[PD] Fwd: chord libraries?

[William Huston]

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 at gmail.com>
Date: Monday, November 2, 2015
Subject: chord libraries?
To: William Huston <williamahuston at 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 at 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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May you, and all beings
be happy and free from suffering :)
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