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'''[[Edt|Division of the third harmonic]] into 73 equal parts''' (73edt) is related to [[46edo|46 edo]], but with the 3/1 rather than the 2/1 being just. The octave is about 1.5078 cents compressed and the step size is about 26.0542 cents. It is consistent to the [[17-odd-limit|18-integer-limit]]. In comparison, 46edo is only consistent up to the [[13-odd-limit|14-integer-limit]].
{{Infobox ET}}
'''[[Edt|Division of the third harmonic]] into 73 equal parts''' (73EDT) is related to [[46edo|46 edo]], but with the 3/1 rather than the 2/1 being just. The octave is about 1.5078 cents compressed and the step size is about 26.0542 cents. It is consistent to the [[17-odd-limit|18-integer-limit]]. In comparison, 46edo is only consistent up to the [[13-odd-limit|14-integer-limit]].


==Harmonics==
{{Harmonics in equal
| steps = 73
| num = 3
| denom = 1
| intervals = integer
}}
{{Harmonics in equal
| steps = 73
| num = 3
| denom = 1
| start = 12
| collapsed = 1
| intervals = integer
}}
==Intervals==
{| class="wikitable"
{| class="wikitable"
|-
|-
! | degree
! | degree
! | cents value
! | cents value
!hekts
! | corresponding <br>JI intervals
! | corresponding <br>JI intervals
! | comments
! | comments
|-
|-
| | 0
! colspan="3" | 0
| | 0.0000
| | '''exact [[1/1]]'''
| | '''exact [[1/1]]'''
| |  
| |  
Line 15: Line 33:
| | 1
| | 1
| | 26.0542
| | 26.0542
|17.8082
| | 66/65
| | 66/65
| |  
| |  
Line 20: Line 39:
| | 2
| | 2
| | 52.1084
| | 52.1084
|35.6164
| | 34/33
| | 34/33
| |  
| |  
Line 25: Line 45:
| | 3
| | 3
| | 78.1625
| | 78.1625
|53.4247
| | 68/65
| | 68/65
| |  
| |  
Line 30: Line 51:
| | 4
| | 4
| | 104.2167
| | 104.2167
|71.2329
| | [[17/16]]
| | [[17/16]]
| |  
| |  
Line 35: Line 57:
| | 5
| | 5
| | 130.2709
| | 130.2709
|89.0411
| | 55/51
| | 55/51
| |  
| |  
Line 40: Line 63:
| | 6
| | 6
| | 156.3251
| | 156.3251
| |  
|106.8493
| | 23/21
| |  
| |  
|-
|-
| | 7
| | 7
| | 182.3792
| | 182.3792
|124.6575
| | [[10/9]]
| | [[10/9]]
| |  
| |  
Line 50: Line 75:
| | 8
| | 8
| | 208.4334
| | 208.4334
|142.46575
| | 44/39
| | 44/39
| |  
| | pseudo-[[9/8]]
|-
|-
| | 9
| | 9
| | 234.4876
| | 234.4876
| |  
|160.274
| |  
| | 63/55
| | pseudo-[[8/7]]
|-
|-
| | 10
| | 10
| | 260.5418
| | 260.5418
|178.0822
| |  
| |  
| |  
| | pseudo-[[7/6]]
|-
|-
| | 11
| | 11
| | 286.5960
| | 286.596
| |  
|195.8904
| | 13/11
| |  
| |  
|-
|-
| | 12
| | 12
| | 312.6501
| | 312.6501
|213.6986
| |  
| |  
| | pseudo-[[6/5]]
| | pseudo-[[6/5]]
Line 75: Line 105:
| | 13
| | 13
| | 338.7043
| | 338.7043
| |  
|231.50685
| | 17/14
| |  
| |  
|-
|-
| | 14
| | 14
| | 364.7585
| | 364.7585
|249.3151
| | 100/81
| | 100/81
| |  
| |  
Line 85: Line 117:
| | 15
| | 15
| | 390.8127
| | 390.8127
|267.1233
| |  
| |  
| | pseudo-[[5/4]]
| | pseudo-[[5/4]]
