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{{Infobox ET}}
{{Infobox ET}}
'''[[Edt|Division of the third harmonic]] into 35 equal parts''' (35edt) is related to [[22edo|22 edo]], but with the 3/1 rather than the 2/1 being just. The octave is about 4.4854 cents compressed and the step size is about 54.3416 cents. It is consistent to the [[11-odd-limit|12-integer-limit]].
{{ED intro}}


==Harmonics==
== Theory ==
{{harmonics in equal|35|3|1|intervals=odd}}
35edt is related to [[22edo]], but with the [[3/1|perfect twelfth]] rather than the [[2/1|octave]] being just. The octave is about 4.4854 cents compressed. Like 22edo, 35edt is [[consistent]] to the [[integer limit|12-integer-limit]].


==Intervals==
=== Harmonics ===
{| class="wikitable"
{{Harmonics in equal|35|3|1|intervals=integer|columns=11}}
{{Harmonics in equal|35|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 35edt (continued)}}
 
=== Subsets and supersets ===
Since 35 factors into primes as {{nowrap| 5 × 7 }}, 35edt has subset edts [[5edt]] and [[7edt]].
 
== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
|-
! | degree
! #
! | cents value
! Cents
!hekts
! Hekts
! | corresponding <br>JI intervals
! Approximate ratios*
! | comments
|-
|-
! colspan="3" | 0
| 0
| | '''exact [[1/1]]'''
| 0.0
| |
| 0.0
| 1/1
|-
|-
| | 1
| 1
| | 54.3416
| 54.3
|37.1429
| 37.1
| | [[33/32]], 32/31
| 33/32, 36/35
| |
|-
|-
| | 2
| 2
| | 108.6831
| 108.7
|74.2857
| 74.3
| | 33/31
| 15/14, 16/15, 17/16, 18/17
| |
|-
|-
| | 3
| 3
| | 163.0247
| 163.0
|111.4286
| 111.4
| |11/10
| 10/9, 11/10, 12/11
| |
|-
|-
| | 4
| 4
| | 217.3663
| 217.4
|148.5714
| 148.6
| | [[17/15]]
| 8/7, 9/8
| |
|-
|-
| | 5
| 5
| | 271.7079
| 271.7
|185.7143
| 185.7
| |7/6
| 7/6
| |
|-
|-
| | 6
| 6
| | 326.0494
| 326.0
|222.8571
| 222.9
| |
| 6/5
| |pseudo-[[6/5]]
|-
|-
| | 7
| 7
| | 380.391
| 380.4
|260
| 260.0
| | 81/65
| 5/4
| | pseudo-[[5/4]]
|-
|-
| | 8
| 8
| | 434.7326
| 434.7
|297.1429
| 297.1
| | [[9/7]]
| 9/7
| |
|-
|-
| | 9
| 9
| | 489.0741
| 489.1
|334.2857
| 334.3
| | 69/52
| 4/3
| |
|-
|-
| | 10
| 10
| | 543.4157
| 543.4
|371.4286
| 371.4
| | [[26/19]]
| 11/8, 15/11, 27/20
| |
|-
|-
| | 11
| 11
| | 597.7573
| 597.8
|408.5714
| 408.6
| | [[24/17]]
| 7/5, 10/7, 17/12, 24/17
| |
|-
|-
| | 12
| 12
| | 652.0989
| 652.1
|445.7143
| 445.7
| | [[35/24]]
| 16/11, 22/15
| |
|-
|-
| | 13
| 13
| | 706.4404
| 706.4
|482.8571
| 482.9
| |
| 3/2
| | pseudo-[[3/2]]
|-
|-
| | 14
| 14
| | 760.782
| 760.8
|520
| 520.0
| | 45/29
| 11/7, 14/9
| |
|-
|-
| | 15
| 15
| | 815.1236
| 815.1
|557.1429
| 557.1
| | [[8/5]]
| 8/5
| |
|-
|-
| | 16
| 16
| | 869.4651
| 869.5
|594.2857
| 594.3
| | 38/23, 81/49
| 5/3, 18/11, 33/20
| |
|-
|-
| | 17
| 17
| | 923.8067
| 923.8
|631.4286
| 631.4
| | 46/27
| 12/7, 17/10
| |
|-
|-
| | 18
| 18
| | 978.1483
| 978.1
|668.5714
| 668.6
| | 81/46
| 7/4, 30/17
| |
|-
|-
| | 19
| 19
| | 1032.4899
| 1032.5
|705.7143
| 705.7
| | 49/27, 69/38
| 9/5, 11/6, 20/11
| |
|-
|-
| | 20
| 20
| | 1086.8314
| 1086.8
|742.8571
| 742.9
| | [[15/8]]
| 15/8
| |
|-
|-
| | 21
| 21
| | 1141.173
| 1141.2
|780
| 780.0
| | 29/15
| 21/11, 27/14
| |
|-
|-
| | 22
| 22
| | 1195.5146
| 1195.5
|817.1429
| 817.1
| |
| 2/1
| | pseudo-[[octave]]
|-
|-
| | 23
| 23
| | 1249.8561
| 1249.9
|854.2857
| 854.3
| | [[36/35|72/35]]
| 33/16, 45/22
| |
|-
|-
| | 24
| 24
| | 1304.1977
| 1304.2
|891.4286
| 891.4
| | [[17/16|17/8]]
| 15/7, 17/8, 21/10, 36/17
| |
|-
|-
| | 25
| 25
| | 1358.5393
| 1358.5
|928.5714
| 928.6
| | 57/26
| 11/5, 20/9, 24/11
| |
|-
|-
| | 26
| 26
| | 1412.8809
| 1412.9
|965.7143
| 965.7
| | 52/23
| 9/4
| |
|-
|-
| | 27
| 27
| | 1467.2224
| 1467.2
|1002.8571
| 1002.9
| | [[7/3]]
| 7/3
| |
|-
|-
| | 28
| 28
| | 1521.564
| 1521.6
|1040
| 1040.0
| | 65/27
| 12/5
| |pseudo-12/5
|-
|-
| | 29
| 29
| | 1575.9056
| 1575.9
|1077.1429
| 1077.1
| |
| 5/2
| |pseudo-5/2
|-
|-
| | 30
| 30
| | 1630.2471
| 1630.2
|1114.2857
| 1114.3
| |18/7
| 18/7
| |
|-
|-
| | 31
| 31
| | 1684.5887
| 1684.6
|1151.4286
| 1151.4
| | 45/17
| 8/3, 21/8
| |
|-
|-
| | 32
| 32
| | 1738.9303
| 1738.9
|1188.5714
| 1188.6
| |03/11
| 11/4, 27/10, 30/11
| |
|-
|-
| | 33
| 33
| | 1793.2719
| 1793.3
|1225.7143
| 1225.7
| | 31/11
| 14/5, 17/6, 45/16, 48/17
| |
|-
|-
| | 34
| 34
| | 1847.6134
| 1847.6
|1262.8571
| 1262.9
| | [[16/11|32/11]], 93/32
| 32/11, 35/12
| |
|-
|-
| | 35
| 35
| | 1901.955
| 1902.0
|1300
| 1300.0
| | '''exact [[3/1]]'''
| 3/1
| | [[3/2|just perfect fifth]] plus an octave
|}
|}
<nowiki/>* As a 2.3.5.7.11.17-subgroup temperament


