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{{Infobox ET}}
{{Infobox ET}}
'''[[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]].
{{ED intro}}


==Intervals==
== Theory ==
{| class="wikitable"
65edt is almost identical to [[41edo]], but with the 3/1 rather than the [[2/1]] being just. The octave is about 0.3053 cents compressed. Like 41edo, 65edt is [[consistent]] to the [[integer limit|16-integer-limit]].
 
=== 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)}}
 
== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
|-
! | Degree
! #
! | Cent 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
| | 29.2608
| 29.3
|20
| 20.0
| | 57/56, 56/55
| [[49/48]], [[50/49]], [[64/63]], [[81/80]]
| |
|-
|-
| | 2
| 2
| | 58.5217
| 58.5
|40
| 40.0
| | 30/29
| [[25/24]], [[28/27]], [[33/32]], [[36/35]]
| |
|-
|-
| | 3
| 3
| | 87.7825
| 87.8
|60
| 60.0
| | [[21/20]], [[20/19]]
| [[19/18]], [[20/19]], [[21/20]], [[22/21]]
| |
|-
|-
| | 4
| 4
| | 117.0434
| 117.0
|80
| 80.0
| | [[15/14]]
| [[14/13]], [[15/14]], [[16/15]]
| |
|-
|-
| | 5
| 5
| | 146.3042
| 146.3
|100
| 100.0
| | 49/45, [[12/11]]
| [[12/11]], [[13/12]]
| |
|-
|-
| | 6
| 6
| | 175.5651
| 175.6
|120
| 120.0
| | [[21/19]], 87/80, 72/65
| [[10/9]], [[11/10]], [[21/19]]
| | pseudo-[[10/9]]
|-
|-
| | 7
| 7
| | 204.8259
| 204.8
|140
| 140.0
| | [[9/8]]
| [[9/8]]
| |
|-
|-
| | 8
| 8
| | 234.0868
| 234.1
|160
| 160.0
| | [[8/7]]
| [[8/7]], [[15/13]]
| |
|-
|-
| | 9
| 9
| | 263.3476
| 263.3
|180
| 180.0
| | [[7/6]]
| [[7/6]], [[22/19]]
| |
|-
|-
| | 10
| 10
| | 292.6085
| 292.6
|200
| 200.0
| | 45/38, [[32/27]]
| [[13/11]], [[19/16]], [[32/27]]
| |
|-
|-
| | 11
| 11
| | 321.8693
| 321.9
|220
| 220.0
| | 65/54
| [[6/5]]
| | pseudo-[[6/5]]
|-
|-
| | 12
| 12
| | 351.1302
| 351.1
|240
| 240.0
| | [[11/9]], 27/22
| [[11/9]], [[16/13]]
| |
|-
|-
| | 13
| 13
| | 380.391
| 380.4
|260
| 260.0
| | 56/45, 81/65
| [[5/4]], [[26/21]]
| | pseudo-[[5/4]]
|-
|-
| | 14
| 14
| | 409.6518
| 409.7
|280
| 280.0
| | [[19/15]]
| [[14/11]], [[19/15]], [[24/19]]
| |
|-
|-
| | 15
| 15
| | 438.9127
| 438.9
|300
| 300.0
| | [[9/7]]
| [[9/7]], [[32/25]]
| |
|-
|-
| | 16
| 16
| | 468.1735
| 468.2
|320
| 320.0
| | [[21/16]]
| [[21/16]], [[13/10]]
| |
|-
|-
| | 17
| 17
| | 497.4344
| 497.4
|340
| 340.0
| | [[4/3]]
| [[4/3]]
| |
|-
|-
| | 18
| 18
| | 526.6952
| 526.7
|360
| 360.0
| | [[19/14]]
| [[15/11]], [[19/14]], [[27/20]]
| |
|-
|-
| | 19
| 19
| | 555.9561
| 556.0
|380
| 380.0
| | 40/29
