Harmony of 23edo

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If you take a look at the intervals of [[23edo]], you'll find that this system does not contain good representations of harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than trivialize 23edo by calling it "atonal" or "nonharmonic," I'd like to consider higher-limit harmonies that could serve as useful sonorities, perhaps even "consonances," in the context of careful composition. [[23edo]] contains intervals which approach harmonics 9, 17, 21, 23, 33, 55, 79, & 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics.

|| degrees of 23edo || cents || nearest harmonic || cents || "error" ||
|| **0** || **0** || **1/1** || **0.000** || **none** ||
|| 1 || 52.174 || 33/32 || 53.273 || -1.099 ||
|| **2** || **104.348** || **17/16** || **104.955** || **-.607** ||
|| **4** || **208.696** || **9/8** || **203.910** || **+4.786** ||
|| 7 || 365.217 || 79/64 || 364.537 || +.680 ||
|| **9** || **469.565** || **21/16** || **470.781** || **-1.216** ||
|| **12** || **626.087** || **23/16** || **628.274** || **-2.187** ||
|| 18 || 939.130 || 55/32 || 937.632 || +1.498 ||
|| 20 || 1043.478 || 117/64 || 1044.438 || -.960 ||
|| **23 (or 0)** || **1200.000** || **2/1** || **1200.000** || **none** ||

You'll see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16; 2 cents of 33/32, 21/16, & 55/32; & 1 cent of 17/16, 79/64, & 117/64. Of course, it also has perfect unisons & octaves, by definition. This means we could potentially build a very strange (& slightly mistuned) harmonic chord which, reduced to within one octave, we could write as frequency ratios 64:66:68:72:79:84:92:110:117:132. I find this cluster a little hard to listen to on its own, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads & tetrads, as a starting point.

I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, & 23, excluding (at least for now) 33, 55, 79, & 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.

Thus we produce ten triads, five tetrads, & one quintad.

==Triads== 

===16:17:18, degrees 0, 2, 4.=== 
17/16 (104.955, error -.607)
9/8 (203.910, error +4.786)
18/17 (98.955, error: +5.393)

===16:17:21, degrees 0, 2, 9.=== 
17/16 (104.955, error -.607)
21/16 (470.781, error -1.216)
21/17 (365.825, error: -.608)

===16:17:23, degrees 0, 2, 12.=== 
17/16 (104.955, error -.607)
23/16 (628.274, error -2.187)
23/17 (523.319, error: -1.578)

===16:18:21, degrees 0, 4, 9.=== 
9/8 (203.910, error +4.786)
21/16 (470.781, error -1.216)
21/18 = 7/6 (266.871, error: -6.001)

===16:18:23, degrees 0, 4, 12.=== 
9/8 (203.910, error +4.786)
23/16 (628.274, error -2.187)
23/18 (424.364, error: -6.973)

===16:21:23, degrees 0, 9, 12.=== 
21/16 (470.781, error -1.216)
23/16 (628.274, error -2.187)
23/21 (157.493, error: -.971)

===17:18:21, degrees 0, 2, 7.=== 
18/17 (98.955, error: +5.393)
21/17 (365.825, error: -.608)
21/18 = 7/6 (266.871, error: -6.001)

===17:18:23, degrees 0, 2, 10.=== 
18/17 (98.955, error: +5.393)
23/17 (523.319, error: -1.578)
23/18 (424.364, error: -6.973)

===17:21:23, degrees 0, 7, 10.=== 
21/17 (365.825, error: -.608)
23/17 (523.319, error: -1.578)
23/21 (157.493, error: -.971)

===18:21:23, degrees 0, 5, 8.=== 
21/18 = 7/6 (266.871, error: -6.001)
23/18 (424.364, error: -6.973)
23/21 (157.493, error: -.971)

==Tetrads== 

===16:17:18:21, degrees 0, 2, 4, 9.=== 
17/16 (104.955, error -.607)
9/8 (203.910, error +4.786)
21/16 (470.781, error -1.216)
18/17 (98.955, error: +5.393)
21/17 (365.825, error: -.608)
21/18 = 7/6 (266.871, error: -6.001)

