Seventeen limit tetrads - Revision history

FloraC: Recategorize; +todo

2023-05-23T12:02:17Z

Recategorize; +todo

← Older revision		Revision as of 12:02, 23 May 2023
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	[[Category:17-limit]]		[[Category:17-limit]]
	[[Category:Tables]]		[[Category:Tables]]
	[[Category:~~Chords~~]]		[[Category:Tetrads]]
	~~[[Category:Tetrad]]~~
			{{Todo\| cleanup \| discuss title }}

BudjarnLambeth: Categorised this uncategorised page

2023-04-23T20:16:40Z

Categorised this uncategorised page

← Older revision		Revision as of 20:16, 23 April 2023
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			[[Category:17-limit]]
			[[Category:Tables]]
			[[Category:Chords]]
			[[Category:Tetrad]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

Wikispaces>Andrew_Heathwaite: Imported revision 314262300 - Original comment:

2012-03-24T18:49:18Z

**Imported revision 314262300 - Original comment: **

← Older revision		Revision as of 18:49, 24 March 2012
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		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>2012-03-24 13:18~~:53~~ UTC</tt>.<br>		: This revision was by author [[User:Andrew_Heathwaite\|Andrew_Heathwaite]] and made on <tt>2012-03-24 18:49:18 UTC</tt>.<br>
	: The original revision id was <tt>~~314219342~~</tt>.<br>		: The original revision id was <tt>314262300</tt>.<br>
	: The revision comment was: <tt></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>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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	<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">The 17th harmonic, octave reduced to the frequency ratio [[17_16\|17/16]], is about 105¢. It is likely to sound dissonant against the fundamental, which is perhaps one reason why [[Just Intonation]] composers usually stop at the [[13-limit]] or lower. Another interval of 17 that can sound just as dissonant is [[18_17\|18/17]], about 99¢. Thus, 17/16 also clashes with [[9_8\|9/8]]. Therefore, one approach for 17-limit harmony is to filter the total list of chords to exclude these small steps, resulting in non-rooted harmonies.		<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">The 17th harmonic, octave reduced to the frequency ratio [[17_16\|17/16]], is about 105¢. It is likely to sound dissonant against the fundamental, which is perhaps one reason why [[Just Intonation]] composers usually stop at the [[13-limit]] or lower. Another interval of 17 that can sound just as dissonant is [[18_17\|18/17]], about 99¢. Thus, 17/16 also clashes with [[9_8\|9/8]]. Therefore, one approach for 17-limit harmony is to filter the total list of chords to exclude these small steps, resulting in non-rooted harmonies.

	Every otonal tetrad within the 17-limit is listed in the table below as the simplest possible subset of harmonics 2-17; there are 56 in total. The columns to the right of the chords provide different cutoffs for filtering out small steps. For example, chords with a 16/15 cutoff do not contain 17/16 or 18/17 among the dyads. Since this eliminates harmonics 2 and 9 right away, it also eliminates chords containing 10/9 and 9/8. (This is why there is a jump from 12/11 to 9/8 in the table below.)		Every otonal tetrad within the 17-limit is listed in the table below as the simplest possible subset of harmonics 2-17; there are 56 in total. The columns to the right of the chords provide different cutoffs for filtering out small steps. For example, chords with a 16/15 cutoff do not contain any interval smaller than or equal to 16/15 (which includes 17/16 and 18/17) among the dyads. Since this eliminates harmonics 2 and 9 right away, it also eliminates chords containing 10/9 and 9/8. (This is why there is a jump from 12/11 to 9/8 in the table below.)
	\|\|~ \|\|~ \|\|~ \|\|~ \|\|~ \|\|~ \|\|~ \|\|~ \|\|		\|\|~ \|\|~ \|\|~ \|\|~ \|\|~ \|\|~ \|\|~ \|\|~ \|\|
	\|\|~ no cutoff		\|\|~ no cutoff
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	<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>seventeen limit tetrads</title></head><body>The 17th harmonic, octave reduced to the frequency ratio <a class="wiki_link" href="/17_16">17/16</a>, is about 105¢. It is likely to sound dissonant against the fundamental, which is perhaps one reason why <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a> composers usually stop at the <a class="wiki_link" href="/13-limit">13-limit</a> or lower. Another interval of 17 that can sound just as dissonant is <a class="wiki_link" href="/18_17">18/17</a>, about 99¢. Thus, 17/16 also clashes with <a class="wiki_link" href="/9_8">9/8</a>. Therefore, one approach for 17-limit harmony is to filter the total list of chords to exclude these small steps, resulting in non-rooted harmonies.<br />		<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>seventeen limit tetrads</title></head><body>The 17th harmonic, octave reduced to the frequency ratio <a class="wiki_link" href="/17_16">17/16</a>, is about 105¢. It is likely to sound dissonant against the fundamental, which is perhaps one reason why <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a> composers usually stop at the <a class="wiki_link" href="/13-limit">13-limit</a> or lower. Another interval of 17 that can sound just as dissonant is <a class="wiki_link" href="/18_17">18/17</a>, about 99¢. Thus, 17/16 also clashes with <a class="wiki_link" href="/9_8">9/8</a>. Therefore, one approach for 17-limit harmony is to filter the total list of chords to exclude these small steps, resulting in non-rooted harmonies.<br />
	<br />		<br />
	Every otonal tetrad within the 17-limit is listed in the table below as the simplest possible subset of harmonics 2-17; there are 56 in total. The columns to the right of the chords provide different cutoffs for filtering out small steps. For example, chords with a 16/15 cutoff do not contain 17/16 or 18/17 among the dyads. Since this eliminates harmonics 2 and 9 right away, it also eliminates chords containing 10/9 and 9/8. (This is why there is a jump from 12/11 to 9/8 in the table below.)<br />		Every otonal tetrad within the 17-limit is listed in the table below as the simplest possible subset of harmonics 2-17; there are 56 in total. The columns to the right of the chords provide different cutoffs for filtering out small steps. For example, chords with a 16/15 cutoff do not contain any interval smaller than or equal to 16/15 (which includes 17/16 and 18/17) among the dyads. Since this eliminates harmonics 2 and 9 right away, it also eliminates chords containing 10/9 and 9/8. (This is why there is a jump from 12/11 to 9/8 in the table below.)<br />

