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'''[[EDF|Division of the just perfect fifth]] into 45 equal parts''' (45EDF) is related to [[77edo|77 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 1.123 cents stretched and the step size is about 15.599 cents. The patent val has a generally sharp tendency for harmonics up to 16, with the exception for 11.
{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 45 equal parts''' (45EDF) is related to [[77edo]], but with the [[3/2]] rather than the [[2/1]] being [[just]]. The octave is [[Octave stretch|stretched]] by about 1.123 [[cents]] and the step size is about 15.599 cents.  
 
The [[patent val]] has a generally sharp tendency for [[harmonic]]s up to 16, with the exception for 11.


Lookalikes: [[77edo]], [[122edt]]
Lookalikes: [[77edo]], [[122edt]]
== Harmonics ==
{{Harmonics in equal|45|3|2|intervals=prime|columns=9}}
{{Harmonics in equal|45|3|2|intervals=prime|columns=9|start=10|collapsed=1}}


== Intervals ==
== Intervals ==
Line 12: Line 19:
in the 13-limit
in the 13-limit
|-
|-
| style="text-align:right;" |0
| colspan="2" style="text-align:right;" |0
| style="text-align:right;" |0.000
| style="text-align:center;" |1/1
| style="text-align:center;" |1/1
|-
|-
| style="text-align:right;" |1
|1
| style="text-align:right;" |15.599
|15.599
| style="text-align:center;" |81/80, 99/98
|81/80, 99/98
|-
|-
| style="text-align:right;" |2
|2
| style="text-align:right;" |31.198
|31.198
| style="text-align:center;" |64/63, 49/48
|64/63, 49/48
|-
|-
| style="text-align:right;" |3
|3
| style="text-align:right;" |46.797
|46.797
| style="text-align:center;" |33/32, 36/35
|33/32, 36/35
|-
|-
| style="text-align:right;" |4
|4
| style="text-align:right;" |62.396
|62.396
| style="text-align:center;" |28/27, 27/26, 26/25
|28/27, 27/26, 26/25
|-
|-
| style="text-align:right;" |5
|5
| style="text-align:right;" |77.995
|77.995
| style="text-align:center;" |21/20, 22/21, 25/24
|21/20, 22/21, 25/24
|-
|-
| style="text-align:right;" |6
|6
| style="text-align:right;" |93.594
|93.594
| style="text-align:center;" |135/128
|135/128
|-
|-
| style="text-align:right;" |7
|7
| style="text-align:right;" |109.193
|109.193
| style="text-align:center;" |16/15
|16/15
|-
|-
| style="text-align:right;" |8
|8
| style="text-align:right;" |124.792
|124.792
| style="text-align:center;" |15/14, 14/13
|15/14, 14/13
|-
|-
| style="text-align:right;" |9
|9
| style="text-align:right;" |140.391
|140.391
| style="text-align:center;" |13/12
|13/12
|-
|-
| style="text-align:right;" |10
|10
| style="text-align:right;" |155.99
|155.99
| style="text-align:center;" |12/11, 11/10
|12/11, 11/10
|-
|-
| style="text-align:right;" |11
|11
| style="text-align:right;" |171.589
|171.589
| style="text-align:center;" |72/65
|72/65
|-
|-
| style="text-align:right;" |12
|12
| style="text-align:right;" |187.188
|187.188
| style="text-align:center;" |10/9
|10/9
|-
|-
| style="text-align:right;" |13
|13
| style="text-align:right;" |202.787
|202.787
| style="text-align:center;" |9/8
|9/8
|-
|-
| style="text-align:right;" |14
|14
| style="text-align:right;" |218.386
|218.386
| style="text-align:center;" |256/225
|256/225
|-
|-
| style="text-align:right;" |15
|15
| style="text-align:right;" |233.985
|233.985
| style="text-align:center;" |8/7
|8/7
|-
|-
| style="text-align:right;" |16
|16
| style="text-align:right;" |249.584
|249.584
| style="text-align:center;" |15/13
|15/13
|-
|-
| style="text-align:right;" |17
|17
| style="text-align:right;" |265.183
|265.183
| style="text-align:center;" |7/6
|7/6
|-
|-
| style="text-align:right;" |18
|18
| style="text-align:right;" |280.782
|280.782
| style="text-align:center;" |33/28
|33/28
|-
|-
| style="text-align:right;" |19
|19
| style="text-align:right;" |296.381
|296.381
| style="text-align:center;" |32/27, 13/11
|32/27, 13/11
|-
|-
| style="text-align:right;" |20
|20
| style="text-align:right;" |311.98