Line 90: Line 123:
| | 16
| | 16
| | 416.8668
| | 416.8668
|284.9315
| | [[14/11]]
| | [[14/11]]
| |  
| |  
|-
|-
| | 17
| | 17
| | 442.9210
| | 442.921
|302.7397
| | 31/24
| | 31/24
| |  
| |  
Line 100: Line 135:
| | 18
| | 18
| | 468.9752
| | 468.9752
| |  
|320.54795
| | 21/16
| |  
| |  
|-
|-
| | 19
| | 19
| | 495.0294
| | 495.0294
|338.3562
| |  
| |  
| |  
| | pseudo-[[4/3]]
|-
|-
| | 20
| | 20
| | 521.0836
| | 521.0836
| |  
|356.1644
| |27/20
| |  
| |  
|-
|-
| | 21
| | 21
| | 547.1377
| | 547.1377
| |  
|373.9726
| | 11/8
| |  
| |  
|-
|-
| | 22
| | 22
| | 573.1919
| | 573.1919
|391.7808
| | 39/28
| | 39/28
| |  
| |pseudo-[[7/5]]
|-
|-
| | 23
| | 23
| | 599.2461
| | 599.2461
|409.589
| | 140/99
| | 140/99
| |  
| |  
Line 130: Line 171:
| | 24
| | 24
| | 625.3003
| | 625.3003
| |  
|427.3973
| |  
| |56/39
| | pseudo-[[10/7]]
|-
|-
| | 25
| | 25
| | 651.3545
| | 651.3545
| |  
|445.2055
| |16/11
| |  
| |  
|-
|-
| | 26
| | 26
| | 677.4086
| | 677.4086
| |  
|463.0137
| |40/27
| |  
| |  
|-
|-
| | 27
| | 27
| | 703.4628
| | 703.4628
|480.8219
| |  
| |  
| | pseudo-[[3/2]]
| | pseudo-[[3/2]]
|-
|-
| | 28
| | 28
| | 729.5170
| | 729.517
|498.6301
| | [[32/21]]
| | [[32/21]]
| |  
| |  
Line 155: Line 201:
| | 29
| | 29
| | 755.5712
| | 755.5712
| |  
|516.4384
| | 48/31
| |  
| |  
|-
|-
| | 30
| | 30
| | 781.6253
| | 781.6253
| |  
|534.2466
| | ([[11/7]])
| |  
| |  
|-
|-
| | 31
| | 31
| | 807.6795
| | 807.6795
|552.0548
| |  
| |  
| |  
| |pseudo-[[8/5]]
|-
|-
| | 32
| | 32
| | 833.7337
| | 833.7337
| |  
|569.863
| | [[34/21]]
| |  
| |  
|-
|-
| | 33
| | 33
| | 859.7879
| | 859.7879
| |  
|587.6712
| | 28/17
| |  
| |  
|-
|-
| | 34
| | 34
| | 885.8421
| | 885.8421
|605.47945
| |  
| |  
| | pseudo-[[5/3]]
| | pseudo-[[5/3]]
Line 185: Line 237:
| | 35
| | 35
| | 911.8962
| | 911.8962
| |  
|623.2877
| | 22/13
| |  
| |  
|-
|-
| | 36
| | 36
| | 937.9504
| | 937.9504
|641.0959
| |  
| |  
| |  
| |pseudo-[[12/7]]
|-
|-
| | 37
| | 37
| | 964.0046
| | 964.0046
| |  
|658.9041
| |  
| | 110/63
| | pseudo-[[7/4]]
|-
|-
| | 38
| | 38
| | 990.0588
| | 990.0588
| |  
|676.7123
| |  
| | 39/22
| | pseudo-[[16/9]]
|-
|-
| | 39
| | 39
| | 1016.1129
| | 1016.1129
| |
|694.52055
| | pseudo-[[9/5]]
| |[[9/5]]
| |
|-
|-
| | 40
| | 40
| | 1042.1671
| | 1042.1671
|712.3288
| | 42/23
| | 42/23
| |  
| |  
Line 215: Line 273:
| | 41
| | 41
| | 1068.2213
| | 1068.2213
| |  
|730.137
| |102/55
| |  
| |  
|-
|-
| | 42
| | 42
| | 1094.2755
| | 1094.2755
| |  
|747.9452
| | 17/8
| |  
| |  
|-
|-
| | 43
| | 43
| | 1120.3297
| | 1120.3297
| |  
|765.7534
| | 65/34
| |  
| |  
|-
|-
| | 44
| | 44
| | 1146.3838
| | 1146.3838
|783.5616
| | 64/33
| | 64/33
| |  
| |  
|-
|-
| | 45
| | 45
| | 1172.4380
| | 1172.438
|801.3699
| | 63/32
| | 63/32
| |  
| |  
Line 240: Line 303:
