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There are 19 trichords (12 prime forms and 7 inversions), and 110 possible three note combinations of notes from any note (11*10=110). Because a trichord has three notes, it has three inversions (in the classical meaning of inversion), and the only trichord that doesn't have an inversion is [0,4,8] (augmented triad), we divide 109/3=36.3, 36.3+1=37.3 (the augmented triad), and 37.3=/=19.

What is the problem with my calculations?

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110 possible three note combinations of notes from any note (11*10=110).

I think you are double counting (or more likely, multiple-counting). If you are saying "if the first note is fixed, say C, there are 11 choices for the second note and 10 for the third" you are counting C-D-E and C-E-D as different "three note combinations" from C, which seems wrong.

Also, choosing C-Bb-D would give the same trichord as Bb-C-D, which is the same as C-D-E.

I suspect your "dividing by 3" idea is also illogical, but given the other problems, it's hard to tell.

A better way to attack the counting exercise might be to work in terms of the intervals (number of semitones) between the notes. The three intervals (notes 1 to 2, 2 to 3, and 3 to an octave above 1) must always sum to 12, and you can then make the trichords unique by putting the smallest interval first.

The 19 sets of intervals are

1 1 10, 1 2 9, ..., 1 9 2 (9 sets)
2 2 8,  2 3 7, ..., 2 7 3 (6 sets)
3 3 6,  3 4 5, 3 5 4      (3 sets)
4 4 4                     (1 set)

Not counting inversions, the 12 sets of intervals are

1 1 10, 1 2 9, ....., 1 5 6 (5 sets)
2 2 8,  2 3 7, 2 4 6, 2 5 5 (4 sets)
3 3 6,  3 4 5               (2 sets)
4 4 4                       (1 set)
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  • So there are 55 (10+9+8+7+6+5+4+3+2+1=55) options. 55-1 (augmented) = 54 which /3 = 18 +1 (augmented triad) = 19, so it makes sense now. Thank you! – user38047 Mar 27 '17 at 13:55
  • Just for reference, the actual mathematical statement for this is 11 choose 2 (11c2). Since one note is fixed you subtract 1 from the pool that you are choosing from and the number you are choosing. – Dom Mar 27 '17 at 14:52

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