35edf: Difference between revisions

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'''[[EDF|Division of the just perfect fifth]] into 35 equal parts''' (35EDF) is related to [[60edo|60 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 20.0559 cents (corresponding to 59.8329 [[edo]], practically identical to every sixth step of [[359edo]]). The patent val has a generally sharp tendency for harmonics up to 18, with the exception for 13. Unlike 60edo, it is only consistent up to the [[7-odd-limit|7-integer-limit]], with discrepancy for the 8th harmonic (three octaves).
{{Infobox ET}}
{{ED intro}}


Lookalikes: [[60edo]], [[95edt]]
== Theory ==
35edf corresponds to 59.8329…[[edo]] and is practically identical to every sixth step of [[359edo]]. It is related to [[60edo]], but with the [[3/2|perfect fifth]] rather than the [[2/1|octave]] being [[just]]. The octave is [[Stretched and compressed tuning|stretched]] by about 3.35 [[cents]].


[[Category:Edf]]
The [[patent val]] has a generally sharp tendency for [[prime harmonic]]s up to 17, with the exception for [[13/1|13]]. Unlike 60edo, which is [[consistent]] to the [[integer limit|10-integer-limit]], 35edf is only consistent up to the 7-integer-limit, with discrepancy for the 8th harmonic (three octaves).
[[Category:Edonoi]]
 
=== Harmonics ===
{{Harmonics in equal|35|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|35|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 35edf (continued)}}
 
=== Subsets and supersets ===
Since 35 factors into primes as {{nowrap| 5 × 7 }}, 35edf has subset edfs [[5edf]] and [[7edf]].
 
== Intervals ==
{| class="wikitable center-1 right-2 mw-collapsible"
|+ Intervals in 35edf
|-
! #
! Cents
! Approximate ratios<br>in the 2.3.5.13 subgroup
! Additional ratios<br>of 7 and 11 (assuming flat values for primes)
|-
| 0
| 0.0
|
|
|-
| 1
| 20.1
| 81/80
|
|-
| 2
| 40.1
|
|
|-
| 3
| 60.2
| 28/27, 27/26
|
|-
| 4
| 80.2
|
| 21/20
|-
| 5
| 100.3
|
|
|-
| 6
| 120.3
| 16/15
|
|-
| 7
| 140.4
|
|
|-
| 8
| 160.4
|
| 12/11, 11/10
|-
| 9
| 180.5
| 10/9
|
|-
| 10
| 200.6
| 9/8
|
|-
| 11
| 220.6
|
|
|-
| 12
| 240.7
| 15/13
| 8/7
|-
| 13
| 260.8
|
| 7/6
|-
| 14
| 280.8
|
|
|-
| 15
| 300.8
|
|
|-
| 16
| 320.9
| 6/5
|
|-
| 17
| 340.9
|
| 11/9
|-
| 18
| 361.0
| 16/13
|
|-
| 19
| 381.1
| 5/4
|
|-
| 20
| 401.1
|
|
|-
| 21
| 421.2
|
| 14/11
|-
| 22
| 441.2
|
| 9/7
|-
| 23
| 461.3
| 13/10
|
|-
| 24
| 481.3
|
|
|-
| 25
| 501.4
| 4/3
|
|-
| 26
| 521.5
|
|
|-
| 27
| 541.5
|
| 11/8, 15/11
|-
| 28
| 561.6
| 18/13
|
|-
| 29
| 581.6
|
| 7/5
|-
| 30
| 601.7
|
|
|-
| 31
| 621.7
|
| 10/7
|-
| 32
| 641.8
| 13/9
|
|-
| 33
| 661.8
|
| 16/11, 22/15
|-
| 34
| 681.9
|
|
|-
| 35
| 702.0
| 3/2
|
|-
| 36
| 722.0
|
|
|-
| 37
| 742.1
| 20/13
|
|-
| 38
| 762.1
|
| 14/9
|-
| 39
| 782.2
|
| 11/7
|-
| 40
| 802.2
|
|
|-
| 41
| 822.3
| 8/5
|
|-
| 42
| 842.3
| 13/8
|
|-
| 43
| 862.4
|
| 18/11
|-
| 44
| 882.5
| 5/3
|
|-
| 45
| 902.5
|
|
|-
| 46
| 922.6
|
|
|-
| 47
| 942.6
|
| 12/7
|-
| 48
| 962.7
| 26/15
| 7/4
|-
| 49
| 982.7
|
|
|-
| 50
| 1002.8
| 16/9
|
|-
| 51
| 1022.8
| 9/5
|
|-
| 52
| 1042.9
|
| 11/6, 20/11
|-
| 53
| 1063.0
|
|
|-
| 54
| 1083.0
| 15/8
|
|-
| 55
| 1103.1
|
|
|-
| 56
| 1123.1
|
|
|-
| 57
| 1143.2
|
|
|-
| 58
| 1163.2
|
|
|-
| 59
| 1183.3
|
|
|-
| 60
| 1203.4
|
|
|-
| 61
| 1223.4
| 81/40
|
|-
| 62
| 1243.5
|
|
|-
| 63
| 1263.5
| 56/27, 27/13
|
|-
| 64
| 1283.6
|
| 21/10
|-
| 65
| 1303.6
|
|
|-
| 66
| 1323.7
| 32/15
|
|-
| 67
| 1343.7
|
|
|-
| 68
| 1363.8
|
| 24/11, 11/5
|-
| 69
| 1383.9
| 20/9
|
|-
| 70
| 1403.9
| 9/4
|
|}
 
