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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]]
| | pseudo-[[9/8]]
Line 55: Line 81:
| | 9
| | 9
| | 234.4876
| | 234.4876
|160.274
| | 63/55
| | 63/55
| | pseudo-[[8/7]]
| | pseudo-[[8/7]]
Line 60: Line 87:
| | 10
| | 10
| | 260.5418
| | 260.5418
|178.0822
| |  
| |  
| | pseudo-[[7/6]]
| | 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]]
| | pseudo-[[4/3]]
Line 110: Line 147:
| | 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]])
| | ([[11/7]])
| |  
| |  
Line 165: Line 213:
| | 31
| | 31
| | 807.6795
| | 807.6795
|552.0548
| |  
| |  
| |  
| |pseudo-[[8/5]]
|-
|-
| | 32
| | 32
| | 833.7337
| | 833.7337
|569.863
| | [[34/21]]
| | [[34/21]]
| |  
| |  
Line 175: Line 225:
| | 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
|}
|}


=Related regular temperament=
==Related regular temperaments==
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.
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.


==46&amp;783 temperament==
===5-limit 46&amp;783===
Comma: |73 -153 73&gt;
Comma: |73 -153 73&gt;


POTE generator: ~|21 -44 21&gt; = 26.0543
POTE generator: ~|21 -44 21&gt; = 26.0543


Map: [&lt;1 0 -1|, &lt;0 73 153|]
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: 46, 783, 829, 1612, 2395, 3128, 4007
EDOs: {{EDOs|46, 737, 783, 829, 875}}


[[Category:Edt]]
== Music ==
[[Category:Edonoi]]
; [[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)
Retrieved from "https://en.xen.wiki/w/73edt"