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{{Infobox ET}}
{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 14 equal parts''' (14EDF) is related to [[24edo|24 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 3.3514 cents stretched and the step size is about 50.1396 cents. The patent val has a generally sharp tendency for harmonics up to 22, with the exception for 7, 14, and 21.
{{ED intro}}


Lookalikes: [[24edo]], [[38edt]]
== Theory ==
==Intervals==
14edf is related to [[24edo]], but with the perfect fifth rather than the [[2/1|octave]] being just, which stretches the octave by about 3.35 cents. The [[patent val]] has a generally sharp tendency for harmonics up to 22, with the exception for [[7/1|7]], [[14/1|14]], and [[21/1|21]].
{| class="wikitable"
 
|+
=== Harmonics ===
!Degree
{{Harmonics in equal|14|3|2|intervals=integer|columns=11}}
!
{{Harmonics in equal|14|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 14edf (continued)}}
 
=== Subsets and supersets ===
Since 14 factors into primes as {{nowrap| 2 × 7 }}, 14edf contains subset edfs [[2edf]] and [[7edf]].
 
== Intervals ==
{{todo|inline=1|complete table|text=Add column with approximated JI ratios and/or notation.}}
 
{| class="wikitable center-1 right-2"
|-
|-
|0
! #
|0
! Cents
|-
|-
|1
| 0
|50.1396
| 0.0
|-
|-
|2
| 1
|100.2793
| 50.1
|-
|-
|3
| 2
|150.4189
| 100.3
|-
|-
|4
| 3
|200.5586
| 150.4
|-
|-
|5
| 4
|250.6982
| 200.6
|-
|-
|6
| 5
|300.8379
| 250.7
|-
|-
|7
| 6
|350.9775
| 300.8
|-
|-
|8
| 7
|401.1171
| 351.0
|-
|-
|9
| 8
|451.2568
| 401.1
|-
|-
|10
| 9
|501.3964
| 451.3
|-
|-
|11
| 10
|551.536
| 501.4
|-
|-
|12
| 11
|601.6757
| 551.5
|-
|-
|13
| 12
|651.8154
| 601.7
|-
|-
|14
| 13
|701.955
| 651.8
|-
|-
|15
| 14
|752.0946
| 702.0
|-
|-
|16
| 15
|802.2343
| 752.1
|-
|-
|17
| 16
|852.3739
| 802.2
|-
|-
|18
| 17
|902.5136
| 852.4
|-
|-
|19
| 18
|952.6532
| 902.5
|-
|-
|20
| 19
|1002.7929
| 952.7
|-
|-
|21
| 20
|1052.9235
| 1002.8
|-
|-
|22
| 21
|1103.0721
| 1052.9
|-
|-
|23
| 22
|1153.2118
| 1103.1
|-
|-
|24
| 23
|1203.3514
| 1153.2
|-
|-
|25
| 24
|1253.4911
| 1203.4
|-
|-
|26
| 25
|1303.6307
| 1253.5
|-
|-
|27
| 26
|1353.7704
| 1303.6
|-
|-
|28
| 27
|1403.91
| 1353.8
|-
| 28
| 1403.9
|}
|}
==Scale tree==
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.


If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.
== See also ==
* [[24edo]] – relative edo
* [[38edt]] – relative edt
* [[56ed5]] – relative ed5
* [[62ed6]] – relative ed6
* [[83ed11]] – relative ed11
* [[86ed12]] – relative ed12
* [[198ed304]] – close to the zeta-optimized tuning for 24edo