===16:17:18:23, degrees 0, 2, 4, 12.=== 
17/16 (104.955, error -.607)
9/8 (203.910, error +4.786)
23/16 (628.274, error -2.187)
18/17 (98.955, error: +5.393)
23/17 (523.319, error: -1.578)
23/18 (424.364, error: -6.973)

===16:17:21:23, degrees 0, 2, 9, 12.=== 
17/16 (104.955, error -.607)
21/16 (470.781, error -1.216)
23/16 (628.274, error -2.187)
21/17 (365.825, error: -.608)
23/17 (523.319, error: -1.578)
23/21 (157.493, error: -.971)

===16:18:21:23, degrees 0, 4, 9, 12.=== 
9/8 (203.910, error +4.786)
21/16 (470.781, error -1.216)
23/16 (628.274, error -2.187)
21/18 = 7/6 (266.871, error: -6.001)
23/18 (424.364, error: -6.973)
23/21 (157.493, error: -.971)

===17:18:21:23, degrees 0, 2, 7, 10.=== 


==Quintad== 

===16:17:18:21:23, degrees 0, 2, 4, 9, 12.=== 
17/16 (104.955, error -.607)
9/8 (203.910, error +4.786)
21/16 (470.781, error -1.216)
23/16 (628.274, error -2.187)
18/17 (98.955, error: +5.393)
21/17 (365.825, error: -.608)
23/17 (523.319, error: -1.578)
21/18 = 7/6 (266.871, error: -6.001)
23/18 (424.364, error: -6.973)
23/21 (157.493, error: -.971)
23/21 (157.493, error: -.971)

Original HTML content:

<html><head><title>Harmony of 23edo</title></head><body>If you take a look at the intervals of <a class="wiki_link" href="/23edo">23edo</a>, you'll find that this system does not contain good representations of harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than trivialize 23edo by calling it &quot;atonal&quot; or &quot;nonharmonic,&quot; I'd like to consider higher-limit harmonies that could serve as useful sonorities, perhaps even &quot;consonances,&quot; in the context of careful composition. <a class="wiki_link" href="/23edo">23edo</a> contains intervals which approach harmonics 9, 17, 21, 23, 33, 55, 79, &amp; 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics.<br />
<br />


<table class="wiki_table">
    <tr>
        <td>degrees of 23edo<br />
</td>
        <td>cents<br />
</td>
        <td>nearest harmonic<br />
</td>
        <td>cents<br />
</td>
        <td>&quot;error&quot;<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td><strong>1/1</strong><br />
</td>
        <td><strong>0.000</strong><br />
</td>
        <td><strong>none</strong><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>52.174<br />
</td>
        <td>33/32<br />
</td>
        <td>53.273<br />
</td>
        <td>-1.099<br />
</td>
    </tr>
    <tr>
        <td><strong>2</strong><br />
</td>
        <td><strong>104.348</strong><br />
</td>
        <td><strong>17/16</strong><br />
</td>
        <td><strong>104.955</strong><br />
</td>
        <td><strong>-.607</strong><br />
</td>
    </tr>
    <tr>
        <td><strong>4</strong><br />
</td>
        <td><strong>208.696</strong><br />
</td>
        <td><strong>9/8</strong><br />
</td>
        <td><strong>203.910</strong><br />
</td>
        <td><strong>+4.786</strong><br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>365.217<br />
</td>
        <td>79/64<br />
</td>
        <td>364.537<br />
</td>
        <td>+.680<br />
</td>
    </tr>
    <tr>
        <td><strong>9</strong><br />
</td>
        <td><strong>469.565</strong><br />
</td>
        <td><strong>21/16</strong><br />
</td>
        <td><strong>470.781</strong><br />
</td>
        <td><strong>-1.216</strong><br />
</td>
    </tr>
    <tr>
        <td><strong>12</strong><br />
</td>
        <td><strong>626.087</strong><br />
</td>
        <td><strong>23/16</strong><br />
</td>
        <td><strong>628.274</strong><br />
</td>
        <td><strong>-2.187</strong><br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>939.130<br />
</td>
        <td>55/32<br />
</td>
        <td>937.632<br />
</td>
        <td>+1.498<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>1043.478<br />
</td>
        <td>117/64<br />
</td>
        <td>1044.438<br />
</td>
        <td>-.960<br />
</td>
    </tr>
    <tr>
        <td><strong>23 (or 0)</strong><br />
</td>
        <td><strong>1200.000</strong><br />
</td>
        <td><strong>2/1</strong><br />
</td>
        <td><strong>1200.000</strong><br />
</td>
        <td><strong>none</strong><br />
</td>
    </tr>
</table>

