Harmony of 23edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:guest|guest]] and made on <tt>2010-05-17 19:30:36 UTC</tt>.

: This revision was by author [[User:guest|guest]] and made on <tt>2010-05-17 20:21:39 UTC</tt>.

: The original revision id was <tt>~~142669403~~</tt>.

: The original revision id was <tt>142682213</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

<h4>Original Wikitext content:</h4>

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

<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 & 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** ||

|| **2** || **104.348** || **17/16** || **104.955** || **-0.607** ||

|| 3 || 156.522 || 23/21 || 157.493 || -0.971 ||

|| **4** || **208.696** || **9/8** || **203.910** || **+4.786** ||

|| **4·** || **208.696** || **9/8** || **203.910** || **+4.786** ||

|| 7 || 365.217 || 21/17 || 365.825 || -.608 ||

|| 5 || 260.869 || 43/37 || 260.174 || +0.695 ||

|| **6** || **313.043** || **6/5** || **315.641** || **-2.598** ||

|| 7· || 365.217 || 21/17 || 365.825 || -0.608 ||

|| 8 || 417.391 || 14/11 || 417.508 || -0.117 ||

|| **9** || **469.565** || **21/16** || **470.781** || **-1.216** ||

|| 10· || 521.739 || 23/17 || 523.319 || -1.58 ||

|| 11 || 573.913 || 32/23 || 571.726 || +2.187 ||

|| **12** || **626.087** || **23/16** || **628.274** || **-2.187** ||

|| 18 || 939.130 || 43/25 || 938.890 || +0.~~240~~ ||

|| **13·** || **678.261** || **34/23** || **676.681** || **+1.58** ||

|| 20 || 1043.478 || 117/64 || 1044.438 || -.~~960~~ ||

|| 14 || 730.435 || 32/21 || 729.219 || +1.216 ||

|| **23 (or 0)** || **1200.000** || **2/1** || **1200.000** || **none** ||

|| 15 || 782.609 || 11/7 || 782.492 || +0.117 ||

|| **16·** || **834.783** || **34/21** || **834.175** || **+0.608** ||

|| 17 || 886.957 || 5/3 || 884.359 || +2.598 ||

|| 18 || 939.130 || 43/25 || 938.890 || +0.24 ||

|| **19·** || **991.304** || **39/22** || **991.165** || **+0.139** ||

|| **20** || **1043.478** || **117/64** || **1044.438** || **-0.96** ||

|| 21 || 1095.652 || 32/17 || 1095.045 || +0.607 ||

|| 22 || 1147.826 || 64/33 || 1146.727 || +1.099 ||

|| **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, 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.

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23/21 (157.493, error: -.971)</pre></div>

<h4>Original HTML content:</h4>

<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"><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, 43 &amp; 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics.

<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"><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.

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<td>104.955

</td>

<td>-.607

<td>-0.607

</td>

</tr>

Line 202:

Line 215:

</tr>

<tr>

<td>4

<td>4·

</td>

<td>208.696

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</tr>

<tr>

<td>7

<td>5

</td>

<td>260.869

</td>

<td>43/37

</td>

<td>260.174

</td>

<td>+0.695

</td>

</tr>

<tr>

<td>6

</td>

<td>313.043

</td>

<td>6/5

</td>

<td>315.641

</td>

<td>-2.598

</td>

</tr>

<tr>

<td>7·

</td>

<td>365.217

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<td>365.825

</td>

<td>-.608

<td>-0.608

</td>

</tr>

<tr>

<td>8

</td>

<td>417.391

</td>

<td>14/11

</td>

<td>417.508

</td>

<td>-0.117

</td>

</tr>

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</td>

<td>-1.216

</td>

</tr>

<tr>

<td>10·

</td>

<td>521.739

</td>

<td>23/17

</td>

<td>523.319

</td>

<td>-1.58

</td>

</tr>

<tr>

<td>11

</td>

<td>573.913

</td>

<td>32/23

</td>

<td>571.726

</td>

<td>+2.187

</td>

</tr>

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</td>

<td>-2.187

</td>

</tr>

<tr>

<td>13·

</td>

<td>678.261

</td>

<td>34/23

</td>

<td>676.681

</td>

<td>+1.58

</td>

</tr>

<tr>

<td>14

</td>

<td>730.435

</td>

<td>32/21

</td>

<td>729.219

</td>

<td>+1.216

</td>

</tr>

<tr>

<td>15

</td>

<td>782.609

</td>

<td>11/7

</td>

<td>782.492

</td>

<td>+0.117

</td>

</tr>

<tr>

<td>16·

</td>

<td>834.783

</td>

<td>34/21

</td>

<td>834.175

</td>

<td>+0.608

</td>

</tr>

<tr>

<td>17

</td>

<td>886.957

</td>

<td>5/3

</td>

<td>884.359

</td>

<td>+2.598

</td>

</tr>

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<td>938.890

</td>

<td>+0.~~240~~

<td>+0.24

</td>

</tr>

<tr>

<td>19·

</td>

<td>991.304

</td>

<td>39/22

</td>

<td>991.165

</td>

<td>+0.139

</td>

</tr>

<tr>

<td>20

</td>

<td>1043.478

</td>

<td>117/64

</td>

<td>1044.438

</td>

<td>-0.96

</td>

</tr>

<tr>

<td>21

</td>

<td>1095.652

</td>

<td>32/17

</td>

<td>1095.045

</td>

<td>+0.607

</td>

</tr>

<tr>

<td>20

<td>22

</td>

<td>~~1043~~.~~478~~

<td>1147.826

</td>

<td>~~117~~/64

<td>64/33

</td>

<td>~~1044~~.~~438~~

<td>1146.727

</td>

<td>-.~~960~~

<td>+1.099

</td>

</tr>

<tr>

<td>23 (or 0)

