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'''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.
{{Infobox ET}}
{{ED intro}} 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 24.606[[edo]]. This implies that each step of 39edt can be approximated by 5 steps of [[123edo]].
39edt can be described as approximately 24.606[[edo]]. This implies that each step of 39edt can be approximated by 5 steps of [[123edo]]. 39edt contains within it a close approximation of [[4ed11/5]]: every seventh step of 39edt equates to a step of 4ed11/5.


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]] (or [[enfactored]]) 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 mapping [{{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]].
== Theory ==
It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]; in fact it has a better no-twos 13-[[odd limit]] relative error than any other edt up to [[914edt]]. Like [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being [[contorted]] in the no-twos 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]], [[1575/1573]], and [[847/845]]. An efficient traversal is therefore given by [[Mintra]] temperament, which in the 13-limit tempers out 275/273 and 1331/1323 alongside 245/243, and is generated by the interval of [[11/7]], which serves as a [[macrodiatonic]] "superpyth" fourth and splits the [[BPS]] generator of [[9/7]], up a tritave, in three.


==Intervals==
If octaves are inserted, 39edt is related to the {{nowrap|49f & 172f}} temperament in the full 13-limit, known as [[Sensamagic clan#Triboh|triboh]], tempering out 245/243, 275/273, 847/845 and 1575/1573, which has mapping [{{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]].


{| class="wikitable"
When treated as an octave-repeating tuning with the sharp octave of 25 steps (about 1219 cents), and the other primes chosen by their best octave-reduced mappings, it functions as a tuning of [[mavila]] temperament, analogous to [[25edo]]'s mavila.
 
Mavila is one of the few places where octave-stretching makes sense, due to how flat the fifth and often the major third are; this fifth of 683 cents is much more recognizable as a perfect fifth of 3/2 than the 672-cent tuning with just octaves.
{{Harmonics in equal|39|3|1|intervals=prime|columns=12}}
 
== Intervals ==
All intervals shown are within the 91-[[odd limit#Nonoctave equaves|throdd limit]] and are consistently represented.
 
