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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]].
{{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.


==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]]. 39edt contains within it a close approximation of [[4ed11/5]]: every seventh step of 39edt equates to a step of 4ed11/5.


{| class="wikitable"
== 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.
 
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]].
 
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"
|-
|-
! | Degrees of 39EDT
! Steps
! | Cents Value
! [[Cent]]s
! | ¢ Octave-Reduced
! [[Hekt]]s
! | Degrees of [[BP|BP]]
! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree
! | Comments
! Corresponding 3.5.7.11.13 subgroup<br />intervals
! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs,<br />{{nowrap|J {{=}} 1/1}})
! Mintaka[7]<br />(E macro-Phrygian)
|-
|-
| | 0
| 0
| | 0.000
| 0
| |
| 0
| | 0
| P1
| | 1/1
| [[1/1]]
| J
| E
|-
|-
| | 1
| 1
| | 48.768
| 48.8
| |  
| 33.3
| |  
| SP1
| | 39th root of 3
| [[77/75]] (+3.2¢); [[65/63]] (&minus;5.3¢)
| ^J
| ^E, vF
|-
|-
| | 2
| 2
| | 97.536
| 97.5
| |  
| 66.7
| |  
| sA1/sm2
| |
| [[35/33]] (&minus;4.3¢); [[81/77]] (+9.9¢)
| vK
| F
|-
|-
| | 3
| 3
| | 146.304
| 146.3
| |  
| 100
| | 1
| A1/m2
| | 13th root of 3
| [[99/91]] (+0.4¢); [[49/45]] (&minus;1.1¢); [[27/25]] (+13.1¢)
| K
| ^F, vGb, Dx
|-
|-
| | 4
| 4
| | 195.072
| 195.1
| |  
| 133.3
| |  
| 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
| |  
| sM2/sd3
| |
| [[15/13]] (&minus;3.9¢); [[63/55]] (+8.7¢)
| vK#, vLb
| ^Gb, E#
|-
|-
| | 6
| 6
| | 292.608
| 292.6
| |  
| 200
| | 2
| M2/d3
| |
| [[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
| |  
| SM2/Sd3
| |
| [[11/9]] (&minus;6.0¢); [[91/75]] (+6.6¢)
| ^K#, ^Lb
| F#
|-
|-
| | 8
| 8
| | 390.145
| 390.1
| |  
| 266.7
| |  
| sA2/sP3/sd4
| |
| [[49/39]] (&minus;5.0¢); [[81/65]] (+9.2¢)
| vL
| vG, ^F#
|-
|-
| | 9
| 9
| | 438.913
| 438.9
| |  
| 300
| | 3
| A2/P3/d4
| |  
| [[9/7]] (+3.8¢); [[35/27]] (&minus;10.3¢)
| L
| G
|-
|-
| | 10
| 10
| | 487.681
| 487.7
| |  
| 333.3
| |  
| SA2/SP3/Sd4
| |
| [[65/49]] (&minus;1.5¢); [[33/25]] (+7.0¢)
| ^L
| ^G, vAb
|-
|-
| | 11
| 11
| | 536.449
| 536.4
| |  
| 366.7
| |  
| sA3/sm4/sd5
| |
| [[15/11]] (&minus;0.5¢)
| vM
| Ab
|-
|-
| | 12
| 12
| | 585.217
| 585.2
| |  
| 400
| | 4
| A3/m4/d5
| |
| [[7/5]] (+2.7¢)
| M
| ^Ab, Fx
|-
|-
| | 13
| 13
| | 633.985
| 634.0
| |  
| 433.3
| |
| SA3/Sm4/Sd5
| | cube root of 3
| [[13/9]] (&minus;2.6¢)
| ^M
| vG#
|-
|-
| | 14
| 14
| | 682.753
| 682.7
| |  
| 466.7
| |  
| 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
| | 5
| M4/m5
| |  
| [[75/49]] (&minus;5.4¢); [[117/77]] (+7.2¢)
| M#, Nb
| vA, ^G#
|-
|-
| | 16
| 16
| | 780.289
| 780.3
| |  
| 533.3
| |  
| SM4/Sm5
| |
| [[11/7]] (&minus;2.2¢); [[39/25]] (+10.4¢)
