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'''[[Edt|Division of the third harmonic]] into 65 equal parts''' (65EDT) is almost identical to [[41edo|41 edo]], but with the 3/1 rather than the 2/1 being just. The octave is about 0.3053 cents compressed and the step size is about 29.2608 cents. It is consistent to the [[15-odd-limit|16-integer-limit]].
{{Infobox ET}}
{{ED intro}}


{| class="wikitable"
== Theory ==
65edt is almost identical to [[41edo]], but with the perfect twelfth rather than the [[2/1|octave]] being just. The octave is about 0.305 cents compressed. Like 41edo, 65edt is [[consistent]] to the [[integer limit|16-integer-limit]], and in comparison, it improves the intonation of primes 3, [[11/1|11]], [[13/1|13]], and [[17/1|17]] at the expense of less accurate intonations of 2, [[5/1|5]], [[7/1|7]], and [[19/1|19]], commending itself as a suitable tuning for [[13-limit|13-]] and [[17-limit]]-focused harmonies.
 
=== Harmonics ===
{{Harmonics in equal|65|3|1|intervals=integer}}
{{Harmonics in equal|65|3|1|intervals=integer|columns=12|start=12|collapsed=true|Approximation of harmonics in 65edt (continued)}}
 
=== Subsets and supersets ===
Since 65 factors into primes as {{nowrap| 5 × 13 }}, 65edt contains [[5edt]] and [[13edt]] as subset edts.
 
== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
|-
! | degree
! #
! | cents value
! Cents
! | corresponding <br>JI intervals
! Hekts
! | comments
! Approximate ratios
|-
|-
| | 0
| 0
| | 0.0000
| 0.0
| | '''exact [[1/1]]'''
| 0.0
| |
| 1/1
|-
|-
| | 1
| 1
| | 29.2608
| 29.3
| | 57/56, 56/55
| 20.0
| |
| [[49/48]], [[50/49]], [[64/63]], [[81/80]]
|-
|-
| | 2
| 2
| | 58.5217
| 58.5
| | 30/29
| 40.0
| |
| [[25/24]], [[28/27]], [[33/32]], [[36/35]]
|-
|-
| | 3
| 3
| | 87.7825
| 87.8
| | [[21/20]], [[20/19]]
| 60.0
| |
| [[19/18]], [[20/19]], [[21/20]], [[22/21]]
|-
|-
| | 4
| 4
| | 117.0434
| 117.0
| | [[15/14]]
| 80.0
| |
| [[14/13]], [[15/14]], [[16/15]]
|-
|-
| | 5
| 5
| | 146.3042
| 146.3
| | 49/45, [[12/11]]
| 100.0
| |
| [[12/11]], [[13/12]]
|-
|-
| | 6
| 6
| | 175.5651
| 175.6
| | [[21/19]], 87/80, 72/65
| 120.0
| | pseudo-[[10/9]]
| [[10/9]], [[11/10]], [[21/19]]
|-
|-
| | 7
| 7
| | 204.8259
| 204.8
| | [[9/8]]
| 140.0
| |
| [[9/8]]
|-
|-
| | 8
| 8
| | 234.0868
| 234.1
| | [[8/7]]
| 160.0
| |
| [[8/7]], [[15/13]]
|-
|-
| | 9
| 9
| | 263.3476
| 263.3
| | [[7/6]]
| 180.0
| |
| [[7/6]], [[22/19]]
|-
|-
| | 10
| 10
| | 292.6085
| 292.6
| | 45/38, [[32/27]]
| 200.0
| |
| [[13/11]], [[19/16]], [[32/27]]
|-
|-
| | 11
| 11
| | 321.8693
| 321.9
| | 65/54
| 220.0
| | pseudo-[[6/5]]
| [[6/5]]
|-
|-
| | 12
| 12
| | 351.1302
| 351.1
| | [[11/9]], 27/22
| 240.0
| |
| [[11/9]], [[16/13]]
|-
|-
| | 13
| 13
| | 380.3910
| 380.4
| | 56/45, 81/65
| 260.0
| | pseudo-[[5/4]]
| [[5/4]], [[26/21]]
|-
|-
| | 14
| 14
| | 409.6518
| 409.7
| | [[19/15]]
| 280.0
| |
| [[14/11]], [[19/15]], [[24/19]]
|-
|-
| | 15
| 15
| | 438.9127
| 438.9
| | [[9/7]]
| 300.0
| |
| [[9/7]], [[32/25]]
|-
|-
| | 16
| 16
| | 468.1735
| 468.2
| | [[21/16]]
| 320.0
| |
| [[21/16]], [[13/10]]
|-
|-
| | 17
| 17
| | 497.4344
| 497.4
| | [[4/3]]
| 340.0
| |
| [[4/3]]
|-
|-
| | 18
| 18
| | 526.6952
| 526.7
| | [[19/14]]
| 360.0
| |
| [[15/11]], [[19/14]], [[27/20]]
|-
|-
| | 19
| 19
| | 555.9561
| 556.0
| | 40/29
| 380.0
| |
| [[11/8]], [[18/13]], [[26/19]]
|-
|-
| | 20
| 20
| | 585.2169
| 585.2
| | [[7/5]]
| 400.0
| |
| [[7/5]], [[45/32]]
|-
|-
| | 21
| 21
| | 614.4778
| 614.5
| | [[10/7]]
| 420.0
| |
| [[10/7]], [[64/45]]
|-
|-
| | 22
| 22
| | 643.7386
| 643.7
| | 29/20
| 440.0
| |
| [[13/9]], [[16/11]], [[19/13]]
|-
|-
| | 23
| 23
| | 672.9995
| 673.0
| | [[28/19]]
| 460.0
| |
| [[22/15]], [[28/19]], [[40/27]]
|-
|-
| | 24
| 24
| | 702.2603
| 702.3
| | [[3/2]]
| 480.0
| |
| [[3/2]]
|-
|-
| | 25
| 25
| | 731.5212
| 731.5
| | [[32/21]]
| 500.0
| |
| [[20/13]], [[32/21]]
|-
|-
| | 26
| 26
| | 760.7820
| 760.8
| | 45/29
| 520.0
| |
| [[14/9]], [[25/16]]
|-
|-
| | 27
| 27
| | 790.0428
| 790.0
| | [[30/19]]
| 540.0
| |
| [[11/7]], [[19/12]], [[30/19]]
|-
|-
| | 28
| 28
| | 819.3037
| 819.3
| | 45/28
| 560.0
| | pseudo-[[8/5]]
| [[8/5]], [[21/13]]
|-
|-
| | 29
| 29
| | 848.5645
| 848.6
| | [[18/11]]
| 580.0
| |
| [[13/8]], [[18/11]]
|-
|-
| | 30
| 30
| | 877.8254
| 877.8
