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'''45EDT''' is the [[Edt|equal division of the third harmonic]] into 45 parts of 42.2657 [[cent|cents]] each, corresponding to 28.3918 [[edo]]. It makes for a strong no-twos 17-limit system, particularly with respect to the tuning of 5, 13, and 17. It tempers out 3125/3087 in the 7-limit, 891/875 and 2475/2401 in the 11-limit, 275/273, 351/343, 847/845 and 2197/2187 in the 13-limit, and 121/119, 459/455 and 2025/2023 in the 17-limit (no-twos subgroup). It is the tenth [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos zeta peak edt]].
{{Infobox ET}}
'''45EDT''' is the [[Edt|equal division of the third harmonic]] into 45 parts of 42.2657 [[cent|cents]] each, corresponding to 28.3918 [[edo]]. It makes for a strong no-twos 17-limit system, particularly with respect to the tuning of 5, 13, and 17. It tempers out 3125/3087 in the 7-limit, 891/875 and 2475/2401 in the 11-limit, 275/273, 351/343, 847/845 and 2197/2187 in the 13-limit, and 121/119, 459/455 and 2025/2023 in the 17-limit (no-twos subgroup). It is the tenth [[the Riemann zeta function and tuning#Removing primes|no-twos zeta peak edt]].


=Intervals of 45EDT=
== Harmonics ==
{{Harmonics in equal
| steps = 45
| num = 3
| denom = 1
| intervals = integer
}}
{{Harmonics in equal
| steps = 45
| num = 3
| denom = 1
| start = 12
| collapsed = 1
| intervals = integer
}}


