Re: [tuning-math] Glumma
- From: Carl Lumma <carl@xxxxxxxxx>
- To: tuning-math@xxxxxxxxxxxxx
- Date: Sat, 13 Jul 2002 12:18:38 -0700
>The six complete tetrads in cubic lattice notation form a 2x3 rectangle:
>
>[0 -1 0] [0 0 0]
>[-1 -1 0] [-1 0 0]
>[-2 -1 0] [-2 0 0]
There you are, Gene!
I meant:
show locations 4:5:6:7
--> Ack! You can't cut and paste from Scala!
"
Locations of 5/4 3/2, 7/4
4-7-10: 0-4-7-10
4-7-10: 2-6-9-12
4-8-10: 5-9-13-15
"
So we see that two distinct scale-degree patterns are needed to
generate the 6 tetrads, and that the scale (like stairs) is not
proper. However, we don't care about propriety from a melodic
standpoint for a 12-note scale, so as long as the pattern of
consonances isn't too crazy...
The only other "bad" thing about this scale is that it only has
two 3:2s. Is there another rotation with more?
>The rule is that the chord [a b c] is major (otonal) if a+b+c is
>even, and minor if it is odd. A major chord has root 3^((-a+b+c)/2)
>5^((a-b+c)/2) 7^(a+b-c)/2), and a minor chord root 3^((-a+b+c-1)/2)
>5^((a-b+c+1)/2) 7^((a+b-c+1)/2). This means that [0 0 0] and [-1 0 0]
>are the major and minor tonic tetrads, and [-1 -1 0] and [-2 -1 0]
>are the major and minor tetrads on 8/7. [0 -1 0] is the minor tetrad
>with root 5/3, and [-2 0 0] the major tetrad with root 48/35.
This is apparently something that would be seen after about .5 seconds
of looking at the lattice.
-Carl
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