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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
|}
|}
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