== Intervals ==
{| class="wikitable"
{| class="wikitable"
|-
|-
| | Degrees
! Degrees
| | Cents
! [[Cent]]s
|hekts
! [[Hekt]]s
| | Approximate Ratios
! Approximate ratios
|-
|-
|  colspan="3"| 0
! colspan="3" | 0
| | <span style="color: #660000;">[[1/1]]</span>
| <span style="color: #660000;">[[1/1]]</span>
|-
|-
| | 1
| 1
| | 42.266
| 42.266
|28.889
| 28.889
| |
|  
|-
|-
| | 2
| 2
| | 84.531
| 84.531
|57.778
| 57.778
| | [[21/20]]
| [[21/20]]
|-
|-
| | 3
| 3
| | 126.797
| 126.797
|86.667
| 86.667
| | [[14/13]], [[15/14]], [[16/15]], 29/27
| [[14/13]], [[15/14]], [[16/15]], 29/27
|-
|-
| | 4
| 4
| | 169.063
| 169.063
|115.556
| 115.556
| | 11/10
| 11/10
|-
|-
| | 5
| 5
| | 211.328
| 211.328
|144.444
| 144.444
| | 9/8
| 9/8
|-
|-
| | 6
| 6
| | 253.594
| 253.594
|173.333
| 173.333
| | [[15/13]]
| [[15/13]]
|-
|-
| | 7
| 7
| | 295.86
| 295.86
|202.222
| 202.222
| | 19/16
| 19/16
|-
|-
| | 8
| 8
| | 338.125
| 338.125
|231.111
| 231.111
| | 17/14
| 17/14
|-
|-
| | 9
| 9
| | 380.391
| 380.391
|260
| 260
| | <span style="color: #660000;">[[5/4]]</span>
| <span style="color: #660000;">[[5/4]]</span>
|-
|-
| | 10
| 10
| | 422.657
| 422.657
|288.889
| 288.889
| |14/11
| 14/11
|-
|-
| | 11
| 11
| | 464.922
| 464.922
|317.778
| 317.778
| | [[21/16]], [[17/13]]
| [[21/16]], [[17/13]]
|-
|-
| | 12
| 12
| | 507.188
| 507.188
|336.667
| 336.667
| | [[4/3]]
| [[4/3]]
|-
|-
| | 13
| 13
| | 549.454
| 549.454
|375.556
| 375.556
| | 11/8
| 11/8
|-
|-
| | 14
| 14
| | 591.719
| 591.719
|304.444
| 304.444
| | 7/5
| 7/5
|-
|-
| | 15
| 15
| | 633.985
| 633.985
|433.333
| 433.333
| | [[13/9]]
| [[13/9]]
|-
|-
| | 16
| 16
| | 676.251
| 676.251
|462.222
| 462.222
| |40/27. 189/128
| 40/27. 189/128
|-
|-
| | 17
| 17
| | 718.516
| 718.516
|491.111
| 491.111
| | 50/33
| 50/33
|-
|-
| | 18
| 18
| | 760.782
| 760.782
|520
| 520
| | <span style="color: #660000;">[[14/9]]</span>
| <span style="color: #660000;">[[14/9]]</span>
|-
|-
| | 19
| 19
| | 803.048
| 803.048
|548.889
| 548.889
| | 8/5
| 8/5
|-
|-
| | 20
| 20
| | 845.313
| 845.313
|577.778
| 577.778
| |13/8
| 13/8
|-
|-
| | 21
| 21
| | 887.579
| 887.579
|606.667
| 606.667
| | [[5/3]], 17/11
| [[5/3]], 17/11
|-
|-
| | 22
| 22
| | 929.845
| 929.845
|635.556
| 635.556
| | 12/7
| 12/7
|-
|-
| | 23
| 23
| | 972.110
| 972.110
|664.444
| 664.444
| | 7/4
| 7/4
|-
|-
| | 24
| 24
| | 1014.376
| 1014.376
|693.333
| 693.333
| | [[9/5]], 33/17
| [[9/5]], 33/17
|-
|-
| | 25
| 25
| | 1056.642
| 1056.642
|722.222
| 722.222
| |24/13
| 24/13
|-
|-
| | 26
| 26
| | 1098.907
| 1098.907
|751.111
| 751.111
| | 17/9
| 17/9
|-
|-
| | 27
| 27
| | 1141.173
| 1141.173
|780
| 780
| | <span style="color: #660000;">[[27/14]]</span>
| <span style="color: #660000;">[[27/14]]</span>
|-
|-
| | 28
| 28
| | 1183.439
| 1183.439
|808.889
| 808.889
| |99/50
| 99/50
|-
|-
| | 29
| 29
| | 1225.704
| 1225.704
|837.778
| 837.778
| |81/40, 128/63
| 81/40, 128/63
|-
|-
| | 30
| 30
| | 1267.97
| 1267.97
|866.667
| 866.667
| | <span style="color: #660000;">[[27/26|27/13]]</span>
| <span style="color: #660000;">[[27/26|27/13]]</span>
|-
|-
| | 31
| 31
| | 1310.236
| 1310.236
|895.556
| 895.556
| | 32/15
| 32/15
|-
|-
| | 32
| 32
| | 1352.501
| 1352.501
|924.444
| 924.444
| | 24/11
| 24/11
|-
|-
| | 33
| 33
| | 1394.767
| 1394.767
|953.333
| 953.333
| | <span style="color: #660000;">[[9/4]]</span> ([[9/8]] plus an octave)
| <span style="color: #660000;">[[9/4]]</span> ([[9/8]] plus an octave)
|-
|-
| | 34
| 34
| | 1437.033
| 1437.033
|982.222
| 982.222
| | 16/7, 39/17
| 16/7, 39/17
|-
|-
| | 35
| 35
| | 1479.298
| 1479.298
|1011.111
| 1011.111
| |33/14
| 33/14
|-
|-
| | 36
| 36
| | 1521.564
| 1521.564
|1040
| 1040
| | <span style="color: #660000;">[[12/5]]</span> (<span style="color: #660000;">[[6/5]]</span> plus an octave)
| <span style="color: #660000;">[[12/5]]</span> (<span style="color: #660000;">[[6/5]]</span> plus an octave)
|-
|-
| | 37
| 37
| | 1563.83
| 1563.83
|1068.889
| 1068.889
| |42/17
| 42/17
|-
|-
| | 38
| 38
| | 1606.095
| 1606.095
|1097.778
| 1097.778
| |48/19
| 48/19
|-
|-
| | 39
| 39
| | 1648.361
| 1648.361
|1126.667
| 1126.667
| | <span style="color: #660000;">[[13/5]]</span> ([[13/10]] plus an octave)
| <span style="color: #660000;">[[13/5]]</span> ([[13/10]] plus an octave)
|-
|-
| | 40
| 40
| | 1690.627
| 1690.627
|1155.556
| 1155.556
| | [[8/3]]
| [[8/3]]
|-
|-
| | 41
| 41
| | 1732.892
| 1732.892
|1184.444
| 1184.444
| | 30/11
| 30/11
|-
|-
| | 42
| 42
| | 1775.158
| 1775.158
|1213.333
| 1213.333
| | <span style="color: #660000;">39/14,[[14/5]]</span> ([[7/5]] plus an octave), 45/16, 81/29
| <span style="color: #660000;">39/14, [[14/5]]</span> ([[7/5]] plus an octave), 45/16, 81/29
|-
|-
| | 43
| 43
| | 1817.424
| 1817.424
|1242.222
| 1242.222
| | [[10/7|20/7]]
| [[10/7|20/7]]
|-
|-
| | 44
| 44
| | 1859.689
| 1859.689
|1271.111
| 1271.111
| |  
|  
|-
|-
| | 45
| 45
| | 1901.955
| 1901.955
|1300
| 1300
| | <span style="color: #660000;">[[3/1]]</span>
| <span style="color: #660000;">[[3/1]]</span>
|}
|}


