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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}}
: This revision was by author [[User:guest|guest]] and made on <tt>2012-06-04 14:05:06 UTC</tt>.<br>
: The original revision id was <tt>342518720</tt>.<br>
: 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>
<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
=&lt;span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;"&gt;15 Equal Divisions of the Tritave&lt;/span&gt;=


=Properties=  
== Theory ==
The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each, corresponding to 9.464 edo, or 18.928 ed4. It has 5 and 13 closely in tune, but does not do so well for 7 and 11, which are quite sharp. It tempers out the comma |0 22 -15&gt; in the 5-limit, which is tempered out by [[19edo]] but has an [[optimal patent val]] of [[303edo]]. As a 3.5.13 subgroup system, it tempers out 2197/2187 and 3159/3125. In the 7-limit it tempers out 375/343 and 6561/6125, and in the 11-limit, 81/77, 125/121 and 363/343. 15edt is related to the 2.3.5.13 subgroup temperament 19&amp;123, which has a mapping [&lt;1 0 0 0|, &lt;0 15 22 35|], where the generator, an approximate 27/25, has a POTE tuning of 126.773, very close to 15edt.
15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1|5]] and [[13/1|13]] closely in tune, but does not do so well for [[11/1|11]], which is quite sharp. The main appeal of 15edt is that it allows for strong tritave equivalency, while supporting more conventional harmony. It achieves this with fantastic approximation of the [[4/1|4th harmonic]], and terrible approximation of the [[2/1|octave]]. In other words; 3:4:5 is available, but 4:5:6 is not. Like the octave, the [[7/1|7th harmonic]] is about halfway between steps, so 6:7:8 is well approximated, but not 4:5:7. It also tempers out the syntonic comma, [[81/80]], in the 3.4.5 subgroup, as the major third is three perfect fourths below a tritave. As a 3.5.13-[[subgroup]] system, it tempers out [[2197/2187]] and [[3159/3125]], and if these commas are added, 15edt is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping {{mapping| 1 0 0 0 | 0 15 22 35 }}, where the generator, an approximate 27/25, has a [[POTE tuning]] of 126.773, very close to 15edt.  


=Intervals of 15edt=
Using the patent val, it tempers out [[375/343]] and [[6561/6125]] in the 7-limit; [[81/77]], [[125/121]], and [[363/343]] in the 11-limit; [[65/63]], [[169/165]], [[585/539]], and [[1287/1225]] in the 13-limit; [[51/49]], [[121/119]], [[125/119]], [[189/187]], and [[195/187]] in the 17-limit (no-twos subgroup). With the patent [[4/1|4]], it tempers out [[36/35]], [[64/63]], and 375/343 in the 3.4.5.7 subgroup; [[45/44]], [[80/77]], 81/77, and 363/343 in the 3.4.5.7.11 subgroup; [[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, [[52/51]], [[85/84]], and 121/119 in the 3.4.5.7.11.13.17 subgroup ( that 15edt treated this way is essentially a retuning of [[19ed4]]). The [[k*N subgroups|2*15 subgroup]] of 15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for [[30edt]].
|| Degrees || Cents || Approximate Ratios ||
|| 0 || 0 || &lt;span style="color: #660000;"&gt;[[1_1|1/1]]&lt;/span&gt; ||
|| 1 || 126.797 || [[14_13|14/13]], [[15_14|15/14]], [[16_15|16/15]], 29/27 ||
|| 2 || 253.594 || [[15_13|15/13]] ||
|| 3 || 380.391 || &lt;span style="color: #660000;"&gt;[[5_4|5/4]]&lt;/span&gt; ||
|| 4 || 507.188 || [[4_3|4/3]] ||
|| 5 || 633.985 || [[13_9|13/9]] ||
|| 6 || 760.782 || &lt;span style="color: #660000;"&gt;[[14_9|14/9]]&lt;/span&gt; ||
|| 7 || 887.579 || [[5_3|5/3]] ||
|| 8 || 1014.376 || [[9_5|9/5]] ||
|| 9 || 1141.173 || &lt;span style="color: #660000;"&gt;[[27_14|27/14]]&lt;/span&gt; ||
|| 10 || 1267.970 || [[27_13|27/13]] ||
|| 11 || 1394.767 || [[9_4|9/4]] ([[9_8|9/8]] plus an octave) ||
|| 12 || 1521.564 || [[12_5|12/5]] (&lt;span style="color: #660000;"&gt;[[6_5|6/5]]&lt;/span&gt; plus an octave) ||
|| 13 || 1648.361 || [[13_5|13/5]] ([[13_10|13/10]] plus an octave) ||
|| 14 || 1775.158 || [[14_5|14/5]] ([[7_5|7/5]] plus an octave) ||
|| 15 || 1901.955 || [[3_1|3/1]] ||