Wikispaces>guest: Imported revision 314219342 - Original comment:

2012-03-24T13:18:53Z

**Imported revision 314219342 - Original comment: **

Show changes

Wikispaces>Andrew_Heathwaite: Imported revision 312094974 - Original comment:

2012-03-18T11:55:34Z

**Imported revision 312094974 - Original comment: **

New page

<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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2012-03-18 11:55:34 UTC</tt>. 
: The original revision id was <tt>312094974</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">The 17th harmonic, octave reduced to the frequency ratio [[17_16|17/16]], is about 105¢. It is likely to sound dissonant against the fundamental, which is perhaps one reason why [[Just Intonation]] composers usually stop at the [[13-limit]] or lower. Another interval of 17 that can sound just as dissonant is [[18_17|18/17]], about 99¢. Thus, 17/16 also clashes with [[9_8|9/8]]. However, if we don't mind some clashing from semitone-sized harmonic intervals, we can make the following list of 56 17-limit tetrads.

||~ Simplest open position voicing with harmonics 2-17 ||
||~ subsets of 2:3:5:7:9:11:13:15:17 ||
|| 2:3:5:17 ||
|| 2:3:7:17 ||
|| 2:3:9:17 ||
|| 2:3:11:17 ||
|| 2:3:13:17 ||
|| 2:3:15:17 ||
|| 2:5:7:17 ||
|| 2:5:9:17 ||
|| 2:5:11:17 ||
|| 2:5:13:17 ||
|| 2:5:15:17 ||
|| 2:7:9:17 ||
|| 2:7:11:17 ||
|| 2:7:13:17 ||
|| 2:7:15:17 ||
|| 2:9:11:17 ||
|| 2:9:13:17 ||
|| 2:9:15:17 ||
|| 2:11:13:17 ||
|| 2:11:15:17 ||
|| 2:13:15:17 ||
|| 3:5:7:17 ||
|| 3:5:9:17 ||
|| 3:5:11:17 ||
|| 3:5:13:17 ||
|| 3:5:15:17 ||
|| 3:7:9:17 ||
|| 3:7:11:17 ||
|| 3:7:13:17 ||
|| 3:7:15:17 ||
|| 3:9:11:17 ||
|| 3:9:13:17 ||
|| 3:9:15:17 ||
|| 3:11:13:17 ||
|| 3:11:15:17 ||
|| 3:13:15:17 ||
|| 5:7:9:17 ||
|| 5:7:11:17 ||
|| 5:7:13:17 ||
|| 5:7:15:17 ||
|| 5:9:11:17 ||
|| 5:9:13:17 ||
|| 5:9:15:17 ||
|| 5:11:13:17 ||
|| 5:11:15:17 ||
|| 5:13:15:17 ||
|| 7:9:11:17 ||
|| 7:9:13:17 ||
|| 7:9:15:17 ||
|| 7:11:13:17 ||
|| 7:11:15:17 ||
|| 7:13:15:17 ||
|| 9:11:13:17 ||
|| 9:11:15:17 ||
|| 9:13:15:17 ||
|| 11:13:15:17 ||
||~ ||</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>seventeen limit tetrads</title></head><body>The 17th harmonic, octave reduced to the frequency ratio <a class="wiki_link" href="/17_16">17/16</a>, is about 105¢. It is likely to sound dissonant against the fundamental, which is perhaps one reason why <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a> composers usually stop at the <a class="wiki_link" href="/13-limit">13-limit</a> or lower. Another interval of 17 that can sound just as dissonant is <a class="wiki_link" href="/18_17">18/17</a>, about 99¢. Thus, 17/16 also clashes with <a class="wiki_link" href="/9_8">9/8</a>. However, if we don't mind some clashing from semitone-sized harmonic intervals, we can make the following list of 56 17-limit tetrads. 
 

