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'''[[Edt|Division of the third harmonic]] into 71 equal parts''' (71edt) is related to [[45edo|45 edo]], but with the 3/1 rather than the 2/1 being just. The octave is about 5.4644 cents stretched and the step size is about 26.7881 cents. It is related to the 13-limit temperament which tempers out 540/539, 1575/1573, 2200/2197, and 4375/4374, which is supported by [[45edo]] (45ef val), [[179edo]] (179ef val), [[224edo]], [[269edo]] (269ce val), and [[403edo]] (403def val).
{{Infobox ET}}
'''71EDT''' is the [[Edt|equal division of the third harmonic]] into 71 parts of 26.7881 [[cent|cents]] each, corresponding to 44.7960 [[edo]] (45edo with 5.4644 cents octave stretch). It is related to the 13-limit temperament which tempers out 540/539, 1575/1573, 2200/2197, and 4375/4374, which is supported by [[45edo]] (45ef val), [[179edo]] (179ef val), [[224edo]], [[269edo]] (269ce val), and [[403edo]] (403def val).


71EDT is the 13th [[the Riemann zeta function and tuning#Removing primes|no-twos zeta peak EDT]].
== Harmonics ==
{{Harmonics in equal
| steps = 71
| num = 3
| denom = 1
| intervals = prime
}}
{{Harmonics in equal
| steps = 71
| num = 3
| denom = 1
| start = 12
| collapsed = 1
| intervals = prime
}}
== Intervals ==
{| class="wikitable"
{| class="wikitable"
|-
|-
! | degree
! Degree
! | cents value
! [[Cent]]s
! | corresponding <br>JI intervals
! [[Hekt]]s
! | comments
! Corresponding<br />JI intervals
! Comments
|-
|-
| | 0
! colspan="3" | 0
| | 0.0000
| '''exact [[1/1]]'''
| | '''exact [[1/1]]'''
|  
| |  
|-
|-
| | 1
| 1
| | 26.7881
| 26.7881
| | 66/65
| 18.3099
| |  
| 66/65
|  
|-
|-
| | 2
| 2
| | 53.5762
| 53.5762
| | 65/63
| 36.6197
| |  
| 65/63
|  
|-
|-
| | 3
| 3
| | 80.3643
| 80.3643
| | [[22/21]]
| 54.9296
| |  
| [[22/21]]
|  
|-
|-
| | 4
| 4
| | 107.1524
| 107.1524
| | 117/110
| 73.2394
| |  
| 117/110
|  
|-
|-
| | 5
| 5
| | 133.9405
| 133.9405
| | [[27/25]]
| 91.5493
| |  
| [[27/25]]
|  
|-
|-
| | 6
| 6
| | 160.7286
| 160.7286
| | 169/154
| 109.85915
| |  
| 169/154
|  
|-
|-
| | 7
| 7
| | 187.5167
| 187.5167
| | 39/35
| 128.169
| |  
| 39/35
|  
|-
|-
| | 8
| 8
| | 214.3048
| 214.3048
| | 147/130, 198/175
| 146.4789
| |  
| 147/130, 198/175
|  
|-
|-
| | 9
| 9
| | 241.0929
| 241.0929
| | 169/147
| 164.7887
| |  
| 169/147
|  
|-
|-
| | 10
| 10
| | 267.8810
| 267.8810
| | [[7/6]]
| 183.0986
| |  
| [[7/6]]
|  
|-
|-
| | 11
| 11
| | 294.6691
| 294.6691
| | 77/65
| 201.40845
| |  
| 77/65
|  
|-
|-
| | 12
| 12
| | 321.4572
| 321.4572
| |  
| 219.7183
| |  
| 65/54
|  
|-
|-
| | 13
| 13
| | 348.2453
| 348.2453
