17ed5: Difference between revisions

Revision as of 11:02, 7 October 2011

IMPORTED REVISION FROM WIKISPACES

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

This revision was by author Kosmorsky and made on 2011-10-07 11:02:47 UTC.

The original revision id was 262580558.

The revision comment was:

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

Original Wikitext content:

=Division of the 5/1 into 17 tones= 

Suppose you go hunting for alternate tonalities. The octave has been pretty well explored, not that there isn't a lot to be done. The tritave beckons - but Bohlen, Pierce and friends have kind of found what you'd be looking for. While there's tremendous ground to cover there, you have the pioneering instinct, so you examine the 5th harmonic as an equivalence.

The first place to look is isoharmonic chords. In diatonic music we take 4:5:6:(7):8 and get the sublime meantone, and assorted other possibilities. In the tritave one finds that 1:2:3 looks back to the octave - although [[11edt]] does it very prettily. But when you crack open that 2/1 shell, you get to the meat of the tritave - [[Bohlen-Pierce]]- based on 3:5:7:9 isoharmony. In the pentave (5/1 and I am perpetuating the technically incorrect terminology, because it works) the first thing you find is 1:2:3:4:5 which is a good lock combination, but not the best pentave-specific chord. But if you pry it open like a clam you'll find a pearl - 5:9:13:17:21:25 a stunning consonance. The full pentad is a little complex, so you might expect to leave out the 21st, just like one does the 7th in diatonicism.

So what does one do with this? Well it took me long enough to stumble on the answer... a year... But the rule is now: Anything worth hearing is worth taking the 17th root of. Because bizarrely enough, the pattern known as "superpyth" in an octave context, is the key to tempering together these cosmic harmonies in the pentave! So I dub it "hyperpyth" but if you want to call it "kosmorsky" that's fine by me too. Musical examples forthcoming.

17ed5 offers a good compromise between 13/5 and 17/5, but the 9/5 of 983 cents is a little bit flat. Nevertheless it's the simplest equal hyperpyth, and quite consonant, like rolling clouds of colorful interstellar dust. However, in hyperpyth, 21/5 isn't necessarily represented, at least not well - which is the case in 17ed5. Luckily, 27, 29, and 39 do a fair job of it. I wonder if there is an alternate pentave system involving 11 and 7 to correspond to octaves' "[[26edo|Orgone]]"? I bet there is and I'll look for it soon.

But wait, an interesting pattern emerges:

22ed5 focuses on 9/5
27ed5 focuses on 13/5
29ed5 focuses on 17/5
(and 34=17*2)

so: 22+27+29=78=39*2
and behold, of the lot, 39ed5 offers the best balance between those intervals.

|| 0: 0.000 cents || 1/1 ||   ||
|| 1: 163.901 ||   ||   ||
|| 2: 327.802 ||   ||   ||
|| 3: 491.702 ||   ||   ||
|| 4: 655.603 ||   ||   ||
|| 5: 819.504 ||   ||   ||
|| 6: 983.405 || 9/5, 16/9, 7/4 || 1017 ||
|| 7: 1147.306 ||   ||   ||
|| 8: 1311.206 ||   ||   ||
|| 9: 1475.107 ||   ||   ||
|| 10: 1639.008 || 13/5 || 1654 ||
|| 11: 1802.909 ||   ||   ||
|| 12: 1966.810 ||   ||   ||
|| 13: 2130.710 || 17/5 || 2118 ||
|| 14: 2294.611 ||   ||   ||
|| 15: 2458.512 || (21/5) || 2486 ||
|| 16: 2622.413 ||   ||   ||
|| 17: 2786.314 || 5/1 ||   ||

Original HTML content:

