Isoharmonic chord: Difference between revisions

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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 ||   || 347 ||   || 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>347<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 12:03:39 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2010-03-30 13:27:54 UTC</tt>.<br>
-: The original revision id was <tt>131087113</tt>.<br>
+: The original revision id was <tt>131380685</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 22: / Line 22: @@
 || 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 ||   ||
+|| cents diff ||   || 1904 ||   || 884 ||   || 583 ||   || 435 ||   || 347 ||   || 289 ||   || 248 ||   || 217 ||   || 193 ||   || 173 ||   || 157 ||   || 144 ||   || 133 ||   || 124 ||   || 115 ||   ||
 ===class iii===
@@ Line 277: / Line 277: @@
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;359&lt;br /&gt;
+         &lt;td&gt;347&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;

Isoharmonic chord: Difference between revisions

Revision as of 13:27, 30 March 2010

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

Original HTML content:

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