Highschool scales: Difference between revisions

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When [[~~Gene_Ward_Smith|~~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:

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.

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Now starting with the famous Ptolemy-Zarlino JI diatonic:

9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2

: 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)

* 9/8 = 15/14 * 21/20 (square numerator to two triangular)

* 10/9 = 16/15 * 25/24 (triangular numerator to two square)

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.

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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 ==

~~[[9highschool|9highschool]]~~

~~[[10highschool1|10highschool1]]~~

~~[[10highschool2|10highschool2]]~~

~~[[12highschool1|12highschool1]]~~

~~[[12highschool2|12highschool2]]~~

~~[[12highschool3|12highschool3]]~~

~~[[12highschool4|12highschool4]]~~

~~[[15highschool1|15highschool1]]~~

~~[[15highschool2|15highschool2]]~~

~~[[19highschool1|19highschool1]]~~

[[19highschool2~~|19highschool2~~]]

* [[9highschool]]

* [[10highschool1]]

* [[10highschool2]]

* [[12highschool1]]

* [[12highschool2]]

* [[12highschool3]]

* [[12highschool4]]

* [[15highschool1]]

* [[15highschool2]]

* [[19highschool1]]

* [[19highschool2]]

* [[22highschool]]

~~[[22highschool|22highschool]]~~ [[Category:7-limit]]

[[Category:7-limit]]

[[Category:highschool]]

[[Category:~~just~~]]

[[Category:Just intonation scales]]

@@ Line 1: / Line 1: @@
-When [[Gene_Ward_Smith|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:
+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]
+ S[n] = T[2n-1] * T[2n]
-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.
@@ Line 9: / Line 8: @@
 Now starting with the famous Ptolemy-Zarlino JI diatonic:
-/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2
+: 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
-/8 = 15/14 * 21/20 (square numerator to two triangular)
+* 9/8 = 15/14 * 21/20 (square numerator to two triangular)
+* 10/9 = 16/15 * 25/24 (triangular numerator to two square)
-/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&amp;135/128 blocks.
@@ Line 31: / Line 29: @@
 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 ==
-[[9highschool|9highschool]]
-[[10highschool1|10highschool1]]
-[[10highschool2|10highschool2]]
-[[12highschool1|12highschool1]]
-[[12highschool2|12highschool2]]
-[[12highschool3|12highschool3]]
-[[12highschool4|12highschool4]]
-[[15highschool1|15highschool1]]
-[[15highschool2|15highschool2]]
-[[19highschool1|19highschool1]]
-[[19highschool2|19highschool2]]
+* [[9highschool]]
+* [[10highschool1]]
+* [[10highschool2]]
+* [[12highschool1]]
+* [[12highschool2]]
+* [[12highschool3]]
+* [[12highschool4]]
+* [[15highschool1]]
+* [[15highschool2]]
+* [[19highschool1]]
+* [[19highschool2]]
+* [[22highschool]]
-[[22highschool|22highschool]]      [[Category:7-limit]]
+[[Category:7-limit]]
 [[Category:highschool]]
-[[Category:just]]
+[[Category:Just intonation scales]]