Highschool scales: Difference between revisions
Wikispaces>genewardsmith **Imported revision 240386625 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 240388975 - Original comment: ** |
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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:genewardsmith|genewardsmith]] and made on <tt>2011-07-07 13: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-07 13:33:42 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>240388975</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> | ||
<h4>Original Wikitext content:</h4> | <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">When [[Gene Ward Smith]] was in high school he made his first venture into the world of scale creation, devising a system of just intonation scales which were therefore given the name //highschool scales//. They were based on the observation that superparticular ratios whose numerators were square or triangular numbers make for good intervals for constructing scales, and that these two were related. If S[n] is n^2/(n^2-1), the superparticular ratio with square numerator n^2, and T[n] is the superparticular ratio with | <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">When [[Gene Ward Smith]] was in high school he made his first venture into the world of scale creation, devising a system of just intonation scales which were therefore given the name //highschool scales//. They were based on the observation that superparticular ratios whose numerators were square or triangular numbers make for good intervals for constructing scales, and that these two were related. If S[n] is n^2/(n^2-1), the superparticular ratio with square numerator n^2, and T[n] is the superparticular ratio with triangular numerator n(n+1)/2, then we have the following relationships: | ||
T[n] = S[n] * S[n+1] | T[n] = S[n] * S[n+1] | ||
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with intervals 9/8 10/9 16/15 9/8 10/9 9/8 16/15, we may split both the 9/8 and 10/9 in half by | with intervals 9/8 10/9 16/15 9/8 10/9 9/8 16/15, we may split both the 9/8 and 10/9 in half by | ||
9/8 = 15/14 * 21/20 (square | 9/8 = 15/14 * 21/20 (square numerator to two triangular) | ||
10/9 = 16/15 * 25/24 (triangular | 10/9 = 16/15 * 25/24 (triangular numerator to two square) | ||
and used them to expand the scale. The choices involving 10/9 are virtally automatic: between 9/8 and 5/4 we choose between 6/5 and 75/64, so we choose 6/5; between 3/2 and 5/3 we choose between 8/5 and 25/16, and so select 8/5. This gives us a natural nine-note extension which is more or less self-recommending, and which showed up in the survey of 9-note, 5-limit Fokker blocks as "mavlim7", one of the 27/25&135/128 blocks. | and used them to expand the scale. The choices involving 10/9 are virtally automatic: between 9/8 and 5/4 we choose between 6/5 and 75/64, so we choose 6/5; between 3/2 and 5/3 we choose between 8/5 and 25/16, and so select 8/5. This gives us a natural nine-note extension which is more or less self-recommending, and which showed up in the survey of 9-note, 5-limit Fokker blocks as "mavlim7", one of the 27/25&135/128 blocks. | ||
Splitting 9/8 and introducing the 7-limit is where it gets more interesting. Between 5/3 and 15/8 we have a choice between 7/4 and 25/14, and so of course we choose 7/4. There seems to be no clear reason to choose either 15/14 or 21/20 between 1 and 9/8, and so we may try both. Then between 4/3 and 3/2, we must decide between 7/5 and 10/7. But clearly 7/5 goes with 21/20, and 10/7 with 15/14, and we are done, having constructed the two 12-note "Highschool" scales. | Splitting 9/8 and introducing the 7-limit is where it gets more interesting. Between 5/3 and 15/8 we have a choice between 7/4 and 25/14, and so of course we choose 7/4. There seems to be no clear reason to choose either 15/14 or 21/20 between 1 and 9/8, and so we may try both. Then between 4/3 and 3/2, we must decide between 7/5 and 10/7. But clearly 7/5 goes with 21/20, and 10/7 with 15/14, and we are done, having constructed the two 12-note "Highschool" scales. | ||