|311.98
| style="text-align:center;" |6/5
|6/5
|-
|-
| style="text-align:right;" |21
|21
| style="text-align:right;" |327.579
|327.579
| style="text-align:center;" |98/81
|98/81
|-
|-
| style="text-align:right;" |22
|22
| style="text-align:right;" |343.178
|343.178
| style="text-align:center;" |11/9, 39/32
|11/9, 39/32
|-
|-
| style="text-align:right;" |23
|23
| style="text-align:right;" |358.777
|358.777
| style="text-align:center;" |16/13
|16/13
|-
|-
| style="text-align:right;" |24
|24
| style="text-align:right;" |374.376
|374.376
| style="text-align:center;" |56/45, 26/21
|56/45, 26/21
|-
|-
| style="text-align:right;" |25
|25
| style="text-align:right;" |389.975
|389.975
| style="text-align:center;" |5/4
|5/4
|-
|-
| style="text-align:right;" |26
|26
| style="text-align:right;" |405.574
|405.574
| style="text-align:center;" |33/26, 81/64
|33/26, 81/64
|-
|-
| style="text-align:right;" |27
|27
| style="text-align:right;" |420.173
|420.173
| style="text-align:center;" |14/11, 32/25
|14/11, 32/25
|-
|-
| style="text-align:right;" |28
|28
| style="text-align:right;" |436.772
|436.772
| style="text-align:center;" |9/7
|9/7
|-
|-
| style="text-align:right;" |29
|29
| style="text-align:right;" |452.371
|452.371
| style="text-align:center;" |13/10
|13/10
|-
|-
| style="text-align:right;" |30
|30
| style="text-align:right;" |467.97
|467.97
| style="text-align:center;" |21/16
|21/16
|-
|-
| style="text-align:right;" |31
|31
| style="text-align:right;" |483.569
|483.569
| style="text-align:center;" |120/91
|120/91
|-
|-
| style="text-align:right;" |32
|32
| style="text-align:right;" |499.168
|499.168
| style="text-align:center;" |4/3
|4/3
|-
|-
| style="text-align:right;" |33
|33
| style="text-align:right;" |514.767
|514.767
| style="text-align:center;" |27/20
|27/20
|-
|-
| style="text-align:right;" |34
|34
| style="text-align:right;" |530.366
|530.366
| style="text-align:center;" |49/36
|49/36
|-
|-
| style="text-align:right;" |35
|35
| style="text-align:right;" |545.965
|545.965
| style="text-align:center;" |11/8, 15/11
|11/8, 15/11
|-
|-
| style="text-align:right;" |36
|36
| style="text-align:right;" |561.564
|561.564
| style="text-align:center;" |18/13
|18/13
|-
|-
| style="text-align:right;" |37
|37
| style="text-align:right;" |577.163
|577.163
| style="text-align:center;" |7/5
|7/5
|-
|-
| style="text-align:right;" |38
|38
| style="text-align:right;" |592.762
|592.762
| style="text-align:center;" |45/32
|45/32
|-
|-
| style="text-align:right;" |39
|39
| style="text-align:right;" |608.361
|608.361
| style="text-align:center;" |64/45
|64/45
|-
|-
| style="text-align:right;" |40
|40
| style="text-align:right;" |623.96
|623.96
| style="text-align:center;" |10/7
|10/7
|-
|-
| style="text-align:right;" |41
|41
| style="text-align:right;" |639.559
|639.559
| style="text-align:center;" |13/9
|13/9
|-
|-
| style="text-align:right;" |42
|42
| style="text-align:right;" |655.158
|655.158
| style="text-align:center;" |16/11, 22/15
|16/11, 22/15
|-
|-
| style="text-align:right;" |43
|43
| style="text-align:right;" |670.757
|670.757
| style="text-align:center;" |72/49
|72/49
|-
|-
| style="text-align:right;" |44
|44
| style="text-align:right;" |686.356
|686.356
| style="text-align:center;" |40/27
|40/27
|-
|-
| style="text-align:right;" |45
|45
| style="text-align:right;" |701.955
|701.955
| style="text-align:center;" |3/2
|3/2
|-
|-
| style="text-align:right;" |46
|46
| style="text-align:right;" |717.554
|717.554
| style="text-align:center;" |91/60
|91/60
|-
|-
| style="text-align:right;" |47
|47
| style="text-align:right;" |733.153
|733.153
| style="text-align:center;" |32/21
|32/21
|-
|-
| style="text-align:right;" |48
|48
| style="text-align:right;" |748.752
|748.752
| style="text-align:center;" |20/13
|20/13
|-
|-
| style="text-align:right;" |49
|49
| style="text-align:right;" |764.351
|764.351
| style="text-align:center;" |14/9
