14edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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14-EDO has quite a bit of xenharmonic appeal, in a similar way to 17-EDO, on account of having three types of 3rd and three types of 6th, rather than the usual two of 12-TET. Since 14-EDO also has a recognizable 4th and 5th, this makes it good for those wishing to explore alternative triadic harmonies without adding significantly more notes. It possesses a triad-rich 9-note MOS scale of 5L4s, wherein 7 of 9 notes are tonic to a subminor, supermajor, and/or neutral triad.

It is also the largest edo whose patent val supports [[titanium]] temperament, tempering out the chromatic semitone (21:20), and falling toward the "brittle" (fifths wider than in 9edo) end of that spectrum. Titanium is one of the simplest 7-limit temperaments, although rather inaccurate (the 7:5 is mapped onto 6\14, over 70 cents flat). Its otonal/major and utonal/minor tetrads are inversions of one another, which allows a greater variety of chord progressions (since different inversions of the same chord may have very different expressive qualities). Despite being so heavily tempered, the tetrads are still recognizable and aren't unpleasant-sounding as long as one uses the right timbres. Titanium forms enneatonic modes which are melodically strong and are very similar to diatonic modes, only with two mediants and submediants instead of one. Titanium[9] has similarities to mavila, slendro, and pelog scales as well.

===Titanium[9]===

14edo is also the largest edo whose patent val supports [[titanium]] temperament, tempering out the chromatic semitone (21:20), and falling toward the "brittle" (fifths wider than in 9edo) end of that spectrum. Titanium is one of the simplest 7-limit temperaments, although rather inaccurate (the 7:5 is mapped onto 6\14, over 70 cents flat). Its otonal/major and utonal/minor tetrads are inversions of one another, which allows a greater variety of chord progressions (since different inversions of the same chord may have very different expressive qualities). Despite being so heavily tempered, the tetrads are still recognizable and aren't unpleasant-sounding as long as one uses the right timbres ("bell-like" or opaque-sounding ones probably work best). Titanium forms enneatonic modes which are melodically strong and are very similar to diatonic modes, only with two mediants and submediants instead of one. Titanium[9] has similarities to mavila, slendro, and pelog scales as well.

Using titanium[9], we could name the intervals of 14edo as follows. The 3, 5, 6, 8, 9, and 11-step intervals are all consonant, while 1, 2, 4, 7, 10, 12, and 13 steps are dissonant. There is no distinction between "perfect" (modulatory) and "imperfect" (major/minor) consonances here; there are enough chords here that root motion may occur by //any// consonant interval, and thus //all// six consonances are "perfect" intervals, rather than just two of them as in the diatonic system. As in the diatonic scale, the perfect intervals come in pairs separated by a major second, and with a characteristic dissonance between them; in titanium[9] there are three such pairs rather than just one.

1\14: Minor 2nd<span style="vertical-align: sub;">9</span>: functions similarly to the diatonic minor second, but is more incisive.

2\14: Major 2nd<span style="vertical-align: sub;">9</span>: functions similarly to the diatonic major second, but is narrower and has a rather different quality.

3\14: Perfect 3rd<span style="vertical-align: sub;">9</span>: the generator, standing in for 8:7, 7:6, //and// 6:5, but closest to 7:6.

4\14: Augmented 3rd<span style="vertical-align: sub;">9</span>/diminished 4th<span style="vertical-align: sub;">9</span>: A dissonance, falling in between two perfect consonances and hence analogous to the tritone.

5\14: Perfect 4th<span style="vertical-align: sub;">9</span>: technically represents 5:4 but is quite a bit wider.

6\14: Perfect 5th<span style="vertical-align: sub;">9</span>: represents 4:3 and 7:5, much closer to the former.

7\14: Augmented 5th<span style="vertical-align: sub;">9</span>/diminished 6th<span style="vertical-align: sub;">9</span>: The so-called "tritone" (but no longer made up of three whole tones). Like 4\14 and 10\14, this is a characteristic dissonance separating a pair of perfect consonances.

8\14: Perfect 6th<span style="vertical-align: sub;">9</span>: represents 10:7 and 3:2, much closer to the latter.