| | 46
| | 46
| | 1198.4922
| | 1198.4922
|819.1781
| |  
| |  
| | pseudo-[[octave]]
| | pseudo-[[octave]]
Line 245: Line 309:
| | 47
| | 47
| | 1224.5464
| | 1224.5464
| |  
|836.9863
| | 81/40
| |  
| |  
|-
|-
| | 48
| | 48
| | 1250.6005
| | 1250.6005
|854.7945
| | 35/17
| | 35/17
| |  
| |  
Line 255: Line 321:
| | 49
| | 49
| | 1276.6547
| | 1276.6547
|872.6027
| | 23/11
| | 23/11
| |  
| |  
Line 260: Line 327:
| | 50
| | 50
| | 1302.7089
| | 1302.7089
| |  
|890.411
| | 17/8
| |  
| |  
|-
|-
| | 51
| | 51
| | 1328.7631
| | 1328.7631
|908.2192
| | [[14/13|28/13]]
| | [[14/13|28/13]]
| |  
| |  
Line 270: Line 339:
| | 52
| | 52
| | 1354.8173
| | 1354.8173
| |  
|926.0274
| | 24/11
| |  
| |  
|-
|-
| | 53
| | 53
| | 1380.8714
| | 1380.8714
| |  
|943.8356
| | 20/9
| |  
| |  
|-
|-
| | 54
| | 54
| | 1406.9256
| | 1406.9256
| |  
|961.6438
| | 9/4
| |  
| |  
|-
|-
| | 55
| | 55
| | 1432.9798
| | 1432.9798
| |  
|979.45205
| | 16/7
| |  
| |  
|-
|-
| | 56
| | 56
| | 1459.0340
| | 1459.034
| |
|997.2603
| |  
| |  
| |pseudo-7/3
|-
|-
| | 57
| | 57
| | 1485.0882
| | 1485.0882
| |  
|1015.0685
| | 26/11
| |  
| |  
|-
|-
| | 58
| | 58
| | 1511.1423
| | 1511.1423
|1032.8767
| |  
| |  
| | pseudo-[[12/5]]
| | pseudo-[[12/5]]
Line 305: Line 381:
| | 59
| | 59
| | 1537.1965
| | 1537.1965
| |  
|1050.6849
| | 17/7
| |  
| |  
|-
|-
| | 60
| | 60
| | 1563.2507
| | 1563.2507
| |  
|1068.49315
| | 42/17
| |  
| |  
|-
|-
| | 61
| | 61
| | 1589.3049
| | 1589.3049
|1086.3014
| |  
| |  
| | pseudo-[[5/2]]
| | pseudo-[[5/2]]
|-
|-
| | 62
| | 62
| | 1615.3590
| | 1615.359
| |  
|1104.1096
| | 28/11
| |  
| |  
|-
|-
| | 63
| | 63
| | 1641.4132
| | 1641.4132
|1121.9178
| |  
| |  
| |  
| |pseudo-18/7
|-
|-
| | 64
| | 64
| | 1667.4674
| | 1667.4674
| |  
|1139.726
| | 21/8
| |  
| |  
|-
|-
| | 65
| | 65
| | 1693.5216
| | 1693.5216
| |  
|1157.53425
| | 8/3
| |  
| |  
|-
|-
| | 66
| | 66
| | 1719.5758
| | 1719.5758
|1175.3425
| | [[27/20|27/10]]
| | [[27/20|27/10]]
| |  
| |  
Line 345: Line 429:
| | 67
| | 67
| | 1745.6299
| | 1745.6299
| |  
|1193.1507
| | 11/4
| |  
| |  
|-
|-
| | 68
| | 68
| | 1771.6841
| | 1771.6841
| |  
|1210.9589
| | 39/14
| |  
| |  
|-
|-
| | 69
| | 69
| | 1797.7383
| | 1797.7383
|1228.7671
| | [[24/17|48/17]]
| | [[24/17|48/17]]
| |  
| |  
Line 360: Line 447:
| | 70
| | 70
| | 1823.7925
| | 1823.7925
| |  
|1246.5753
| |112/39
| |  
| |  
|-
|-
| | 71
| | 71
| | 1849.8466
| | 1849.8466
|1264.3836
| | 99/34
| | 99/34
| |  
| |  
Line 370: Line 459:
| | 72
| | 72
| | 1875.9008
| | 1875.9008
|1282.1918
| | 65/22
| | 65/22
| |  
| |  
Line 375: Line 465:
| | 73
| | 73
| | 1901.9550
| | 1901.9550
|1300
| | '''exact [[3/1]]'''
| | '''exact [[3/1]]'''
| | [[3/2|just perfect fifth]] plus an octave
| | [[3/2|just perfect fifth]] plus an octave
|}
|}


