Isoharmonic chord: Difference between revisions

Revision as of 12:03, 29 March 2010

IMPORTED REVISION FROM WIKISPACES

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=isoharmonic chords= 

In [[JustIntonation|just intonation]], Isoharmonic chords are build by successive jumps up the [[OverToneSeries|harmonic series]] by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape -- with diminishing step size as one ascends. It happens that all isoharmonic chords are equal-hertz chords (but not all equal-hertz chords are isoharmonic chords). An isoharmonic "chord" may function more like a "scale" than a chord (depending on the composition of course), but I will use the word "chord" on this page for consistency.

===class i=== 
The simplest isoharmonic chords are built by stepping up the harmonic series by single steps (adjacent steps in the harmonic series). Take, for instance, 4:5:6:7, the harmonic seventh chord. I call these class i isoharmonic chords. There is one class i series (the harmonic series), which looks like this:

|| harmonic || 1 ||   || 2 ||   || 3 ||   || 4 ||   || 5 ||   || 6 ||   || 7 ||   || 8 ||   || 9 ||   || 10 ||   || 11 ||   || 12 ||   || 13 ||   || 14 ||   || 15 ||   || 16 ||
|| cents diff ||   || 1200 ||   || 702 ||   || 498 ||   || 386 ||   || 316 ||   || 267 ||   || 231 ||   || 204 ||   || 182 ||   || 165 ||   || 151 ||   || 139 ||   || 128 ||   || 119 ||   || 112 ||   ||

Some "scales" built this way: [[otones12-24]], [[otones20-40]]...

===class ii=== 
The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the [[BP|Bohlen-Pierce]] tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):

|| harmonic || 1 ||   || 3 ||   || 5 ||   || 7 ||   || 9 ||   || 11 ||   || 13 ||   || 15 ||   || 17 ||   || 19 ||   || 21 ||   || 23 ||   || 25 ||   || 27 ||   || 29 ||   || 31 ||
|| cents diff ||   || 1904 ||   || 884 ||   || 583 ||   || 435 ||   || 359 ||   || 289 ||   || 248 ||   || 217 ||   || 193 ||   || 173 ||   || 157 ||   || 144 ||   || 133 ||   || 124 ||   || 115 ||   ||

===class iii=== 
Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (eg. 9:12:15 = 3:4:5). There are two series for class iii:

|| harmonic || 1 ||   || 4 ||   || 7 ||   || 10 ||   || 13 ||   || 16 ||   || 19 ||   || 22 ||   || 25 ||   || 28 ||   || 31 ||   || 34 ||   || 37 ||   || 40 ||   || 43 ||   || 46 ||
|| cents diff ||   || 2400 ||   || 969 ||   || 617 ||   || 454 ||   || 359 ||   || 298 ||   || 254 ||   || 221 ||   || 196 ||   || 176 ||   || 160 ||   || 146 ||   || 135 ||   || 125 ||   || 117 ||   ||

|| harmonic || 2 ||   || 5 ||   || 8 ||   || 11 ||   || 14 ||   || 17 ||   || 20 ||   || 23 ||   || 26 ||   || 29 ||   || 32 ||   || 35 ||   || 38 ||   || 41 ||   || 44 ||   || 47 ||
|| cents diff ||   || 1586 ||   || 814 ||   || 551 ||   || 418 ||   || 336 ||   || 281 ||   || 242 ||   || 212 ||   || 189 ||   || 170 ||   || 155 ||   || 142 ||   || 132 ||   || 122 ||   || 114 ||   ||


===class iv and beyond=== 
...explore for yourself!

Original HTML content:

<html><head><title>isoharmonic chords</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="isoharmonic chords"></a><!-- ws:end:WikiTextHeadingRule:0 -->isoharmonic chords</h1>
 <br />
In <a class="wiki_link" href="/JustIntonation">just intonation</a>, Isoharmonic chords are build by successive jumps up the <a class="wiki_link" href="/OverToneSeries">harmonic series</a> by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape -- with diminishing step size as one ascends. It happens that all isoharmonic chords are equal-hertz chords (but not all equal-hertz chords are isoharmonic chords). An isoharmonic &quot;chord&quot; may function more like a &quot;scale&quot; than a chord (depending on the composition of course), but I will use the word &quot;chord&quot; on this page for consistency.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="isoharmonic chords--class i"></a><!-- ws:end:WikiTextHeadingRule:2 -->class i</h3>
 The simplest isoharmonic chords are built by stepping up the harmonic series by single steps (adjacent steps in the harmonic series). Take, for instance, 4:5:6:7, the harmonic seventh chord. I call these class i isoharmonic chords. There is one class i series (the harmonic series), which looks like this:<br />
<br />