9\14: Perfect 7th<span style="vertical-align: sub;">9</span>: technically represents 5:8 but noticeably narrower.

10\14: Augmented 7th<span style="vertical-align: sub;">9</span>/diminished 8th<span style="vertical-align: sub;">9</span>: The third and final characteristic dissonance, analogous to the tritone.

11\14: Perfect 8th<span style="vertical-align: sub;">9</span>: Represents 5:3, 12:7 and 7:4.

12\14: Minor 9th<span style="vertical-align: sub;">9</span>: Analogous to the diatonic minor seventh, but sharper than usual.

13\14: Major 9th<span style="vertical-align: sub;">9</span>: A high, incisive leading tone.

14\14: The 10th<span style="vertical-align: sub;">9 </span>or "enneatonic decave", (i. e., the octave, 2:1).

=Notation=

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</span><br />

<a href="#Intervals">Intervals</a> | <a href="#Rank two temperaments">Rank two temperaments</a> | <a href="#Harmony">Harmony</a> | <a href="#Notation">Notation</a> | <a href="#Images">Images</a> | <a href="#Commas">Commas</a> | <a href="#Modes">Modes</a> | <a href="#Books">Books</a> | <a href="#Compositions">Compositions</a>

<a href="#Intervals">Intervals</a> | <a href="#Rank two temperaments">Rank two temperaments</a> | <a href="#Harmony">Harmony</a> | <a href="#Notation">Notation</a> | <a href="#Images">Images</a> | <a href="#Commas">Commas</a> | <a href="#Modes">Modes</a> | <a href="#Books">Books</a> | <a href="#Compositions">Compositions</a>

<h1 id="toc0"><a name="Intervals"></a><strong>Intervals</strong></h1>

<h1 id="toc0"><a name="Intervals"></a><strong>Intervals</strong></h1>

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<object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/14ed2-001.svg">alt : Your browser has no SVG support.</object><br />

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<br />

<h1 id="toc1"><a name="Rank two temperaments"></a>Rank two temperaments</h1>

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14-EDO has quite a bit of xenharmonic appeal, in a similar way to 17-EDO, on account of having three types of 3rd and three types of 6th, rather than the usual two of 12-TET. Since 14-EDO also has a recognizable 4th and 5th, this makes it good for those wishing to explore alternative triadic harmonies without adding significantly more notes. It possesses a triad-rich 9-note MOS scale of 5L4s, wherein 7 of 9 notes are tonic to a subminor, supermajor, and/or neutral triad.<br />

<br />

It is also the largest edo whose patent val supports <a class="wiki_link" href="/titanium">titanium</a> temperament, tempering out the chromatic semitone (21:20), and falling toward the &quot;brittle&quot; (fifths wider than in 9edo) end of that spectrum. Titanium is one of the simplest 7-limit temperaments, although rather inaccurate (the 7:5 is mapped onto 6\14, over 70 cents flat). Its otonal/major and utonal/minor tetrads are inversions of one another, which allows a greater variety of chord progressions (since different inversions of the same chord may have very different expressive qualities). Despite being so heavily tempered, the tetrads are still recognizable and aren't unpleasant-sounding as long as one uses the right timbres. Titanium forms enneatonic modes which are melodically strong and are very similar to diatonic modes, only with two mediants and submediants instead of one. Titanium[9] has similarities to mavila, slendro, and pelog scales as well.<br />

<h3 id="toc3"><a name="Harmony--Titanium[9]"></a>Titanium[9]</h3>

<br />

14edo is also the largest edo whose patent val supports <a class="wiki_link" href="/titanium">titanium</a> temperament, tempering out the chromatic semitone (21:20), and falling toward the &quot;brittle&quot; (fifths wider than in 9edo) end of that spectrum. Titanium is one of the simplest 7-limit temperaments, although rather inaccurate (the 7:5 is mapped onto 6\14, over 70 cents flat). Its otonal/major and utonal/minor tetrads are inversions of one another, which allows a greater variety of chord progressions (since different inversions of the same chord may have very different expressive qualities). Despite being so heavily tempered, the tetrads are still recognizable and aren't unpleasant-sounding as long as one uses the right timbres (&quot;bell-like&quot; or opaque-sounding ones probably work best). Titanium forms enneatonic modes which are melodically strong and are very similar to diatonic modes, only with two mediants and submediants instead of one. Titanium[9] has similarities to mavila, slendro, and pelog scales as well.<br />

