39edt: Difference between revisions

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The 39 equal division of 3, the tritave, divides it into 39 equal parts of 48.678 cents each, a [[nonoctave|nonoctave]] tuning corresponding to 24.606 edo. It is a strong no-twos 13-limit system, a fact first noted by [[Paul_Erlich|Paul Erlich]], and like [[26edt|26edt]] and [[52edt|52edt]] it is a multiple of [[13edt|13edt]] and so contains the [[Bohlen-Pierce|Bohlen-Pierce]] scale. It is contorted in the 7-limit, tempering out the same BP commas 245/243 and 3125/3087 as 13edt. In the 11-limit it tempers out 1331/1323 and in the 13-limit 275/273, 847/845 and 1575/1573. It is related to the 49f&172f temperament tempering out 245/243, 275/273, 847/845 and 1575/1573, which has map [<1 0 0 0 0 0|, <0 39 57 69 85 91|]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos zeta peak edt]].
'''39 equal divisions of the tritave''' ('''39edt''') is the [[nonoctave]] [[tuning system]] derived by dividing the [[tritave]] (3/1) into 39 equal steps of approximately 48.7 [[cent]]s each, or the 39th root of 3. It is also known as the '''Triple Bohlen-Pierce scale''' ('''Triple BP'''), since it divides each step of the equal-tempered [[Bohlen-Pierce]] scale ([[13edt]]) into three equal parts.


==Intervals of 39EDT==
39edt can be described as approximately 24.606[[edo]]. This implies that each step of 39edt can be approximated by 5 steps of [[123edo]].
 
It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]], and like [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale. It is contorted in the 7-limit, tempering out the same BP commas 245/243 and 3125/3087 as 13edt. In the 11-limit it tempers out 1331/1323 and in the 13-limit 275/273, 847/845 and 1575/1573. It is related to the 49f&172f temperament tempering out 245/243, 275/273, 847/845 and 1575/1573, which has map [{{val|1 0 0 0 0 0}}, {{val|0 39 57 69 85 91}}]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos zeta peak edt]].
 