[[Category:Edt]]
==Related regular temperaments==
[[Category:Edonoi]]
73edt is also related to the microtemperament which tempers out |73 -153 73&gt; in the 5-limit, which is supported by 46, 783, 829, 1612, 2395, 3128, and 4007 EDOs.
 
===5-limit 46&amp;783===
Comma: |73 -153 73&gt;
 
POTE generator: ~|21 -44 21&gt; = 26.0543
 
Mapping: [&lt;1 0 -1|, &lt;0 73 153|]
 
EDOs: {{EDOs|46, 737, 783, 829, 875, 1612, 2395, 2441, 3128, 4007, 5573, 6402}}
 
===7-limit 46&amp;783===
Commas: 4375/4374, |-92 20 3 19&gt;
 
POTE generator: ~335544320/330812181 = 26.0533
 
Mapping: [&lt;1 0 -1 5|, &lt;0 73 153 -101|]
 
EDOs: {{EDOs|46, 691, 737, 783, 829, 1520, 1612}}
 
===11-limit 46&amp;783===
Commas: 4375/4374, 806736/805255, 2097152/2096325
 
POTE generator: ~3072/3025 = 26.0542
 
Mapping: [&lt;1 0 -1 5 6|, &lt;0 73 153 -101 -117|]
 
EDOs: {{EDOs|46, 737, 783, 829, 875}}
 
== Music ==
; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=N7inULohKQI ''Fugue for that other kind of string quartet in 73EDT BPS<nowiki>[</nowiki>9<nowiki>]</nowiki> (Sensi Extension) in LsLssLsLs "Moll I"''] (2024)

Latest revision as of 19:23, 1 August 2025

← 72edt 73edt 74edt →
Prime factorization 73 (prime)
Step size 26.0542 ¢ 
Octave 46\73edt (1198.49 ¢)
Consistency limit 18
Distinct consistency limit 11

Division of the third harmonic into 73 equal parts (73EDT) is related to 46 edo, but with the 3/1 rather than the 2/1 being just. The octave is about 1.5078 cents compressed and the step size is about 26.0542 cents. It is consistent to the 18-integer-limit. In comparison, 46edo is only consistent up to the 14-integer-limit.

Harmonics

Approximation of harmonics in 73edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.5 +0.0 -3.0 +1.5 -1.5 -7.8 -4.5 +0.0 -0.0 -8.7 -3.0
Relative (%) -5.8 +0.0 -11.6 +5.7 -5.8 -30.1 -17.4 +0.0 -0.1 -33.4 -11.6
Steps
(reduced) 		46
(46) 		73
(0) 		92
(19) 		107
(34) 		119
(46) 		129
(56) 		138
(65) 		146
(0) 		153
(7) 		159
(13) 		165
(19)
Approximation of harmonics in 73edt
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -11.3 -9.3 +1.5 -6.0 -6.8 -1.5 +9.1 -1.5 -7.8 -10.2 -9.0
Relative (%) -43.4 -35.9 +5.7 -23.1 -26.0 -5.8 +34.9 -5.9 -30.1 -39.2 -34.6
Steps
(reduced) 		170
(24) 		175
(29) 		180
(34) 		184
(38) 		188
(42) 		192
(46) 		196
(50) 		199
(53) 		202
(56) 		205
(59) 		208
(62)

