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 triangular 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 numerator to two 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.

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.