==Intervals==


{| class="wikitable"
{| class="wikitable"
|-
|-
! | Degrees of 39EDT
! |Degree
! | Cents Value
! | Cents
!hekts
! |Cents<br>([[octave reduction|octave-reduced]])
! | ¢ Octave-Reduced
! |Hekts
! | Degrees of [[BP|BP]]
! |Degree of [[Bohlen-Pierce|BP]]
! | Comments
! |Comments
|-
|-
! colspan="3" | 0
| |
| | 0
| | 0
| | 1/1
| |0
| |
| |0
| | 0
| |1/1
|-
|-
| | 1
| |1
| | 48.768
| | 48.768
| |
|33.333
|33.333
| |  
| |  
| |
| |39th root of 3
| | 39th root of 3
|-
|-
| | 2
| |2
| | 97.536
| |97.536
| |
|66.667
|66.667
| |
| |
| |  
| |  
| |  
|-
|-
| | 3
| |3
| | 146.304
| |146.304
| |
|100
|100
| |
| |1
| | 1
| |13th root of 3
| | 13th root of 3
|-
|-
| | 4
| | 4
| | 195.072
| |195.072
| |
|133.333
|133.333
| |
| |
| |  
| |  
| |  
|-
|-
| | 5
| | 5
| | 243.840
| |243.840
| |
|166.667
|166.667
| |
| |
| |  
| |
| |  
|-
|-
| | 6
| |6
| | 292.608
| | 292.608
| |
|200
|200
| |
| |2
| | 2
| |
| |  
|-
|-
| | 7
| |7
| | 341.377
| | 341.377
| |
|233.333
|233.333
| |
| |
| |  
| |
| |  
|-
|-
| | 8
| |8
| | 390.145
| |390.145
| |
|266.667
|266.667
| |
| |
| |  
| |
| |  
|-
|-
| | 9
| | 9
| | 438.913
| |438.913
|300
| |
| | 3
| |  
| |  
| 300
| |3
| |
|-
|-
| | 10
| |10
| | 487.681
| | 487.681
| |
|333.333
|333.333
| |
| |
| |  
| |
| |  
|-
|-
| | 11
| | 11
| | 536.449
| |536.449
|366.667
| |
| |
| 366.667
| |  
| |
| |  
| |
|-
|-
| | 12
| |12
| | 585.217
| |585.217
|400
| |
| | 4
| |  
| |  
| 400
| |4
| |
|-
|-
| | 13
| |13
| | 633.985
| | 633.985
| |
|433.333
|433.333
| |
| |
| |  
| |cube root of 3
| | cube root of 3
|-
|-
| | 14
| |14
| | 682.753
| |682.753
| |
|466.667
|466.667
| |
| |
| |  
| |
| |  
|-
|-
| | 15
| |15
| | 731.521
| |731.521
| |
|500
|500
| |
| |5
| | 5
| |  
| |  
|-
|-
| | 16
| | 16
| | 780.289
| |780.289
|533.333
| |
| |
| 533.333
| |  
| |  
| |  
| |  
|-
|-
| | 17
| | 17
| | 829.057
| |829.057
| |
|566.667
|566.667
| |
| |
| |  
| |
| |  
|-
|-
| | 18
| | 18
| | 877.825
| | 877.825
| |
|600
|600
| |
| |6
| | 6
| |
| |  
|-
|-
| | 19
| | 19
| | 926.593
| |926.593
| |
|633.333
|633.333
| |
| |
| |  
| |
| |  
|-
|-
| | 20
| | 20
| | 975.362
| |975.362
| |
|666.667
|666.667
| |  
| |  
| |
| |
| |  
|-
|-
| | 21
| |21
| | 1024.13
| |1024.13
| |
|700
|700
| |
| |7
| | 7
| |
| |  
|-
|-
| | 22
| | 22
| | 1072.898
| |1072.898
| |
|733.333
|733.333
| |
| |  
| |  
| |  
| |  
Line 173: Line 179:
| | 23
| | 23
| | 1121.666
| | 1121.666
| |
|766.667
|766.667
| |
| |
| |  
| |
| |  
|-
|-
| | 24
| |24
| | 1170.434
| |1170.434
| |
|800
|800
| |
| |8
| | 8
| |
| |  
|-
|-
| | 25
| | 25
| | 1219.202
| | 1219.202
| |19.202
|833.333
|833.333
| | 19.202
| |
| |  
| |  
| |  
|-
|-
| | 26
| |26
| | 1267.97
| | 1267.97
|866.667
| |67.97
| | 67.97
| 866.667
| |  
| |
| |  
| |  
|-
|-
| | 27
