=<span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;">15 Equal Divisions of the Tritave</span>=
== Theory ==
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.
=Properties=
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]].
The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each, corresponding to 9.464 edo. 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> 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&123, which has a 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=
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]].
|| 11 || 1394.767 || 9/4 ([[9_8|9/8]] plus an octave) ||
|| 12 || 1521.564 || 12/5 (<span style="color: #660000;">[[6_5|6/5]]</span> plus an octave) ||
|| 13 || 1648.361 || 13/5 ([[13_10|13/10]] plus an octave) ||
|| 14 || 1775.158 || 14/5 ([[7_5|7/5]] plus an octave) ||
|| 15 || 1901.955 || 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...
=== 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 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.
== 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
|}
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.
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…
=Z function=
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.
Below is a plot of the [[The Riemann Zeta Function and Tuning#Removing primes|no-twos Z function]] in the vicinity of 15edt:
[[image:15edt.png]]</pre></div>
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.
<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"><html><head><title>15edt</title></head><body><!-- ws:start:WikiTextTocRule:8:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#x15 Equal Divisions of the Tritave">15 Equal Divisions of the Tritave</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#Properties">Properties</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#Intervals of 15edt">Intervals of 15edt</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Z function">Z function</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: -->
The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each, corresponding to 9.464 edo. 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 <a class="wiki_link" href="/19edo">19edo</a> but has an <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> of <a class="wiki_link" href="/303edo">303edo</a>. 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals of 15edt"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals of 15edt</h1>
== JI approximation ==
=== Z function ===
Below is a plot of the [[The Riemann zeta function and tuning #Removing primes|no-twos Z function]] in the vicinity of 15edt:
<td>9/4 (<a class="wiki_link" href="/9_8">9/8</a> plus an octave)<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>1521.564<br />
</td>
<td>12/5 (<span style="color: #660000;"><a class="wiki_link" href="/6_5">6/5</a></span> plus an octave)<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>1648.361<br />
</td>
<td>13/5 (<a class="wiki_link" href="/13_10">13/10</a> plus an octave)<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>1775.158<br />
</td>
<td>14/5 (<a class="wiki_link" href="/7_5">7/5</a> plus an octave)<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>1901.955<br />
</td>
<td>3/1<br />
</td>
</tr>
</table>
<br />
== Audio examples ==
15edt contains 4 intervals from <a class="wiki_link" href="/5edt">5edt</a> and 2 intervals from <a class="wiki_link" href="/3edt">3edt</a>, 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...<br />
[[File:Mus_northstar_lossless.flac]]
<br />
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.<br />
A short composition by [[User:Unque|Unque]].
<br />
15edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L+3s MOS &quot;augmented scale&quot;, in which three 13/9 intervals close to a tritave, and another three are set 5/3 away.<br />
Below is a plot of the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing primes">no-twos Z function</a> in the vicinity of 15edt:<br />
15 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 15edt or 15ed3), is a nonoctavetuning system that divides the interval of 3/1 into 15 equal parts of about 127 ¢ each. Each step represents a frequency ratio of 31/15, or the 15th root of 3.
15edt corresponds to 9.4639…edo. It has harmonics5 and 13 closely in tune, but does not do so well for 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 4th harmonic, and terrible approximation of the octave. In other words; 3:4:5 is available, but 4:5:6 is not. Like the octave, the 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 [⟨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.
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, 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 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.
15edt is also associated with tempering out the mowgli comma, [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 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 also contains a 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.
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.
