17edf: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''17EDF''' is the [[EDF|Division of the just perfect fifth]] into 17 equal parts. It is related to [[29edo|29 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 2.5474 cents compressed and the step size is about 41.2915 cents. Unlike 29edo, it is only consistent up to the [[6-integer-limit]], with discrepancy for the 7th harmonic.
'''17EDF''' is the [[EDF|Division of the just perfect fifth]] into 17 equal parts. It is related to [[29edo]], but with the [[3/2]] rather than the [[2/1]] being [[just]]. The [[octave]] is [[Octave shrinking|compressed]] by about 2.5474 [[cents]] and the step size is about 41.2915 cents. Unlike 29edo, it is only consistent up to the 6-[[integer-limit]], with discrepancy for the 7th harmonic.


Lookalikes: [[29edo]], [[46edt]]
Lookalikes: [[29edo]], [[46edt]]
== Harmonics ==
{{Harmonics in equal|17|3|2|intervals=prime|columns=8}}
{{Harmonics in equal|17|3|2|start=9|intervals=prime|columns=8}}


== Intervals ==
== Intervals ==
 
{| class="wikitable center-all right-2 mw-collapsible"
{| class="wikitable center-all right-2"
|+ Intervals of 17edf
|-
|-
! Degree
! Degree
Line 152: Line 156:
| 9/4
| 9/4
|}
|}
==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.
{{todo|expand}}
 
Generator range: 40.33613 cents (4\7/17 = 4\119) to 42.35294 cents (3\5/17 = 3\85)
{| class="wikitable center-all"
! colspan="7" | Fifth
!Cents
!Comments
|-
|4\7||  || || || || ||  ||40.3361||
|-
| || || || ||  || ||27\47 ||40.5507||
|-
| || || || || || 23\40|| ||40.5882||
|-
| || || || || || || 42\73||40.6124 ||
|-
| ||  || ||  ||19\33|| || || 40.6417||
|-
| || || || || || || 53\92||40.6650||
|-
| || || || || ||34\59|| ||40.6780||
|-
|  || ||  || || || ||49\85||40.6920||
|-
| || || ||15\26|| || || ||40.7240||
|-
| || || || || || ||56\97|| 40.7520||
|-
| || || || ||  ||41\71|| ||40.7622||
|-
| || ||  || || || ||67\116||40.7708 ||
|-
| || || || ||26\45|| || ||40.7843||[[Flattone]] is in this region
|-
| ||  || || || || || 63\109||40.7897||
|-
| || || || || ||37\64|| || 40.8088||
|-
| || || || || || ||48\83||40.8221||
|-
| || || 11\19|| || || || ||40.8669||
|-
| || || || || || ||51\88||40.{{Overline|90}}||
|-
| || || || || ||40\69||  ||40.9207||
|-
| || || || || || ||69\119||40.9293||
|-
| || || || ||29\50|| ||  ||40.9412||
|-
| || || || || || ||76\131||40.95195||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||40.9586||
|-
| || || || || || ||65\112||40.6994||
|-
| || || ||18\31|| || || ||40.9867||[[Meantone]] is in this region
|-
| || || || || || ||61\105||41.0084||
|-
| || || || || ||43\74|| ||41.1075||
|-
| || || || || || ||68\117||41.0256||
|-
| || || || ||25\43|| || ||41.0397||
|-
| || || || || || ||57\98||41.0564||
|-
| || || || || ||32\55|| ||41.0695||
|-
| || || || || || ||39\67||41.0887||
|-
| ||7\12|| || || || || ||41.1765||
|-
| || || || || || ||38\65||41.2670||
|-
| || || || || ||31\53|| ||41.2875||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||41.3016||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||41.3199||
|-
| || || || || || ||65\111||41.33545||
|-
| || || || || ||41\70|| ||41.3445||
|-
| || || || || || ||58\99||41.3547||
|-
| || || ||17\29|| || || ||41.3793||
|-
| || || || || || ||61\104||41.4027||
|-
| || || || || ||44\75|| ||41.4118||
|-
| || || || || || ||71\121||41.4195||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||41.4322||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||41.4463||
|-
| || || || || ||37\63|| ||41.4566||
|-
| || || || || || ||47\80||41.4706||
|-
| || ||10\17|| || || || ||41.5225||
|-
| || || || || || ||43\73||41.5764||
|-
| || || || || ||33\56|| ||41.5966||
|-
| || || || || || ||56\95||41.6099||
|-
| || || || ||23\39|| || ||41.6290||
|-
| || || || || || ||59\100||41.6471||
|-
| || || || || ||36\61|| ||41.6586||
|-
| || || || || || ||49\83||41.6726||
|-
| || || ||13\22|| || || ||41.7112||[[Archy]] is in this region
|-
| || || || || || ||42\71||41.7564||The generator closest to a just 14/11 for EDOs less than 3400
|-
| || || || || ||29\49|| ||41.7768||
|-
| || || || || || ||45\76||41.7957||
|-
| || || || ||16\27|| || ||41.8301||
|-
| || || || || || ||35\59||41.8744||
|-
| || || || || ||19\32|| ||41.9118||
|-
| || || || || || ||22\37||41.9714||
|-
|3\5|| || || || || || ||42.3529||
|}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 08:44, 20 December 2024

