Highschool scales
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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 triagular numerator n(n+1)/2, then we have the following relationships: T[n] = S[n] * S[n+1] S[n] = T[2n-1] * T[2n] Using these, we can break apart every scale step with a square numerator into two with trangular numerators, and every step with a triangular numerator into two with square numerators. Now starting with the famous Ptolemy-Zarlino JI diatonic: 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2 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 denominator to two triangular) 10/9 = 16/15 * 25/24 (triangular denominator 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. 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. =Scales= [[9highschool]] [[10highschool1]] [[10highschool2]] [[12highschool1]] [[12highschool2]] [[12highschool3]] [[12highschool4]] [[15highschool1]] [[15highschool2]] [[19highschool1]] [[19highschool2]]
Original HTML content:
<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 triagular numerator n(n+1)/2, then we have the following relationships:<br /> <br /> T[n] = S[n] * S[n+1]<br /> S[n] = T[2n-1] * T[2n] <br /> <br /> Using these, we can break apart every scale step with a square numerator into two with trangular numerators, and every step with a triangular numerator into two with square numerators.<br /> <br /> Now starting with the famous Ptolemy-Zarlino JI diatonic:<br /> <br /> 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2<br /> <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 /> 9/8 = 15/14 * 21/20 (square denominator to two triangular)<br /> <br /> 10/9 = 16/15 * 25/24 (triangular denominator to two square)<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 "mavlim7", one of the 27/25&135/128 blocks.<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 "Highschool" scales.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scales</h1> <a class="wiki_link" href="/9highschool">9highschool</a><br /> <a class="wiki_link" href="/10highschool1">10highschool1</a><br /> <a class="wiki_link" href="/10highschool2">10highschool2</a><br /> <a class="wiki_link" href="/12highschool1">12highschool1</a><br /> <a class="wiki_link" href="/12highschool2">12highschool2</a><br /> <a class="wiki_link" href="/12highschool3">12highschool3</a><br /> <a class="wiki_link" href="/12highschool4">12highschool4</a><br /> <a class="wiki_link" href="/15highschool1">15highschool1</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="/19highschool2">19highschool2</a></body></html>