| |27
| | 1316.738
| |1316.738
| |116.738
|900
|900
| | 116.738
| |9
| | 9
| |
| |  
|-
|-
| | 28
| |28
| | 1365.506
| |1365.506
| |165.506
|933.333
|933.333
| | 165.506
| |
| |  
| |
| |  
|-
|-
| | 29
| |29
| | 1414.274
| |1414.274
| | 214.274
|966.667
|966.667
| | 214.274
| |
| |  
| |
| |  
|-
|-
| | 30
| |30
| | 1463.042
| |1463.042
| |263.042
|1000
|1000
| | 263.042
| |10
| | 10
| |
| |  
|-
|-
| | 31
| | 31
| | 1511.81
| |1511.81
|1033.333
| |311.81
| | 311.81
| 1033.333
| |  
| |
| |  
| |
|-
|-
| | 32
| | 32
| | 1560.578
| |1560.578
| |360.578
|1066.667
|1066.667
| | 360.578
| |
| |  
| |
| |  
|-
|-
| | 33
| |33
| | 1609.347
| |1609.347
| |409.347
|1100
|1100
| | 409.347
| | 11
| | 11
| |  
| |  
|-
|-
| | 34
| |34
| | 1658.115
| |1658.115
| | 458.115
|1133.333
|1133.333
| | 458.115
| |
| |  
| |  
| |  
|-
|-
| | 35
| |35
| | 1706.883
| |1706.883
| | 506.883
|1166.667
|1166.667
| | 506.883
| |
| |  
| |  
| |  
|-
|-
| | 36
| |36
| | 1755.651
| |1755.651
| |555.651
|1200
|1200
| | 555.651
| | 12
| | 12
| |  
| |  
|-
|-
| | 37
| |37
| | 1804.419
| |1804.419
| | 604.419
|1233.333
|1233.333
| | 604.419
| |
| |  
| |  
| |  
|-
|-
| | 38
| |38
| | 1853.187
| |1853.187
| | 653.187
|1266.667
|1266.667
| | 653.187
| |
| |  
| |  
| |  
|-
|-
| | 39
| |39
| | 1901.955
| |1901.955
| |701.955
|1300
|1300
| | 701.955
| | 13
| | 13
| | 3/1 (tritave)
| | 3/1 (tritave)
|-
|-
| | 40
| | 40
| | 1950.723
| |1950.723
|1333.333
| |750.723
| | 750.723
| 1333.333
| |  
| |
| |  
| |
|-
|-
| | 41
| |41
| | 1999.491
| |1999.491
| |799.491
|1366.667
|1366.667
| | 799.491
| |
| |  
| |
| |  
|-
|-
| | 42
| | 42
| | 2048.259
| |2048.259
|1400
| | 848.259
| | 848.259
| | 14
| 1400
| |  
| |14
| |
|-
|-
| | 43
| |43
| | 2097.027
| |2097.027
|1433.333
| |897.027
| | 897.027
| 1433.333
| |  
| |
| |  
| |
|-
|-
| | 44
| |44
| | 2145.795
| |2145.795
| | 945.795
|1466.667
|1466.667
| | 945.795
| |
| |  
| |
| |  
|-
|-
| | 45
| |45
| | 2194.563
| |2194.563
| |994.563
|1500
|1500
| | 994.563
| |15
| | 15
| |
| |  
|-
|-
| | 46
| |46
| | 2243.332
| |2243.332
| |1043.332
|1533.333
|1533.333
| | 1043.332
| |
| |  
| |
| |  
|-
|-
| | 47
| |47
| | 2292.100
| |2292.100
|1566.667
| |1092.100
| | 1092.100
| 1566.667
| |  
| |
| |  
| |
|-
|-
| | 48
| |48
| | 2340.868
| |2340.868
| |1140.868
|1600
|1600
| | 1140.868
| |16
| | 16
| |
| |  
|-
|-
| | 49
| |49
| | 2389.636
| |2389.636
|1633.333
| |1189.636
| | 1189.636
| 1633.333
| |  
| |
| |  
| |
|-
|-
| | 50
| | 50
| | 2438.404
| |2438.404
| | 38.404
|1666.667
|1666.667
| | 38.404
| |
| |  
| |
| |  
|-
|-
| | 51
| |51
| | 2487.172
| | 2487.172
| |87.172
|1700
|1700
| | 87.172
| |17
| | 17
| |
| |  
|-
|-
| | 52
| | 52
| | 2535.940
| |2535.940
| | 135.940
|1733.333
|1733.333
| | 135.940
| |
| |  
