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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 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
! #
! | 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
| [[49/25]], [[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
| [[45/22]], [[49/24]], [[55/27]], [[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
| [[44/15]]
| |
|-
|-
| | 65
| 65
| | 1901.9550
| 1902.0
|1300
| 1300.0
| | '''exact [[3/1]]'''
| [[3/1]]
| | [[3/2|just perfect fifth]] plus an octave
|}
|}


==Harmonics==
== See also ==
{{Harmonics in equal
* [[24edf]] – relative edf
| steps = 65
* [[41edo]] – relative edo
| num = 3
* [[95ed5]] – relative ed5
| denom = 1
* [[106ed6]] – relative ed6
| intervals = integer
* [[147ed12]] – relative ed12
}}
* [[361ed448]] – close to the zeta-optimized tuning for 41edo
{{Harmonics in equal
| steps = 65
| num = 3
| denom = 1
| start = 12
| collapsed = 1
| intervals = integer
}}


[[Category:Edt]]
[[Category:41edo]]
[[Category:Edonoi]]
Retrieved from "https://en.xen.wiki/w/65edt"