|14/9
|-
|-
| style="text-align:right;" |50
|50
| style="text-align:right;" |779.95
|779.95
| style="text-align:center;" |11/7, 25/16
|11/7, 25/16
|-
|-
| style="text-align:right;" |51
|51
| style="text-align:right;" |795.549
|795.549
| style="text-align:center;" |52/33, 128/81
|52/33, 128/81
|-
|-
| style="text-align:right;" |52
|52
| style="text-align:right;" |818.148
|818.148
| style="text-align:center;" |8/5
|8/5
|-
|-
| style="text-align:right;" |53
|53
| style="text-align:right;" |826.747
|826.747
| style="text-align:center;" |45/28, 21/13
|45/28, 21/13
|-
|-
| style="text-align:right;" |54
|54
| style="text-align:right;" |842.346
|842.346
| style="text-align:center;" |13/8
|13/8
|-
|-
| style="text-align:right;" |55
|55
| style="text-align:right;" |857.945
|857.945
| style="text-align:center;" |18/11, 64/39
|18/11, 64/39
|-
|-
| style="text-align:right;" |56
|56
| style="text-align:right;" |873.544
|873.544
| style="text-align:center;" |81/49
|81/49
|-
|-
| style="text-align:right;" |57
|57
| style="text-align:right;" |889.143
|889.143
| style="text-align:center;" |5/3
|5/3
|-
|-
| style="text-align:right;" |58
|58
| style="text-align:right;" |904.742
|904.742
| style="text-align:center;" |27/16, 22/13
|27/16, 22/13
|-
|-
| style="text-align:right;" |59
|59
| style="text-align:right;" |920.341
|920.341
| style="text-align:center;" |56/33
|56/33
|-
|-
| style="text-align:right;" |60
|60
| style="text-align:right;" |935.84
|935.84
| style="text-align:center;" |12/7
|12/7
|-
|-
| style="text-align:right;" |61
|61
| style="text-align:right;" |951.539
|951.539
| style="text-align:center;" |26/15
|26/15
|-
|-
| style="text-align:right;" |62
|62
| style="text-align:right;" |967.138
|967.138
| style="text-align:center;" |7/4
|7/4
|-
|-
| style="text-align:right;" |63
|63
| style="text-align:right;" |982.737
|982.737
| style="text-align:center;" |225/128
|225/128
|-
|-
| style="text-align:right;" |64
|64
| style="text-align:right;" |998.336
|998.336
| style="text-align:center;" |16/9
|16/9
|-
|-
| style="text-align:right;" |65
|65
| style="text-align:right;" |1013.935
|1013.935
| style="text-align:center;" |9/5
|9/5
|-
|-
| style="text-align:right;" |66
|66
| style="text-align:right;" |1029.534
|1029.534
| style="text-align:center;" |65/36
|65/36
|-
|-
| style="text-align:right;" |67
|67
| style="text-align:right;" |1045.133
|1045.133
| style="text-align:center;" |11/6, 20/11
|11/6, 20/11
|-
|-
| style="text-align:right;" |68
|68
| style="text-align:right;" |1060.732
|1060.732
| style="text-align:center;" |24/13
|24/13
|-
|-
| style="text-align:right;" |69
|69
| style="text-align:right;" |1076.331
|1076.331
| style="text-align:center;" |28/15
|28/15
|-
|-
| style="text-align:right;" |70
|70
| style="text-align:right;" |1091.93
|1091.93
| style="text-align:center;" |15/8
|15/8
|-
|-
| style="text-align:right;" |71
|71
| style="text-align:right;" |1107.529
|1107.529
| style="text-align:center;" |256/135
|256/135
|-
|-
| style="text-align:right;" |72
|72
| style="text-align:right;" |1123.128
|1123.128
| style="text-align:center;" |40/21, 48/25
|40/21, 48/25
|-
|-
| style="text-align:right;" |73
|73
| style="text-align:right;" |1138.727
|1138.727
| style="text-align:center;" |27/14, 25/13
|27/14, 25/13
|-
|-
| style="text-align:right;" |74
|74
| style="text-align:right;" |1154.326
|1154.326
| style="text-align:center;" |64/33, 35/18
|64/33, 35/18
|-
|-
| style="text-align:right;" |75
|75
| style="text-align:right;" |1169.925
|1169.925
| style="text-align:center;" |63/32, 96/49
|63/32, 96/49
|-
|-
| style="text-align:right;" |76
|76
| style="text-align:right;" |1185.524
|1185.524
| style="text-align:center;" |160/81, 196/99
|160/81, 196/99
|-
|-
| style="text-align:right;" |77
|77
| style="text-align:right;" |1201.123
|1201.123
| style="text-align:center;" |2/1
|/1
|-
|-
|78
|78
Line 376: Line 382:
|9/4
|9/4
|}
|}
[[Category:Edf]]
 
