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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{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.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-04 14:05:29 UTC</tt>.<br>
 
: The original revision id was <tt>250636156</tt>.<br>
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.
: The revision comment was: <tt></tt><br>
 
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
== Theory ==
<h4>Original Wikitext content:</h4>
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.
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The 39 equal division of 3, the tritave, divides it into 39 equal parts of 48.678 cents each, corresponding to 24.606 edo. 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&amp;172f temperament tempering out 245/243, 275/273, 847/845 and 1575/1573, which has map [&lt;1 0 0 0 0 0|, &lt;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]].  
 
</pre></div>
If octaves are inserted, 39edt is related to the {{nowrap|49f &amp; 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]].
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;39edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 39 equal division of 3, the tritave, divides it into 39 equal parts of 48.678 cents each, corresponding to 24.606 edo. It is a strong no-twos 13-limit system, a fact first noted by &lt;a class="wiki_link" href="/Paul%20Erlich"&gt;Paul Erlich&lt;/a&gt;, and like &lt;a class="wiki_link" href="/26edt"&gt;26edt&lt;/a&gt; and &lt;a class="wiki_link" href="/52edt"&gt;52edt&lt;/a&gt; it is a multiple of &lt;a class="wiki_link" href="/13edt"&gt;13edt&lt;/a&gt; and so contains the &lt;a class="wiki_link" href="/Bohlen-Pierce"&gt;Bohlen-Pierce&lt;/a&gt; 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&amp;amp;172f temperament tempering out 245/243, 275/273, 847/845 and 1575/1573, which has map [&amp;lt;1 0 0 0 0 0|, &amp;lt;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 &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing primes"&gt;no-twos zeta peak edt&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
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"
|-
! Steps
! [[Cent]]s
! [[Hekt]]s
! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree
! 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
| P1
| [[1/1]]
| J
| E
|-
| 1
| 48.8
| 33.3
| SP1
| [[77/75]] (+3.2¢); [[65/63]] (&minus;5.3¢)
| ^J
| ^E, vF
|-
| 2
| 97.5
| 66.7
| sA1/sm2
| [[35/33]] (&minus;4.3¢); [[81/77]] (+9.9¢)
| vK
| F
|-
| 3
| 146.3
| 100
| A1/m2
| [[99/91]] (+0.4¢); [[49/45]] (&minus;1.1¢); [[27/25]] (+13.1¢)
| K
| ^F, vGb, Dx
|-
| 4
| 195.1
| 133.3
| SA1/Sm2
| [[55/49]] (&minus;4.9¢); [[91/81]] (&minus;6.5¢); [[39/35]] (+7.7¢)
| ^K
| Gb, vE#
|-
| 5
| 243.8
| 166.7
| sM2/sd3
| [[15/13]] (&minus;3.9¢); [[63/55]] (+8.7¢)
| vK#, vLb
| ^Gb, E#
|-
| 6
| 292.6
| 200
| M2/d3
| [[77/65]] (&minus;0.7¢); [[13/11]] (+3.4¢); [[25/21]] (&minus;9.2¢)
| K#, Lb
| vF#, ^E#
|-
| 7
| 341.4
| 233.3
| SM2/Sd3
| [[11/9]] (&minus;6.0¢); [[91/75]] (+6.6¢)
