Harmony of 23edo: Difference between revisions

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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, 39, 43 & 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics.

|| degrees || note || cents || nearest harmonic || cents || "error" ||
|| **0** || **1** || **0** || **1/1** || **0.000** || **none** ||
|| 1 || 1# || 52.174 || 33/32 || 53.273 || -1.099 ||
|| **2** || **2b** || **104.348** || **17/16** || **104.955** || **-0.607** ||
|| 3 || 2 || 156.522 || 23/21 || 157.493 || -0.971 ||
|| **4·** || **2#** || **208.696** || **9/8** || **203.910** || **+4.786** ||
|| 5 || 3b || 260.869 || 43/37 || 260.174 || +0.695 ||
|| **6** || **3** || **313.043** || **6/5** || **315.641** || **-2.598** ||
|| **7·** || **3#** || **365.217** || **21/17** || **365.825** || **-0.608** ||
|| 8 || 4b || 417.391 || 14/11 || 417.508 || -0.117 ||
|| **9** || **4** || **469.565** || **21/16** || **470.781** || **-1.216** ||
|| **10·** || **5** || **521.739** || **23/17** || **523.319** || **-1.58** ||
|| 11 || 5# || 573.913 || 32/23 || 571.726 || +2.187 ||
|| **12** || **6b** || **626.087** || **23/16** || **628.274** || **-2.187** ||
|| **13·** || **6** || **678.261** || **34/23** || **676.681** || **+1.58** ||
|| 14 || 6# || 730.435 || 32/21 || 729.219 || +1.216 ||
|| 15 || 7b || 782.609 || 11/7 || 782.492 || +0.117 ||
|| **16·** || **7(A)** || **834.783** || **34/21** || **834.175** || **+0.608** ||
|| 17 || 7# || 886.957 || 5/3 || 884.359 || +2.598 ||
|| 18 || 8b || 939.130 || 43/25 || 938.890 || +0.24 ||
|| **19·** || **8** || **991.304** || **39/22** || **991.165** || **+0.139** ||
|| **20** || **8#** || **1043.478** || **117/64** || **1044.438** || **-0.96** ||
|| 21 || 9b || 1095.652 || 32/17 || 1095.045 || +0.607 ||
|| 22 || 9 || 1147.826 || 64/33 || 1146.727 || +1.099 ||
|| **23·· (or 0)** || **1** || **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, 35/32, & 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 64:66:68:70:72:79:84:92:110:117. I find this cluster a little hard to listen to, 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, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), & I recommend trying different voicings. Moving chord tones up & down by octaves, you can unmuddy a muddy chord.

==Triads== 

===16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21).=== 
17/16 (104.955, error -.607)
18/16 = 9/8 (203.910, error +4.786)
18/17 (98.955, error: +5.393)

===16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).=== 
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 (inversion 0, 11, 21).=== 
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 (inversion 0, 14, 19).=== 
18/16 = 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 (inversion 0, 11, 19).=== 
18/16 = 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 (inversion 0, 11, 14).=== 
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 (inversion 0, 16, 21).=== 
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 (inversion 0, 13, 21).=== 
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 (inversion 0, 13, 16).=== 
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 (inversion 0, 15, 18).=== 
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 (inversion 0, 14, 19, 21).=== 
17/16 (104.955, error -.607)
18/16 = 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 (inversion 0, 11 19, 21).=== 
17/16 (104.955, error -.607)
18/16 = 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 (inversion 0, 11, 14, 21).=== 
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 (inversion 0, 11, 14, 19).=== 
18/16 = 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 (inversion 0, 13, 16, 21).=== 
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)