{| class="wikitable center-all right-2 right-3"
|-
|-
! |Degree
! Steps
! | Cents
! [[Cent]]s
! |Cents<br>([[octave reduction|octave-reduced]])
! [[Hekt]]s
! |Hekts
! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree
! |Degree of [[Bohlen-Pierce|BP]]
! Corresponding 3.5.7.11.13 subgroup<br />intervals
! |Comments
! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs,<br />{{nowrap|J {{=}} 1/1}})
! Mintaka[7]<br />(E macro-Phrygian)
|-
|-
| | 0
| 0
| |0
| 0
| |
| 0
| |0
| P1
| | 0
| [[1/1]]
| |1/1
| J
| E
|-
|-
| |1
| 1
| | 48.768
| 48.8
| |
| 33.3
|33.333
| SP1
| |
| [[77/75]] (+3.2¢); [[65/63]] (&minus;5.3¢)
| |39th root of 3
| ^J
| ^E, vF
|-
|-
| |2
| 2
| |97.536
| 97.5
| |
| 66.7
|66.667
| sA1/sm2
| |
| [[35/33]] (&minus;4.3¢); [[81/77]] (+9.9¢)
| |
| vK
| F
|-
|-
| |3
| 3
| |146.304
| 146.3
| |
| 100
|100
| A1/m2
| |1
| [[99/91]] (+0.4¢); [[49/45]] (&minus;1.1¢); [[27/25]] (+13.1¢)
| |13th root of 3
| K
| ^F, vGb, Dx
|-
|-
| | 4
| 4
| |195.072
| 195.1
| |
| 133.3
|133.333
| SA1/Sm2
| |
| [[55/49]] (&minus;4.9¢); [[91/81]] (&minus;6.5¢); [[39/35]] (+7.7¢)
| |
| ^K
| Gb, vE#
|-
|-
| | 5
| 5
| |243.840
| 243.8
| |
| 166.7
|166.667
| sM2/sd3
| |
| [[15/13]] (&minus;3.9¢); [[63/55]] (+8.7¢)
| |
| vK#, vLb
| ^Gb, E#
|-
|-
| |6
| 6
| | 292.608
| 292.6
| |
| 200
|200
| M2/d3
| |2
| [[77/65]] (&minus;0.7¢); [[13/11]] (+3.4¢); [[25/21]] (&minus;9.2¢)
| |
| K#, Lb
| vF#, ^E#
|-
|-
| |7
| 7
| | 341.377
| 341.4
| |
| 233.3
|233.333
| SM2/Sd3
| |
| [[11/9]] (&minus;6.0¢); [[91/75]] (+6.6¢)
| |
| ^K#, ^Lb
| F#
|-
|-
| |8
| 8
| |390.145
| 390.1
| |
| 266.7
|266.667
| sA2/sP3/sd4
| |
| [[49/39]] (&minus;5.0¢); [[81/65]] (+9.2¢)
| |
| vL
| vG, ^F#
|-
|-
| | 9
| 9
| |438.913
| 438.9
| |
| 300
| 300
| |3
| A2/P3/d4
| |
| [[9/7]] (+3.8¢); [[35/27]] (&minus;10.3¢)
| L
| G
|-
|-
| |10
| 10
| | 487.681
| 487.7
| |
| 333.3
|333.333
| SA2/SP3/Sd4
| |
| [[65/49]] (&minus;1.5¢); [[33/25]] (+7.0¢)
| |
| ^L
| ^G, vAb
|-
|-
| | 11
| 11
| |536.449
| 536.4
| |
| 366.7
| 366.667
| sA3/sm4/sd5
| |
| [[15/11]] (&minus;0.5¢)
| |
| vM
| Ab
|-
|-
| |12
| 12
| |585.217
| 585.2
| |
| 400
| 400
| |4
| A3/m4/d5
| |
| [[7/5]] (+2.7¢)
| M
| ^Ab, Fx
|-
|-
| |13
| 13
| | 633.985
| 634.0
| |
| 433.3
|433.333
| SA3/Sm4/Sd5
| |
| [[13/9]] (&minus;2.6¢)
| |cube root of 3
| ^M
| vG#
|-
|-
| |14
| 14
| |682.753
| 682.7
| |
| 466.7
|466.667
| sM4/sm5
| |
| [[135/91]] (+0.07¢); [[49/33]] (&minus;1.6¢); [[81/55]] (+12.6¢)
| |
| vM#, vNb
| G#
|-
|-
| |15
| 15
| |731.521
| 731.5
| |
| 500
|500
| M4/m5
| |5
| [[75/49]] (&minus;5.4¢); [[117/77]] (+7.2¢)
| |  
| M#, Nb
| vA, ^G#
|-
|-
| | 16
| 16
| |780.289
| 780.3
| |
| 533.3
| 533.333
| SM4/Sm5
| |
| [[11/7]] (&minus;2.2¢); [[39/25]] (+10.4¢)
| |
| ^M#, ^Nb
| A
|-
|-
| | 17
| 17
| |829.057
| 829.0
| |
| 566.7
|566.667
| sA4/sM5
| |
| [[21/13]] (&minus;1.2¢)
| |
| vN
| ^A, vBb
|-
|-
| | 18
| 18
| | 877.825
| 877.8
| |
| 600
|600
| A4/M5
| |6
| [[91/55]] (+6.1¢); [[5/3]] (&minus;6.5¢); [[81/49]] (+7.7¢)
| |
| N
| Bb
|-
|-
| | 19
| 19
| |926.593
| 926.6
| |  
| 633.3
|633.333
| SA4/SM5
| |
| [[77/45]] (&minus;3.3¢)
| |
| ^N
| ^Bb, vCb, Gx
|-
|-
| | 20
| 20
| |975.362
| 975.3
| |  
| 666.7
|666.667
| sA5/sm6/sd7
| |
| [[135/77]] (+3.3¢)
| |
| vO
| vA#, Cb
|-
|-
| |21
| 21
| |1024.13
| 1024.1
| |
| 700
|700
| A5/m6/d7
| |7
| [[165/91]] (&minus;6.1¢); [[9/5]] (+6.5¢); [[49/27]] (&minus;7.7¢)
| |
| O
| A#, ^Cb
|-
|-
| | 22
| 22
| |1072.898
| 1072.9
| |