[[Category:Edt]]
== See also ==
[[Category:Edonoi]]
* [[22edo]] – relative edo
* [[57ed6]] – relative ed6
* [[79ed12]] – relative ed12

Latest revision as of 09:37, 27 May 2025

← 34edt 35edt 36edt →
Prime factorization 5 × 7
Step size 54.3416 ¢ 
Octave 22\35edt (1195.51 ¢)
Consistency limit 12
Distinct consistency limit 8

35 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 35edt or 35ed3), is a nonoctave tuning system that divides the interval of 3/1 into 35 equal parts of about 54.3 ¢ each. Each step represents a frequency ratio of 31/35, or the 35th root of 3.

Theory

35edt is related to 22edo, but with the perfect twelfth rather than the octave being just. The octave is about 4.4854 cents compressed. Like 22edo, 35edt is consistent to the 12-integer-limit.

Harmonics

Approximation of harmonics in 35edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.5 +0.0 -9.0 -14.9 -4.5 +0.4 -13.5 +0.0 -19.4 -21.4 -9.0
Relative (%) -8.3 +0.0 -16.5 -27.4 -8.3 +0.6 -24.8 +0.0 -35.7 -39.3 -16.5
Steps
(reduced) 		22
(22) 		35
(0) 		44
(9) 		51
(16) 		57
(22) 		62
(27) 		66
(31) 		70
(0) 		73
(3) 		76
(6) 		79
(9)
Approximation of harmonics in 35edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +15.5 -4.1 -14.9 -17.9 -14.2 -4.5 +10.6 -23.9 +0.4 -25.8 +5.9 -13.5
Relative (%) +28.5 -7.6 -27.4 -33.0 -26.2 -8.3 +19.5 -43.9 +0.6 -47.6 +10.8 -24.8
Steps
(reduced) 		82
(12) 		84
(14) 		86
(16) 		88
(18) 		90
(20) 		92
(22) 		94
(24) 		95
(25) 		97
(27) 		98
(28) 		100
(30) 		101
(31)

Subsets and supersets

Since 35 factors into primes as 5 × 7, 35edt has subset edts 5edt and 7edt.

Intervals

# Cents Hekts Approximate ratios*
0 0.0 0.0 1/1
1 54.3 37.1 33/32, 36/35
2 108.7 74.3 15/14, 16/15, 17/16, 18/17
3 163.0 111.4 10/9, 11/10, 12/11
4 217.4 148.6 8/7, 9/8
5 271.7 185.7 7/6
6 326.0 222.9 6/5
7 380.4 260.0 5/4
8 434.7 297.1 9/7
9 489.1 334.3 4/3
10 543.4 371.4 11/8, 15/11, 27/20
11 597.8 408.6 7/5, 10/7, 17/12, 24/17
12 652.1 445.7 16/11, 22/15
13 706.4 482.9 3/2
14 760.8 520.0 11/7, 14/9
15 815.1 557.1 8/5
16 869.5 594.3 5/3, 18/11, 33/20
17 923.8 631.4 12/7, 17/10
18 978.1 668.6 7/4, 30/17
19 1032.5 705.7 9/5, 11/6, 20/11
20 1086.8 742.9 15/8
21 1141.2 780.0 21/11, 27/14
22 1195.5 817.1 2/1
23 1249.9 854.3 33/16, 45/22
24 1304.2 891.4 15/7, 17/8, 21/10, 36/17
25 1358.5 928.6 11/5, 20/9, 24/11
26 1412.9 965.7 9/4
27 1467.2 1002.9 7/3
28 1521.6 1040.0 12/5
29 1575.9 1077.1 5/2
30 1630.2 1114.3 18/7
31 1684.6 1151.4 8/3, 21/8
32 1738.9 1188.6 11/4, 27/10, 30/11
33 1793.3 1225.7 14/5, 17/6, 45/16, 48/17
34 1847.6 1262.9 32/11, 35/12
35 1902.0 1300.0 3/1

* As a 2.3.5.7.11.17-subgroup temperament

See also

Retrieved from "https://en.xen.wiki/index.php?title=35edt&oldid=197821"