observations on 12:7 (was: [tuning] u/otonality and major/minor)

> From: jpehrson2 <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, December 27, 2001 11:14 AM
> Subject: [tuning] Re: u/otonality and major/minor
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> /tuning/topicId_31809.html#31814
>
> >
> > The dichotomy becomes even more clear when we progress from
> > triads to tetrads. Traditional theory adds a "7th", continuing
> > to stack the "minor chord" by ascending "3rds" the same as with
> > the "major chord" -- so here the it's a "minor 7th" where with
> > the "major chord" it's a "major 7th". But the dualistic theory
> > continues the construction *downward* by adding a fourth
> > chord-identity *below* the utonal triad given above, thus:
> >
> > C Ab F D
> > 1/4 1/5 1/6 1/7
> >
> >
>
> ****So, essentially, that would turn into a *sixth* rather than a
> *seventh* yes??

Yup. If you invert the chord so that the 1/7 is at the top,
it forms an interval ~933.1290944 cents above the "root" 1/6.

In my JustMusic harmonic analysis notation, I'd write that:

1 n^0
-- = -------
7 2^0 * 7
8 2^3 * 1
10 2^1 * 5
12 2^2 * 3

Rewritten as ordinary ratios, from the top note down:

16/7 = 2/1 * 8/7
16/8 = 2/1
16/10 = 8/5
16/12 = 4/3

In other words, it's almost the same amount wider than
the 12-EDO "major 6th", as the 7:4 is narrower than
the 12-EDO "minor 7th".

In fact, the difference between these two differences is
exactly the same as the amount of tempering of the 3:2 in
12-EDO, a tiny unit of interval measurement known as a "grad".
http://www.ixpres.com/interval/dict/grad.htm

I found this coincidence interesting, so I pursued it
further for this post:

Where
"a" = excess of 12:7 "septimal major 6th" over 12-EDO
"b" = deficit of 7:4 "septimal minor 7th" under 12-EDO

(3/2) / ( 2^(7/12) ) = a - b

[Note: "septimal minor 7th" is more frequently called
"harmonic 7th".]

PROOF:

(3/2) / ( 2^(7/12) )

= [-12/12 1] 3:2 ratio = Pythagorean "perfect 5th"
- [ 7/12 0] 12-EDO "perfect 5th"
-------------
[ 19/12 1] = ~1.955000865 cents = 1 grad

a = ( (16/7) / (4/3) ) / ( 2^(9/12) )

(16/7) / (4/3)
=
[ 4 0 0 -1] 8:7 ratio + "8ve" = "septimal 9th"
- [ 2 -1 0 0] 4:3 ratio = Pythagorean "perfect 4th"
---------------
[ 2 1 0 -1] = [ 24/12 1 0 -1] 12:7 = "septimal M6th"
- [ 9/12 0 0 0] 12-EDO "major 6th"
-------------------
[ 15/12 1 0 -1]

= [ 5/4 1 0 -1] = ~33.1290944 cents

b = ( 2^(10/12) ) / (7/4)

= [ 10/12 0 0 0] 12-EDO "minor 7th"
- [-24/12 0 0 1] 7:4 ratio = "septimal minor 7th"
-------------------
[ 34/12 0 0 -1]

[ 17/6 0 0 -1] = ~31.17409353 cents

a - b =

[ 15/12 1 0 -1] a
- [ 34/12 0 0 -1] b
-------------------
[ 19/12 1 0 0] = ~1.955000865 cents = 1 grad

I would love to see what this looks like in elegant algebra.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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