==Quintad== 

===16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21).=== 
17/16 (104.955, error -.607)
18/16 = 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, 39, 43 &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<br />
</td>
        <td>note<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><strong>1</strong><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>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>2b</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>-0.607</strong><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>2<br />
</td>
        <td>156.522<br />
</td>
        <td>23/21<br />
</td>
        <td>157.493<br />
</td>
        <td>-0.971<br />
</td>
    </tr>
    <tr>
        <td><strong>4·</strong><br />
</td>
        <td><strong>2#</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>5<br />
</td>
        <td>3b<br />
</td>
        <td>260.869<br />
</td>
        <td>43/37<br />
</td>
        <td>260.174<br />
</td>
        <td>+0.695<br />
</td>
    </tr>
    <tr>
        <td><strong>6</strong><br />
</td>
        <td><strong>3</strong><br />
</td>
        <td><strong>313.043</strong><br />
</td>
        <td><strong>6/5</strong><br />
</td>
        <td><strong>315.641</strong><br />
</td>
        <td><strong>-2.598</strong><br />
</td>
    </tr>
    <tr>
        <td><strong>7·</strong><br />
</td>
        <td><strong>3#</strong><br />
</td>
        <td><strong>365.217</strong><br />
</td>
        <td><strong>21/17</strong><br />
</td>
        <td><strong>365.825</strong><br />
</td>
        <td><strong>-0.608</strong><br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>4b<br />
</td>
        <td>417.391<br />
</td>
        <td>14/11<br />
</td>
        <td>417.508<br />
</td>
        <td>-0.117<br />
</td>
    </tr>
    <tr>
        <td><strong>9</strong><br />
</td>
        <td><strong>4</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>10·</strong><br />
</td>
        <td><strong>5</strong><br />
</td>
        <td><strong>521.739</strong><br />
</td>
        <td><strong>23/17</strong><br />
</td>
        <td><strong>523.319</strong><br />
</td>
        <td><strong>-1.58</strong><br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>5#<br />
</td>
        <td>573.913<br />
</td>
        <td>32/23<br />
</td>
        <td>571.726<br />
</td>
        <td>+2.187<br />
</td>
    </tr>
    <tr>
        <td><strong>12</strong><br />
</td>
        <td><strong>6b</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><strong>13·</strong><br />
</td>
        <td><strong>6</strong><br />
</td>
        <td><strong>678.261</strong><br />
</td>
        <td><strong>34/23</strong><br />
</td>
        <td><strong>676.681</strong><br />
</td>
        <td><strong>+1.58</strong><br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>6#<br />
</td>
        <td>730.435<br />
</td>
        <td>32/21<br />
</td>
        <td>729.219<br />
</td>
        <td>+1.216<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>7b<br />
</td>
        <td>782.609<br />
</td>
        <td>11/7<br />
</td>
        <td>782.492<br />
</td>
        <td>+0.117<br />
</td>
    </tr>
    <tr>
        <td><strong>16·</strong><br />
</td>
        <td><strong>7(A)</strong><br />
</td>
        <td><strong>834.783</strong><br />
</td>
        <td><strong>34/21</strong><br />
</td>
        <td><strong>834.175</strong><br />
</td>
        <td><strong>+0.608</strong><br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>7#<br />
</td>
        <td>886.957<br />
</td>
        <td>5/3<br />
</td>
        <td>884.359<br />
</td>
        <td>+2.598<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>8b<br />
</td>
        <td>939.130<br />
</td>
        <td>43/25<br />
</td>
        <td>938.890<br />
</td>
        <td>+0.24<br />
</td>
    </tr>
    <tr>
        <td><strong>19·</strong><br />
</td>
        <td><strong>8</strong><br />
</td>
        <td><strong>991.304</strong><br />
</td>
        <td><strong>39/22</strong><br />
</td>
        <td><strong>991.165</strong><br />
</td>
        <td><strong>+0.139</strong><br />
</td>
    </tr>
    <tr>
        <td><strong>20</strong><br />
</td>
        <td><strong>8#</strong><br />
</td>
        <td><strong>1043.478</strong><br />
</td>