| |
| |  
|-
|-
| | 53
| | 53
| | 2584.708
| |2584.708
| | 184.708
|1766.667
|1766.667
| | 184.708
| |
| |  
| |
| |  
|-
|-
| | 54
| |54
| | 2633.476
| |2633.476
| |233.476
|1800
|1800
| | 233.476
| |18
| | 18
| |
| |  
|-
|-
| | 55
| | 55
| | 2682.244
| |2682.244
| | 282.244
|1833.333
|1833.333
| | 282.244
| |
| |  
| |
| |  
|-
|-
| | 56
| | 56
| | 2731.012
| |2731.012
| | 331.012
|1866.667
|1866.667
| | 331.012
| |
| |  
| |
| |  
|-
|-
| | 57
| |57
| | 2779.78
| |2779.78
| |379.78
|1900
|1900
| | 379.78
| |19
| | 19
| |
| |  
|-
|-
| | 58
| | 58
| | 2828.548
| |2828.548
| |428.548
|1933.333
|1933.333
| | 428.548
| |
| |  
| |
| |  
|-
|-
| | 59
| | 59
| | 2877.317
| |2877.317
| |477.317
|1966.667
|1966.667
| | 477.317
| |
| |  
| |
| |  
|-
|-
| | 60
| |60
| | 2926.085
| |2926.085
| |526.085
|2000
|2000
| | 526.085
| |20
| | 20
| |
| |  
|-
|-
| | 61
| | 61
| | 2974.853
| |2974.853
| |574.853
|2033.333
|2033.333
| | 574.853
| |
| |  
| |
| |  
|-
|-
| | 62
| | 62
| | 3023.621
| |3023.621
| |623.621
|2066.667
|2066.667
| | 623.621
| |
| |  
| |
| |  
|-
|-
| | 63
| |63
| | 3072.389
| |3072.389
| |672.389
|2100
|2100
| | 672.389
| |21
| | 21
| |
| |  
|-
|-
| | 64
| | 64
| | 3121.157
| |3121.157
|2133.333
| |721.157
| | 721.157
| 2133.333
| |  
| |
| |  
| |
|-
|-
| | 65
| |65
| | 3169.925
| |3169.925
| |769.925
|2166.667
|2166.667
| | 769.925
| |
| |  
| |
| |  
|-
|-
| | 66
| | 66
| | 3218.693
| |3218.693
|2200
| | 818.693
| | 818.693
| | 22
| 2200
| |  
| |22
| |
|-
|-
| | 67
| |67
| | 3267.461
| |3267.461
| |867.461
|2233.333
|2233.333
| | 867.461
| |
| |  
| |
| |  
|-
|-
| | 68
| |68
| | 3316.229
| |3316.229
| |916.229
|2266.667
|2266.667
| | 916.229
| |
| |  
| |
| |  
|-
|-
| | 69
| |69
| | 3364.997
| | 3364.997
| |964.997
|2300
|2300
| | 964.997
| |23
| | 23
| |
| |  
|-
|-
| | 70
| |70
| | 3413.765
| |3413.765
|2333.333
| |1013.765
| | 1013.765
| 2333.333
| |  
| |
| |  
| |
|-
|-
| | 71
| | 71
| | 3462.533
| |3462.533
| | 1062.533
|2366.667
|2366.667
| | 1062.533
| |
| |  
| |  
| |  
|-
|-
| | 72
| |72
| | 3511.302
| |3511.302
| |1111.302
|2400
|2400
| | 1111.302
| |24
| | 24
| |
| |  
|-
|-
| | 73
| |73
| | 3560.07
| |3560.07
| |1160.07
|2433.333
|2433.333
| | 1160.07
| |
| |  
| |
| |  
|-
|-
| | 74
| |74
| | 3608.838
| |3608.838
| |8.838
|2466.667
|2466.667
| | 8.838
| |
| |  
| |
| |  
|-
|-
| | 75
| |75
| | 3657.606
| |3657.606
| |57.606
|2500
|2500
| | 57.606
| |25
| | 25
| |
| |  
|-
|-
| | 76
| |76
| | 3706.374
| |3706.374
| |106.374
|2533.333
|2533.333
| | 106.374
| |
| |  
| |
| |  
|-
|-
| | 77
| |77
| | 3755.142
| |3755.142
| |155.142
|2566.667
|2566.667
| | 155.142
| |
| |  
| |
| |  
|-
|-
| | 78
| |78
| | 3803.91
| |3803.91
| |203.91
|2600
|2600
| | 203.91
| |26
| | 26
| |9/1
| | 9/1
|}
|}