← 16edf 17edf 18edf →
Prime factorization 17 (prime)
Step size 41.2915 ¢ 
Octave 29\17edf (1197.45 ¢)
(semiconvergent)
Twelfth 46\17edf (1899.41 ¢)
(semiconvergent)
Consistency limit 6
Distinct consistency limit 6

17EDF is the Division of the just perfect fifth into 17 equal parts. It is related to 29edo, but with the 3/2 rather than the 2/1 being just. The octave is compressed by about 2.5474 cents and the step size is about 41.2915 cents. Unlike 29edo, it is only consistent up to the 6-integer-limit, with discrepancy for the 7th harmonic.

Lookalikes: 29edo, 46edt

Harmonics

Approximation of prime harmonics in 17edf
Harmonic 2 3 5 7 11 13 17 19
Error Absolute (¢) -2.5 -2.5 -19.8 +17.1 +19.1 +19.0 +8.7 -18.7
Relative (%) -6.2 -6.2 -47.9 +41.4 +46.3 +45.9 +21.1 -45.2
Steps
(reduced) 		29
(12) 		46
(12) 		67
(16) 		82
(14) 		101
(16) 		108
(6) 		119
(0) 		123
(4)
Approximation of prime harmonics in 17edf
Harmonic 23 29 31 37 41 43 47 53
Error Absolute (¢) -19.1 -7.5 +0.9 -16.3 +12.4 +12.5 -17.6 -19.1
Relative (%) -46.2 -18.1 +2.3 -39.6 +30.0 +30.4 -42.6 -46.3
Steps
(reduced) 		131
(12) 		141
(5) 		144
(8) 		151
(15) 		156
(3) 		158
(5) 		161
(8) 		166
(13)

Intervals

Intervals of 17edf
Degree Cents Approx. ratios of the 15-odd-limit
0 0.0000 1/1
1 41.2915 25/24~33/32~56/55~81/80
2 82.5829 21/20
3 123.8744 16/15, 15/14, 14/13, 13/12
4 165.1659 12/11, 11/10
5 206.4574 9/8
6 248.7488 8/7, 7/6, 15/13
7· 289.0403 13/11
8 330.3318 6/5, 11/9
9 371.6232 5/4, 16/13
10 412.9147 14/11
11 455.2062 9/7, 13/10
12· 495.4976 4/3
13 536.7891 11/8, 15/11
14 578.0806 7/5, 18/13
15 619.3721 10/7, 13/9
16 660.6635 16/11, 22/15
17· 701.9550 3/2
18 743.2465 14/9, 20/13
19 784.5379 11/7
20 825.8294 8/5, 13/8
21 867.1209 5/3, 18/11
22· 908.4124 22/13
23 949.7038 7/4, 12/7, 26/15
24 990.9952 16/9
25 1032.3287 11/6, 20/11
26 1073.5782 15/8, 28/15, 13/7, 24/13
27 1114.8697 40/21
28 1156.1612 48/25~64/33~55/28 ~160/81
29 1197.4526 2/1
30 1238.7441 25/12~33/16~112/55~81/40
31 1280.0356 21/10
32 1321.3271 32/15, 15/7, 28/13, 13/6
33 1362.6185 24/11, 11/5
34 1403.9100 9/4
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