        <td><strong>117/64</strong><br />
</td>
        <td><strong>1044.438</strong><br />
</td>
        <td><strong>-0.96</strong><br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>9b<br />
</td>
        <td>1095.652<br />
</td>
        <td>32/17<br />
</td>
        <td>1095.045<br />
</td>
        <td>+0.607<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>9<br />
</td>
        <td>1147.826<br />
</td>
        <td>64/33<br />
</td>
        <td>1146.727<br />
</td>
        <td>+1.099<br />
</td>
    </tr>
    <tr>
        <td><strong>23·· (or 0)</strong><br />
</td>
        <td><strong>1</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, 35/32, &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 64:66:68:70:72:79:84:92:110:117. I find this cluster a little hard to listen to, 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, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), &amp; I recommend trying different voicings. Moving chord tones up &amp; down by octaves, you can unmuddy a muddy chord.<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 (inversion 0, 19, 21)."></a><!-- ws:end:WikiTextHeadingRule:2 -->16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21).</h3>
 17/16 (104.955, error -.607)<br />
18/16 = 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 (inversion 0, 14, 21)."></a><!-- ws:end:WikiTextHeadingRule:4 -->16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).</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 (inversion 0, 11, 21)."></a><!-- ws:end:WikiTextHeadingRule:6 -->16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21).</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 (inversion 0, 14, 19)."></a><!-- ws:end:WikiTextHeadingRule:8 -->16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19).</h3>
 18/16 = 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 (inversion 0, 11, 19)."></a><!-- ws:end:WikiTextHeadingRule:10 -->16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19).</h3>
 18/16 = 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 (inversion 0, 11, 14)."></a><!-- ws:end:WikiTextHeadingRule:12 -->16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14).</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 (inversion 0, 16, 21)."></a><!-- ws:end:WikiTextHeadingRule:14 -->17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21).</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 (inversion 0, 13, 21)."></a><!-- ws:end:WikiTextHeadingRule:16 -->17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21).</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 (inversion 0, 13, 16)."></a><!-- ws:end:WikiTextHeadingRule:18 -->17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16).</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 (inversion 0, 15, 18)."></a><!-- ws:end:WikiTextHeadingRule:20 -->18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18).</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 (inversion 0, 14, 19, 21)."></a><!-- ws:end:WikiTextHeadingRule:24 -->16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21).</h3>
 17/16 (104.955, error -.607)<br />
18/16 = 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 (inversion 0, 11 19, 21)."></a><!-- ws:end:WikiTextHeadingRule:26 -->16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21).</h3>
 17/16 (104.955, error -.607)<br />
18/16 = 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 (inversion 0, 11, 14, 21)."></a><!-- ws:end:WikiTextHeadingRule:28 -->16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21).</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 (inversion 0, 11, 14, 19)."></a><!-- ws:end:WikiTextHeadingRule:30 -->16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19).</h3>
 18/16 = 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 (inversion 0, 13, 16, 21)."></a><!-- ws:end:WikiTextHeadingRule:32 -->17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21).</h3>
 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 />
<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 (inversion 0, 11, 14, 19, 21)."></a><!-- ws:end:WikiTextHeadingRule:36 -->16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21).</h3>
 17/16 (104.955, error -.607)<br />
18/16 = 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>