<br />

Using titanium[9], we could name the intervals of 14edo as follows. The 3, 5, 6, 8, 9, and 11-step intervals are all consonant, while 1, 2, 4, 7, 10, 12, and 13 steps are dissonant. There is no distinction between &quot;perfect&quot; (modulatory) and &quot;imperfect&quot; (major/minor) consonances here; there are enough chords here that root motion may occur by <em>any</em> consonant interval, and thus <em>all</em> six consonances are &quot;perfect&quot; intervals, rather than just two of them as in the diatonic system. As in the diatonic scale, the perfect intervals come in pairs separated by a major second, and with a characteristic dissonance between them; in titanium[9] there are three such pairs rather than just one.<br />

<br />

1\14: Minor 2nd<span style="vertical-align: sub;">9</span>: functions similarly to the diatonic minor second, but is more incisive.<br />

2\14: Major 2nd<span style="vertical-align: sub;">9</span>: functions similarly to the diatonic major second, but is narrower and has a rather different quality.<br />

3\14: Perfect 3rd<span style="vertical-align: sub;">9</span>: the generator, standing in for 8:7, 7:6, <em>and</em> 6:5, but closest to 7:6.<br />

4\14: Augmented 3rd<span style="vertical-align: sub;">9</span>/diminished 4th<span style="vertical-align: sub;">9</span>: A dissonance, falling in between two perfect consonances and hence analogous to the tritone.<br />

5\14: Perfect 4th<span style="vertical-align: sub;">9</span>: technically represents 5:4 but is quite a bit wider.<br />

6\14: Perfect 5th<span style="vertical-align: sub;">9</span>: represents 4:3 and 7:5, much closer to the former.<br />

7\14: Augmented 5th<span style="vertical-align: sub;">9</span>/diminished 6th<span style="vertical-align: sub;">9</span>: The so-called &quot;tritone&quot; (but no longer made up of three whole tones). Like 4\14 and 10\14, this is a characteristic dissonance separating a pair of perfect consonances.<br />

8\14: Perfect 6th<span style="vertical-align: sub;">9</span>: represents 10:7 and 3:2, much closer to the latter.<br />

9\14: Perfect 7th<span style="vertical-align: sub;">9</span>: technically represents 5:8 but noticeably narrower.<br />

10\14: Augmented 7th<span style="vertical-align: sub;">9</span>/diminished 8th<span style="vertical-align: sub;">9</span>: The third and final characteristic dissonance, analogous to the tritone.<br />

11\14: Perfect 8th<span style="vertical-align: sub;">9</span>: Represents 5:3, 12:7 and 7:4.<br />

12\14: Minor 9th<span style="vertical-align: sub;">9</span>: Analogous to the diatonic minor seventh, but sharper than usual. <br />

13\14: Major 9th<span style="vertical-align: sub;">9</span>: A high, incisive leading tone.<br />

14\14: The 10th<span style="vertical-align: sub;">9 </span>or &quot;enneatonic decave&quot;, (i. e., the octave, 2:1).<br />

<br />

<h1 id="~~toc3~~"><a name="Notation"></a>Notation</h1>

<h1 id="toc4"><a name="Notation"></a>Notation</h1>

<a class="wiki_link" href="/Ivor%20Darreg">Ivor Darreg</a> wrote in <a class="wiki_link_ext" href="http://sonic-arts.org/darreg/dar15.htm" rel="nofollow">this article</a>:<br />

<br />

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The following chart (made by <a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil">TDW</a>) shows this recommendation as &quot;standard notation&quot; as well as a proposed alternative.<br />