| ^M#, ^Nb
| A
|-
|-
| | 17
| 17
| | 829.057
| 829.0
| |  
| 566.7
| |  
| sA4/sM5
| |
| [[21/13]] (&minus;1.2¢)
| vN
| ^A, vBb
|-
|-
| | 18
| 18
| | 877.825
| 877.8
| |  
| 600
| | 6
| A4/M5
| |  
| [[91/55]] (+6.1¢); [[5/3]] (&minus;6.5¢); [[81/49]] (+7.7¢)
| N
| Bb
|-
|-
| | 19
| 19
| | 926.593
| 926.6
| |  
| 633.3
| |  
| SA4/SM5
| |
| [[77/45]] (&minus;3.3¢)
| ^N
| ^Bb, vCb, Gx
|-
|-
| | 20
| 20
| | 975.362
| 975.3
| |  
| 666.7
| |  
| sA5/sm6/sd7
| |
| [[135/77]] (+3.3¢)
| vO
| vA#, Cb
|-
|-
| | 21
| 21
| | 1024.130
| 1024.1
| |  
| 700
| | 7
| A5/m6/d7
| |  
| [[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
| |  
| SA5/Sm6/Sd7
| |
| [[13/7]] (+1.2¢)
| ^O
| vB, ^A#
|-
|-
| | 23
| 23
| | 1121.666
| 1121.6
| |  
| 766.7
| |  
| sM6/sm7
| |
| [[21/11]] (+2.2¢); [[25/13]] (&minus;10.4¢)
| vO#, vPb
| B
|-
|-
| | 24
| 24
| | 1170.434
| 1170.4
| |  
| 800
| | 8
| M6/m7
| |
| [[49/25]] (+5.4¢); [[77/39]] (&minus;7.2¢)
| O#, Pb
| ^B, vC
|-
|-
| | 25
| 25
| | 1219.202
| 1219.2
| | 19.202
| 833.3
| |  
| SM6/Sm7
| |  
| [[91/45]] (+0.07¢); [[99/49]] (+1.6¢); [[55/27]] (&minus;12.6¢)
| ^O#, ^Pb
| C
|-
|-
| | 26
| 26
| | 1267.970
| 1267.9
| | 67.970
| 866.7
| |  
| sA6/sM7/sd8
| |  
| [[27/13]] (+2.6¢)
| vP
| ^C, vDb
|-
|-
| | 27
| 27
| | 1316.738
| 1316.7
| | 116.738
| 900
| | 9
| A6/M7/d8
| |
| [[15/7]] (&minus;2.7¢)
| P
| Db, vB#
|-
|-
| | 28
| 28
| | 1365.506
| 1365.5
| | 165.506
| 933.3
| |  
| SA6/SM7/Sd8
| |  
| [[11/5]] (+0.5¢)
| ^P
| ^Db, B#
|-
|-
| | 29
| 29
| | 1414.274
| 1414.2
| | 214.274
| 966.7
| |  
| sP8/sd9
| |  
| [[147/65]] (+1.5¢); [[25/11]] (&minus;7.0¢)
| vQ
| vC#, ^B#
|-
|-
| | 30
| 30
| | 1463.042
| 1463.0
| | 263.042
| 1000
| | 10
| P8/d9
| |  
| [[7/3]] (&minus;3.8¢); [[81/35]] (+10.3¢)
| Q
| C#
|-
|-
| | 31
| 31
| | 1511.810
| 1511.8
| | 311.810
| 1033.3
| |  
| SP8/Sd9
| |  
| [[117/49]] (+5.0¢); [[65/27]] (&minus;9.2¢)
| ^Q
| vD, ^C#
|-
|-
| | 32
| 32
| | 1560.578
| 1560.5
| | 360.578
| 1066.7
| |  
| sA8/sm9
| |  
| [[27/11]] (+6.0¢); [[225/91]] (+6.6¢)
| vQ#, vRb
| D
|-
|-
| | 33
| 33
| | 1609.347
| 1609.3
| | 409.347
| 1100
| | 11
| A8/m9
| |
| [[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
| |  
| SA8/Sm9
| |  
| [[13/5]] (+3.9¢); [[55/21]] (&minus;8.7¢)
| ^Q#, ^Rb
| Eb
|-
|-
| | 35
| 35
| | 1706.883
| 1706.9
| | 506.883
| 1166.7
| |  
| 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
| | 12
| M9/d10
| |
| [[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
| |  
| SM9/Sd10
| |  
| [[99/35]] (+4.3¢); [[77/27]] (&minus;9.9¢)
| ^R
| D#, ^Fb
|-
|-
| | 38
| 38
| | 1853.187
| 1853.2
| | 653.187
| 1266.7
| |  
| sA9/sP10
| |  
| [[225/77]] (&minus;3.2¢); [[189/65]] (+5.3¢)
| vJ
| vE, ^D#
|-
|-
| | 39
| 39
| | 1901.955
| 1902.0
| | 701.955
| 1300
| | 13
| A9/P10
| | 3/1 (tritave)
| [[3/1]]
|-
| J
| | 40
| E
| | 1950.723
|}
| | 750.723
 