@@ 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:guest|guest]] and made on <tt>2010-05-18 01:02:17 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2010-05-18 02:09:46 UTC</tt>.<br>
-: The original revision id was <tt>142736149</tt>.<br>
+: The original revision id was <tt>142743943</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">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, 39, 43 &amp; 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" ||
+|| degrees || note || cents || nearest harmonic || cents || "error" ||
-|| **0** || **0** || **1/1** || **0.000** || **none** ||
+|| **0** || **1** || **0** || **1/1** || **0.000** || **none** ||
-|| 1 || 52.174 || 33/32 || 53.273 || -1.099 ||
+|| 1 || 1# || 52.174 || 33/32 || 53.273 || -1.099 ||
-|| **2** || **104.348** || **17/16** || **104.955** || **-0.607** ||
+|| **2** || **2b** || **104.348** || **17/16** || **104.955** || **-0.607** ||
-|| 3 || 156.522 || 23/21 || 157.493 || -0.971 ||
+|| 3 || 2 || 156.522 || 23/21 || 157.493 || -0.971 ||
-|| **4·** || **208.696** || **9/8** || **203.910** || **+4.786** ||
+|| **4·** || **2#** || **208.696** || **9/8** || **203.910** || **+4.786** ||
-|| 5 || 260.869 || 43/37 || 260.174 || +0.695 ||
+|| 5 || 3b || 260.869 || 43/37 || 260.174 || +0.695 ||
-|| **6** || **313.043** || **6/5** || **315.641** || **-2.598** ||
+|| **6** || **3** || **313.043** || **6/5** || **315.641** || **-2.598** ||
-|| **7·** || **365.217** || **21/17** || **365.825** || **-0.608** ||
+|| **7·** || **3#** || **365.217** || **21/17** || **365.825** || **-0.608** ||
-|| 8 || 417.391 || 14/11 || 417.508 || -0.117 ||
+|| 8 || 4b || 417.391 || 14/11 || 417.508 || -0.117 ||
-|| **9** || **469.565** || **21/16** || **470.781** || **-1.216** ||
+|| **9** || **4** || **469.565** || **21/16** || **470.781** || **-1.216** ||
-|| **10·** || **521.739** || **23/17** || **523.319** || **-1.58** ||
+|| **10·** || **5** || **521.739** || **23/17** || **523.319** || **-1.58** ||
-|| 11 || 573.913 || 32/23 || 571.726 || +2.187 ||
+|| 11 || 5# || 573.913 || 32/23 || 571.726 || +2.187 ||
-|| **12** || **626.087** || **23/16** || **628.274** || **-2.187** ||
+|| **12** || **6b** || **626.087** || **23/16** || **628.274** || **-2.187** ||
-|| **13·** || **678.261** || **34/23** || **676.681** || **+1.58** ||
+|| **13·** || **6** || **678.261** || **34/23** || **676.681** || **+1.58** ||
-|| 14 || 730.435 || 32/21 || 729.219 || +1.216 ||
+|| 14 || 6# || 730.435 || 32/21 || 729.219 || +1.216 ||
-|| 15 || 782.609 || 11/7 || 782.492 || +0.117 ||
+|| 15 || 7b || 782.609 || 11/7 || 782.492 || +0.117 ||
-|| **16·** || **834.783** || **34/21** || **834.175** || **+0.608** ||
+|| **16·** || **7(A)** || **834.783** || **34/21** || **834.175** || **+0.608** ||
-|| 17 || 886.957 || 5/3 || 884.359 || +2.598 ||
+|| 17 || 7# || 886.957 || 5/3 || 884.359 || +2.598 ||
-|| 18 || 939.130 || 43/25 || 938.890 || +0.24 ||
+|| 18 || 8b || 939.130 || 43/25 || 938.890 || +0.24 ||
-|| **19·** || **991.304** || **39/22** || **991.165** || **+0.139** ||
+|| **19·** || **8** || **991.304** || **39/22** || **991.165** || **+0.139** ||
-|| **20** || **1043.478** || **117/64** || **1044.438** || **-0.96** ||
+|| **20** || **8#** || **1043.478** || **117/64** || **1044.438** || **-0.96** ||
-|| 21 || 1095.652 || 32/17 || 1095.045 || +0.607 ||
+|| 21 || 9b || 1095.652 || 32/17 || 1095.045 || +0.607 ||
-|| 22 || 1147.826 || 64/33 || 1146.727 || +1.099 ||
+|| 22 || 9 || 1147.826 || 64/33 || 1146.727 || +1.099 ||
-|| **23·· (or 0)** || **1200.000** || **2/1** || **1200.000** || **none** ||
+|| **23·· (or 0)** || **1** || **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, 35/32, &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 64:66:68:70:72:79:84:92:110:117. I find this cluster a little hard to listen to, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads &amp; tetrads, as a starting point.
@@ Line 155: / Line 155: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;td&gt;degrees of 23edo&lt;br /&gt;
+         &lt;td&gt;degrees&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;note&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;cents&lt;br /&gt;
@@ Line 168: / Line 170: @@
      &lt;tr&gt;
          &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;1&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;0&lt;br /&gt;
@@ Line 180: / Line 184: @@
      &lt;tr&gt;
          &lt;td&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;52.174&lt;br /&gt;
@@ Line 192: / Line 198: @@
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;2&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;2b&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;104.348&lt;/strong&gt;&lt;br /&gt;