<html><head><title>17ed5</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Division of the 5/1 into 17 tones"></a><!-- ws:end:WikiTextHeadingRule:0 -->Division of the 5/1 into 17 tones</h1>
 <br />
Suppose you go hunting for alternate tonalities. The octave has been pretty well explored, not that there isn't a lot to be done. The tritave beckons - but Bohlen, Pierce and friends have kind of found what you'd be looking for. While there's tremendous ground to cover there, you have the pioneering instinct, so you examine the 5th harmonic as an equivalence.<br />
<br />
The first place to look is isoharmonic chords. In diatonic music we take 4:5:6:(7):8 and get the sublime meantone, and assorted other possibilities. In the tritave one finds that 1:2:3 looks back to the octave - although <a class="wiki_link" href="/11edt">11edt</a> does it very prettily. But when you crack open that 2/1 shell, you get to the meat of the tritave - <a class="wiki_link" href="/Bohlen-Pierce">Bohlen-Pierce</a>- based on 3:5:7:9 isoharmony. In the pentave (5/1 and I am perpetuating the technically incorrect terminology, because it works) the first thing you find is 1:2:3:4:5 which is a good lock combination, but not the best pentave-specific chord. But if you pry it open like a clam you'll find a pearl - 5:9:13:17:21:25 a stunning consonance. The full pentad is a little complex, so you might expect to leave out the 21st, just like one does the 7th in diatonicism.<br />
<br />
So what does one do with this? Well it took me long enough to stumble on the answer... a year... But the rule is now: Anything worth hearing is worth taking the 17th root of. Because bizarrely enough, the pattern known as &quot;superpyth&quot; in an octave context, is the key to tempering together these cosmic harmonies in the pentave! So I dub it &quot;hyperpyth&quot; but if you want to call it &quot;kosmorsky&quot; that's fine by me too. Musical examples forthcoming.<br />
<br />
17ed5 offers a good compromise between 13/5 and 17/5, but the 9/5 of 983 cents is a little bit flat. Nevertheless it's the simplest equal hyperpyth, and quite consonant, like rolling clouds of colorful interstellar dust. However, in hyperpyth, 21/5 isn't necessarily represented, at least not well - which is the case in 17ed5. Luckily, 27, 29, and 39 do a fair job of it. I wonder if there is an alternate pentave system involving 11 and 7 to correspond to octaves' &quot;<a class="wiki_link" href="/26edo">Orgone</a>&quot;? I bet there is and I'll look for it soon.<br />
<br />
But wait, an interesting pattern emerges:<br />
<br />
22ed5 focuses on 9/5<br />
27ed5 focuses on 13/5<br />
29ed5 focuses on 17/5<br />
(and 34=17*2)<br />
<br />
so: 22+27+29=78=39*2<br />
and behold, of the lot, 39ed5 offers the best balance between those intervals.<br />
<br />


<table class="wiki_table">
    <tr>
        <td>0: 0.000 cents<br />
</td>
        <td>1/1<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1: 163.901<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2: 327.802<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>3: 491.702<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>4: 655.603<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>5: 819.504<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>6: 983.405<br />
</td>
        <td>9/5, 16/9, 7/4<br />
</td>
        <td>1017<br />
</td>
    </tr>
    <tr>
        <td>7: 1147.306<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>8: 1311.206<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>9: 1475.107<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>10: 1639.008<br />
</td>
        <td>13/5<br />
</td>
        <td>1654<br />
</td>
    </tr>
    <tr>
        <td>11: 1802.909<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>12: 1966.810<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>13: 2130.710<br />
</td>
        <td>17/5<br />
</td>
        <td>2118<br />
</td>
    </tr>
    <tr>
        <td>14: 2294.611<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>15: 2458.512<br />
</td>
        <td>(21/5)<br />
</td>
        <td>2486<br />
</td>
    </tr>
    <tr>
        <td>16: 2622.413<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>17: 2786.314<br />
</td>
        <td>5/1<br />
</td>
        <td><br />
</td>
    </tr>
</table>

