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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be happy and free from suffering :)
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