/12
|
|
|
|-
|-
| 30
| 30
| 1316.1656
| 1316.2
| 15/7
| 15/7
|
|
|
|-
|-
| 31
| 31
| 1360.0378
| 1360.0
| 57/26
| 57/26
|
|
|
|-
|-
| 32
| 32
| 1403.9100
| 1403.9
| [[9/4]] (exact)
| [[9/4]]
| pythagorean ninth
|
| Pythagorean major ninth
|}
|}


== Scale tree ==
== Music ==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.
; [[Nae Ayy]]
* [https://www.youtube.com/watch?v=8YegsoiO1Co ''Neptune''] (2021)


If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.
; [[nationalsolipsism]]
 
* [https://www.youtube.com/watch?v=-RUeO6hJLBY ''schizophrenic lullaby fugue''] (2011)
Generator range: 42.85714 cents (4\7/16 = 1\28) to 45 cents (3\5/16 = 3\80)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || || || || ||42.8571||
|-
| ||  || || || || || 27\47||43.0851||
|-
| || || || ||  ||23\40|| ||43.1250||
|-
| || || || ||  || ||42\73||43.1507||
|-
| || || || ||19\33|| || ||43.{{Overline|18}}||
|-
| || || || || || ||53\92 ||43.2065||
|-
| || || || || ||34\59|| ||43.2203||
|-
| || || || || ||  || 49\85||43.2353||
|-
|  || || ||15\26||  || || ||43.2692||
|-
| || || || || || ||56\97||43.2990||
|-
|  || ||  || || ||41\71|| ||43.3099||
|-
|  || || || || || ||67\116|| 43.3190||
|-
| ||  ||  || || 26\45|| || ||43.{{Overline|3}}||[[Flattone]] is in this region
|-
|  || || || ||  || ||63\109||43.3486||
|-
| || || || || ||37\64 || ||43.3594 ||
|-
|  || || || || || ||48\83 ||43.3735||
|-
| || ||11\19|| || || || ||43.42105||
|-
| || || || || || || 51\88||43.465{{Overline|90}}||
|-
| || || || || ||40\69|| ||43.4783||
|-
| || || || || || ||69\119||43.4874||
|-
| || || || ||29\50|| || ||43.5000||
|-
| || || || || || ||76\131||43.51145||[[Golden meantone]] (696.2145¢)
The generator closest to a just [[9/7]] for EDOs less than 800
|-
| || || || || ||47\81|| ||43.{{Overline|518}}||
|-
| || || || || || ||65\112||43.5268||
|-
| || || ||18\31|| || || ||43.5484||[[Meantone]] is in this region
|-
| || || || || || ||61\105||43.5714||
|-
| || || || || ||43\74|| ||43.5{{Overline|810}}||
|-
| || || || || || ||68\117||43.5897||
|-
| || || || ||25\43|| || ||43.60465||
|-
| || || || || || ||57\98||43.62245||
|-
| || || || || ||32\55|| ||43.{{Overline|63}}||
|-
| || || || || || ||39\67||43.6567||
|-
| ||7\12|| || || || || ||43.7500||
|-
| || || || || || ||38\65||43.84615||
|-
| || || || || ||31\53|| ||43.8679||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||43.8830||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||43.9024||
|-
| || || || || || ||65\111||43.{{Overline|1=918}}||
|-
| || || || || ||41\70|| ||43.9286||
|-
| || || || || || ||58\99||43.{{Overline|1=93}}||
|-
| || || ||17\29|| || || ||43.9655||
|-
| || || || || || ||61\104||43.9904||
|-
| || || || || ||44\75|| ||44.0000||
|-
| || || || || || ||71\121||44.0083||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||44.0217||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||44.0367||
|-
| || || || || ||37\63|| ||44.0476||
|-
| || || || || || ||47\80||44.0625||
|-
| || ||10\17|| || || || ||44.11765||
|-
| || || || || || ||43\73||44.1781||
|-
| || || || || ||33\56|| ||44.1964||
|-
| || || || || || ||56\95||44.2105||
|-
| || || || ||23\39|| || ||44.3208||
|-
| || || || || || ||59\100||43.2500||
|-
| || || || || ||36\61|| ||44.2623||
|-
| || || || || || ||49\83||44.2771||
|-
| || || ||13\22|| || || ||44.3{{Overline|18}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||44.3662||
|-
| || || || || ||29\49|| ||44.3878||
|-
| || || || || || ||45\76||44.4079||
|-
| || || || ||16\27|| || ||44.{{Overline|4}}||
|-
| || || || || || ||35\59||44.4915||
|-
| || || || || ||19\32|| ||44.53125||
|-
| || || || || || ||22\37||44.{{Overline|594}}||
|-
|3\5|| || || || || || ||45.0000||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.


Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
== See also ==
== Compositions ==
* [[27edo]] – relative edo
* [[43edt]] – relative edt
* [[70ed6]] – relative ed6
* [[90ed10]] – relative ed10
* [[97ed12]] – relative ed12


[https://www.youtube.com/watch?v=-RUeO6hJLBY schizophrenic lullaby fugue]
{{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

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