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Tuesday, January 14, 2014

Ernst Bacon Redux (3)


Continued from Ernst Bacon Redux (2)

Here is Bacon's "Table III; N=4" from "Our Musical Idiom" tracing his "calculation" of tetrachords:


Take the eight "combinations" (listed in the left-most column as 2,3,4,6,9,10,12,14). These all have one thing in common relative to this discussion. Each of them represents an interval string (our terminology, not Bacon's) that contains two intervals that are the same along with two intervals that are different – e.g., C.2 represents all interval strings (in 12tET) that are made up of two 1's (minor seconds), one 2 (major second), and one 8 (minor sixth). The H column tells us that there are just three distinct "harmonies" (cycle-class interval strings) that can be made from those intervals by permutation. Similarly, C.3 shows that there are three "harmonies" that can be derived from permutations of 1,1,3,7. And the same idea follows for C.4,6,9,10,12,14. All 8 of these represent solutions of 2a + b + c = 12.

Here is how the algorithm introduced in the previous post (which can be viewed as a "completion" of Bacon's method) gets to all 43 (Tn) "harmonies" in Z12:

1. List the partitions of 4:
  1.          4
  2.         31
  3.         22
  4.       211
  5.     1111
2. Generate equations from step 1:
  1.                4a = m
  2.         3a + bm
  3.        2a + 2b = m
  4.    2a + b + m
  5.b + c + dm
3. Pull out the equivalence classes (distinct cyclic permutations) from step 2 (or call from a template "library" of k-length cyclic permutation classes, not shown here):
  1. (a,a,a,a
  2. (a,a,a,b
  3. (a,b,b,a⤸, (a,b,a,b
  4. (a,a,b,c⤸, (b,a,a,c⤸, (a,b,a,c
  5. (a,b,c,d⤸, (c,b,a,d⤸, (b,c,a,d⤸,(a,c,b,d⤸,(c,a,b,d⤸,(b,a,c,d
4. Set m=12 & list solutions to step 2:
  1. (3)
  2. (1,9),(2,6)
  3. (1,5),(2,4)
  4. (1,2,8),(1,3,7),(1,4,6),(2,1,7),(2,3,5),(3,1,5),(3,2,4),(4,1,3)
  5. (1,2,3,6),(1,2,4,5)
5. Substitute results of step 4 into the templates listed in step 3:
  1. 3333
  2. 1119 2226
  3. 1551 1515 2442 2424
  4. 1128 2118 1218 1137 3117 1317 1146 4116 1416 2217 1227 2127 2235 3225 2325
      3315 1335 3135 3324 2334 3234 4413 1443 4143
  5. 1236 3216 2316 1326 3126 2136 1245 4215 2415 1425 4125 2145

The most significant step in the algorithm is no. 3, because once you have worked it out for one chromatic universe (the template library alluded to), it is the same for any other. While the list of strings grows dramatically as m increases (the number of partitions of m increase as m grows larger), the number of templates in step 3 remains the same because k, the length of any string in the list, is constant. 

Eliminating mirror-equivalent strings from step 3 will result in the usual TnI set-class list. For the results of setting k=4 and running m from 4 to 24, see "Tetrads mod 4 through 24."