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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
! 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
|  
|  
|-
|-
| 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
|}
|}


== 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]]
Retrieved from "https://en.xen.wiki/w/16edf"