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{{Infobox ET}}
{{Infobox ET}}
'''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 48.8 [[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.
{{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 contains within it a close approximation of [[4ed11/5]]: every seventh step of 39edt equates to a step of 4ed11/5.
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.


== Theory ==
== 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 1575/1573 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.
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 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]].
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]].


{{Harmonics in equal|39|3|1|intervals=prime}}
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 ==
== Intervals ==
Line 20: Line 22:
! [[Cent]]s
! [[Cent]]s
! [[Hekt]]s
! [[Hekt]]s
! [[4L 5s (3/1-equivalent)|Enneatonic]] degree
! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree
! Corresponding<br />3.5.7.11.13 subgroup<br />intervals
! Corresponding 3.5.7.11.13 subgroup<br />intervals
! [[Lambda ups and downs notation|Lambda]] <br />(sLsLsLsLs,<br />J = 1/1)
! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs,<br />{{nowrap|J {{=}} 1/1}})
! Mintaka[7]<br />(E macro-Phrygian)
! Mintaka[7]<br />(E macro-Phrygian)
|-
|-
Line 37: Line 39:
| 33.3
| 33.3
| SP1
| SP1
| [[77/75]] (+3.2c); [[65/63]] (-5.3c)
| [[77/75]] (+3.2¢); [[65/63]] (&minus;5.3¢)
| ^J
| ^J
| ^E, vF
| ^E, vF
Line 45: Line 47:
| 66.7
| 66.7
| sA1/sm2
| sA1/sm2
| [[35/33]] (-4.3c); [[81/77]] (+9.9c)
| [[35/33]] (&minus;4.3¢); [[81/77]] (+9.9¢)
| vK
| vK
| F
| F
Line 53: Line 55:
| 100
| 100
| A1/m2
| A1/m2
| [[49/45]] (-1.1c); [[27/25]] (+13.1c)
| [[99/91]] (+0.4¢); [[49/45]] (&minus;1.1¢); [[27/25]] (+13.1¢)
| K
| K
| ^F, vGb, Dx
| ^F, vGb, Dx
Line 61: Line 63:
| 133.3
| 133.3
| SA1/Sm2
| SA1/Sm2
| [[55/49]] (-4.9c); [[39/35]] (+7.7c)
| [[55/49]] (&minus;4.9¢); [[91/81]] (&minus;6.5¢); [[39/35]] (+7.7¢)
| ^K
| ^K
| Gb, vE#
| Gb, vE#
Line 69: Line 71:
| 166.7
| 166.7
| sM2/sd3
| sM2/sd3
| [[15/13]] (-3.9c); [[63/55]] (+8.7c)
| [[15/13]] (&minus;3.9¢); [[63/55]] (+8.7¢)
| vK#, vLb
| vK#, vLb
| ^Gb, E#
| ^Gb, E#
Line 77: Line 79:
| 200
| 200
| M2/d3
| M2/d3
| [[77/65]] (-0.7c); [[13/11]] (+3.4c); [[25/21]] (-9.2c)
| [[77/65]] (&minus;0.7¢); [[13/11]] (+3.4¢); [[25/21]] (&minus;9.2¢)
| K#, Lb
| K#, Lb
| vF#, ^E#
| vF#, ^E#
Line 85: Line 87:
| 233.3
| 233.3
| SM2/Sd3
| SM2/Sd3
| [[11/9]] (-6.0c)
| [[11/9]] (&minus;6.0¢); [[91/75]] (+6.6¢)
| ^K#, ^Lb
| ^K#, ^Lb
| F#
| F#
Line 93: Line 95:
| 266.7
| 266.7
| sA2/sP3/sd4
| sA2/sP3/sd4