[[Category:Edonoi]]
{{todo|expand}}

Latest revision as of 19:23, 1 August 2025

← 44edf 45edf 46edf →
Prime factorization 32 × 5
Step size 15.599 ¢ 
Octave 77\45edf (1201.12 ¢)
Twelfth 122\45edf (1903.08 ¢)
Consistency limit 10
Distinct consistency limit 10

Division of the just perfect fifth into 45 equal parts (45EDF) is related to 77edo, but with the 3/2 rather than the 2/1 being just. The octave is stretched by about 1.123 cents and the step size is about 15.599 cents.

The patent val has a generally sharp tendency for harmonics up to 16, with the exception for 11.

Lookalikes: 77edo, 122edt

Harmonics

Approximation of prime harmonics in 45edf
Harmonic 2 3 5 7 11 13 17 19 23
Error Absolute (¢) +1.12 +1.12 +5.91 +0.56 -1.98 +5.19 -6.87 +3.36 +0.18
Relative (%) +7.2 +7.2 +37.9 +3.6 -12.7 +33.3 -44.0 +21.5 +1.1
Steps
(reduced) 		77
(32) 		122
(32) 		179
(44) 		216
(36) 		266
(41) 		285
(15) 		314
(44) 		327
(12) 		348
(33)
Approximation of prime harmonics in 45edf
Harmonic 29 31 37 41 43 47 53 59 61
Error Absolute (¢) +4.45 -1.82 +3.85 -2.27 -6.73 -4.73 +5.65 +7.18 -3.74
Relative (%) +28.5 -11.6 +24.7 -14.6 -43.2 -30.3 +36.2 +46.0 -24.0
Steps
(reduced) 		374
(14) 		381
(21) 		401
(41) 		412
(7) 		417
(12) 		427
(22) 		441
(36) 		453
(3) 		456
(6)

Intervals

Degree Cents

Value

Approximate Ratios

in the 13-limit

0 1/1
1 15.599 81/80, 99/98
2 31.198 64/63, 49/48
3 46.797 33/32, 36/35
4 62.396 28/27, 27/26, 26/25
5 77.995 21/20, 22/21, 25/24
6 93.594 135/128
7 109.193 16/15
8 124.792 15/14, 14/13
9 140.391 13/12
10 155.99 12/11, 11/10
11 171.589 72/65
12 187.188 10/9
13 202.787 9/8
14 218.386 256/225
15 233.985 8/7
16 249.584 15/13
17 265.183 7/6
18 280.782 33/28
19 296.381 32/27, 13/11
20 311.98 6/5
21 327.579 98/81
22 343.178 11/9, 39/32
23 358.777 16/13
24 374.376 56/45, 26/21
25 389.975 5/4
26 405.574 33/26, 81/64
27 420.173 14/11, 32/25
28 436.772 9/7
29 452.371 13/10
30 467.97 21/16
31 483.569 120/91
32 499.168 4/3
33 514.767 27/20
34 530.366 49/36
35 545.965 11/8, 15/11
36 561.564 18/13
37 577.163 7/5
38 592.762 45/32
39 608.361 64/45
40 623.96 10/7
41 639.559 13/9
42 655.158 16/11, 22/15
43 670.757 72/49
44 686.356 40/27
45 701.955 3/2
46 717.554 91/60
47 733.153 32/21
48 748.752 20/13
49 764.351 14/9
50 779.95 11/7, 25/16
51 795.549 52/33, 128/81
52 818.148 8/5
53 826.747 45/28, 21/13
54 842.346 13/8
55 857.945 18/11, 64/39
56 873.544 81/49
57 889.143 5/3
58 904.742 27/16, 22/13
59 920.341 56/33
60 935.84 12/7
61 951.539 26/15
62 967.138 7/4
63 982.737 225/128
64 998.336 16/9
65 1013.935 9/5
66 1029.534 65/36
67 1045.133 11/6, 20/11
68 1060.732 24/13
69 1076.331 28/15
70 1091.93 15/8
71 1107.529 256/135
72 1123.128 40/21, 48/25
73 1138.727 27/14, 25/13
74 1154.326 64/33, 35/18
75 1169.925 63/32, 96/49
76 1185.524 160/81, 196/99
77 1201.123 /1
78 1216.722 81/40, 99/49
79 1232.321 128/63, 49/24
80 1247.92 33/16, 72/35
81 1263.519 56/27, 27/13, 52/25
82 1279.118 21/10, 44/21,25/12
83 1294.717 135/64
84 1310.316 32/15
85 1325.915 15/7, 28/13
86 1341.514 13/6
87 1357.113 24/11, 11/5
88 1372.712 144/65
89 1388.311 20/9
90 1403.91 9/4
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