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 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 ===
{{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)}}
 
=== 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 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 31:
|-
|-
| 1
| 1
| 43.8722
| 43.9
| 40/39, 39/38
| 40/39, 39/38
| ^C
| ^C
Line 33: Line 37:
|-
|-
| 2
| 2
| 87.7444
| 87.7
| [[20/19]]
| [[20/19]]
| Db
| Db
Line 39: Line 43:
|-
|-
| 3
| 3
| 131.6166
| 131.6
| 55/51, ([[27/25]])
| 55/51, ([[27/25]])
| vD
| vD
Line 45: Line 49:
|-
|-
| 4
| 4
| 175.4888
| 175.5
| ([[21/19]])
| ([[21/19]])
| D
| D
Line 51: Line 55:
|-
|-
| 5
| 5
| 219.3609
| 219.4
|  
|  
| vE
| vE
Line 57: Line 61:
|-
|-
| 6
| 6
| 263.2331
| 263.2
| ([[7/6]])
| ([[7/6]])
| E
| E
Line 63: Line 67:
|-
|-
| 7
| 7
| 307.1053
| 307.1
|  
|  
| Fb
| Fb
Line 69: Line 73:
|-
|-
| 8
| 8
| 350.9775
| 351.0
| 60/49, 49/40
| 60/49, 49/40
| vF
| vF
Line 75: Line 79:
|-
|-
| 9
| 9
| 394.8497
| 394.8
| (44/35)
| (44/35)
| F
| F
Line 81: Line 85:
|-
|-
| 10
| 10
| 438.7219
| 438.7
| ([[9/7]])
| ([[9/7]])
| Ab
| Ab
Line 87: Line 91:
|-
|-
| 11
| 11
| 482.5941
| 482.6
|  
|  
| vA
| vA
Line 93: Line 97:
|-
|-
| 12
| 12
| 526.4663
| 526.5
| ([[19/14]])
| ([[19/14]])
| A
| A
Line 99: Line 103:
|-
|-
| 13
| 13
| 570.3384
| 570.3
| ([[25/18]]), 153/110, 112/81
| ([[25/18]]), 153/110, 112/81
| B
| B
Line 105: Line 109:
|-
|-
| 14
| 14
| 614.2106
| 614.2
| ([[10/7]])
| ([[10/7]])
| Cb
| Cb
Line 111: Line 115:
|-
|-
| 15
| 15
| 658.0828
| 658.1
| [[19/13]]
| [[19/13]]
| vC
| vC
Line 117: 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 129: Line 133:
|-
|-
| 18
| 18
| 789.6994
| 789.7
| [[30/19]]
| [[30/19]]
|
|
Line 135: Line 139:
|-
|-
| 19
| 19
| 833.5716
| 833.6
| 55/34
| 55/34
|
|
Line 141: Line 145:
|-
|-
| 20
| 20
| 877.4438
| 877.4
|
|
|  
|  
Line 147: Line 151:
|-
|-
| 21
| 21
| 921.3159
| 921.3
|
|
|  
|  
Line 153: Line 157:
|-
|-
| 22
| 22
| 965.1881
| 965.2
|
| [[7/4]]
| [[7/4]]
|
|  
|  
|-
|-
| 23
| 23
| 1009.0603
| 1009.0
|
|
|  
|  
Line 165: Line 169:
|-
|-
| 24
| 24
| 1052.9325
| 1052.9
| 90/49, ([[11/6]])
| 90/49, ([[11/6]])
|
|
Line 171: Line 175:
|-
|-
| 25
| 25
| 1096.8047
| 1096.8
| (66/35)
| (66/35)
|
|
Line 177: Line 181:
|-
|-
| 26
| 26
| 1140.6769
| 1140.7
|  
|  
|
|
Line 183: Line 187:
|-
|-
| 27
| 27
| 1184.5491
| 1184.5
|  
|  
|
|
Line 189: Line 193:
|-
|-
| 28
| 28
| 1228.4213
| 1228.4
| 128/63
| 128/63
|  
|  
Line 195: Line 199:
|-
|-
| 29
| 29
| 1272.2934
| 1272.3
| 25/12
| 25/12
|
|
Line 201: Line 205:
|-
|-
| 30
| 30
| 1316.1656
| 1316.2
| 15/7
| 15/7
|
|
Line 207: Line 211:
|-
|-
| 31
| 31
| 1360.0378
| 1360.0
| 57/26
| 57/26
|
|
Line 213: 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]]
{{todo|expand}}

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"