| [[49/39]] (-5.0c); [[81/65]] (+9.2c)
| [[49/39]] (&minus;5.0¢); [[81/65]] (+9.2¢)
| vL
| vL
| vG, ^F#
| vG, ^F#
Line 101: Line 103:
| 300
| 300
| A2/P3/d4
| A2/P3/d4
| [[9/7]] (+3.8c); [[35/27]] (-10.3c)
| [[9/7]] (+3.8¢); [[35/27]] (&minus;10.3¢)
| L
| L
| G
| G
Line 109: Line 111:
| 333.3
| 333.3
| SA2/SP3/Sd4
| SA2/SP3/Sd4
| [[65/49]] (-1.5c); [[33/25]] (+7.0c)
| [[65/49]] (&minus;1.5¢); [[33/25]] (+7.0¢)
| ^L
| ^L
| ^G, vAb
| ^G, vAb
Line 117: Line 119:
| 366.7
| 366.7
| sA3/sm4/sd5
| sA3/sm4/sd5
| [[15/11]] (-0.5c)
| [[15/11]] (&minus;0.5¢)
| vM
| vM
| Ab
| Ab
Line 125: Line 127:
| 400
| 400
| A3/m4/d5
| A3/m4/d5
| [[7/5]] (+2.7c)
| [[7/5]] (+2.7¢)
| M
| M
| ^Ab, Fx
| ^Ab, Fx
Line 133: Line 135:
| 433.3
| 433.3
| SA3/Sm4/Sd5
| SA3/Sm4/Sd5
| [[13/9]] (-2.6c)
| [[13/9]] (&minus;2.6¢)
| ^M
| ^M
| vG#
| vG#
Line 141: Line 143:
| 466.7
| 466.7
| sM4/sm5
| sM4/sm5
| [[49/33]] (-1.6c); [[81/55]] (+12.6c)
| [[135/91]] (+0.07¢); [[49/33]] (&minus;1.6¢); [[81/55]] (+12.6¢)
| vM#, vNb
| vM#, vNb
| G#
| G#
Line 149: Line 151:
| 500
| 500
| M4/m5
| M4/m5
| [[75/49]] (-5.4c); [[117/77]] (+7.2c)
| [[75/49]] (&minus;5.4¢); [[117/77]] (+7.2¢)
| M#, Nb
| M#, Nb
| vA, ^G#
| vA, ^G#
Line 157: Line 159:
| 533.3
| 533.3
| SM4/Sm5
| SM4/Sm5
| [[11/7]] (-2.2c); [[39/25]] (+10.4c)
| [[11/7]] (&minus;2.2¢); [[39/25]] (+10.4¢)
| ^M#, ^Nb
| ^M#, ^Nb
| A
| A
Line 165: Line 167:
| 566.7
| 566.7
| sA4/sM5
| sA4/sM5
| [[21/13]] (-1.2c)
| [[21/13]] (&minus;1.2¢)
| vN
| vN
| ^A, vBb
| ^A, vBb
Line 173: Line 175:
| 600
| 600
| A4/M5
| A4/M5
| [[5/3]] (-6.5c); [[81/49]] (+7.7c)
| [[91/55]] (+6.1¢); [[5/3]] (&minus;6.5¢); [[81/49]] (+7.7¢)
| N
| N
| Bb
| Bb
Line 181: Line 183:
| 633.3
| 633.3
| SA4/SM5
| SA4/SM5
| [[77/45]] (-3.3c)
| [[77/45]] (&minus;3.3¢)
| ^N
| ^N
| ^Bb, vCb, Gx
| ^Bb, vCb, Gx
Line 189: Line 191:
| 666.7
| 666.7
| sA5/sm6/sd7
| sA5/sm6/sd7
| [[135/77]] (+3.3c)
| [[135/77]] (+3.3¢)
| vO
| vO
| vA#, Cb
| vA#, Cb
Line 197: Line 199:
| 700
| 700
| A5/m6/d7
| A5/m6/d7
| [[9/5]] (+6.5c); [[49/27]] (-7.7c)
| [[165/91]] (&minus;6.1¢); [[9/5]] (+6.5¢); [[49/27]] (&minus;7.7¢)
| O
| O
| A#, ^Cb
| A#, ^Cb
Line 205: Line 207:
| 733.3
| 733.3
| SA5/Sm6/Sd7
| SA5/Sm6/Sd7
| [[13/7]] (+1.2c)
| [[13/7]] (+1.2¢)
| ^O
| ^O
| vB, ^A#
| vB, ^A#
Line 213: Line 215:
| 766.7
| 766.7
| sM6/sm7
| sM6/sm7
| [[21/11]] (+2.2c); [[25/13]] (-10.4c)
| [[21/11]] (+2.2¢); [[25/13]] (&minus;10.4¢)
| vO#, vPb
| vO#, vPb
| B
| B
Line 221: Line 223:
| 800
| 800
| M6/m7
| M6/m7
| [[49/25]] (+5.4c); [[77/39]] (-7.2c)
| [[49/25]] (+5.4¢); [[77/39]] (&minus;7.2¢)
| O#, Pb
| O#, Pb
| ^B, vC
| ^B, vC
Line 229: Line 231:
| 833.3
| 833.3
| SM6/Sm7
| SM6/Sm7
| [[99/49]] (+1.6c); [[55/27]] (-12.6c)