<br />
You'll see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16; 2 cents of 33/32, 21/16, &amp; 55/32; &amp; 1 cent of 17/16, 79/64, &amp; 117/64. Of course, it also has perfect unisons &amp; octaves, by definition. This means we could potentially build a very strange (&amp; slightly mistuned) harmonic chord which, reduced to within one octave, we could write as frequency ratios 64:66:68:72:79:84:92:110:117:132. I find this cluster a little hard to listen to on its own, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads &amp; tetrads, as a starting point.<br />
<br />
I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, &amp; 23, excluding (at least for now) 33, 55, 79, &amp; 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.<br />
<br />
Thus we produce ten triads, five tetrads, &amp; one quintad.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Triads"></a><!-- ws:end:WikiTextHeadingRule:0 -->Triads</h2>
 <br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Triads-16:17:18, degrees 0, 2, 4."></a><!-- ws:end:WikiTextHeadingRule:2 -->16:17:18, degrees 0, 2, 4.</h3>
 17/16 (104.955, error -.607)<br />
9/8 (203.910, error +4.786)<br />
18/17 (98.955, error: +5.393)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-Triads-16:17:21, degrees 0, 2, 9."></a><!-- ws:end:WikiTextHeadingRule:4 -->16:17:21, degrees 0, 2, 9.</h3>
 17/16 (104.955, error -.607)<br />
21/16 (470.781, error -1.216)<br />
21/17 (365.825, error: -.608)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x-Triads-16:17:23, degrees 0, 2, 12."></a><!-- ws:end:WikiTextHeadingRule:6 -->16:17:23, degrees 0, 2, 12.</h3>
 17/16 (104.955, error -.607)<br />
23/16 (628.274, error -2.187)<br />
23/17 (523.319, error: -1.578)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="x-Triads-16:18:21, degrees 0, 4, 9."></a><!-- ws:end:WikiTextHeadingRule:8 -->16:18:21, degrees 0, 4, 9.</h3>
 9/8 (203.910, error +4.786)<br />
21/16 (470.781, error -1.216)<br />
21/18 = 7/6 (266.871, error: -6.001)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Triads-16:18:23, degrees 0, 4, 12."></a><!-- ws:end:WikiTextHeadingRule:10 -->16:18:23, degrees 0, 4, 12.</h3>
 9/8 (203.910, error +4.786)<br />
23/16 (628.274, error -2.187)<br />
23/18 (424.364, error: -6.973)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x-Triads-16:21:23, degrees 0, 9, 12."></a><!-- ws:end:WikiTextHeadingRule:12 -->16:21:23, degrees 0, 9, 12.</h3>
 21/16 (470.781, error -1.216)<br />
23/16 (628.274, error -2.187)<br />
23/21 (157.493, error: -.971)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x-Triads-17:18:21, degrees 0, 2, 7."></a><!-- ws:end:WikiTextHeadingRule:14 -->17:18:21, degrees 0, 2, 7.</h3>
 18/17 (98.955, error: +5.393)<br />
21/17 (365.825, error: -.608)<br />
21/18 = 7/6 (266.871, error: -6.001)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x-Triads-17:18:23, degrees 0, 2, 10."></a><!-- ws:end:WikiTextHeadingRule:16 -->17:18:23, degrees 0, 2, 10.</h3>
 18/17 (98.955, error: +5.393)<br />
23/17 (523.319, error: -1.578)<br />
23/18 (424.364, error: -6.973)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="x-Triads-17:21:23, degrees 0, 7, 10."></a><!-- ws:end:WikiTextHeadingRule:18 -->17:21:23, degrees 0, 7, 10.</h3>