<td>23·· (or 0)

</td>

<td>1200.000

@@ 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>2010-05-17 19:30:36 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2010-05-17 20:21:39 UTC</tt>.<br>
-: The original revision id was <tt>142669403</tt>.<br>
+: The original revision id was <tt>142682213</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>
 <h4>Original Wikitext content:</h4>
-<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, 43 &amp; 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics.
+<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" ||
 || **0** || **0** || **1/1** || **0.000** || **none** ||
 || 1 || 52.174 || 33/32 || 53.273 || -1.099 ||
-|| **2** || **104.348** || **17/16** || **104.955** || **-.607** ||
+|| **2** || **104.348** || **17/16** || **104.955** || **-0.607** ||
 || 3 || 156.522 || 23/21 || 157.493 || -0.971 ||
-|| **4** || **208.696** || **9/8** || **203.910** || **+4.786** ||
+|| **4·** || **208.696** || **9/8** || **203.910** || **+4.786** ||
-|| 7 || 365.217 || 21/17 || 365.825 || -.608 ||
+|| 5 || 260.869 || 43/37 || 260.174 || +0.695 ||
+|| **6** || **313.043** || **6/5** || **315.641** || **-2.598** ||
+|| 7· || 365.217 || 21/17 || 365.825 || -0.608 ||
+|| 8 || 417.391 || 14/11 || 417.508 || -0.117 ||
 || **9** || **469.565** || **21/16** || **470.781** || **-1.216** ||
+|| 10· || 521.739 || 23/17 || 523.319 || -1.58 ||
+|| 11 || 573.913 || 32/23 || 571.726 || +2.187 ||
 || **12** || **626.087** || **23/16** || **628.274** || **-2.187** ||
-|| 18 || 939.130 || 43/25 || 938.890 || +0.240 ||
+|| **13·** || **678.261** || **34/23** || **676.681** || **+1.58** ||
-|| 20 || 1043.478 || 117/64 || 1044.438 || -.960 ||
+|| 14 || 730.435 || 32/21 || 729.219 || +1.216 ||
-|| **23 (or 0)** || **1200.000** || **2/1** || **1200.000** || **none** ||
+|| 15 || 782.609 || 11/7 || 782.492 || +0.117 ||
+|| **16·** || **834.783** || **34/21** || **834.175** || **+0.608** ||
+|| 17 || 886.957 || 5/3 || 884.359 || +2.598 ||
+|| 18 || 939.130 || 43/25 || 938.890 || +0.24 ||
+|| **19·** || **991.304** || **39/22** || **991.165** || **+0.139** ||
+|| **20** || **1043.478** || **117/64** || **1044.438** || **-0.96** ||
+|| 21 || 1095.652 || 32/17 || 1095.045 || +0.607 ||
+|| 22 || 1147.826 || 64/33 || 1146.727 || +1.099 ||
+|| **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, 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 136: / Line 149: @@
 /21 (157.493, error: -.971)</pre></div>
 <h4>Original HTML content:</h4>
-<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;Harmony of 23edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;If you take a look at the intervals of &lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&gt;, 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 &amp;quot;atonal&amp;quot; or &amp;quot;nonharmonic,&amp;quot; I'd like to consider higher-limit harmonies that could serve as useful sonorities, perhaps even &amp;quot;consonances,&amp;quot; in the context of careful composition. &lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&gt; contains intervals which approach harmonics 9, 17, 21, 23, 33, 43 &amp;amp; 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics.&lt;br /&gt;
+<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;Harmony of 23edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;If you take a look at the intervals of &lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&gt;, 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 &amp;quot;atonal&amp;quot; or &amp;quot;nonharmonic,&amp;quot; I'd like to consider higher-limit harmonies that could serve as useful sonorities, perhaps even &amp;quot;consonances,&amp;quot; in the context of careful composition. &lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&gt; contains intervals which approach harmonics 9, 17, 21, 23, 33, 39, 43 &amp;amp; 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 186: / Line 199: @@
          &lt;td&gt;&lt;strong&gt;104.955&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;-.607&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;-0.607&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 202: / Line 215: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;strong&gt;4&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;4·&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;208.696&lt;/strong&gt;&lt;br /&gt;
@@ Line 214: / Line 227: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;7&lt;br /&gt;
+         &lt;td&gt;5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;260.869&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;43/37&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;260.174&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;+0.695&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &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;313.043&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;6/5&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;315.641&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;-2.598&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;7·&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;365.217&lt;br /&gt;
@@ Line 222: / Line 259: @@
          &lt;td&gt;365.825&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;-.608&lt;br /&gt;
+         &lt;td&gt;-0.608&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;417.391&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;14/11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;417.508&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;-0.117&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 235: / Line 284: @@
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;-1.216&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;10·&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;521.739&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;23/17&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;523.319&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;-1.58&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;573.913&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;32/23&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;571.726&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;+2.187&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 247: / Line 320: @@
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;-2.187&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;strong&gt;13·&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
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