<table class="wiki_table">
    <tr>
        <td>harmonic<br />
</td>
        <td>1<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td>10<br />
</td>
        <td><br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td>12<br />
</td>
        <td><br />
</td>
        <td>13<br />
</td>
        <td><br />
</td>
        <td>14<br />
</td>
        <td><br />
</td>
        <td>15<br />
</td>
        <td><br />
</td>
        <td>16<br />
</td>
    </tr>
    <tr>
        <td>cents diff<br />
</td>
        <td><br />
</td>
        <td>1200<br />
</td>
        <td><br />
</td>
        <td>702<br />
</td>
        <td><br />
</td>
        <td>498<br />
</td>
        <td><br />
</td>
        <td>386<br />
</td>
        <td><br />
</td>
        <td>316<br />
</td>
        <td><br />
</td>
        <td>267<br />
</td>
        <td><br />
</td>
        <td>231<br />
</td>
        <td><br />
</td>
        <td>204<br />
</td>
        <td><br />
</td>
        <td>182<br />
</td>
        <td><br />
</td>
        <td>165<br />
</td>
        <td><br />
</td>
        <td>151<br />
</td>
        <td><br />
</td>
        <td>139<br />
</td>
        <td><br />
</td>
        <td>128<br />
</td>
        <td><br />
</td>
        <td>119<br />
</td>
        <td><br />
</td>
        <td>112<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
Some &quot;scales&quot; built this way: <a class="wiki_link" href="/otones12-24">otones12-24</a>, <a class="wiki_link" href="/otones20-40">otones20-40</a>...<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="isoharmonic chords--class ii"></a><!-- ws:end:WikiTextHeadingRule:4 -->class ii</h3>
 The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the <a class="wiki_link" href="/BP">Bohlen-Pierce</a> tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):<br />
<br />


<table class="wiki_table">
    <tr>
        <td>harmonic<br />
</td>
        <td>1<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td>13<br />
</td>
        <td><br />
</td>
        <td>15<br />
</td>
        <td><br />
</td>
        <td>17<br />
</td>
        <td><br />
</td>
        <td>19<br />
</td>
        <td><br />
</td>
        <td>21<br />
</td>
        <td><br />
</td>
        <td>23<br />
</td>
        <td><br />
</td>
        <td>25<br />
</td>
        <td><br />
</td>
        <td>27<br />
</td>
        <td><br />
</td>
        <td>29<br />
</td>
        <td><br />
</td>
        <td>31<br />
</td>
    </tr>
    <tr>
        <td>cents diff<br />
</td>
        <td><br />
</td>
        <td>1904<br />
</td>
        <td><br />
</td>
        <td>884<br />
</td>
        <td><br />
</td>
        <td>583<br />
</td>
        <td><br />
</td>
        <td>435<br />
</td>
        <td><br />
</td>
        <td>359<br />
</td>
        <td><br />
</td>
        <td>289<br />
</td>
        <td><br />
</td>
        <td>248<br />
</td>
        <td><br />
</td>
        <td>217<br />
</td>
        <td><br />
</td>
        <td>193<br />
</td>
        <td><br />
</td>
        <td>173<br />
</td>
        <td><br />
</td>
        <td>157<br />
</td>
        <td><br />
</td>
        <td>144<br />
</td>
        <td><br />
</td>
        <td>133<br />
</td>
        <td><br />
</td>
        <td>124<br />
</td>
        <td><br />
</td>
        <td>115<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="isoharmonic chords--class iii"></a><!-- ws:end:WikiTextHeadingRule:6 -->class iii</h3>
 Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (eg. 9:12:15 = 3:4:5). There are two series for class iii:<br />
<br />