<table class="wiki_table">
<tr>
<th>Simplest open position voicing with harmonics 2-17 
</th>
</tr>
<tr>
<th>subsets of 2:3:5:7:9:11:13:15:17 
</th>
</tr>
<tr>
<td>2:3:5:17 
</td>
</tr>
<tr>
<td>2:3:7:17 
</td>
</tr>
<tr>
<td>2:3:9:17 
</td>
</tr>
<tr>
<td>2:3:11:17 
</td>
</tr>
<tr>
<td>2:3:13:17 
</td>
</tr>
<tr>
<td>2:3:15:17 
</td>
</tr>
<tr>
<td>2:5:7:17 
</td>
</tr>
<tr>
<td>2:5:9:17 
</td>
</tr>
<tr>
<td>2:5:11:17 
</td>
</tr>
<tr>
<td>2:5:13:17 
</td>
</tr>
<tr>
<td>2:5:15:17 
</td>
</tr>
<tr>
<td>2:7:9:17 
</td>
</tr>
<tr>
<td>2:7:11:17 
</td>
</tr>
<tr>
<td>2:7:13:17 
</td>
</tr>
<tr>
<td>2:7:15:17 
</td>
</tr>
<tr>
<td>2:9:11:17 
</td>
</tr>
<tr>
<td>2:9:13:17 
</td>
</tr>
<tr>
<td>2:9:15:17 
</td>
</tr>
<tr>
<td>2:11:13:17 
</td>
</tr>
<tr>
<td>2:11:15:17 
</td>
</tr>
<tr>
<td>2:13:15:17 
</td>
</tr>
<tr>
<td>3:5:7:17 
</td>
</tr>
<tr>
<td>3:5:9:17 
</td>
</tr>
<tr>
<td>3:5:11:17 
</td>
</tr>
<tr>
<td>3:5:13:17 
</td>
</tr>
<tr>
<td>3:5:15:17 
</td>
</tr>
<tr>
<td>3:7:9:17 
</td>
</tr>
<tr>
<td>3:7:11:17 
</td>
</tr>
<tr>
<td>3:7:13:17 
</td>
</tr>
<tr>
<td>3:7:15:17 
</td>
</tr>
<tr>
<td>3:9:11:17 
</td>
</tr>
<tr>
<td>3:9:13:17 
</td>
</tr>
<tr>
<td>3:9:15:17 
</td>
</tr>
<tr>
<td>3:11:13:17 
</td>
</tr>
<tr>
<td>3:11:15:17 
</td>
</tr>
<tr>
<td>3:13:15:17 
</td>
</tr>
<tr>
<td>5:7:9:17 
</td>
</tr>
<tr>
<td>5:7:11:17 
</td>
</tr>
<tr>
<td>5:7:13:17 
</td>
</tr>
<tr>
<td>5:7:15:17 
</td>
</tr>
<tr>
<td>5:9:11:17 
</td>
</tr>
<tr>
<td>5:9:13:17 
</td>
</tr>
<tr>
<td>5:9:15:17 
</td>
</tr>
<tr>
<td>5:11:13:17 
</td>
</tr>
<tr>
<td>5:11:15:17 
</td>
</tr>
<tr>
<td>5:13:15:17 
</td>
</tr>
<tr>
<td>7:9:11:17 
</td>
</tr>
<tr>
<td>7:9:13:17 
</td>
</tr>
<tr>
<td>7:9:15:17 
</td>
</tr>
<tr>
<td>7:11:13:17 
</td>
</tr>
<tr>
<td>7:11:15:17 
</td>
</tr>
<tr>
<td>7:13:15:17 
</td>
</tr>
<tr>
<td>9:11:13:17 
</td>
</tr>
<tr>
<td>9:11:15:17 
</td>
</tr>
<tr>
<td>9:13:15:17 
</td>
</tr>
<tr>
<td>11:13:15:17 
</td>
</tr>
<tr>
<th> 
</th>
</tr>
</table>

</body></html></pre></div>