| [[91/45]] (+0.07¢); [[99/49]] (+1.6¢); [[55/27]] (&minus;12.6¢)
| ^O#, ^Pb
| ^O#, ^Pb
| C
| C
Line 237: Line 239:
| 866.7
| 866.7
| sA6/sM7/sd8
| sA6/sM7/sd8
| [[27/13]] (+2.6c)
| [[27/13]] (+2.6¢)
| vP
| vP
| ^C, vDb
| ^C, vDb
Line 245: Line 247:
| 900
| 900
| A6/M7/d8
| A6/M7/d8
| [[15/7]] (-2.7c)
| [[15/7]] (&minus;2.7¢)
| P
| P
| Db, vB#
| Db, vB#
Line 253: Line 255:
| 933.3
| 933.3
| SA6/SM7/Sd8
| SA6/SM7/Sd8
| [[11/5]] (+0.5c)
| [[11/5]] (+0.5¢)
| ^P
| ^P
| ^Db, B#
| ^Db, B#
Line 261: Line 263:
| 966.7
| 966.7
| sP8/sd9
| sP8/sd9
| [[147/65]] (+1.5c); [[25/11]] (-7.0c)
| [[147/65]] (+1.5¢); [[25/11]] (&minus;7.0¢)
| vQ
| vQ
| vC#, ^B#
| vC#, ^B#
Line 269: Line 271:
| 1000
| 1000
| P8/d9
| P8/d9
| [[7/3]] (-3.8c); [[81/35]] (+10.3c)
| [[7/3]] (&minus;3.8¢); [[81/35]] (+10.3¢)
| Q
| Q
| C#
| C#
Line 277: Line 279:
| 1033.3
| 1033.3
| SP8/Sd9
| SP8/Sd9
| [[117/49]] (+5.0c); [[65/27]] (-9.2c)
| [[117/49]] (+5.0¢); [[65/27]] (&minus;9.2¢)
| ^Q
| ^Q
| vD, ^C#
| vD, ^C#
Line 285: Line 287:
| 1066.7
| 1066.7
| sA8/sm9
| sA8/sm9
| [[27/11]] (+6.0c)
| [[27/11]] (+6.0¢); [[225/91]] (+6.6¢)
| vQ#, vRb
| vQ#, vRb
| D
| D
Line 293: Line 295:
| 1100
| 1100
| A8/m9
| A8/m9
| [[195/77]] (-0.7c); [[33/13]] (-3.4c); [[63/25]] (+9.2c)
| [[195/77]] (&minus;0.7¢); [[33/13]] (&minus;3.4¢); [[63/25]] (+9.2¢)
| Q#, Rb
| Q#, Rb
| ^D, vEb
| ^D, vEb
Line 301: Line 303:
| 1133.3
| 1133.3
| SA8/Sm9
| SA8/Sm9
| [[13/5]] (+3.9c); [[55/21]] (-8.7c)
| [[13/5]] (+3.9¢); [[55/21]] (&minus;8.7¢)
| ^Q#, ^Rb
| ^Q#, ^Rb
| Eb
| Eb
Line 309: Line 311:
| 1166.7
| 1166.7
| sM9/sd10
| sM9/sd10
| [[147/55]] (+4.9c); [[35/13]] (-7.7c)
| [[147/55]] (+4.9¢); [[243/91]] (+6.5¢); [[35/13]] (&minus;7.7¢)
| vR
| vR
| ^Eb, vFb, Cx
| ^Eb, vFb, Cx
Line 317: Line 319:
| 1200
| 1200
| M9/d10
| M9/d10
| [[135/49]] (+1.1c); [[25/9]] (-13.1c)
| [[91/33]] (+0.4¢); [[135/49]] (+1.1¢); [[25/9]] (&minus;13.1¢)
| R
| R
| vD#, Fb
| vD#, Fb
Line 325: Line 327:
| 1233.3
| 1233.3
| SM9/Sd10
| SM9/Sd10
| [[99/35]] (+4.3c); [[77/27]] (-9.9c)
| [[99/35]] (+4.3¢); [[77/27]] (&minus;9.9¢)
| ^R
| ^R
| D#, ^Fb
| D#, ^Fb
Line 333: Line 335:
| 1266.7
| 1266.7
| sA9/sP10
| sA9/sP10
| [[225/77]] (-3.2c); [[189/65]] (+5.3c)
| [[225/77]] (&minus;3.2¢); [[189/65]] (+5.3¢)
| vJ
| vJ
| vE, ^D#
| vE, ^D#
Line 345: Line 347:
| E
| E
|}
|}
== Approximation to JI ==
=== No-2 zeta peak ===
{| class="wikitable"
|+
!Steps
per octave
!Steps
per tritave
!Step size
(cents)
!Height
!Tritave size
(cents)
!Tritave stretch
(cents)
|-
|24.573831630
|38.948601633
|48.832433543
|4.665720
|1904.464908194
|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)
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