<br />

<table class="captionBox"><tr><td class="captionedImage"><img src="/file/view/Ciclo_Tetradecafon%C3%ADa.png/233692646/456x468/Ciclo_Tetradecafon%C3%ADa.png" alt="Ciclo_Tetradecafonía.png" title="Ciclo_Tetradecafonía.png" style="height: 468px; width: 456px;" /></td></tr><tr><td class="imageCaption">Intervallic Cycle of 14 steps Equal per Octave</td></tr></table><br />

<table class="captionBox"><tr><td class="captionedImage"><img src="/file/view/Ciclo_Tetradecafon%C3%ADa.png/233692646/456x468/Ciclo_Tetradecafon%C3%ADa.png" alt="Ciclo_Tetradecafonía.png" title="Ciclo_Tetradecafonía.png" style="height: 468px; width: 456px;" /></td></tr><tr><td class="imageCaption">Intervallic Cycle of 14 steps Equal per Octave</td></tr></table><br />

<br />

<h1 id="~~toc4~~"><a name="Images"></a>Images</h1>

<h1 id="toc5"><a name="Images"></a>Images</h1>

<br />

<img src="/file/view/14edo%20wheel.png/516971016/343x343/14edo%20wheel.png" alt="14edo wheel.png" title="14edo wheel.png" style="height: 343px; width: 343px;" /><br />

<img src="/file/view/14edo%20wheel.png/516971016/343x343/14edo%20wheel.png" alt="14edo wheel.png" title="14edo wheel.png" style="height: 343px; width: 343px;" /><br />

<br />

<h1 id="~~toc5~~"><a name="Commas"></a>Commas</h1>

<h1 id="toc6"><a name="Commas"></a>Commas</h1>

14 EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes the val &lt; 14 22 33 39 48 52 |.)<br />

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<br />

<h1 id="~~toc6~~"><a name="Modes"></a><strong><a class="wiki_link" href="/modes">Modes</a></strong></h1>

<h1 id="toc7"><a name="Modes"></a><strong><a class="wiki_link" href="/modes">Modes</a></strong></h1>

6 6 2 - <a class="wiki_link" href="/MOSScales">MOS</a> of <a class="wiki_link" href="/2L1s">2L1s</a><br />

5 5 4 - <span style="cursor: pointer;"><a class="wiki_link" href="/MOSScales">MOS</a></span> of <span style="cursor: pointer;"><a class="wiki_link" href="/2L1s">2L1s</a></span><br />

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<br />

<h1 id="~~toc7~~"><a name="Books"></a>Books</h1>

<h1 id="toc8"><a name="Books"></a>Books</h1>

<img src="/file/view/Libro_Tetradecaf%C3%B3nico.PNG/163539671/Libro_Tetradecaf%C3%B3nico.PNG" alt="Libro_Tetradecafónico.PNG" title="Libro_Tetradecafónico.PNG" /><br />

<img src="/file/view/Libro_Tetradecaf%C3%B3nico.PNG/163539671/Libro_Tetradecaf%C3%B3nico.PNG" alt="Libro_Tetradecafónico.PNG" title="Libro_Tetradecafónico.PNG" /><br />

<em>Sword, Ron. &quot;Tetradecaphonic Scales for Guitar&quot; IAAA Press. First Ed: June 2009.</em><br />

<br />

<h1 id="~~toc8~~"><a name="Compositions"></a><strong>Compositions</strong></h1>

<h1 id="toc9"><a name="Compositions"></a><strong>Compositions</strong></h1>

<a class="wiki_link_ext" href="http://split-notes.com/004/" rel="nofollow">NANA WODORI</a> by knowsur<br />

