I just remembered this Julian Hook article on Music Theory Online, if you want to find out more about how to find the number of chords of a given size:
http://www.mtosmt.org/issues/mto.07.13.4/mto.07.13.4.hook.html
On Mon, Nov 2, 2015 at 3:00 PM, William Huston williamahuston@gmail.com wrote:
Matt suggested I forward this side discussion (and attachment) to the list.
N.B., Matt is referring by number to the list of 55 intervals I identified in my original, quoted below. However, it was an HTML list, and the numbering got munched during quoting. Sorry if that makes this slightly unclear. --BH
---------- Forwarded message ---------- From: Matt Barber brbrofsvl@gmail.com Date: Monday, November 2, 2015 Subject: chord libraries? To: William Huston williamahuston@gmail.com
We also eliminate transpositions, so in this case 55 is the same chord as 1, 18 and 12 are the same, and so forth. If we eliminate all transpositions and inversions, we end up with 12 trichords: 012* 013 014 015 016 024* 025 026 027* 036* 037 048* I marked the 5 that are self-inversional with a star. They're all "musical," but again it depends on the type of music you're looking at. All 19 trichords (eliminating transpositions and respacing) have been in use as simultaneities since the late 1800s, but some more than others. 012 didn't get a huge foothold until the first decade or two of the 20th century. See the attached; it does eliminate inversions, but it wouldn't be that hard to make that a user preference. [list-setclass] outputs the normal form as I described above, and the interval-vector one outputs a catalog of the chromatic intervals contained in the chord (eliminating unisons, octaves and inversions). In mod 12 there are 6 chromatic intervals, counting minor seconds and major sevenths as the same -- the output just gives you the number of each interval in the chord from 1 to 6 (or whatever if you choose a different modulus). Matt PS -- if you like, you can bump this response up to the list. On Mon, Nov 2, 2015 at 12:28 PM, William Huston williamahuston@gmail.com wrote:
Thanks Matt. Yes interested.
FTR, if we do not eliminate inversions, I count 55 3-tone chords in
12TET:
111000000000 110100000000 110010000000 110001000000 110000100000 110000010000 110000001000 110000000100 110000000010 110000000001 101100000000 101010000000 101001000000 101000100000 101000010000 101000001000 101000000100 101000000010 101000000001 100110000000 100101000000 100100100000 100100010000 100100001000 100100000100 100100000010 100100000001 100011000000 100010100000 100010010000 100010001000 100010000100 100010000010 100010000001 100001100000 100001010000 100001001000 100001000100 100001000010 100001000001 100000110000 100000101000 100000100100 100000100010 100000100001 100000011000 100000010100 100000010010 100000010001 100000001100 100000001010 100000001001 100000000110 100000000101 100000000011
But roughly 2/3's are inversions, so if we exclude inversions as being
the same chord, then yes we get 55*.3333 =~ 19. I'm guessing at least one of these is an inversion of itself which is why 55 is not evenly divisible by 3.
Anyway, I'm not sure how many of these are really musical. I would think
of those 19, only about 8-10 are commonly used.
Thanks, interested in whatever you have :)
BH
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