| [[11/8]], [[18/13]], [[26/19]]
| |
|-
|-
| | 20
| 20
| | 585.2169
| 585.2
|400
| 400.0
| | [[7/5]]
| [[7/5]], [[45/32]]
| |
|-
|-
| | 21
| 21
| | 614.4778
| 614.5
|420
| 420.0
| | [[10/7]]
| [[10/7]], [[64/45]]
| |
|-
|-
| | 22
| 22
| | 643.7386
| 643.7
|440
| 440.0
| | 29/20
| [[13/9]], [[16/11]], [[19/13]]
| |
|-
|-
| | 23
| 23
| | 672.9995
| 673.0
|460
| 460.0
| | [[28/19]]
| [[22/15]], [[28/19]], [[40/27]]
| |
|-
|-
| | 24
| 24
| | 702.2603
| 702.3
|480
| 480.0
| | [[3/2]]
| [[3/2]]
| |
|-
|-
| | 25
| 25
| | 731.5212
| 731.5
|500
| 500.0
| | [[32/21]]
| [[20/13]], [[32/21]]
| |
|-
|-
| | 26
| 26
| | 760.782
| 760.8
|520
| 520.0
| | 45/29
| [[14/9]], [[25/16]]
| |
|-
|-
| | 27
| 27
| | 790.0428
| 790.0
|540
| 540.0
| | [[30/19]]
| [[11/7]], [[19/12]], [[30/19]]
| |
|-
|-
| | 28
| 28
| | 819.3037
| 819.3
|560
| 560.0
| | 45/28
| [[8/5]], [[21/13]]
| | pseudo-[[8/5]]
|-
|-
| | 29
| 29
| | 848.5645
| 848.6
|580
| 580.0
| | [[18/11]]
| [[13/8]], [[18/11]]
| |
|-
|-
| | 30
| 30
| | 877.8254
| 877.8
|600
| 600.0
| | 108/65
| [[5/3]]
| | pseudo-[[5/3]]
|-
|-
| | 31
| 31
| | 907.0862
| 907.1
|620
| 620.0
| | [[27/16]]
| [[22/13]], [[27/16]], [[32/19]]
| |
|-
|-
| | 32
| 32
| | 936.3471
| 936.3
|640
| 640.0
| | [[12/7]]
| [[12/7]], [[19/11]]
| |
|-
|-
| | 33
| 33
| | 965.6079
| 965.6
|660
| 660.0
| | [[7/4]]
| [[7/4]], [[26/15]]
| |
|-
|-
| | 34
| 34
| | 994.8688
| 994.9
|680
| 680.0
| | [[16/9]]
| [[16/9]]
| |
|-
|-
| | 35
| 35
| | 1024.1296
| 1024.1
|700
| 700.0
| | 65/36
| [[9/5]]
| | pseudo-[[9/5]]
|-
|-
| | 36
| 36
| | 1053.3905
| 1053.4
|720
| 720.0
| | [[11/6]]
| [[11/6]]
| |
|-
|-
| | 37
| 37
| | 1082.6513
| 1082.7
|740
| 740.0
| | [[28/15]]
| [[13/7]], [[15/8]]
| |
|-
|-
| | 38
| 38
| | 1111.9122
| 1111.9
|760
| 760.0
| | [[19/10]]
| [[19/10]], [[21/11]]
| |
|-
|-
| | 39
| 39
| | 1141.173
| 1141.2
|780
| 780.0
| | 29/15
| [[27/14]], [[35/18]]
| |
|-
|-
| | 40
| 40
| | 1170.4338
| 1170.4
|800
| 800.0
| | 55/28
| [[55/28]], [[63/32]]
| |
|-
|-
| | 41
| 41
| | 1199.6947
| 1199.7
|820
| 820.0
| | [[Octave|2/1]]
| [[2/1]]
| |
|-
|-
| | 42
| 42
| | 1228.9555
| 1229.0
|840
| 840.0
| | 57/28
| [[49/24]], [[81/40]]
| |
|-
|-
| | 43
| 43
| | 1258.2164
| 1258.2
|860
| 860.0
| | 60/29
| [[25/12]], [[33/16]]
| |
|-
|-
| | 44
| 44
| | 1287.4772
| 1287.5
|880
| 880.0
| | 21/10
| [[19/9]], [[21/10]]
| |
|-
|-
| | 45
| 45
| | 1316.7381
| 1316.7
|900
| 900.0
| | [[15/7]]
| [[15/7]]
| |
|-
|-
| | 46
| 46
| | 1345.9989
| 1346.0
|920