15edt contains 4 intervals from [[5edt]] and 2 intervals from [[3edt]], meaning that it contains 6 redundant intervals and 8 new intervals. The new intervals introduced include good approximations to 15/14, 15/13, 4/3, 5/3 and their tritave inverses. This allows for new chord possibilities such as 1:3:4:5:9:12:13:14:15:16...
15edt is also associated with [[tempering out]] the mowgli comma, {{monzo| 0 22 -15 }} in the [[5-limit]], which fixes [[5/3]] to 7\15edt; in an octave context, this temperament is supported by [[19edo]] but has an [[optimal patent val]] of [[303edo]].  


15edt also contains a 5L5s MOS similar to Blackwood Decatonic, which I call Ebony. This MOS has a period of 1/5 of the tritave and the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.
=== Harmonics ===
{{Harmonics in equal|15|3|1|prec=2}}
{{Harmonics in equal|15|3|1|prec=2|columns=12|start=12|collapsed=true|Approximation of harmonics in 15edt (continued)}}


15edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L+3s MOS "augmented scale", in which three 13/9 intervals close to a tritave, and another three are set 5/3 away.
== Intervals ==
{| class="wikitable center-1 center-2 center-3"
|-
! #
! Cents
! Hekts
! Approximate ratios
! [[Polaris]] nonatonic notation
|-
| 0
| 0.0
| 0.0
| [[1/1]]
| H
|-
| 1
| 126.8
| 86.7
| [[14/13]], [[15/14]], [[16/15]], 29/27
| Ib
|-
| 2
| 253.6
| 173.3
| [[15/13]]
| vH#, ^Ib
|-
| 3
| 380.4
| 260.0
| [[5/4]]
| H#
|-
| 4
| 507.2
| 346.7
| [[4/3]]
| I
|-
| 5
| 634.0
| 433.3
| [[13/9]]
| J
|-
| 6
| 760.8
| 520.0
| [[14/9]]
| K
|-
| 7
| 887.6
| 606.7
| [[5/3]]
| L
|-
| 8
| 1014.4
| 793.3
| [[9/5]]
| Mb
|-
| 9
| 1141.2
| 780.0
| [[27/14]]
| vL#, ^Mb
|-
| 10
| 1268.0
| 866.7
| [[27/13]]
| L#
|-
| 11
| 1394.8
| 953.3
| [[9/4]]
| M
|-
| 12
| 1521.6
| 1040.0
| [[12/5]]
| N
|-
| 13
| 1648.4
| 1126.7
| [[13/5]]
| O
|-
| 14
| 1775.2
| 1213.3
| [[14/5]]
| P
|-
| 15
| 1902.0
| 1300.0
| [[3/1]]
| H
|}


=Z function=
15edt contains 4 intervals from [[5edt]] and 2 intervals from [[3edt]], meaning that it contains 6 redundant intervals and 8 new intervals. The new intervals introduced include good approximations to 15/14, 15/13, 4/3, 5/3 and their tritave inverses. This allows for new chord possibilities such as 1:3:4:5:9:12:13:14:15:16…
Below is a plot of the [[The Riemann Zeta Function and Tuning#Removing%20primes|no-twos Z function]] in the vicinity of 15edt:


[[image:15edt.png]]
15edt also contains a [[5L 5s (3/1-equivalent)|5L 5s]] mos similar to Blackwood Decatonic, which I{{who}} call Ebony. This mos has a period of 1/5 of the tritave and the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.


Music:
15edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L 3s mos "augmented scale", in which three 13/9 intervals close to a tritave, and another three are set 5/3 away.
http://www.youtube.com/watch?v=bC_Pc4jKm2k</pre></div>
<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;15edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:8:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#x15 Equal Divisions of the Tritave"&gt;15 Equal Divisions of the Tritave&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt; | &lt;a href="#Properties"&gt;Properties&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt; | &lt;a href="#Intervals of 15edt"&gt;Intervals of 15edt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Z function"&gt;Z function&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;
&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x15 Equal Divisions of the Tritave"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;"&gt;15 Equal Divisions of the Tritave&lt;/span&gt;&lt;/h1&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Properties&lt;/h1&gt;
The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each, corresponding to 9.464 edo, or 18.928 ed4. It has 5 and 13 closely in tune, but does not do so well for 7 and 11, which are quite sharp. It tempers out the comma |0 22 -15&amp;gt; in the 5-limit, which is tempered out by &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt; but has an &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; of &lt;a class="wiki_link" href="/303edo"&gt;303edo&lt;/a&gt;. As a 3.5.13 subgroup system, it tempers out 2197/2187 and 3159/3125. In the 7-limit it tempers out 375/343 and 6561/6125, and in the 11-limit, 81/77, 125/121 and 363/343. 15edt is related to the 2.3.5.13 subgroup temperament 19&amp;amp;123, which has a mapping [&amp;lt;1 0 0 0|, &amp;lt;0 15 22 35|], where the generator, an approximate 27/25, has a POTE tuning of 126.773, very close to 15edt.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Intervals of 15edt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Intervals of 15edt&lt;/h1&gt;