Revision as of 03:51, 26 August 2021

39 equal divisions of the tritave (39edt) is the nonoctave tuning system derived by dividing the tritave (3/1) into 39 equal steps of approximately 48.7 cents each, or the 39th root of 3. It is also known as the Triple Bohlen-Pierce scale (Triple BP), since it divides each step of the equal-tempered Bohlen-Pierce scale (13edt) into three equal parts.

39edt can be described as approximately 24.606edo. This implies that each step of 39edt can be approximated by 5 steps of 123edo.

It is a strong no-twos 13-limit system, a fact first noted by Paul Erlich, and like 26edt and 52edt, it is a multiple of 13edt and so contains the Bohlen-Pierce scale. It is contorted in the 7-limit, tempering out the same BP commas 245/243 and 3125/3087 as 13edt. In the 11-limit it tempers out 1331/1323 and in the 13-limit 275/273, 847/845 and 1575/1573. It is related to the 49f&172f temperament tempering out 245/243, 275/273, 847/845 and 1575/1573, which has map [⟨1 0 0 0 0 0], ⟨0 39 57 69 85 91]]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth no-twos zeta peak edt.

Intervals

Degree Cents Cents
(octave-reduced) 		Hekts Degree of BP Comments
0 0 0 0 1/1
1 48.768 33.333 39th root of 3
2 97.536 66.667
3 146.304 100 1 13th root of 3
4 195.072 133.333
5 243.840 166.667
6 292.608 200 2
7 341.377 233.333
8 390.145 266.667
9 438.913 300 3
10 487.681 333.333
11 536.449 366.667
12 585.217 400 4
13 633.985 433.333 cube root of 3
14 682.753 466.667
15 731.521 500 5
16 780.289 533.333
17 829.057 566.667
18 877.825 600 6
19 926.593 633.333
20 975.362 666.667
21 1024.13 700 7
22 1072.898 733.333
23 1121.666 766.667
24 1170.434 800 8
25 1219.202 19.202 833.333
26 1267.97 67.97 866.667
27 1316.738 116.738 900 9
28 1365.506 165.506 933.333
29 1414.274 214.274 966.667
30 1463.042 263.042 1000 10
31 1511.81 311.81 1033.333
32 1560.578 360.578 1066.667
33 1609.347 409.347 1100 11
34 1658.115 458.115 1133.333
35 1706.883 506.883 1166.667
36 1755.651 555.651 1200 12
37 1804.419 604.419 1233.333
38 1853.187 653.187 1266.667
39 1901.955 701.955 1300 13 3/1 (tritave)
40 1950.723 750.723 1333.333
41 1999.491 799.491 1366.667
42 2048.259 848.259 1400 14
43 2097.027 897.027 1433.333
44 2145.795 945.795 1466.667
45 2194.563 994.563 1500 15
46 2243.332 1043.332 1533.333
47 2292.100 1092.100 1566.667
48 2340.868 1140.868 1600 16
49 2389.636 1189.636 1633.333
50 2438.404 38.404 1666.667
51 2487.172 87.172 1700 17
52 2535.940 135.940 1733.333
53 2584.708 184.708 1766.667
54 2633.476 233.476 1800 18
55 2682.244 282.244 1833.333
56 2731.012 331.012 1866.667
57 2779.78 379.78 1900 19
58 2828.548 428.548 1933.333
59 2877.317 477.317 1966.667
60 2926.085 526.085 2000 20
61 2974.853 574.853 2033.333
62 3023.621 623.621 2066.667
63 3072.389 672.389 2100 21
64 3121.157 721.157 2133.333
65 3169.925 769.925 2166.667
66 3218.693 818.693 2200 22
67 3267.461 867.461 2233.333
68 3316.229 916.229 2266.667
69 3364.997 964.997 2300 23
70 3413.765 1013.765 2333.333
71 3462.533 1062.533 2366.667
72 3511.302 1111.302 2400 24
73 3560.07 1160.07 2433.333
74 3608.838 8.838 2466.667
75 3657.606 57.606 2500 25
76 3706.374 106.374 2533.333
77 3755.142 155.142 2566.667
78 3803.91 203.91 2600 26 9/1
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