| | [[11/9]]
| 238.0282
| |  
| [[11/9]]
|  
|-
|-
| | 14
| 14
| | 375.0334
| 375.0334
| | 273/220
| 256.338
| |  
| 273/220
|  
|-
|-
| | 15
| 15
| | 401.8215
| 401.8215
| | 63/50
| 274.6479
| |  
| 63/50
|  
|-
|-
| | 16
| 16
| | 428.6096
| 428.6096
| |  
| 292.95775
| |  
| 169/132
|  
|-
|-
| | 17
| 17
| | 455.3977
| 455.3977
| | [[13/10]]
| 311.2676
| |  
| [[13/10]]
|  
|-
|-
| | 18
| 18
| | 482.1858
| 482.1858
| | 33/25
| 329.5775
| |  
| 33/25
|  
|-
|-
| | 19
| 19
| | 508.9739
| 508.9739
| |  
| 347.8873
| |  
| 169/126
|  
|-
|-
| | 20
| 20
| | 535.7620
| 535.7620
| | [[15/11]]
| 366.1972
| |  
| [[15/11]]
|  
|-
|-
| | 21
| 21
| | 562.5501
| 562.5501
| | [[18/13]]
| 384.507
| |  
| [[18/13]]
|  
|-
|-
| | 22
| 22
| | 589.3382
| 589.3382
| |  
| 402.8169
| |  
| ([[45/32]])
|  
|-
|-
| | 23
| 23
| | 616.1263
| 616.1263
| | [[10/7]]
| 421.1268
| |  
| [[10/7]]
|  
|-
|-
| | 24
| 24
| | 642.9144
| 642.9144
| | 132/91
| 439.4366
| |  
| 132/91
|  
|-
|-
| | 25
| 25
| | 669.7025
| 669.7025
| |  
| 457.7465
| |  
| 22/15
|  
|-
|-
| | 26
| 26
| | 696.4906
| 696.4906
| | 486/325, 220/147
| 476.0563
| |
| 486/325, 220/147
| pseudo-[[3/2]]
|-
|-
| | 27
| 27
| | 723.2787
| 723.2787
| |  
| 494.3662
| |  
| 50/33
|  
|-
|-
| | 28
| 28
| | 750.0668
| 750.0668
| | 54/35
| 512.6761
| |  
| 54/35
|  
|-
|-
| | 29
| 29
| | 776.8549
| 776.8549
| |  
| 530.9859
| |  
| 264/169
|  
|-
|-
| | 30
| 30
| | 803.6430
| 803.643
| | 35/22
| 549.2958
| |  
| 35/22
|  
|-
|-
| | 31
| 31
| | 830.4311
| 830.4311
| | [[21/13]]
| 567.6056
| |  
| [[21/13]]
|  
|-
|-
| | 32
| 32
| | 857.2192
| 857.2192
| |  
| 585.9155
| |  
| 18/11
|  
|-
|-
| | 33
| 33
| | 884.0073
| 884.0073
| | [[5/3]]
| 604.22535
| |  
| [[5/3]]
|  
|-
|-
| | 34
| 34
| | 910.7954
| 910.7954
| | [[22/13]]
| 622.5352
| |  
| [[22/13]]
|  
|-
|-
| | 35
| 35
| | 937.5835
| 937.5835
| | 189/110
| 640.8451
| |  
| 12/7
|  
|-
|-
| | 36
| 36
| | 964.3715
| 964.3715
| | 110/63
| 659.1549
| |  
| 7/4
|  
|-
|-
| | 37
| 37
| | 991.1596
| 991.1596
| | 39/22
| 677.4648
| |  
| 39/22
|  
|-
|-
| | 38
| 38
| | 1017.9477
| 1017.9477
| | [[9/5]]
| 695.77465
| |  
| [[9/5]]
|  
|-
|-
| | 39
| 39
| | 1044.7358
| 1044.7358
| |  
| 714.0845
| |  
| 11/6
|  
|-
|-
| | 40
| 40
| | 1071.5239
| 1071.5239
| | [[13/7]]
| 732.3944
| |  
| [[13/7]]
|  
|-
|-
| | 41
| 41
| | 1098.3120
| 1098.312
| | 66/35
| 750.7042
| |  
| 66/35
|  
|-
|-
| | 42
| 42
| | 1125.1001
| 1125.1001
| |  
| 769.0141
| |  
| 21/11
|  
|-
|-
| | 43
| 43
| | 1151.8882
| 1151.8882
| | 35/18
| 787.3239
| |  
| 35/18
|  
|-
|-
| | 44
| 44
| | 1178.6763
| 1178.6763
| |  