| 920.0
| | 87/40
| [[13/6]]
| |
|-
|-
| | 47
| 47
| | 1375.2598
| 1375.3
|940
| 940.0
| | 42/19
| [[11/5]]
| |
|-
|-
| | 48
| 48
| | 1404.5206
| 1404.5
|960
| 960.0
| | [[9/4]]
| [[9/4]]
| |
|-
|-
| | 49
| 49
| | 1433.7815
| 1433.8
|980
| 980.0
| | [[16/7]]
| [[16/7]]
| |
|-
|-
| | 50
| 50
| | 1463.0423
| 1463.0
|1000
| 1000.0
| | [[7/3]]
| [[7/3]]
| |
|-
|-
| | 51
| 51
| | 1492.3032
| 1492.3
|1020
| 1020.0
| | 45/19
| [[19/8]]
| |
|-
|-
| | 52
| 52
| | 1521.564
| 1521.6
|1040
| 1040.0
| | 65/27
| [[12/5]]
| | pseudo-[[12/5]]
|-
|-
| | 53
| 53
| | 1550.8248
| 1550.8
|1060
| 1060.0
| | 22/9, 27/11
| [[22/9]], [[27/11]]
| |
|-
|-
| | 54
| 54
| | 1580.0857
| 1580.1
|1080
| 1080.0
| | 162/65
| [[5/2]]
| | pseudo-[[5/2]]
|-
|-
| | 55
| 55
| | 1609.3465
| 1609.3
|1100
| 1100.0
| | 38/15
| [[28/11]], [[33/13]]
| |
|-
|-
| | 56
| 56
| | 1638.6074
| 1638.6
|1120
| 1120.0
| | 18/7
| [[18/7]]
| |
|-
|-
| | 57
| 57
| | 1667.8682
| 1667.9
|1140
| 1140.0
| | 21/8
| [[21/8]]
| |
|-
|-
| | 58
| 58
| | 1697.1291
| 1697.1
|1160
| 1160.0
| | [[8/3]]
| [[8/3]]
| |
|-
|-
| | 59
| 59
| | 1726.3899
| 1726.4
|1180
| 1180.0
| | 19/7
| [[19/7]]
| |
|-
|-
| | 60
| 60
| | 1755.6508
| 1755.7
|1200
| 1200.0
| | [[11/4]]
| [[11/4]]
| |
|-
|-
| | 61
| 61
| | 1784.9116
| 1784.9
|1220
| 1220.0
| | [[14/5]]
| [[14/5]]
| |
|-
|-
| | 62
| 62
| | 1814.1725
| 1814.2
|1240
| 1240.0
| | 20/7
| [[20/7]]
| |
|-
|-
| | 63
| 63
| | 1843.4333
| 1843.4
|1260
| 1260.0
| | 29/10
| [[26/9]]
| |
|-
|-
| | 64
| 64
| | 1872.6942
| 1872.7
|1280
| 1280.0
| | 56/19
| [[27/10]]
| |
|-
|-
| | 65
| 65
| | 1901.9550
| 1902.0
|1300
| 1300.0
| | '''exact [[3/1]]'''
| [[3/1]]
| | [[3/2|just perfect fifth]] plus an octave
|}
|}
==Harmonics==
{{Harmonics in equal
| steps = 65
| num = 3
| denom = 1
| intervals = integer
}}
{{Harmonics in equal
| steps = 65
| num = 3
| denom = 1
| start = 12
| collapsed = 1
| intervals = integer
}}
[[Category:Edt]]
[[Category:Edonoi]]

Revision as of 16:24, 20 March 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 3/1 rather than the 2/1 being just. The octave is about 0.3053 cents compressed. Like 41edo, 65edt is consistent to the 16-integer-limit.

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)

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 55/28, 63/32
41 1199.7 820.0 2/1
42 1229.0 840.0 49/24, 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 27/10
65 1902.0 1300.0 3/1
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