Eventually this process breaks down (for on thing, 36 is both square and triangular) but it can be continued to larger scales. The next step involves breaking 15/14 as 15/14 = 25/24 * 36/35. | |||
Starting from the 12highschool1 scale, it isn't obvious what to do between 21/20 and 9/8, or between 7/5 and 3/2. But between 7/4 and 15/8, it's clear we should go with 9/5 = 36/35 * 7/4 rather than 175/96 = 25/24 * 7/4. Hence we pick 36/35 * 25/24 rather than the reverse in all three cases, so that they correspond, 15highschool1. Similar reasoning applies to 12highschool2, giving 15highschool2. | |||
At this point we might take note of the fact that 15highschool1 is a better scale from the point of view of harmony than 15highschool2, and is a highly recommendable 15-note, 7-limit JI scale. | |||
Continuing on, the next stage is to break apart 16/15 inside 15highschool1 by 16/15 = 28/27 * 36/35. Putting a 7/6 between 9/8 and 6/5, a 9/7 between 5/4 and 4/3 and a 14/9 between 3/2 and 8/5 are all obvious, leaving the question of what to do between 15/8 and 2. The two choices lead to the two scales 19highschool1 and 19highschool2, with 19highschool1 having the fuller harmony. | |||
Now we have three 21/20 intervals to break as 21/20 = 36/35 * 49/48 in 19highschool1 to get a 22-note scale. Two of them are not obvious, but it's clear we put 12/7 between 5/4 and 7/4, and this tells us how to break the other two, giving 22highschool. | |||
=Scales= | =Scales= | ||
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[[15highschool2]] | [[15highschool2]] | ||
[[19highschool1]] | [[19highschool1]] | ||
[[19highschool2]]</pre></div> | [[19highschool2]] | ||
[[22highschool]]</pre></div> | |||
<h4>Original HTML content:</h4> | <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>Highschool scales</title></head><body>When <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a> was in high school he made his first venture into the world of scale creation, devising a system of just intonation scales which were therefore given the name <em>highschool scales</em>. They were based on the observation that superparticular ratios whose numerators were square or triangular numbers make for good intervals for constructing scales, and that these two were related. If S[n] is n^2/(n^2-1), the superparticular ratio with square numerator n^2, and T[n] is the superparticular ratio with | <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>Highschool scales</title></head><body>When <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a> was in high school he made his first venture into the world of scale creation, devising a system of just intonation scales which were therefore given the name <em>highschool scales</em>. They were based on the observation that superparticular ratios whose numerators were square or triangular numbers make for good intervals for constructing scales, and that these two were related. If S[n] is n^2/(n^2-1), the superparticular ratio with square numerator n^2, and T[n] is the superparticular ratio with triangular numerator n(n+1)/2, then we have the following relationships:<br /> | ||
<br /> | <br /> | ||
T[n] = S[n] * S[n+1]<br /> | T[n] = S[n] * S[n+1]<br /> | ||
| Line 53: | Line 64: | ||
with intervals 9/8 10/9 16/15 9/8 10/9 9/8 16/15, we may split both the 9/8 and 10/9 in half by<br /> | with intervals 9/8 10/9 16/15 9/8 10/9 9/8 16/15, we may split both the 9/8 and 10/9 in half by<br /> | ||
<br /> | <br /> | ||
9/8 = 15/14 * 21/20 (square | 9/8 = 15/14 * 21/20 (square numerator to two triangular)<br /> | ||
<br /> | <br /> | ||
10/9 = 16/15 * 25/24 (triangular | 10/9 = 16/15 * 25/24 (triangular numerator to two square)<br /> | ||
<br /> | <br /> | ||