| 733.3
|733.333
| SA5/Sm6/Sd7
| |
| [[13/7]] (+1.2¢)
| |
| ^O
| vB, ^A#
|-
|-
| | 23
| 23
| | 1121.666
| 1121.6
| |
| 766.7
|766.667
| sM6/sm7
| |
| [[21/11]] (+2.2¢); [[25/13]] (&minus;10.4¢)
| |
| vO#, vPb
| B
|-
|-
| |24
| 24
| |1170.434
| 1170.4
| |
| 800
|800
| M6/m7
| |8
| [[49/25]] (+5.4¢); [[77/39]] (&minus;7.2¢)
| |
| O#, Pb
| ^B, vC
|-
|-
| | 25
| 25
| | 1219.202
| 1219.2
| |19.202
| 833.3
|833.333
| SM6/Sm7
| |
| [[91/45]] (+0.07¢); [[99/49]] (+1.6¢); [[55/27]] (&minus;12.6¢)
| |
| ^O#, ^Pb
| C
|-
|-
| |26
| 26
| | 1267.97
| 1267.9
| |67.97
| 866.7
| 866.667
| sA6/sM7/sd8
| |
| [[27/13]] (+2.6¢)
| |
| vP
| ^C, vDb
|-
|-
| |27
| 27
| |1316.738
| 1316.7
| |116.738
| 900
|900
| A6/M7/d8
| |9
| [[15/7]] (&minus;2.7¢)
| |
| P
| Db, vB#
|-
|-
| |28
| 28
| |1365.506
| 1365.5
| |165.506
| 933.3
|933.333
| SA6/SM7/Sd8
| |
| [[11/5]] (+0.5¢)
| |
| ^P
| ^Db, B#
|-
|-
| |29
| 29
| |1414.274
| 1414.2
| | 214.274
| 966.7
|966.667
| sP8/sd9
| |
| [[147/65]] (+1.5¢); [[25/11]] (&minus;7.0¢)
| |
| vQ
| vC#, ^B#
|-
|-
| |30
| 30
| |1463.042
| 1463.0
| |263.042
| 1000
|1000
| P8/d9
| |10
| [[7/3]] (&minus;3.8¢); [[81/35]] (+10.3¢)
| |
| Q
| C#
|-
|-
| | 31
| 31
| |1511.81
| 1511.8
| |311.81
| 1033.3
| 1033.333
| SP8/Sd9
| |
| [[117/49]] (+5.0¢); [[65/27]] (&minus;9.2¢)
| |
| ^Q
| vD, ^C#
|-
|-
| | 32
| 32
| |1560.578
| 1560.5
| |360.578
| 1066.7
|1066.667
| sA8/sm9
| |
| [[27/11]] (+6.0¢); [[225/91]] (+6.6¢)
| |
| vQ#, vRb
| D
|-
|-
| |33
| 33
| |1609.347
| 1609.3
| |409.347
| 1100
|1100
| A8/m9
| | 11
| [[195/77]] (&minus;0.7¢); [[33/13]] (&minus;3.4¢); [[63/25]] (+9.2¢)
| |
| Q#, Rb
| ^D, vEb
|-
|-
| |34
| 34
| |1658.115
| 1658.1
| | 458.115
| 1133.3
|1133.333
| SA8/Sm9
| |
| [[13/5]] (+3.9¢); [[55/21]] (&minus;8.7¢)
| |
| ^Q#, ^Rb
| Eb
|-
|-
| |35
| 35
| |1706.883
| 1706.9
| | 506.883
| 1166.7
|1166.667
| sM9/sd10
| |
| [[147/55]] (+4.9¢); [[243/91]] (+6.5¢); [[35/13]] (&minus;7.7¢)
| |
| vR
| ^Eb, vFb, Cx
|-
|-
| |36
| 36
| |1755.651
| 1755.7
| |555.651
| 1200
|1200
| M9/d10
| | 12
| [[91/33]] (+0.4¢); [[135/49]] (+1.1¢); [[25/9]] (&minus;13.1¢)
| |
| R
| vD#, Fb
|-
|-
| |37
| 37
| |1804.419
| 1804.5
| | 604.419
| 1233.3
|1233.333
| SM9/Sd10
| |
| [[99/35]] (+4.3¢); [[77/27]] (&minus;9.9¢)
| |
| ^R
| D#, ^Fb
|-
|-
| |38
| 38
| |1853.187
| 1853.2
| | 653.187
| 1266.7
|1266.667
| sA9/sP10
| |
| [[225/77]] (&minus;3.2¢); [[189/65]] (+5.3¢)
| |
| vJ
| vE, ^D#
|-
|-
| |39
| 39
| |1901.955
| 1902.0
| |701.955
| 1300
|1300
| A9/P10
| | 13
| [[3/1]]
| | 3/1 (tritave)
| J
|-
| E
| | 40
|}
| |1950.723
 
| |750.723
== Approximation to JI ==
| 1333.333
 
| |
=== No-2 zeta peak ===
| |
{| class="wikitable"
|-
|+
| |41
!Steps
| |1999.491
per octave
| |799.491
!Steps
|1366.667
per tritave
| |
!Step size
| |
(cents)
|-
!Height
| | 42
!Tritave size
| |2048.259
(cents)
| | 848.259
!Tritave stretch
| 1400
(cents)
| |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
|24.573831630
| |3803.91
|38.948601633
| |203.91
|48.832433543
|2600
|4.665720
| |26
|1904.464908194
| |9/1
|2.509907328
|}
|}
Every 7 steps of the [[172edo|172f]] val is an excellent approximation of the ninth no-2 zeta peak in the 15-limit.
== Music ==
; [[Francium]]
* [https://www.youtube.com/watch?v=jstg4_B0jfY ''Strange Juice''] (2025)
;[https://www.youtube.com/@PhanomiumMusic Phanomium]
* ''[https://www.youtube.com/watch?v=GX79ZX1Z8C8 Polygonal]'' (2025)
Retrieved from "https://en.xen.wiki/w/39edt"