| |
== Approximation to JI ==
| |
 
|-
=== No-2 zeta peak ===
| | 41
{| class="wikitable"
| | 1999.491
|+
| | 799.491
!Steps
| |
per octave
| |
!Steps
|-
per tritave
| | 42
!Step size
| | 2048.259
(cents)
| | 848.259
!Height
| | 14
!Tritave size
| |
(cents)
|-
!Tritave stretch
| | 43
(cents)
| | 2097.027
| | 897.027
| |
| |
|-
| | 44
| | 2145.795
| | 945.795
| |
| |
|-
| | 45
| | 2194.563
| | 994.563
| | 15
| |
|-
| | 46
| | 2243.332
| | 1043.332
| |
| |
|-
| | 47
| | 2292.100
| | 1092.100
| |
| |
|-
| | 48
| | 2340.868
| | 1140.868
| | 16
| |
|-
| | 49
| | 2389.636
| | 1189.636
| |
| |
|-
| | 50
| | 2438.404
| | 38.404
| |
| |
|-
| | 51
| | 2487.172
| | 87.172
| | 17
| |
|-
| | 52
| | 2535.940
| | 135.940
| |
| |
|-
| | 53
| | 2584.708
| | 184.708
| |
| |
|-
| | 54
| | 2633.476
| | 233.476
| | 18
| |
|-
| | 55
| | 2682.244
| | 282.244
| |
| |
|-
| | 56
| | 2731.012
| | 331.012
| |
| |
|-
| | 57
| | 2779.780
| | 379.780
| | 19
| |
|-
|-
| | 58
|24.573831630
| | 2828.548
|38.948601633
| | 428.548
|48.832433543
| |
|4.665720
| |
|1904.464908194
|-
|2.509907328
| | 59
| | 2877.317
| | 477.317
| |
| |
|-
| | 60
| | 2926.085
| | 526.085
| | 20
| |
|-
| | 61
| | 2974.853
| | 574.853
| |
| |
|-
| | 62
| | 3023.621
| | 623.621
| |
| |
|-
| | 63
| | 3072.389
| | 672.389
| | 21
| |
|-
| | 64
| | 3121.157
| | 721.157
| |
| |
|-
| | 65
| | 3169.925
| | 769.925
| |
| |
|-
| | 66
| | 3218.693
| | 818.693
| | 22
| |
|-
| | 67
| | 3267.461
| | 867.461
| |
| |
|-
| | 68
| | 3316.229
| | 916.229
| |
| |
|-
| | 69
| | 3364.997
| | 964.997
| | 23
| |
|-
| | 70
| | 3413.765
| | 1013.765
| |
| |
|-
| | 71
| | 3462.533
| | 1062.533
| |
| |
|-
| | 72
| | 3511.302
| | 1111.302
| | 24
| |
|-
| | 73
| | 3560.070
| | 1160.070
| |
| |
|-
| | 74
| | 3608.838
| | 8.838
| |
| |
|-
| | 75
| | 3657.606
| | 57.606
| | 25
| |
|-
| | 76
| | 3706.374
| | 106.374
| |
| |
|-
| | 77
| | 3755.142
| | 155.142
| |
| |
|-
| | 78
| | 3803.910
| | 203.910
| | 26
| | 9/1
|}
|}
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"