&lt;table class="wiki_table"&gt;
== JI approximation ==
    &lt;tr&gt;
=== Z function ===
        &lt;td&gt;Degrees&lt;br /&gt;
Below is a plot of the [[The Riemann zeta function and tuning #Removing primes|no-twos Z function]] in the vicinity of 15edt:
&lt;/td&gt;
        &lt;td&gt;Cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Approximate Ratios&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;span style="color: #660000;"&gt;&lt;a class="wiki_link" href="/1_1"&gt;1/1&lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;126.797&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/14_13"&gt;14/13&lt;/a&gt;, &lt;a class="wiki_link" href="/15_14"&gt;15/14&lt;/a&gt;, &lt;a class="wiki_link" href="/16_15"&gt;16/15&lt;/a&gt;, 29/27&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;253.594&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/15_13"&gt;15/13&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;380.391&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;span style="color: #660000;"&gt;&lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;507.188&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;633.985&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/13_9"&gt;13/9&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;760.782&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;span style="color: #660000;"&gt;&lt;a class="wiki_link" href="/14_9"&gt;14/9&lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;887.579&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/5_3"&gt;5/3&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1014.376&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/9_5"&gt;9/5&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1141.173&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;span style="color: #660000;"&gt;&lt;a class="wiki_link" href="/27_14"&gt;27/14&lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1267.970&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/27_13"&gt;27/13&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1394.767&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/9_4"&gt;9/4&lt;/a&gt; (&lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt; plus an octave)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1521.564&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/12_5"&gt;12/5&lt;/a&gt; (&lt;span style="color: #660000;"&gt;&lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;&lt;/span&gt; plus an octave)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1648.361&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/13_5"&gt;13/5&lt;/a&gt; (&lt;a class="wiki_link" href="/13_10"&gt;13/10&lt;/a&gt; plus an octave)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1775.158&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/14_5"&gt;14/5&lt;/a&gt; (&lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt; plus an octave)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1901.955&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/3_1"&gt;3/1&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
[[File:15edt.png|alt=15edt.png|15edt.png]]
15edt contains 4 intervals from &lt;a class="wiki_link" href="/5edt"&gt;5edt&lt;/a&gt; and 2 intervals from &lt;a class="wiki_link" href="/3edt"&gt;3edt&lt;/a&gt;, meaning that it contains 6 redundant intervals and 8 new intervals. The new intervals introduced include good approximations to 15/14, 15/13, 4/3, 5/3 and their tritave inverses. This allows for new chord possibilities such as 1:3:4:5:9:12:13:14:15:16...&lt;br /&gt;
 
&lt;br /&gt;
== Audio examples ==
15edt also contains a 5L5s MOS similar to Blackwood Decatonic, which I call Ebony. This MOS has a period of 1/5 of the tritave and the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.&lt;br /&gt;
[[File:Mus_northstar_lossless.flac]]
&lt;br /&gt;
 
15edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L+3s MOS &amp;quot;augmented scale&amp;quot;, in which three 13/9 intervals close to a tritave, and another three are set 5/3 away.&lt;br /&gt;
A short composition by [[User:Unque|Unque]].
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Z function"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Z function&lt;/h1&gt;
== Music ==
Below is a plot of the &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing%20primes"&gt;no-twos Z function&lt;/a&gt; in the vicinity of 15edt:&lt;br /&gt;
; [[nationalsolipsism]]
&lt;br /&gt;
* [https://www.youtube.com/watch?v=bC_Pc4jKm2k ''ox-idation''] (2012)
&lt;!-- ws:start:WikiTextLocalImageRule:152:&amp;lt;img src=&amp;quot;/file/view/15edt.png/250617832/15edt.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/15edt.png/250617832/15edt.png" alt="15edt.png" title="15edt.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:152 --&gt;&lt;br /&gt;
 
&lt;br /&gt;
[[Category:Macrotonal]]
Music:&lt;br /&gt;
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