@@ Line 204: / Line 212: @@
      &lt;tr&gt;
          &lt;td&gt;3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;156.522&lt;br /&gt;
@@ Line 216: / Line 226: @@
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;4·&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;2#&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;208.696&lt;/strong&gt;&lt;br /&gt;
@@ Line 228: / Line 240: @@
      &lt;tr&gt;
          &lt;td&gt;5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;3b&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;260.869&lt;br /&gt;
@@ Line 240: / Line 254: @@
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;6&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;3&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;313.043&lt;/strong&gt;&lt;br /&gt;
@@ Line 252: / Line 268: @@
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;7·&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;3#&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;365.217&lt;/strong&gt;&lt;br /&gt;
@@ Line 264: / Line 282: @@
      &lt;tr&gt;
          &lt;td&gt;8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;4b&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;417.391&lt;br /&gt;
@@ Line 276: / Line 296: @@
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;9&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;4&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;469.565&lt;/strong&gt;&lt;br /&gt;
@@ Line 288: / Line 310: @@
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;10·&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;5&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;521.739&lt;/strong&gt;&lt;br /&gt;
@@ Line 300: / Line 324: @@
      &lt;tr&gt;
          &lt;td&gt;11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;5#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;573.913&lt;br /&gt;
@@ Line 312: / Line 338: @@
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;12&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;6b&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;626.087&lt;/strong&gt;&lt;br /&gt;
@@ Line 324: / Line 352: @@
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;13·&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;6&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;678.261&lt;/strong&gt;&lt;br /&gt;
@@ Line 336: / Line 366: @@
      &lt;tr&gt;
          &lt;td&gt;14&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;6#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;730.435&lt;br /&gt;
@@ Line 348: / Line 380: @@
      &lt;tr&gt;
          &lt;td&gt;15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7b&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;782.609&lt;br /&gt;
@@ Line 360: / Line 394: @@
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;16·&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;7(A)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;834.783&lt;/strong&gt;&lt;br /&gt;
@@ Line 372: / Line 408: @@
      &lt;tr&gt;
          &lt;td&gt;17&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;886.957&lt;br /&gt;
@@ Line 384: / Line 422: @@
      &lt;tr&gt;
          &lt;td&gt;18&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;8b&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;939.130&lt;br /&gt;
@@ Line 396: / Line 436: @@
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;19·&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;8&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;991.304&lt;/strong&gt;&lt;br /&gt;
@@ Line 408: / Line 450: @@
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;20&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;8#&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;1043.478&lt;/strong&gt;&lt;br /&gt;
@@ Line 420: / Line 464: @@
      &lt;tr&gt;
          &lt;td&gt;21&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;9b&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;1095.652&lt;br /&gt;
@@ Line 432: / Line 478: @@
      &lt;tr&gt;
          &lt;td&gt;22&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;9&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;1147.826&lt;br /&gt;
@@ Line 444: / Line 492: @@
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;23·· (or 0)&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;1&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;1200.000&lt;/strong&gt;&lt;br /&gt;

Harmony of 23edo: Difference between revisions

Revision as of 02:09, 18 May 2010

IMPORTED REVISION FROM WIKISPACES

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Original HTML content:

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