<a class="wiki_link_ext" href="http://micro.soonlabel.com/0-praxis/audio/August/august_03_thereminnards.mp3" rel="nofollow">Thereminnards</a> by Ralph Lewis<br />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2016-07-13 17:54:53 UTC</tt>.<br>
+: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2016-07-13 23:42:11 UTC</tt>.<br>
-: The original revision id was <tt>586884073</tt>.<br>
+: The original revision id was <tt>586893795</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>
@@ Line 64: / Line 64: @@
 -EDO has quite a bit of xenharmonic appeal, in a similar way to 17-EDO, on account of having three types of 3rd and three types of 6th, rather than the usual two of 12-TET. Since 14-EDO also has a recognizable 4th and 5th, this makes it good for those wishing to explore alternative triadic harmonies without adding significantly more notes. It possesses a triad-rich 9-note MOS scale of 5L4s, wherein 7 of 9 notes are tonic to a subminor, supermajor, and/or neutral triad.
-It is also the largest edo whose patent val supports [[titanium]] temperament, tempering out the chromatic semitone (21:20), and falling toward the "brittle" (fifths wider than in 9edo) end of that spectrum. Titanium is one of the simplest 7-limit temperaments, although rather inaccurate (the 7:5 is mapped onto 6\14, over 70 cents flat). Its otonal/major and utonal/minor tetrads are inversions of one another, which allows a greater variety of chord progressions (since different inversions of the same chord may have very different expressive qualities). Despite being so heavily tempered, the tetrads are still recognizable and aren't unpleasant-sounding as long as one uses the right timbres. Titanium forms enneatonic modes which are melodically strong and are very similar to diatonic modes, only with two mediants and submediants instead of one. Titanium[9] has similarities to mavila, slendro, and pelog scales as well.
+===Titanium[9]===
+edo is also the largest edo whose patent val supports [[titanium]] temperament, tempering out the chromatic semitone (21:20), and falling toward the "brittle" (fifths wider than in 9edo) end of that spectrum. Titanium is one of the simplest 7-limit temperaments, although rather inaccurate (the 7:5 is mapped onto 6\14, over 70 cents flat). Its otonal/major and utonal/minor tetrads are inversions of one another, which allows a greater variety of chord progressions (since different inversions of the same chord may have very different expressive qualities). Despite being so heavily tempered, the tetrads are still recognizable and aren't unpleasant-sounding as long as one uses the right timbres ("bell-like" or opaque-sounding ones probably work best). Titanium forms enneatonic modes which are melodically strong and are very similar to diatonic modes, only with two mediants and submediants instead of one. Titanium[9] has similarities to mavila, slendro, and pelog scales as well.
+Using titanium[9], we could name the intervals of 14edo as follows. The 3, 5, 6, 8, 9, and 11-step intervals are all consonant, while 1, 2, 4, 7, 10, 12, and 13 steps are dissonant. There is no distinction between "perfect" (modulatory) and "imperfect" (major/minor) consonances here; there are enough chords here that root motion may occur by //any// consonant interval, and thus //all// six consonances are "perfect" intervals, rather than just two of them as in the diatonic system. As in the diatonic scale, the perfect intervals come in pairs separated by a major second, and with a characteristic dissonance between them; in titanium[9] there are three such pairs rather than just one.
+\14: Minor 2nd&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: functions similarly to the diatonic minor second, but is more incisive.
+\14: Major 2nd&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: functions similarly to the diatonic major second, but is narrower and has a rather different quality.
+\14: Perfect 3rd&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: the generator, standing in for 8:7, 7:6, //and// 6:5, but closest to 7:6.
+\14: Augmented 3rd&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;/diminished 4th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: A dissonance, falling in between two perfect consonances and hence analogous to the tritone.
+\14: Perfect 4th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: technically represents 5:4 but is quite a bit wider.
+\14: Perfect 5th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: represents 4:3 and 7:5, much closer to the former.
+\14: Augmented 5th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;/diminished 6th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: The so-called "tritone" (but no longer made up of three whole tones). Like 4\14 and 10\14, this is a characteristic dissonance separating a pair of perfect consonances.