| ^K#, ^Lb
| F#
|-
| 8
| 390.1
| 266.7
| sA2/sP3/sd4
| [[49/39]] (&minus;5.0¢); [[81/65]] (+9.2¢)
| vL
| vG, ^F#
|-
| 9
| 438.9
| 300
| A2/P3/d4
| [[9/7]] (+3.8¢); [[35/27]] (&minus;10.3¢)
| L
| G
|-
| 10
| 487.7
| 333.3
| SA2/SP3/Sd4
| [[65/49]] (&minus;1.5¢); [[33/25]] (+7.0¢)
| ^L
| ^G, vAb
|-
| 11
| 536.4
| 366.7
| sA3/sm4/sd5
| [[15/11]] (&minus;0.5¢)
| vM
| Ab
|-
| 12
| 585.2
| 400
| A3/m4/d5
| [[7/5]] (+2.7¢)
| M
| ^Ab, Fx
|-
| 13
| 634.0
| 433.3
| SA3/Sm4/Sd5
| [[13/9]] (&minus;2.6¢)
| ^M
| vG#
|-
| 14
| 682.7
| 466.7
| sM4/sm5
| [[135/91]] (+0.07¢); [[49/33]] (&minus;1.6¢); [[81/55]] (+12.6¢)
| vM#, vNb
| G#
|-
| 15
| 731.5
| 500
| M4/m5
| [[75/49]] (&minus;5.4¢); [[117/77]] (+7.2¢)
| M#, Nb
| vA, ^G#
|-
| 16
| 780.3
| 533.3
| SM4/Sm5
| [[11/7]] (&minus;2.2¢); [[39/25]] (+10.4¢)
| ^M#, ^Nb
| A
|-
| 17
| 829.0
| 566.7
| sA4/sM5
| [[21/13]] (&minus;1.2¢)
| vN
| ^A, vBb
|-
| 18
| 877.8
| 600
| A4/M5
| [[91/55]] (+6.1¢); [[5/3]] (&minus;6.5¢); [[81/49]] (+7.7¢)
| N
| Bb
|-
| 19
| 926.6
| 633.3
| SA4/SM5
| [[77/45]] (&minus;3.3¢)
| ^N
| ^Bb, vCb, Gx
|-
| 20
| 975.3
| 666.7
| sA5/sm6/sd7
| [[135/77]] (+3.3¢)
| vO
| vA#, Cb
|-
| 21
| 1024.1
| 700
| A5/m6/d7
| [[165/91]] (&minus;6.1¢); [[9/5]] (+6.5¢); [[49/27]] (&minus;7.7¢)
| O
| A#, ^Cb
|-
| 22
| 1072.9
| 733.3
| SA5/Sm6/Sd7
| [[13/7]] (+1.2¢)
| ^O
| vB, ^A#
|-
| 23
| 1121.6
| 766.7
| sM6/sm7
| [[21/11]] (+2.2¢); [[25/13]] (&minus;10.4¢)
| vO#, vPb
| B
|-
| 24
| 1170.4
| 800
| M6/m7
| [[49/25]] (+5.4¢); [[77/39]] (&minus;7.2¢)
| O#, Pb
| ^B, vC
|-
| 25
| 1219.2
| 833.3
| SM6/Sm7
| [[91/45]] (+0.07¢); [[99/49]] (+1.6¢); [[55/27]] (&minus;12.6¢)
| ^O#, ^Pb
| C
|-
| 26
| 1267.9
| 866.7
| sA6/sM7/sd8
| [[27/13]] (+2.6¢)
| vP
| ^C, vDb
|-
| 27
| 1316.7
| 900
| A6/M7/d8
| [[15/7]] (&minus;2.7¢)
| P
| Db, vB#
|-
| 28
| 1365.5
| 933.3
| SA6/SM7/Sd8
| [[11/5]] (+0.5¢)
| ^P
| ^Db, B#
|-
| 29
| 1414.2
| 966.7
| sP8/sd9
| [[147/65]] (+1.5¢); [[25/11]] (&minus;7.0¢)
| vQ
| vC#, ^B#
|-
| 30
| 1463.0
| 1000
| P8/d9
| [[7/3]] (&minus;3.8¢); [[81/35]] (+10.3¢)
| Q
| C#
|-
| 31
| 1511.8
| 1033.3
| SP8/Sd9
| [[117/49]] (+5.0¢); [[65/27]] (&minus;9.2¢)
| ^Q
| vD, ^C#
|-
| 32
| 1560.5
| 1066.7
| sA8/sm9
| [[27/11]] (+6.0¢); [[225/91]] (+6.6¢)
| vQ#, vRb
| D
|-
| 33
| 1609.3
| 1100
| A8/m9
| [[195/77]] (&minus;0.7¢); [[33/13]] (&minus;3.4¢); [[63/25]] (+9.2¢)
| Q#, Rb
| ^D, vEb
|-
| 34
| 1658.1
| 1133.3
| SA8/Sm9
| [[13/5]] (+3.9¢); [[55/21]] (&minus;8.7¢)
| ^Q#, ^Rb
| Eb
|-
| 35
| 1706.9
| 1166.7
| sM9/sd10
| [[147/55]] (+4.9¢); [[243/91]] (+6.5¢); [[35/13]] (&minus;7.7¢)
| vR
| ^Eb, vFb, Cx
|-
| 36
| 1755.7
| 1200
| M9/d10
| [[91/33]] (+0.4¢); [[135/49]] (+1.1¢); [[25/9]] (&minus;13.1¢)
| R
| vD#, Fb
|-
| 37
| 1804.5
| 1233.3
| SM9/Sd10
| [[99/35]] (+4.3¢); [[77/27]] (&minus;9.9¢)
| ^R
| D#, ^Fb
|-
| 38
| 1853.2
| 1266.7
| sA9/sP10
| [[225/77]] (&minus;3.2¢); [[189/65]] (+5.3¢)
| vJ
| vE, ^D#
|-
| 39
| 1902.0
| 1300
| A9/P10
| [[3/1]]
| J
| 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)
Retrieved from "https://en.xen.wiki/w/39edt"