and used them to expand the scale. The choices involving 10/9 are virtally automatic: between 9/8 and 5/4 we choose between 6/5 and 75/64, so we choose 6/5; between 3/2 and 5/3 we choose between 8/5 and 25/16, and so select 8/5. This gives us a natural nine-note extension which is more or less self-recommending, and which showed up in the survey of 9-note, 5-limit Fokker blocks as &quot;mavlim7&quot;, one of the 27/25&amp;135/128 blocks.<br /> | and used them to expand the scale. The choices involving 10/9 are virtally automatic: between 9/8 and 5/4 we choose between 6/5 and 75/64, so we choose 6/5; between 3/2 and 5/3 we choose between 8/5 and 25/16, and so select 8/5. This gives us a natural nine-note extension which is more or less self-recommending, and which showed up in the survey of 9-note, 5-limit Fokker blocks as &quot;mavlim7&quot;, one of the 27/25&amp;135/128 blocks.<br /> | ||
<br /> | <br /> | ||
Splitting 9/8 and introducing the 7-limit is where it gets more interesting. Between 5/3 and 15/8 we have a choice between 7/4 and 25/14, and so of course we choose 7/4. There seems to be no clear reason to choose either 15/14 or 21/20 between 1 and 9/8, and so we may try both. Then between 4/3 and 3/2, we must decide between 7/5 and 10/7. But clearly 7/5 goes with 21/20, and 10/7 with 15/14, and we are done, having constructed the two 12-note &quot;Highschool&quot; scales.<br /> | Splitting 9/8 and introducing the 7-limit is where it gets more interesting. Between 5/3 and 15/8 we have a choice between 7/4 and 25/14, and so of course we choose 7/4. There seems to be no clear reason to choose either 15/14 or 21/20 between 1 and 9/8, and so we may try both. Then between 4/3 and 3/2, we must decide between 7/5 and 10/7. But clearly 7/5 goes with 21/20, and 10/7 with 15/14, and we are done, having constructed the two 12-note &quot;Highschool&quot; scales.<br /> | ||
<br /> | |||
Eventually this process breaks down (for on thing, 36 is both square and triangular) but it can be continued to larger scales. The next step involves breaking 15/14 as 15/14 = 25/24 * 36/35.<br /> | |||
<br /> | |||
Starting from the 12highschool1 scale, it isn't obvious what to do between 21/20 and 9/8, or between 7/5 and 3/2. But between 7/4 and 15/8, it's clear we should go with 9/5 = 36/35 * 7/4 rather than 175/96 = 25/24 * 7/4. Hence we pick 36/35 * 25/24 rather than the reverse in all three cases, so that they correspond, 15highschool1. Similar reasoning applies to 12highschool2, giving 15highschool2.<br /> | |||
<br /> | |||
At this point we might take note of the fact that 15highschool1 is a better scale from the point of view of harmony than 15highschool2, and is a highly recommendable 15-note, 7-limit JI scale.<br /> | |||
<br /> | |||
Continuing on, the next stage is to break apart 16/15 inside 15highschool1 by 16/15 = 28/27 * 36/35. Putting a 7/6 between 9/8 and 6/5, a 9/7 between 5/4 and 4/3 and a 14/9 between 3/2 and 8/5 are all obvious, leaving the question of what to do between 15/8 and 2. The two choices lead to the two scales 19highschool1 and 19highschool2, with 19highschool1 having the fuller harmony.<br /> | |||
<br /> | |||
Now we have three 21/20 intervals to break as 21/20 = 36/35 * 49/48 in 19highschool1 to get a 22-note scale. Two of them are not obvious, but it's clear we put 12/7 between 5/4 and 7/4, and this tells us how to break the other two, giving 22highschool.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scales</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scales</h1> | ||
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<a class="wiki_link" href="/15highschool2">15highschool2</a><br /> | <a class="wiki_link" href="/15highschool2">15highschool2</a><br /> | ||
<a class="wiki_link" href="/19highschool1">19highschool1</a><br /> | <a class="wiki_link" href="/19highschool1">19highschool1</a><br /> | ||
<a class="wiki_link" href="/19highschool2">19highschool2</a></body></html></pre></div> | <a class="wiki_link" href="/19highschool2">19highschool2</a><br /> | ||
<a class="wiki_link" href="/22highschool">22highschool</a></body></html></pre></div> | |||