</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:Kosmorsky|Kosmorsky]] and made on <tt>2011-10-07 02:54:36 UTC</tt>.<br>
+: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on <tt>2011-10-07 11:02:47 UTC</tt>.<br>
-: The original revision id was <tt>262471362</tt>.<br>
+: The original revision id was <tt>262580558</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 26: / Line 26: @@
 and behold, of the lot, 39ed5 offers the best balance between those intervals.
-|| 0: 1/1 0.000 unison, perfect prime ||   ||   ||
+|| 0: 0.000 cents || 1/1 ||   ||
-|| 1: 163.901 cents 163.901 ||   ||   ||
+|| 1: 163.901 ||   ||   ||
-|| 2: 327.802 cents 327.802 ||   ||   ||
+|| 2: 327.802 ||   ||   ||
-|| 3: 491.702 cents 491.702 ||   ||   ||
+|| 3: 491.702 ||   ||   ||
-|| 4: 655.603 cents 655.603 ||   ||   ||
+|| 4: 655.603 ||   ||   ||
-|| 5: 819.504 cents 819.504 ||   ||   ||
+|| 5: 819.504 ||   ||   ||
-|| 6: 983.405 cents 983.405 ||   ||   ||
+|| 6: 983.405 || 9/5, 16/9, 7/4 || 1017 ||
-|| 7: 1147.306 cents 1147.306 ||   ||   ||
+|| 7: 1147.306 ||   ||   ||
-|| 8: 1311.206 cents 1311.206 ||   ||   ||
+|| 8: 1311.206 ||   ||   ||
-|| 9: 1475.107 cents 1475.107 ||   ||   ||
+|| 9: 1475.107 ||   ||   ||
-|| 10: 1639.008 cents 1639.008 ||   ||   ||
+|| 10: 1639.008 || 13/5 || 1654 ||
-|| 11: 1802.909 cents 1802.909 ||   ||   ||
+|| 11: 1802.909 ||   ||   ||
-|| 12: 1966.810 cents 1966.810 ||   ||   ||
+|| 12: 1966.810 ||   ||   ||
-|| 13: 2130.710 cents 2130.710 ||   ||   ||
+|| 13: 2130.710 || 17/5 || 2118 ||
-|| 14: 2294.611 cents 2294.611 ||   ||   ||
+|| 14: 2294.611 ||   ||   ||
-|| 15: 2458.512 cents 2458.512 ||   ||   ||
+|| 15: 2458.512 || (21/5) || 2486 ||
-|| 16: 2622.413 cents 2622.413 ||   ||   ||
+|| 16: 2622.413 ||   ||   ||
-|| 17: 5/1 2786.314 pentave ||   ||   ||</pre></div>
+|| 17: 2786.314 || 5/1 ||   ||</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;17ed5&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Division of the 5/1 into 17 tones"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Division of the 5/1 into 17 tones&lt;/h1&gt;
@@ Line 69: / Line 69: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;td&gt;0: 1/1 0.000 unison, perfect prime&lt;br /&gt;
+         &lt;td&gt;0: 0.000 cents&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;1/1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 77: / Line 77: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;1: 163.901 cents 163.901&lt;br /&gt;
+         &lt;td&gt;1: 163.901&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 85: / Line 85: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;2: 327.802 cents 327.802&lt;br /&gt;
+         &lt;td&gt;2: 327.802&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 93: / Line 93: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;3: 491.702 cents 491.702&lt;br /&gt;
+         &lt;td&gt;3: 491.702&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 101: / Line 101: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;4: 655.603 cents 655.603&lt;br /&gt;
+         &lt;td&gt;4: 655.603&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 109: / Line 109: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;5: 819.504 cents 819.504&lt;br /&gt;
+         &lt;td&gt;5: 819.504&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 117: / Line 117: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;6: 983.405 cents 983.405&lt;br /&gt;
+         &lt;td&gt;6: 983.405&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;9/5, 16/9, 7/4&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;1017&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;7: 1147.306 cents 1147.306&lt;br /&gt;
+         &lt;td&gt;7: 1147.306&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 133: / Line 133: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;8: 1311.206 cents 1311.206&lt;br /&gt;
+         &lt;td&gt;8: 1311.206&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 141: / Line 141: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;9: 1475.107 cents 1475.107&lt;br /&gt;
+         &lt;td&gt;9: 1475.107&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 149: / Line 149: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;10: 1639.008 cents 1639.008&lt;br /&gt;
+         &lt;td&gt;10: 1639.008&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;13/5&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;1654&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;11: 1802.909 cents 1802.909&lt;br /&gt;
+         &lt;td&gt;11: 1802.909&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 165: / Line 165: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;12: 1966.810 cents 1966.810&lt;br /&gt;
+         &lt;td&gt;12: 1966.810&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 173: / Line 173: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;13: 2130.710 cents 2130.710&lt;br /&gt;
+         &lt;td&gt;13: 2130.710&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;17/5&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;2118&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;14: 2294.611 cents 2294.611&lt;br /&gt;
+         &lt;td&gt;14: 2294.611&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 189: / Line 189: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;15: 2458.512 cents 2458.512&lt;br /&gt;
+         &lt;td&gt;15: 2458.512&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;(21/5)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;2486&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;16: 2622.413 cents 2622.413&lt;br /&gt;
+         &lt;td&gt;16: 2622.413&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 205: / Line 205: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;17: 5/1 2786.314 pentave&lt;br /&gt;
+         &lt;td&gt;17: 2786.314&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;5/1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;

17ed5: Difference between revisions

Revision as of 11:02, 7 October 2011

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

Original HTML content:

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