== See also ==
* [[60edo]] – relative edo
* [[95edt]] – relative edt
* [[139ed5]] – relative ed5
* [[155ed6]] – relative ed6

Latest revision as of 12:27, 28 May 2025

← 34edf 35edf 36edf →
Prime factorization 5 × 7
Step size 20.0559 ¢ 
Octave 60\35edf (1203.35 ¢) (→ 12\7edf)
Twelfth 95\35edf (1905.31 ¢) (→ 19\7edf)
Consistency limit 7
Distinct consistency limit 7

35 equal divisions of the perfect fifth (abbreviated 35edf or 35ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 35 equal parts of about 20.1 ¢ each. Each step represents a frequency ratio of (3/2)1/35, or the 35th root of 3/2.

Theory

35edf corresponds to 59.8329…edo and is practically identical to every sixth step of 359edo. It is related to 60edo, but with the perfect fifth rather than the octave being just. The octave is stretched by about 3.35 cents.

The patent val has a generally sharp tendency for prime harmonics up to 17, with the exception for 13. Unlike 60edo, which is consistent to the 10-integer-limit, 35edf is only consistent up to the 7-integer-limit, with discrepancy for the 8th harmonic (three octaves).

Harmonics

Approximation of harmonics in 35edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.35 +3.35 +6.70 +1.45 +6.70 +0.56 -10.00 +6.70 +4.80 +0.24 -10.00
Relative (%) +16.7 +16.7 +33.4 +7.2 +33.4 +2.8 -49.9 +33.4 +23.9 +1.2 -49.9
Steps
(reduced) 		60
(25) 		95
(25) 		120
(15) 		139
(34) 		155
(15) 		168
(28) 		179
(4) 		190
(15) 		199
(24) 		207
(32) 		214
(4)
Approximation of harmonics in 35edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -8.18 +3.91 +4.80 -6.65 +8.73 -10.00 -3.33 +8.15 +3.91 +3.60 +6.86 -6.65
Relative (%) -40.8 +19.5 +23.9 -33.2 +43.5 -49.9 -16.6 +40.7 +19.5 +17.9 +34.2 -33.2
Steps
(reduced) 		221
(11) 		228
(18) 		234
(24) 		239
(29) 		245
(0) 		249
(4) 		254
(9) 		259
(14) 		263
(18) 		267
(22) 		271
(26) 		274
(29)

Subsets and supersets

Since 35 factors into primes as 5 × 7, 35edf has subset edfs 5edf and 7edf.

Intervals

Intervals in 35edf
# Cents Approximate ratios
in the 2.3.5.13 subgroup 		Additional ratios
of 7 and 11 (assuming flat values for primes)
0 0.0
1 20.1 81/80
2 40.1
3 60.2 28/27, 27/26
4 80.2 21/20
5 100.3
6 120.3 16/15
7 140.4
8 160.4 12/11, 11/10
9 180.5 10/9
10 200.6 9/8
11 220.6
12 240.7 15/13 8/7
13 260.8 7/6
14 280.8
15 300.8
16 320.9 6/5
17 340.9 11/9
18 361.0 16/13
19 381.1 5/4
20 401.1
21 421.2 14/11
22 441.2 9/7
23 461.3 13/10
24 481.3
25 501.4 4/3
26 521.5
27 541.5 11/8, 15/11
28 561.6 18/13
29 581.6 7/5
30 601.7
31 621.7 10/7
32 641.8 13/9
33 661.8 16/11, 22/15
34 681.9
35 702.0 3/2
36 722.0
37 742.1 20/13
38 762.1 14/9
39 782.2 11/7
40 802.2
41 822.3 8/5
42 842.3 13/8
43 862.4 18/11
44 882.5 5/3
45 902.5
46 922.6
47 942.6 12/7
48 962.7 26/15 7/4
49 982.7
50 1002.8 16/9
51 1022.8 9/5
52 1042.9 11/6, 20/11
53 1063.0
54 1083.0 15/8
55 1103.1
56 1123.1
57 1143.2
58 1163.2
59 1183.3
60 1203.4
61 1223.4 81/40
62 1243.5
63 1263.5 56/27, 27/13
64 1283.6 21/10
65 1303.6
66 1323.7 32/15
67 1343.7
68 1363.8 24/11, 11/5
69 1383.9 20/9
70 1403.9 9/4

See also

Retrieved from "https://en.xen.wiki/index.php?title=35edf&oldid=197915"