<table class="wiki_table">
    <tr>
        <td>harmonic<br />
</td>
        <td>1<br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td>10<br />
</td>
        <td><br />
</td>
        <td>13<br />
</td>
        <td><br />
</td>
        <td>16<br />
</td>
        <td><br />
</td>
        <td>19<br />
</td>
        <td><br />
</td>
        <td>22<br />
</td>
        <td><br />
</td>
        <td>25<br />
</td>
        <td><br />
</td>
        <td>28<br />
</td>
        <td><br />
</td>
        <td>31<br />
</td>
        <td><br />
</td>
        <td>34<br />
</td>
        <td><br />
</td>
        <td>37<br />
</td>
        <td><br />
</td>
        <td>40<br />
</td>
        <td><br />
</td>
        <td>43<br />
</td>
        <td><br />
</td>
        <td>46<br />
</td>
    </tr>
    <tr>
        <td>cents diff<br />
</td>
        <td><br />
</td>
        <td>2400<br />
</td>
        <td><br />
</td>
        <td>969<br />
</td>
        <td><br />
</td>
        <td>617<br />
</td>
        <td><br />
</td>
        <td>454<br />
</td>
        <td><br />
</td>
        <td>359<br />
</td>
        <td><br />
</td>
        <td>298<br />
</td>
        <td><br />
</td>
        <td>254<br />
</td>
        <td><br />
</td>
        <td>221<br />
</td>
        <td><br />
</td>
        <td>196<br />
</td>
        <td><br />
</td>
        <td>176<br />
</td>
        <td><br />
</td>
        <td>160<br />
</td>
        <td><br />
</td>
        <td>146<br />
</td>
        <td><br />
</td>
        <td>135<br />
</td>
        <td><br />
</td>
        <td>125<br />
</td>
        <td><br />
</td>
        <td>117<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />


<table class="wiki_table">
    <tr>
        <td>harmonic<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td>14<br />
</td>
        <td><br />
</td>
        <td>17<br />
</td>
        <td><br />
</td>
        <td>20<br />
</td>
        <td><br />
</td>
        <td>23<br />
</td>
        <td><br />
</td>
        <td>26<br />
</td>
        <td><br />
</td>
        <td>29<br />
</td>
        <td><br />
</td>
        <td>32<br />
</td>
        <td><br />
</td>
        <td>35<br />
</td>
        <td><br />
</td>
        <td>38<br />
</td>
        <td><br />
</td>
        <td>41<br />
</td>
        <td><br />
</td>
        <td>44<br />
</td>
        <td><br />
</td>
        <td>47<br />
</td>
    </tr>
    <tr>
        <td>cents diff<br />
</td>
        <td><br />
</td>
        <td>1586<br />
</td>
        <td><br />
</td>
        <td>814<br />
</td>
        <td><br />
</td>
        <td>551<br />
</td>
        <td><br />
</td>
        <td>418<br />
</td>
        <td><br />
</td>
        <td>336<br />
</td>
        <td><br />
</td>
        <td>281<br />
</td>
        <td><br />
</td>
        <td>242<br />
</td>
        <td><br />
</td>
        <td>212<br />
</td>
        <td><br />
</td>
        <td>189<br />
</td>
        <td><br />
</td>
        <td>170<br />
</td>
        <td><br />
</td>
        <td>155<br />
</td>
        <td><br />
</td>
        <td>142<br />
</td>
        <td><br />
</td>
        <td>132<br />
</td>
        <td><br />
</td>
        <td>122<br />
</td>
        <td><br />
</td>
        <td>114<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="isoharmonic chords--class iv and beyond"></a><!-- ws:end:WikiTextHeadingRule:8 -->class iv and beyond</h3>
 ...explore for yourself!</body></html>