+\14: Perfect 6th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: represents 10:7 and 3:2, much closer to the latter.
+\14: Perfect 7th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: technically represents 5:8 but noticeably narrower.
+\14: Augmented 7th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;/diminished 8th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: The third and final characteristic dissonance, analogous to the tritone.
+\14: Perfect 8th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: Represents 5:3, 12:7 and 7:4.
+\14: Minor 9th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: Analogous to the diatonic minor seventh, but sharper than usual.
+\14: Major 9th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: A high, incisive leading tone.
+\14: The 10th&lt;span style="vertical-align: sub;"&gt;9 &lt;/span&gt;or "enneatonic decave", (i. e., the octave, 2:1).
 =Notation=
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 <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;14edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; text-align: right;"&gt;&lt;a class="wiki_link" href="/14%E5%B9%B3%E5%9D%87%E5%BE%8B"&gt;日本語&lt;/a&gt;&lt;br /&gt;
 &lt;/span&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextTocRule:23:&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:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Rank two temperaments"&gt;Rank two temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt; | &lt;a href="#Harmony"&gt;Harmony&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Notation"&gt;Notation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Images"&gt;Images&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt; | &lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt; | &lt;a href="#Modes"&gt;Modes&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt; | &lt;a href="#Books"&gt;Books&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;
+&lt;!-- ws:start:WikiTextTocRule:25:&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:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Rank two temperaments"&gt;Rank two temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Harmony"&gt;Harmony&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt; | &lt;a href="#Notation"&gt;Notation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt; | &lt;a href="#Images"&gt;Images&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt; | &lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt; | &lt;a href="#Modes"&gt;Modes&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt; | &lt;a href="#Books"&gt;Books&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;
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+&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;&lt;strong&gt;Intervals&lt;/strong&gt;&lt;/h1&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Rank two temperaments&lt;/h1&gt;
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 -EDO has quite a bit of xenharmonic appeal, in a similar way to 17-EDO, on account of having three types of 3rd and three types of 6th, rather than the usual two of 12-TET. Since 14-EDO also has a recognizable 4th and 5th, this makes it good for those wishing to explore alternative triadic harmonies without adding significantly more notes. It possesses a triad-rich 9-note MOS scale of 5L4s, wherein 7 of 9 notes are tonic to a subminor, supermajor, and/or neutral triad.&lt;br /&gt;
 &lt;br /&gt;
-It is also the largest edo whose patent val supports &lt;a class="wiki_link" href="/titanium"&gt;titanium&lt;/a&gt; temperament, tempering out the chromatic semitone (21:20), and falling toward the &amp;quot;brittle&amp;quot; (fifths wider than in 9edo) end of that spectrum. Titanium is one of the simplest 7-limit temperaments, although rather inaccurate (the 7:5 is mapped onto 6\14, over 70 cents flat). Its otonal/major and utonal/minor tetrads are inversions of one another, which allows a greater variety of chord progressions (since different inversions of the same chord may have very different expressive qualities). Despite being so heavily tempered, the tetrads are still recognizable and aren't unpleasant-sounding as long as one uses the right timbres. Titanium forms enneatonic modes which are melodically strong and are very similar to diatonic modes, only with two mediants and submediants instead of one. Titanium[9] has similarities to mavila, slendro, and pelog scales as well.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:11:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="Harmony--Titanium[9]"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:11 --&gt;Titanium[9]&lt;/h3&gt;
+ &lt;br /&gt;