| | 108/65
| 600.0
| | pseudo-[[5/3]]
| [[5/3]]
|-
|-
| | 31
| 31
| | 907.0862
| 907.1
| | [[27/16]]
| 620.0
| |
| [[22/13]], [[27/16]], [[32/19]]
|-
|-
| | 32
| 32
| | 936.3471
| 936.3
| | [[12/7]]
| 640.0
| |
| [[12/7]], [[19/11]]
|-
|-
| | 33
| 33
| | 965.6079
| 965.6
| | [[7/4]]
| 660.0
| |
| [[7/4]], [[26/15]]
|-
|-
| | 34
| 34
| | 994.8688
| 994.9
| | [[16/9]]
| 680.0
| |
| [[16/9]]
|-
|-
| | 35
| 35
| | 1024.1296
| 1024.1
| | 65/36
| 700.0
| | pseudo-[[9/5]]
| [[9/5]]
|-
|-
| | 36
| 36
| | 1053.3905
| 1053.4
| | [[11/6]]
| 720.0
| |
| [[11/6]]
|-
|-
| | 37
| 37
| | 1082.6513
| 1082.7
| | [[28/15]]
| 740.0
| |
| [[13/7]], [[15/8]]
|-
|-
| | 38
| 38
| | 1111.9122
| 1111.9
| | [[19/10]]
| 760.0
| |
| [[19/10]], [[21/11]]
|-
|-
| | 39
| 39
| | 1141.1730
| 1141.2
| | 29/15
| 780.0
| |
| [[27/14]], [[35/18]]
|-
|-
| | 40
| 40
| | 1170.4338
| 1170.4
| | 55/28
| 800.0
| |
| [[49/25]], [[55/28]], [[63/32]]
|-
|-
| | 41
| 41
| | 1199.6947
| 1199.7
| | [[Octave|2/1]]
| 820.0
| |
| [[2/1]]
|-
|-
| | 42
| 42
| | 1228.9555
| 1229.0
| | 57/28
| 840.0
| |
| [[45/22]], [[49/24]], [[55/27]], [[81/40]]
|-
|-
| | 43
| 43
| | 1258.2164
| 1258.2
| | 60/29
| 860.0
| |
| [[25/12]], [[33/16]]
|-
|-
| | 44
| 44
| | 1287.4772
| 1287.5
| | 21/10
| 880.0
| |
| [[19/9]], [[21/10]]
|-
|-
| | 45
| 45
| | 1316.7381
| 1316.7
| | [[15/7]]
| 900.0
| |
| [[15/7]]
|-
|-
| | 46
| 46
| | 1345.9989
| 1346.0
| | 87/40
| 920.0
| |
| [[13/6]]
|-
|-
| | 47
| 47
| | 1375.2598
| 1375.3
| | 42/19
| 940.0
| |
| [[11/5]]
|-
|-
| | 48
| 48
| | 1404.5206
| 1404.5
| | [[9/4]]
| 960.0
| |
| [[9/4]]
|-
|-
| | 49
| 49
| | 1433.7815
| 1433.8
| | [[16/7]]
| 980.0
| |
| [[16/7]]
|-
|-
| | 50
| 50
| | 1463.0423
| 1463.0
| | [[7/3]]
| 1000.0
| |
| [[7/3]]
|-
|-
| | 51
| 51
| | 1492.3032
| 1492.3
| | 45/19
| 1020.0
| |
| [[19/8]]
|-
|-
| | 52
| 52
| | 1521.5640
| 1521.6
| | 65/27
| 1040.0
| | pseudo-[[12/5]]
| [[12/5]]
|-
|-
| | 53
| 53
| | 1550.8248
| 1550.8
| | 22/9, 27/11
| 1060.0
| |
| [[22/9]], [[27/11]]
|-
|-
| | 54
| 54
| | 1580.0857
| 1580.1
| | 162/65
| 1080.0
| | pseudo-[[5/2]]
| [[5/2]]
|-
|-
| | 55
| 55
| | 1609.3465
| 1609.3
| | 38/15
| 1100.0
| |
| [[28/11]], [[33/13]]
|-
|-
| | 56
| 56
| | 1638.6074
| 1638.6
| | 18/7
| 1120.0
| |
| [[18/7]]
|-
|-
| | 57
| 57
| | 1667.8682
| 1667.9
| | 21/8
| 1140.0
| |
| [[21/8]]
|-
|-
| | 58
| 58
| | 1697.1291
| 1697.1
| | [[8/3]]
| 1160.0
| |
| [[8/3]]
|-
|-
| | 59
| 59
| | 1726.3899
| 1726.4
| | 19/7
| 1180.0
| |
| [[19/7]]
|-
|-
| | 60
| 60
| | 1755.6508
| 1755.7
| | [[11/4]]
| 1200.0
| |
| [[11/4]]
|-
|-
| | 61
| 61
| | 1784.9116
| 1784.9
| | [[14/5]]
| 1220.0
| |
| [[14/5]]
|-
|-
| | 62
| 62
| | 1814.1725
| 1814.2
| | 20/7
| 1240.0
| |
| [[20/7]]
|-
|-
| | 63
| 63
| | 1843.4333
| 1843.4
| | 29/10
| 1260.0
| |
| [[26/9]]
|-
|-
| | 64
| 64
| | 1872.6942
| 1872.7
| | 56/19
| 1280.0
| |
| [[44/15]]
|-
|-
| | 65
| 65
| | 1901.9550
| 1902.0
| | '''exact [[3/1]]'''
| 1300.0
| | [[3/2|just perfect fifth]] plus an octave
| [[3/1]]
|}
|}


