35ed4: Difference between revisions

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{{Stub}}
{{Infobox ET}}
{{Infobox ET}}
{{ED intro}}
{{ED intro}} [[Željko Stojanović]] recommends the use of 35ed4.


== Intervals ==
== Intervals ==
{{Interval table}}
{| class="wikitable mw-collapsible"
! Steps
! Cents
! Approximate ratios
|-
| 0
| 0
| 1/1
|-
| 1
| 68.6
| 26/25, 27/26, 28/27
|-
| 2
| 137.1
| 13/12
|-
| 3
| 205.7
|9/8
|-
| 4
| 274.3
|7/6
|-
| 5
| 342.9
| 11/9
|-
| 6
| 411.4
| 14/11
|-
| 7
| 480
|4/3
|-
| 8
| 548.6
|11/8
|-
| 9
| 617.1
|10/7
|-
| 10
| 685.7
|3/2
|-
| 11
| 754.3
| 14/9
|-
| 12
| 822.9
| 8/5
|-
| 13
| 891.4
| 5/3
|-
| 14
| 960
| 7/4
|-
| 15
| 1028.6
|9/5, 11/6
|-
| 16
| 1097.1
| 15/8, 17/9
|-
| 17
| 1165.7
|35/18
|-
| 18
| 1234.3
|
|-
| 19
| 1302.9
|17/8
|-
| 20
| 1371.4
| 11/5
|-
| 21
| 1440
|16/7, 7/3
|-
| 22
| 1508.6
|12/5
|-
| 23
| 1577.1
|5/2
|-
| 24
| 1645.7
|13/5
|-
| 25
| 1714.3
| 8/3
|-
| 26
| 1782.9
| 14/5
|-
| 27
| 1851.4
|35/12
|-
| 28
| 1920
|3/1
|-
| 29
| 1988.6
|28/9
|-
| 30
| 2057.1
| 23/7
|-
| 31
| 2125.7
| 17/5
|-
| 32
| 2194.3
|13/4, 10/3
|-
| 33
| 2262.9
|11/3, 15/4
|-
| 34
| 2331.4
|
|-
| 35
| 2400
|4/1
|}
''Ratios above were based on direct approximation, not subgroup interpretation.''


== Harmonics ==
== Harmonics ==
35ed4 can be approached as a 4.3.10.7 [[subgroup]] tuning.
35ed4 can be approached as a 4.3.10.7 [[subgroup]] tuning. Alternatively it can be approached as a 2.3.5.11.13.17 tuning because the error on all those [[primes]] is in the same direction and by similar amounts, so intervals built from them turn out better than one would expect from the raw numbers.
{{Harmonics in equal
{{Harmonics in equal
| steps = 35
| steps = 35
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| collapsed = 1
| collapsed = 1
}}
}}
{{todo|expand}}

Latest revision as of 14:25, 30 September 2026

← 33ed4 35ed4 37ed4 →
Prime factorization 5 × 7
Step size 68.5714 ¢ 
Octave 18\35ed4 (1234.29 ¢)
Twelfth 28\35ed4 (1920 ¢) (→ 4\5ed4)
Consistency limit 1
Distinct consistency limit 1

35 equal divisions of the 4th harmonic (abbreviated 35ed4) is a nonoctave tuning system that divides the interval of 4/1 into 35 equal parts of about 68.6 ¢ each. Each step represents a frequency ratio of 41/35, or the 35th root of 4. Željko Stojanović recommends the use of 35ed4.

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 68.6 26/25, 27/26, 28/27
2 137.1 13/12
3 205.7 9/8
4 274.3 7/6
5 342.9 11/9
6 411.4 14/11
7 480 4/3
8 548.6 11/8
9 617.1 10/7
10 685.7 3/2
11 754.3 14/9
12 822.9 8/5
13 891.4 5/3
14 960 7/4
15 1028.6 9/5, 11/6
16 1097.1 15/8, 17/9
17 1165.7 35/18
18 1234.3
19 1302.9 17/8
20 1371.4 11/5
21 1440 16/7, 7/3
22 1508.6 12/5
23 1577.1 5/2
24 1645.7 13/5
25 1714.3 8/3
26 1782.9 14/5
27 1851.4 35/12
28 1920 3/1
29 1988.6 28/9
30 2057.1 23/7
31 2125.7 17/5
32 2194.3 13/4, 10/3
33 2262.9 11/3, 15/4
34 2331.4
35 2400 4/1

Ratios above were based on direct approximation, not subgroup interpretation.

Harmonics

35ed4 can be approached as a 4.3.10.7 subgroup tuning. Alternatively it can be approached as a 2.3.5.11.13.17 tuning because the error on all those primes is in the same direction and by similar amounts, so intervals built from them turn out better than one would expect from the raw numbers.

Approximation of harmonics in 35ed4
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +34.3 +18.0 +0.0 +25.1 -16.2 -8.8 +34.3 -32.5 -9.2 +31.5 +18.0
Relative (%) +50.0 +26.3 +0.0 +36.6 -23.7 -12.9 +50.0 -47.4 -13.4 +46.0 +26.3
Steps
(reduced) 		18
(18) 		28
(28) 		35
(0) 		41
(6) 		45
(10) 		49
(14) 		53
(18) 		55
(20) 		58
(23) 		61
(26) 		63
(28)
Approximation of harmonics in 35ed4
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +16.6 +25.5 -25.4 +0.0 +32.2 +1.8 -23.2 +25.1 +9.2 -2.7 -11.1
Relative (%) +24.2 +37.1 -37.1 +0.0 +46.9 +2.6 -33.9 +36.6 +13.4 -4.0 -16.2
Steps
(reduced) 		65
(30) 		67
(32) 		68
(33) 		70
(0) 		72
(2) 		73
(3) 		74
(4) 		76
(6) 		77
(7) 		78
(8) 		79
(9)
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