Intervals

degree cents value hekts corresponding
JI intervals 		comments
0 exact 1/1
1 26.0542 17.8082 66/65
2 52.1084 35.6164 34/33
3 78.1625 53.4247 68/65
4 104.2167 71.2329 17/16
5 130.2709 89.0411 55/51
6 156.3251 106.8493 23/21
7 182.3792 124.6575 10/9
8 208.4334 142.46575 44/39 pseudo-9/8
9 234.4876 160.274 63/55 pseudo-8/7
10 260.5418 178.0822 pseudo-7/6
11 286.596 195.8904 13/11
12 312.6501 213.6986 pseudo-6/5
13 338.7043 231.50685 17/14
14 364.7585 249.3151 100/81
15 390.8127 267.1233 pseudo-5/4
16 416.8668 284.9315 14/11
17 442.921 302.7397 31/24
18 468.9752 320.54795 21/16
19 495.0294 338.3562 pseudo-4/3
20 521.0836 356.1644 27/20
21 547.1377 373.9726 11/8
22 573.1919 391.7808 39/28 pseudo-7/5
23 599.2461 409.589 140/99
24 625.3003 427.3973 56/39 pseudo-10/7
25 651.3545 445.2055 16/11
26 677.4086 463.0137 40/27
27 703.4628 480.8219 pseudo-3/2
28 729.517 498.6301 32/21
29 755.5712 516.4384 48/31
30 781.6253 534.2466 (11/7)
31 807.6795 552.0548 pseudo-8/5
32 833.7337 569.863 34/21
33 859.7879 587.6712 28/17
34 885.8421 605.47945 pseudo-5/3
35 911.8962 623.2877 22/13
36 937.9504 641.0959 pseudo-12/7
37 964.0046 658.9041 110/63 pseudo-7/4
38 990.0588 676.7123 39/22 pseudo-16/9
39 1016.1129 694.52055 9/5
40 1042.1671 712.3288 42/23
41 1068.2213 730.137 102/55
42 1094.2755 747.9452 17/8
43 1120.3297 765.7534 65/34
44 1146.3838 783.5616 64/33
45 1172.438 801.3699 63/32
46 1198.4922 819.1781 pseudo-octave
47 1224.5464 836.9863 81/40
48 1250.6005 854.7945 35/17
49 1276.6547 872.6027 23/11
50 1302.7089 890.411 17/8
51 1328.7631 908.2192 28/13
52 1354.8173 926.0274 24/11
53 1380.8714 943.8356 20/9
54 1406.9256 961.6438 9/4
55 1432.9798 979.45205 16/7
56 1459.034 997.2603 pseudo-7/3
57 1485.0882 1015.0685 26/11
58 1511.1423 1032.8767 pseudo-12/5
59 1537.1965 1050.6849 17/7
60 1563.2507 1068.49315 42/17
61 1589.3049 1086.3014 pseudo-5/2
62 1615.359 1104.1096 28/11
63 1641.4132 1121.9178 pseudo-18/7
64 1667.4674 1139.726 21/8
65 1693.5216 1157.53425 8/3
66 1719.5758 1175.3425 27/10
67 1745.6299 1193.1507 11/4
68 1771.6841 1210.9589 39/14
69 1797.7383 1228.7671 48/17
70 1823.7925 1246.5753 112/39
71 1849.8466 1264.3836 99/34
72 1875.9008 1282.1918 65/22
73 1901.9550 1300 exact 3/1 just perfect fifth plus an octave

Related regular temperaments

73edt is also related to the microtemperament which tempers out |73 -153 73> in the 5-limit, which is supported by 46, 783, 829, 1612, 2395, 3128, and 4007 EDOs.

5-limit 46&783

Comma: |73 -153 73>

POTE generator: ~|21 -44 21> = 26.0543

Mapping: [<1 0 -1|, <0 73 153|]

EDOs: 46, 737, 783, 829, 875, 1612, 2395, 2441, 3128, 4007, 5573, 6402

7-limit 46&783

Commas: 4375/4374, |-92 20 3 19>

POTE generator: ~335544320/330812181 = 26.0533

Mapping: [<1 0 -1 5|, <0 73 153 -101|]

EDOs: 46, 691, 737, 783, 829, 1520, 1612

11-limit 46&783

Commas: 4375/4374, 806736/805255, 2097152/2096325

POTE generator: ~3072/3025 = 26.0542

Mapping: [<1 0 -1 5 6|, <0 73 153 -101 -117|]

EDOs: 46, 737, 783, 829, 875

Music

Ray Perlner
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