| 805.6338
| |  
| 22/13
|  
|-
|-
| | 45
| 45
| | 1205.4644
| 1205.4644
| | 441/220, 325/162
| 823.9437
| |
| 441/220, 325/162
| pseudo-[[octave]]
|-
|-
| | 46
| 46
| | 1232.2525
| 1232.2525
| |  
| 842.2535
| |  
| 45/22
|  
|-
|-
| | 47
| 47
| | 1259.0406
| 1259.0406
| | 91/44
| 860.5634
| |  
| 91/44
|  
|-
|-
| | 48
| 48
| | 1285.8287
| 1285.8287
| | [[21/20|21/10]]
| 878.8732
| |  
| [[21/20|21/10]]
|  
|-
|-
| | 49
| 49
| | 1312.6168
| 1312.6168
| |  
| 897.1831
| |  
| ([[16/15|32/15]])
|  
|-
|-
| | 50
| 50
| | 1339.4049
| 1339.4049
| | [[13/6]]
| 915.493
| |  
| [[13/6]]
|  
|-
|-
| | 51
| 51
| | 1366.1930
| 1366.193
| | [[11/5]]
| 933.8028
| |  
| [[11/5]]
|  
|-
|-
| | 52
| 52
| | 1392.9811
| 1392.9811
| |  
| 952.1127
| |  
| 378/169
|  
|-
|-
| | 53
| 53
| | 1419.7692
| 1419.7692
| | 25/11
| 970.4225
| |  
| [[25/11]]
|  
|-
|-
| | 54
| 54
| | 1446.5573
| 1446.5573
| | [[15/13|30/13]]
| 988.7324
| |  
| [[15/13|30/13]]
|  
|-
|-
| | 55
| 55
| | 1473.3454
| 1473.3454
| |  
| 1007.04225
| |  
| 396/169
|  
|-
|-
| | 56
| 56
| | 1500.1335
| 1500.1335
| | 50/21
| 1025.3521
| |  
| 50/21
|  
|-
|-
| | 57
| 57
| | 1526.9216
| 1526.9216
| | 220/91
| 1043.662
| |  
| 220/91
|  
|-
|-
| | 58
| 58
| | 1553.7097
| 1553.7097
| | [[27/22|27/11]]
| 1061.9718
| |  
| [[27/22|27/11]]
|  
|-
|-
| | 59
| 59
| | 1580.4978
| 1580.4978
| |  
| 1080.2817
| |  
| 162/65
|  
|-
|-
| | 60
| 60
| | 1607.2859
| 1607.2859
| | 195/77
| 1098.59155
| |  
| 195/77
|  
|-
|-
| | 61
| 61
| | 1634.0740
| 1634.0740
| | [[9/7|18/7]]
| 1161.9014
| |  
| [[9/7|18/7]]
|  
|-
|-
| | 62
| 62
| | 1660.8621
| 1660.8621
| | 441/169
| 1135.2113
| |  
| 441/169
|  
|-
|-
| | 63
| 63
| | 1687.6502
| 1687.6502
| | 175/66, 130/49
| 1153.5211
| |  
| 175/66, 130/49
|  
|-
|-
| | 64
| 64
| | 1714.4383
| 1714.4383
| | 35/13, 132/49
| 1171.831
| |  
| 35/13, 132/49
|  
|-
|-
| | 65
| 65
| | 1741.2264
| 1741.2264
| | 462/169
| 1190.14085
| |  
| 462/169
|  
|-
|-
| | 66
| 66
| | 1768.0145
| 1768.0145
| | [[25/18|25/9]]
| 1208.4507
| |  
| [[25/18|25/9]]
|  
|-
|-
| | 67
| 67
| | 1794.8026
| 1794.8026
| | 110/39
| 1226.7606
| |  
| 110/39
|  
|-
|-
| | 68
| 68
| | 1821.5907
| 1821.5907
| | 63/22
| 1245.0704
| |  
| 63/22
|  
|-
|-
| | 69
| 69
| | 1848.3788
| 1848.3788
| | 189/65
| 1263.3803
| |  
| 189/65
|  
|-
|-
| | 70
| 70
| | 1875.1669
| 1875.1669
| | 65/22
| 1281.6901
| |  
| 65/22
|  
|-
|-
| | 71
| 71
| | 1901.9550
| 1901.9550
| | '''exact [[3/1]]'''
| 1300
| | [[3/2|just perfect fifth]] plus an octave
| '''exact [[3/1]]'''
| [[3/2|just perfect fifth]] plus an octave
|}
|}
[[Category:Edt]]
[[Category:Edonoi]]