+edo is also the largest edo whose patent val supports &lt;a class="wiki_link" href="/titanium"&gt;titanium&lt;/a&gt; temperament, tempering out the chromatic semitone (21:20), and falling toward the &amp;quot;brittle&amp;quot; (fifths wider than in 9edo) end of that spectrum. Titanium is one of the simplest 7-limit temperaments, although rather inaccurate (the 7:5 is mapped onto 6\14, over 70 cents flat). Its otonal/major and utonal/minor tetrads are inversions of one another, which allows a greater variety of chord progressions (since different inversions of the same chord may have very different expressive qualities). Despite being so heavily tempered, the tetrads are still recognizable and aren't unpleasant-sounding as long as one uses the right timbres (&amp;quot;bell-like&amp;quot; or opaque-sounding ones probably work best). Titanium forms enneatonic modes which are melodically strong and are very similar to diatonic modes, only with two mediants and submediants instead of one. Titanium[9] has similarities to mavila, slendro, and pelog scales as well.&lt;br /&gt;
+&lt;br /&gt;
+Using titanium[9], we could name the intervals of 14edo as follows. The 3, 5, 6, 8, 9, and 11-step intervals are all consonant, while 1, 2, 4, 7, 10, 12, and 13 steps are dissonant. There is no distinction between &amp;quot;perfect&amp;quot; (modulatory) and &amp;quot;imperfect&amp;quot; (major/minor) consonances here; there are enough chords here that root motion may occur by &lt;em&gt;any&lt;/em&gt; consonant interval, and thus &lt;em&gt;all&lt;/em&gt; six consonances are &amp;quot;perfect&amp;quot; intervals, rather than just two of them as in the diatonic system. As in the diatonic scale, the perfect intervals come in pairs separated by a major second, and with a characteristic dissonance between them; in titanium[9] there are three such pairs rather than just one.&lt;br /&gt;
+&lt;br /&gt;
+\14: Minor 2nd&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: functions similarly to the diatonic minor second, but is more incisive.&lt;br /&gt;
+\14: Major 2nd&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: functions similarly to the diatonic major second, but is narrower and has a rather different quality.&lt;br /&gt;
+\14: Perfect 3rd&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: the generator, standing in for 8:7, 7:6, &lt;em&gt;and&lt;/em&gt; 6:5, but closest to 7:6.&lt;br /&gt;
+\14: Augmented 3rd&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;/diminished 4th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: A dissonance, falling in between two perfect consonances and hence analogous to the tritone.&lt;br /&gt;
+\14: Perfect 4th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: technically represents 5:4 but is quite a bit wider.&lt;br /&gt;
+\14: Perfect 5th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: represents 4:3 and 7:5, much closer to the former.&lt;br /&gt;
+\14: Augmented 5th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;/diminished 6th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: The so-called &amp;quot;tritone&amp;quot; (but no longer made up of three whole tones). Like 4\14 and 10\14, this is a characteristic dissonance separating a pair of perfect consonances.&lt;br /&gt;
+\14: Perfect 6th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: represents 10:7 and 3:2, much closer to the latter.&lt;br /&gt;
+\14: Perfect 7th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: technically represents 5:8 but noticeably narrower.&lt;br /&gt;
+\14: Augmented 7th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;/diminished 8th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: The third and final characteristic dissonance, analogous to the tritone.&lt;br /&gt;
+\14: Perfect 8th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: Represents 5:3, 12:7 and 7:4.&lt;br /&gt;
+\14: Minor 9th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: Analogous to the diatonic minor seventh, but sharper than usual. &lt;br /&gt;
+\14: Major 9th&lt;span style="vertical-align: sub;"&gt;9&lt;/span&gt;: A high, incisive leading tone.&lt;br /&gt;
+\14: The 10th&lt;span style="vertical-align: sub;"&gt;9 &lt;/span&gt;or &amp;quot;enneatonic decave&amp;quot;, (i. e., the octave, 2:1).&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:11:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:11 --&gt;Notation&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt;Notation&lt;/h1&gt;
   &lt;a class="wiki_link" href="/Ivor%20Darreg"&gt;Ivor Darreg&lt;/a&gt; wrote in &lt;a class="wiki_link_ext" href="http://sonic-arts.org/darreg/dar15.htm" rel="nofollow"&gt;this article&lt;/a&gt;:&lt;br /&gt;
 &lt;br /&gt;
@@ Line 455: / Line 493: @@