@@ 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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-03-29 11:54:50 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-03-29 12:03:39 UTC</tt>.<br>
-: The original revision id was <tt>131084543</tt>.<br>
+: The original revision id was <tt>131087113</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 8: / Line 8: @@
 <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">=isoharmonic chords=
-Isoharmonic chords are build by successive jumps up the harmonic series by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape -- with diminishing step size as one ascends. It happens that all isoharmonic chords are equal-hertz chords (but not all equal-hertz chords are isoharmonic chords). An isoharmonic "chord" may function more like a "scale" than a chord (depending on the composition of course), but I will use the word "chord" on this page for consistency.
+In [[JustIntonation|just intonation]], Isoharmonic chords are build by successive jumps up the [[OverToneSeries|harmonic series]] by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape -- with diminishing step size as one ascends. It happens that all isoharmonic chords are equal-hertz chords (but not all equal-hertz chords are isoharmonic chords). An isoharmonic "chord" may function more like a "scale" than a chord (depending on the composition of course), but I will use the word "chord" on this page for consistency.
 ===class i===
@@ Line 15: / Line 15: @@
 || harmonic || 1 ||   || 2 ||   || 3 ||   || 4 ||   || 5 ||   || 6 ||   || 7 ||   || 8 ||   || 9 ||   || 10 ||   || 11 ||   || 12 ||   || 13 ||   || 14 ||   || 15 ||   || 16 ||
 || cents diff ||   || 1200 ||   || 702 ||   || 498 ||   || 386 ||   || 316 ||   || 267 ||   || 231 ||   || 204 ||   || 182 ||   || 165 ||   || 151 ||   || 139 ||   || 128 ||   || 119 ||   || 112 ||   ||
+Some "scales" built this way: [[otones12-24]], [[otones20-40]]...
 ===class ii===
-The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the Bohlen-Pierce tuning system) and 7:9:11:13. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):
+The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the [[BP|Bohlen-Pierce]] tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):
 || harmonic || 1 ||   || 3 ||   || 5 ||   || 7 ||   || 9 ||   || 11 ||   || 13 ||   || 15 ||   || 17 ||   || 19 ||   || 21 ||   || 23 ||   || 25 ||   || 27 ||   || 29 ||   || 31 ||
@@ Line 23: / Line 25: @@
 ===class iii===
-Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 11:14:17:20. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (eg. 9:12:15 = 3:4:5). There are two series for class iii:
+Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (eg. 9:12:15 = 3:4:5). There are two series for class iii:
 || harmonic || 1 ||   || 4 ||   || 7 ||   || 10 ||   || 13 ||   || 16 ||   || 19 ||   || 22 ||   || 25 ||   || 28 ||   || 31 ||   || 34 ||   || 37 ||   || 40 ||   || 43 ||   || 46 ||
@@ Line 37: / Line 39: @@
 <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;isoharmonic chords&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="isoharmonic chords"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;isoharmonic chords&lt;/h1&gt;
   &lt;br /&gt;
-Isoharmonic chords are build by successive jumps up the harmonic series by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape -- with diminishing step size as one ascends. It happens that all isoharmonic chords are equal-hertz chords (but not all equal-hertz chords are isoharmonic chords). An isoharmonic &amp;quot;chord&amp;quot; may function more like a &amp;quot;scale&amp;quot; than a chord (depending on the composition of course), but I will use the word &amp;quot;chord&amp;quot; on this page for consistency.&lt;br /&gt;
+In &lt;a class="wiki_link" href="/JustIntonation"&gt;just intonation&lt;/a&gt;, Isoharmonic chords are build by successive jumps up the &lt;a class="wiki_link" href="/OverToneSeries"&gt;harmonic series&lt;/a&gt; by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape -- with diminishing step size as one ascends. It happens that all isoharmonic chords are equal-hertz chords (but not all equal-hertz chords are isoharmonic chords). An isoharmonic &amp;quot;chord&amp;quot; may function more like a &amp;quot;scale&amp;quot; than a chord (depending on the composition of course), but I will use the word &amp;quot;chord&amp;quot; on this page for consistency.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="isoharmonic chords--class i"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;class i&lt;/h3&gt;
@@ Line 179: / Line 181: @@
 &lt;/table&gt;
+&lt;br /&gt;
+Some &amp;quot;scales&amp;quot; built this way: &lt;a class="wiki_link" href="/otones12-24"&gt;otones12-24&lt;/a&gt;, &lt;a class="wiki_link" href="/otones20-40"&gt;otones20-40&lt;/a&gt;...&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="isoharmonic chords--class ii"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;class ii&lt;/h3&gt;
-  The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the Bohlen-Pierce tuning system) and 7:9:11:13. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):&lt;br /&gt;
+  The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the &lt;a class="wiki_link" href="/BP"&gt;Bohlen-Pierce&lt;/a&gt; tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):&lt;br /&gt;
 &lt;br /&gt;
@@ Line 322: / Line 326: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="isoharmonic chords--class iii"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;class iii&lt;/h3&gt;
-  Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 11:14:17:20. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (eg. 9:12:15 = 3:4:5). There are two series for class iii:&lt;br /&gt;
+  Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (eg. 9:12:15 = 3:4:5). There are two series for class iii:&lt;br /&gt;
 &lt;br /&gt;

Isoharmonic chord: Difference between revisions

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