Latest revision as of 19:23, 1 August 2025

← 70edt 71edt 72edt →
Prime factorization 71 (prime)
Step size 26.7881 ¢ 
Octave 45\71edt (1205.46 ¢)
Consistency limit 7
Distinct consistency limit 7

71EDT is the equal division of the third harmonic into 71 parts of 26.7881 cents each, corresponding to 44.7960 edo (45edo with 5.4644 cents octave stretch). It is related to the 13-limit temperament which tempers out 540/539, 1575/1573, 2200/2197, and 4375/4374, which is supported by 45edo (45ef val), 179edo (179ef val), 224edo, 269edo (269ce val), and 403edo (403def val).

71EDT is the 13th no-twos zeta peak EDT.

Harmonics

Approximation of prime harmonics in 71edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +5.5 +0.0 -0.4 +6.5 +0.8 +6.3 -2.7 -7.8 +9.7 +10.2 +1.9
Relative (%) +20.4 +0.0 -1.3 +24.2 +3.1 +23.5 -10.2 -29.0 +36.2 +38.2 +7.2
Steps
(reduced) 		45
(45) 		71
(0) 		104
(33) 		126
(55) 		155
(13) 		166
(24) 		183
(41) 		190
(48) 		203
(61) 		218
(5) 		222
(9)
Approximation of prime harmonics in 71edt
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -9.7 +0.1 -2.0 +4.7 +11.0 +12.9 +8.7 +7.1 -13.0 -7.5 -10.3
Relative (%) -36.3 +0.3 -7.5 +17.7 +41.2 +48.1 +32.7 +26.3 -48.4 -27.9 -38.4
Steps
(reduced) 		233
(20) 		240
(27) 		243
(30) 		249
(36) 		257
(44) 		264
(51) 		266
(53) 		272
(59) 		275
(62) 		277
(64) 		282
(69)

Intervals

Degree Cents Hekts Corresponding
JI intervals 		Comments
0 exact 1/1
1 26.7881 18.3099 66/65
2 53.5762 36.6197 65/63
3 80.3643 54.9296 22/21
4 107.1524 73.2394 117/110
5 133.9405 91.5493 27/25
6 160.7286 109.85915 169/154
7 187.5167 128.169 39/35
8 214.3048 146.4789 147/130, 198/175
9 241.0929 164.7887 169/147
10 267.8810 183.0986 7/6
11 294.6691 201.40845 77/65
12 321.4572 219.7183 65/54
13 348.2453 238.0282 11/9
14 375.0334 256.338 273/220
15 401.8215 274.6479 63/50
16 428.6096 292.95775 169/132
17 455.3977 311.2676 13/10
18 482.1858 329.5775 33/25
19 508.9739 347.8873 169/126
20 535.7620 366.1972 15/11
21 562.5501 384.507 18/13
22 589.3382 402.8169 (45/32)
23 616.1263 421.1268 10/7
24 642.9144 439.4366 132/91
25 669.7025 457.7465 22/15
26 696.4906 476.0563 486/325, 220/147 pseudo-3/2
27 723.2787 494.3662 50/33
28 750.0668 512.6761 54/35
29 776.8549 530.9859 264/169
30 803.643 549.2958 35/22
31 830.4311 567.6056 21/13
32 857.2192 585.9155 18/11
33 884.0073 604.22535 5/3
34 910.7954 622.5352 22/13
35 937.5835 640.8451 12/7
36 964.3715 659.1549 7/4
37 991.1596 677.4648 39/22
38 1017.9477 695.77465 9/5
39 1044.7358 714.0845 11/6
40 1071.5239 732.3944 13/7
41 1098.312 750.7042 66/35
42 1125.1001 769.0141 21/11
43 1151.8882 787.3239 35/18
44 1178.6763 805.6338 22/13
45 1205.4644 823.9437 441/220, 325/162 pseudo-octave
46 1232.2525 842.2535 45/22
47 1259.0406 860.5634 91/44
48 1285.8287 878.8732 21/10
49 1312.6168 897.1831 (32/15)
50 1339.4049 915.493 13/6
51 1366.193 933.8028 11/5
52 1392.9811 952.1127 378/169
53 1419.7692 970.4225 25/11
54 1446.5573 988.7324 30/13
55 1473.3454 1007.04225 396/169
56 1500.1335 1025.3521 50/21
57 1526.9216 1043.662 220/91
58 1553.7097 1061.9718 27/11
59 1580.4978 1080.2817 162/65
60 1607.2859 1098.59155 195/77
61 1634.0740 1161.9014 18/7
62 1660.8621 1135.2113 441/169
63 1687.6502 1153.5211 175/66, 130/49
64 1714.4383 1171.831 35/13, 132/49
65 1741.2264 1190.14085 462/169
66 1768.0145 1208.4507 25/9
67 1794.8026 1226.7606 110/39
68 1821.5907 1245.0704 63/22
69 1848.3788 1263.3803 189/65
70 1875.1669 1281.6901 65/22
71 1901.9550 1300 exact 3/1 just perfect fifth plus an octave
Retrieved from "https://en.xen.wiki/index.php?title=71edt&oldid=206050"