[[Category:Edt]]
== See also ==
[[Category:Edonoi]]
* [[24edf]] – relative edf
* [[41edo]] – relative edo
* [[95ed5]] – relative ed5
* [[106ed6]] – relative ed6
* [[147ed12]] – relative ed12
* [[361ed448]] – close to the zeta-optimized tuning for 41edo
 
[[Category:41edo]]

Latest revision as of 13:08, 20 June 2025

← 64edt 65edt 66edt →
Prime factorization 5 × 13
Step size 29.2608 ¢ 
Octave 41\65edt (1199.69 ¢)
(convergent)
Consistency limit 16
Distinct consistency limit 10

65 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 65edt or 65ed3), is a nonoctave tuning system that divides the interval of 3/1 into 65 equal parts of about 29.3 ¢ each. Each step represents a frequency ratio of 31/65, or the 65th root of 3.

Theory

65edt is almost identical to 41edo, but with the perfect twelfth rather than the octave being just. The octave is about 0.305 cents compressed. Like 41edo, 65edt is consistent to the 16-integer-limit, and in comparison, it improves the intonation of primes 3, 11, 13, and 17 at the expense of less accurate intonations of 2, 5, 7, and 19, commending itself as a suitable tuning for 13- and 17-limit-focused harmonies.

Harmonics

Approximation of harmonics in 65edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.3 +0.0 -0.6 -6.5 -0.3 -3.8 -0.9 +0.0 -6.8 +3.7 -0.6
Relative (%) -1.0 +0.0 -2.1 -22.3 -1.0 -13.1 -3.1 +0.0 -23.4 +12.7 -2.1
Steps
(reduced) 		41
(41) 		65
(0) 		82
(17) 		95
(30) 		106
(41) 		115
(50) 		123
(58) 		130
(0) 		136
(6) 		142
(12) 		147
(17)
Approximation of harmonics in 65edt
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +7.1 -4.1 -6.5 -1.2 +10.9 -0.3 -6.1 -7.1 -3.8 +3.4 +14.2 -0.9
Relative (%) +24.3 -14.1 -22.3 -4.2 +37.1 -1.0 -20.9 -24.4 -13.1 +11.7 +48.7 -3.1
Steps
(reduced) 		152
(22) 		156
(26) 		160
(30) 		164
(34) 		168
(38) 		171
(41) 		174
(44) 		177
(47) 		180
(50) 		183
(53) 		186
(56) 		188
(58)

Subsets and supersets

Since 65 factors into primes as 5 × 13, 65edt contains 5edt and 13edt as subset edts.

Intervals

# Cents Hekts Approximate ratios
0 0.0 0.0 1/1
1 29.3 20.0 49/48, 50/49, 64/63, 81/80
2 58.5 40.0 25/24, 28/27, 33/32, 36/35
3 87.8 60.0 19/18, 20/19, 21/20, 22/21
4 117.0 80.0 14/13, 15/14, 16/15
5 146.3 100.0 12/11, 13/12
6 175.6 120.0 10/9, 11/10, 21/19
7 204.8 140.0 9/8
8 234.1 160.0 8/7, 15/13
9 263.3 180.0 7/6, 22/19
10 292.6 200.0 13/11, 19/16, 32/27
11 321.9 220.0 6/5
12 351.1 240.0 11/9, 16/13
13 380.4 260.0 5/4, 26/21
14 409.7 280.0 14/11, 19/15, 24/19
15 438.9 300.0 9/7, 32/25
16 468.2 320.0 21/16, 13/10
17 497.4 340.0 4/3
18 526.7 360.0 15/11, 19/14, 27/20
19 556.0 380.0 11/8, 18/13, 26/19
20 585.2 400.0 7/5, 45/32
21 614.5 420.0 10/7, 64/45
22 643.7 440.0 13/9, 16/11, 19/13
23 673.0 460.0 22/15, 28/19, 40/27
24 702.3 480.0 3/2
25 731.5 500.0 20/13, 32/21
26 760.8 520.0 14/9, 25/16
27 790.0 540.0 11/7, 19/12, 30/19
28 819.3 560.0 8/5, 21/13
29 848.6 580.0 13/8, 18/11
30 877.8 600.0 5/3
31 907.1 620.0 22/13, 27/16, 32/19
32 936.3 640.0 12/7, 19/11
33 965.6 660.0 7/4, 26/15
34 994.9 680.0 16/9
35 1024.1 700.0 9/5
36 1053.4 720.0 11/6
37 1082.7 740.0 13/7, 15/8
38 1111.9 760.0 19/10, 21/11
39 1141.2 780.0 27/14, 35/18
40 1170.4 800.0 49/25, 55/28, 63/32
41 1199.7 820.0 2/1
42 1229.0 840.0 45/22, 49/24, 55/27, 81/40
43 1258.2 860.0 25/12, 33/16
44 1287.5 880.0 19/9, 21/10
45 1316.7 900.0 15/7
46 1346.0 920.0 13/6
47 1375.3 940.0 11/5
48 1404.5 960.0 9/4
49 1433.8 980.0 16/7
50 1463.0 1000.0 7/3
51 1492.3 1020.0 19/8
52 1521.6 1040.0 12/5
53 1550.8 1060.0 22/9, 27/11
54 1580.1 1080.0 5/2
55 1609.3 1100.0 28/11, 33/13
56 1638.6 1120.0 18/7
57 1667.9 1140.0 21/8
58 1697.1 1160.0 8/3
59 1726.4 1180.0 19/7
60 1755.7 1200.0 11/4
61 1784.9 1220.0 14/5
62 1814.2 1240.0 20/7
63 1843.4 1260.0 26/9
64 1872.7 1280.0 44/15
65 1902.0 1300.0 3/1

See also

Retrieved from "https://en.xen.wiki/index.php?title=65edt&oldid=200064"