 21/17 (365.825, error: -.608)<br />
23/17 (523.319, error: -1.578)<br />
23/21 (157.493, error: -.971)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="x-Triads-18:21:23, degrees 0, 5, 8."></a><!-- ws:end:WikiTextHeadingRule:20 -->18:21:23, degrees 0, 5, 8.</h3>
 21/18 = 7/6 (266.871, error: -6.001)<br />
23/18 (424.364, error: -6.973)<br />
23/21 (157.493, error: -.971)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="x-Tetrads"></a><!-- ws:end:WikiTextHeadingRule:22 -->Tetrads</h2>
 <br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="x-Tetrads-16:17:18:21, degrees 0, 2, 4, 9."></a><!-- ws:end:WikiTextHeadingRule:24 -->16:17:18:21, degrees 0, 2, 4, 9.</h3>
 17/16 (104.955, error -.607)<br />
9/8 (203.910, error +4.786)<br />
21/16 (470.781, error -1.216)<br />
18/17 (98.955, error: +5.393)<br />
21/17 (365.825, error: -.608)<br />
21/18 = 7/6 (266.871, error: -6.001)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="x-Tetrads-16:17:18:23, degrees 0, 2, 4, 12."></a><!-- ws:end:WikiTextHeadingRule:26 -->16:17:18:23, degrees 0, 2, 4, 12.</h3>
 17/16 (104.955, error -.607)<br />
9/8 (203.910, error +4.786)<br />
23/16 (628.274, error -2.187)<br />
18/17 (98.955, error: +5.393)<br />
23/17 (523.319, error: -1.578)<br />
23/18 (424.364, error: -6.973)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="x-Tetrads-16:17:21:23, degrees 0, 2, 9, 12."></a><!-- ws:end:WikiTextHeadingRule:28 -->16:17:21:23, degrees 0, 2, 9, 12.</h3>
 17/16 (104.955, error -.607)<br />
21/16 (470.781, error -1.216)<br />
23/16 (628.274, error -2.187)<br />
21/17 (365.825, error: -.608)<br />
23/17 (523.319, error: -1.578)<br />
23/21 (157.493, error: -.971)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="x-Tetrads-16:18:21:23, degrees 0, 4, 9, 12."></a><!-- ws:end:WikiTextHeadingRule:30 -->16:18:21:23, degrees 0, 4, 9, 12.</h3>
 9/8 (203.910, error +4.786)<br />
21/16 (470.781, error -1.216)<br />
23/16 (628.274, error -2.187)<br />
21/18 = 7/6 (266.871, error: -6.001)<br />
23/18 (424.364, error: -6.973)<br />
23/21 (157.493, error: -.971)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="x-Tetrads-17:18:21:23, degrees 0, 2, 7, 10."></a><!-- ws:end:WikiTextHeadingRule:32 -->17:18:21:23, degrees 0, 2, 7, 10.</h3>
 <br />
<br />
<!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="x-Quintad"></a><!-- ws:end:WikiTextHeadingRule:34 -->Quintad</h2>
 <br />
<!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="x-Quintad-16:17:18:21:23, degrees 0, 2, 4, 9, 12."></a><!-- ws:end:WikiTextHeadingRule:36 -->16:17:18:21:23, degrees 0, 2, 4, 9, 12.</h3>
 17/16 (104.955, error -.607)<br />
9/8 (203.910, error +4.786)<br />
21/16 (470.781, error -1.216)<br />
23/16 (628.274, error -2.187)<br />
18/17 (98.955, error: +5.393)<br />
21/17 (365.825, error: -.608)<br />
23/17 (523.319, error: -1.578)<br />
21/18 = 7/6 (266.871, error: -6.001)<br />
23/18 (424.364, error: -6.973)<br />
23/21 (157.493, error: -.971)<br />
23/21 (157.493, error: -.971)</body></html>
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