 The following chart (made by &lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&gt;TDW&lt;/a&gt;) shows this recommendation as &amp;quot;standard notation&amp;quot; as well as a proposed alternative.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:450:&amp;lt;img src=&amp;quot;/file/view/Ciclo_Tetradecafon%C3%ADa.png/233692646/456x468/Ciclo_Tetradecafon%C3%ADa.png&amp;quot; alt=&amp;quot;Intervallic Cycle of 14 steps Equal per Octave&amp;quot; title=&amp;quot;Intervallic Cycle of 14 steps Equal per Octave&amp;quot; style=&amp;quot;height: 468px; width: 456px;&amp;quot; /&amp;gt; --&gt;&lt;table class="captionBox"&gt;&lt;tr&gt;&lt;td class="captionedImage"&gt;&lt;img src="/file/view/Ciclo_Tetradecafon%C3%ADa.png/233692646/456x468/Ciclo_Tetradecafon%C3%ADa.png" alt="Ciclo_Tetradecafonía.png" title="Ciclo_Tetradecafonía.png" style="height: 468px; width: 456px;" /&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td class="imageCaption"&gt;Intervallic Cycle of 14 steps Equal per Octave&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;&lt;!-- ws:end:WikiTextLocalImageRule:450 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:453:&amp;lt;img src=&amp;quot;/file/view/Ciclo_Tetradecafon%C3%ADa.png/233692646/456x468/Ciclo_Tetradecafon%C3%ADa.png&amp;quot; alt=&amp;quot;Intervallic Cycle of 14 steps Equal per Octave&amp;quot; title=&amp;quot;Intervallic Cycle of 14 steps Equal per Octave&amp;quot; style=&amp;quot;height: 468px; width: 456px;&amp;quot; /&amp;gt; --&gt;&lt;table class="captionBox"&gt;&lt;tr&gt;&lt;td class="captionedImage"&gt;&lt;img src="/file/view/Ciclo_Tetradecafon%C3%ADa.png/233692646/456x468/Ciclo_Tetradecafon%C3%ADa.png" alt="Ciclo_Tetradecafonía.png" title="Ciclo_Tetradecafonía.png" style="height: 468px; width: 456px;" /&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td class="imageCaption"&gt;Intervallic Cycle of 14 steps Equal per Octave&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;&lt;!-- ws:end:WikiTextLocalImageRule:453 --&gt;&lt;br /&gt;
 &lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:15:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Images"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:15 --&gt;Images&lt;/h1&gt;
   &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:451:&amp;lt;img src=&amp;quot;/file/view/14edo%20wheel.png/516971016/343x343/14edo%20wheel.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 343px; width: 343px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/14edo%20wheel.png/516971016/343x343/14edo%20wheel.png" alt="14edo wheel.png" title="14edo wheel.png" style="height: 343px; width: 343px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:451 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:454:&amp;lt;img src=&amp;quot;/file/view/14edo%20wheel.png/516971016/343x343/14edo%20wheel.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 343px; width: 343px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/14edo%20wheel.png/516971016/343x343/14edo%20wheel.png" alt="14edo wheel.png" title="14edo wheel.png" style="height: 343px; width: 343px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:454 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:15:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:15 --&gt;Commas&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:17:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:17 --&gt;Commas&lt;/h1&gt;
 EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes the val &amp;lt; 14 22 33 39 48 52 |.)&lt;br /&gt;
@@ Line 625: / Line 663: @@
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:17:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Modes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:17 --&gt;&lt;strong&gt;&lt;a class="wiki_link" href="/modes"&gt;Modes&lt;/a&gt;&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:19:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Modes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:19 --&gt;&lt;strong&gt;&lt;a class="wiki_link" href="/modes"&gt;Modes&lt;/a&gt;&lt;/strong&gt;&lt;/h1&gt;
 6 2 - &lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt; of &lt;a class="wiki_link" href="/2L1s"&gt;2L1s&lt;/a&gt;&lt;br /&gt;
 5 4 - &lt;span style="cursor: pointer;"&gt;&lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt;&lt;/span&gt; of &lt;span style="cursor: pointer;"&gt;&lt;a class="wiki_link" href="/2L1s"&gt;2L1s&lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
@@ Line 653: / Line 691: @@
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-&lt;!-- ws:start:WikiTextHeadingRule:19:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Books"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:19 --&gt;Books&lt;/h1&gt;
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 &lt;em&gt;Sword, Ron. &amp;quot;Tetradecaphonic Scales for Guitar&amp;quot; IAAA Press. First Ed: June 2009.&lt;/em&gt;&lt;br /&gt;
 &lt;br /&gt;
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   &lt;a class="wiki_link_ext" href="http://split-notes.com/004/" rel="nofollow"&gt;NANA WODORI&lt;/a&gt; by knowsur&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/0-praxis/audio/August/august_03_thereminnards.mp3" rel="nofollow"&gt;Thereminnards&lt;/a&gt; by Ralph Lewis&lt;br /&gt;