Highschool scales: Difference between revisions

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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.

Eventually this process breaks down (for one 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.

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[[Category:highschool]]

[[Category:Just intonation scales]]

~~[[Category:Pages with mostly numerical content]]~~

@@ Line 19: / Line 19: @@
 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.
+Eventually this process breaks down (for one 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.
@@ Line 47: / Line 47: @@
 [[Category:highschool]]
 [[Category:Just intonation scales]]
-[[Category:Pages with mostly numerical content]]