Generator range: 48.97959 cents (4\7/14 = 2\49) to 51.42857 cents (3\5/14 = 3\70)
[[Category:24edo]]
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || || || || ||48.9796
|
|-
| || || || || || ||27\47||49.2401||
|-
| || || || || ||23\40|| ||49.2857||
|-
| || || || || || ||42\73||49.3151||
|-
| || || || ||19\33|| || ||49.35065||
|-
| || || || || || ||53\92||49.3789||
|-
| || || || || ||34\59|| ||49.3947||
|-
| || || || || || ||49\85||49.4118||
|-
| || || ||15\26|| || || ||49.45055||
|-
| || || || || || ||56\97||49.4845||
|-
| || || || || ||41\71|| ||49.4970||
|-
| || || || || || ||67\116||49.5074||
|-
| || || || ||26\45|| || ||49.5238||[[Flattone]] is in this region
|-
| || || || || || ||63\109||49.5413||
|-
| || || || || ||37\64|| ||49.5535||
|-
| || || || || || ||48\83||49.5697||
|-
| || ||11\19|| || || || ||49.6241||
|-
| || || || || || ||51\88||49.6753||
|-
| || || || || ||40\69|| ||49.6894||
|-
| || || || || || ||69\119||49.6999||
|-
| || || || ||29\50|| || ||49.7143||
|-
| || || || || || ||76\131||49.7274||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||49.73545||
|-
| || || || || || ||65\112||49.7449||
|-
| || || ||18\31|| || || ||49.7696||[[Meantone]] is in this region
|-
| || || || || || ||61\105||49.7959||
|-
| || || || || ||43\74|| ||49.8070||The generator closest to a just [[4/3]] for EDOs less than 2800
|-
| || || || || || ||68\117||49.81685||
|-
| || || || ||25\43|| || ||49.8339||
|-
| || || || || || ||57\98||49.8542||
|-
| || || || || ||32\55|| ||49.8701||
|-
| || || || || || ||39\67||49.8934||
|-
| ||7\12|| || || || || ||50.0000||
|-
| || || || || || ||38\65||50.1099||
|-
| || || || || ||31\53|| ||50.1348||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||50.1520||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||50.1742||
|-
| || || || || || ||65\111||50.19305||
|-
| || || || || ||41\70|| ||50.2041||
|-
| || || || || || ||58\99||50.21645||
|-
| || || ||17\29|| || || ||50.2463||
|-
| || || || || || ||61\104||50.2747||
|-
| || || || || ||44\75|| ||50.2857||
|-
| || || || || || ||71\121||50.2952||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||50.3106||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||50.32765||
|-
| || || || || ||37\63|| ||50.3401||
|-
| || || || || || ||47\80||50.3571||
|-
| || ||10\17|| || || || ||50.4202||
|-
| || || || || || ||43\73||50.4892||
|-
| || || || || ||33\56|| ||50.5102||
|-
| || || || || || ||56\95||50.5263||
|-
| || || || ||23\39|| || ||50.54945||
|-
| || || || || || ||59\100||50.5714||
|-
| || || || || ||36\61|| ||50.5855||
|-
| || || || || || ||49\83||50.6024||
|-
| || || ||13\22|| || || ||50.64935||[[Archy]] is in this region
|-
| || || || || || ||42\71||50.7042||
|-
| || || || || ||29\49|| ||50.7289||
|-
| || || || || || ||45\76||50.7519||
|-
| || || || ||16\27|| || ||50.79365||
|-
| || || || || || ||35\59||50.8475||
|-
| || || || || ||19\32|| ||50.8929||
|-
| || || || || || ||22\37||50.96525||
|-
|3\5|| || || || || || ||51.4286||
|}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.
[[Category:Edf]]
[[Category:Edonoi]]
[[Category:todo:improve synopsis]]

Latest revision as of 19:18, 25 June 2025

← 13edf 14edf 15edf →
Prime factorization 2 × 7
Step size 50.1396 ¢ 
Octave 24\14edf (1203.35 ¢) (→ 12\7edf)
Twelfth 38\14edf (1905.31 ¢) (→ 19\7edf)
Consistency limit 6
Distinct consistency limit 6

14 equal divisions of the perfect fifth (abbreviated 14edf or 14ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 14 equal parts of about 50.1 ¢ each. Each step represents a frequency ratio of (3/2)1/14, or the 14th root of 3/2.

Theory

14edf is related to 24edo, but with the perfect fifth rather than the octave being just, which stretches the octave by about 3.35 cents. The patent val has a generally sharp tendency for harmonics up to 22, with the exception for 7, 14, and 21.

Harmonics

Approximation of harmonics in 14edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.4 +3.4 +6.7 +21.5 +6.7 -9.5 +10.1 +6.7 +24.9 +10.3 +10.1
Relative (%) +6.7 +6.7 +13.4 +42.9 +13.4 -18.9 +20.1 +13.4 +49.6 +20.5 +20.1
Steps
(reduced) 		24
(10) 		38
(10) 		48
(6) 		56
(0) 		62
(6) 		67
(11) 		72
(2) 		76
(6) 		80
(10) 		83
(13) 		86
(2)
Approximation of harmonics in 14edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +21.9 -6.1 +24.9 +13.4 +8.7 +10.1 +16.7 -21.9 -6.1 +13.6 -13.2 +13.4
Relative (%) +43.7 -12.2 +49.6 +26.7 +17.4 +20.1 +33.4 -43.7 -12.2 +27.2 -26.3 +26.7
Steps
(reduced) 		89
(5) 		91
(7) 		94
(10) 		96
(12) 		98
(0) 		100
(2) 		102
(4) 		103
(5) 		105
(7) 		107
(9) 		108
(10) 		110
(12)

Subsets and supersets

Since 14 factors into primes as 2 × 7, 14edf contains subset edfs 2edf and 7edf.

Intervals

Todo: complete table

Add column with approximated JI ratios and/or notation.

# Cents
0 0.0
1 50.1
2 100.3
3 150.4
4 200.6
5 250.7
6 300.8
7 351.0
8 401.1
9 451.3
10 501.4
11 551.5
12 601.7
13 651.8
14 702.0
15 752.1
16 802.2
17 852.4
18 902.5
19 952.7
20 1002.8
21 1052.9
22 1103.1
23 1153.2
24 1203.4
25 1253.5
26 1303.6
27 1353.8
28 1403.9

See also

Retrieved from "https://en.xen.wiki/index.php?title=14edf&oldid=203642"