Isoharmonic chord
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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:<h1> --><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 "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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h3> --><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 "scales" 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:<h3> --><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:<h3> --><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:<h3> --><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>