[[Category:Edt]]
{{todo|expand}}
[[Category:Edonoi]]

Latest revision as of 19:23, 1 August 2025

← 44edt 45edt 46edt →
Prime factorization 32 × 5
Step size 42.2657 ¢ 
Octave 28\45edt (1183.44 ¢)
Consistency limit 3
Distinct consistency limit 3

45EDT is the equal division of the third harmonic into 45 parts of 42.2657 cents each, corresponding to 28.3918 edo. It makes for a strong no-twos 17-limit system, particularly with respect to the tuning of 5, 13, and 17. It tempers out 3125/3087 in the 7-limit, 891/875 and 2475/2401 in the 11-limit, 275/273, 351/343, 847/845 and 2197/2187 in the 13-limit, and 121/119, 459/455 and 2025/2023 in the 17-limit (no-twos subgroup). It is the tenth no-twos zeta peak edt.

Harmonics

Approximation of harmonics in 45edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -16.6 +0.0 +9.1 +3.2 -16.6 +12.4 -7.4 +0.0 -13.3 -9.3 +9.1
Relative (%) -39.2 +0.0 +21.6 +7.6 -39.2 +29.4 -17.6 +0.0 -31.6 -22.0 +21.6
Steps
(reduced) 		28
(28) 		45
(0) 		57
(12) 		66
(21) 		73
(28) 		80
(35) 		85
(40) 		90
(0) 		94
(4) 		98
(8) 		102
(12)
Approximation of harmonics in 45edt
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -2.6 -4.1 +3.2 +18.3 -2.1 -16.6 +16.6 +12.4 +12.4 +16.4 -18.3
Relative (%) -6.2 -9.8 +7.6 +43.3 -5.1 -39.2 +39.4 +29.3 +29.4 +38.9 -43.2
Steps
(reduced) 		105
(15) 		108
(18) 		111
(21) 		114
(24) 		116
(26) 		118
(28) 		121
(31) 		123
(33) 		125
(35) 		127
(37) 		128
(38)

Intervals

Degrees Cents Hekts Approximate ratios
0 1/1
1 42.266 28.889
2 84.531 57.778 21/20
3 126.797 86.667 14/13, 15/14, 16/15, 29/27
4 169.063 115.556 11/10
5 211.328 144.444 9/8
6 253.594 173.333 15/13
7 295.86 202.222 19/16
8 338.125 231.111 17/14
9 380.391 260 5/4
10 422.657 288.889 14/11
11 464.922 317.778 21/16, 17/13
12 507.188 336.667 4/3
13 549.454 375.556 11/8
14 591.719 304.444 7/5
15 633.985 433.333 13/9
16 676.251 462.222 40/27. 189/128
17 718.516 491.111 50/33
18 760.782 520 14/9
19 803.048 548.889 8/5
20 845.313 577.778 13/8
21 887.579 606.667 5/3, 17/11
22 929.845 635.556 12/7
23 972.110 664.444 7/4
24 1014.376 693.333 9/5, 33/17
25 1056.642 722.222 24/13
26 1098.907 751.111 17/9
27 1141.173 780 27/14
28 1183.439 808.889 99/50
29 1225.704 837.778 81/40, 128/63
30 1267.97 866.667 27/13
31 1310.236 895.556 32/15
32 1352.501 924.444 24/11
33 1394.767 953.333 9/4 (9/8 plus an octave)
34 1437.033 982.222 16/7, 39/17
35 1479.298 1011.111 33/14
36 1521.564 1040 12/5 (6/5 plus an octave)
37 1563.83 1068.889 42/17
38 1606.095 1097.778 48/19
39 1648.361 1126.667 13/5 (13/10 plus an octave)
40 1690.627 1155.556 8/3
41 1732.892 1184.444 30/11
42 1775.158 1213.333 39/14, 14/5 (7/5 plus an octave), 45/16, 81/29
43 1817.424 1242.222 20/7
44 1859.689 1271.111
45 1901.955 1300 3/1
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