a 3-Voci, doublestacked necklaces

Below are the interval vectors for I,ii,iii,IV,V,vi against
I,ii,iii,IV,V and vi. (I am working on all white-key trichords
and also the full set of trichords (double-stacked necklaces))

Here, there are only 8 unique ones. I've labelled the first appearance
of each one. (In this case it makes sense that, for example, I-IV and
I-V and ii-vi and iii-vi all have the same vectors. There are no Z-
relations)

3,0,0,2,2,2,0 I-I
0,1,4,1,0,3,0 I-ii
2,1,0,2,2,2,0 I-iii
0,1,4,1,0,2,1 ii-iii
1,1,2,1,1,3,0 I-IV
0,2,3,0,1,2,1 iii-IV
1,0,3,2,0,2,1 ii-V
2,0,1,2,2,2,0 I-vi

This is so easy I could have put it on "tuning", but since I am doing
the others here I put it here also. So these are the energy states
between chords an ordinary garage-band might use, (power chords!) for
what it's worth.

It does show a little of the voice-leading potentialities, I guess.
I don't know if "energy-states between two necklaces" really
carries over from physics into music, that would be a question for
a psychoacoustician. EIS A045612 only considers cases where
the number of "positive and negative charges" are the same on the
two necklaces so I am already stretching things a bit.

-- PHj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> Below are the interval vectors for I,ii,iii,IV,V,vi against
> I,ii,iii,IV,V and vi. (I am working on all white-key trichords
> and also the full set of trichords (double-stacked necklaces))
>
> Here, there are only 8 unique ones. I've labelled the first
appearance
> of each one. (In this case it makes sense that, for example, I-IV
and
> I-V and ii-vi and iii-vi all have the same vectors. There are no Z-
> relations)
>
> 3,0,0,2,2,2,0 I-I
> 0,1,4,1,0,3,0 I-ii
> 2,1,0,2,2,2,0 I-iii
> 0,1,4,1,0,2,1 ii-iii
> 1,1,2,1,1,3,0 I-IV
> 0,2,3,0,1,2,1 iii-IV
> 1,0,3,2,0,2,1 ii-V
> 2,0,1,2,2,2,0 I-vi
>
> This is so easy I could have put it on "tuning", but since I am
doing
> the others here I put it here also. So these are the energy states
> between chords an ordinary garage-band might use, (power chords!)
for
> what it's worth.
>
> It does show a little of the voice-leading potentialities, I guess.
> I don't know if "energy-states between two necklaces" really
> carries over from physics into music, that would be a question for
> a psychoacoustician. EIS A045612 only considers cases where
> the number of "positive and negative charges" are the same on the
> two necklaces so I am already stretching things a bit.
>
> -- PHj

What the last sentence means is that EIS A045612 considers 2n elements
on each necklace, n of one set and n of its complement. (On both
necklaces). It is easy to show that the energy state (interval vector)
between n1 and n2 is the same as the one comparing n1' and n2', (even
when n1 and n1' are z-related.) So counts for A045612 combine a set
and its complement into a single count.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> Below are the interval vectors for I,ii,iii,IV,V,vi against
> I,ii,iii,IV,V and vi. (I am working on all white-key trichords
> and also the full set of trichords (double-stacked necklaces))
>
> Here, there are only 8 unique ones. I've labelled the first
appearance
> of each one. (In this case it makes sense that, for example, I-IV
and
> I-V and ii-vi and iii-vi all have the same vectors. There are no Z-
> relations)
>
> 3,0,0,2,2,2,0 I-I
> 0,1,4,1,0,3,0 I-ii
> 2,1,0,2,2,2,0 I-iii
> 0,1,4,1,0,2,1 ii-iii
> 1,1,2,1,1,3,0 I-IV
> 0,2,3,0,1,2,1 iii-IV
> 1,0,3,2,0,2,1 ii-V
> 2,0,1,2,2,2,0 I-vi
>
> This is so easy I could have put it on "tuning", but since I am
doing
> the others here I put it here also. So these are the energy states
> between chords an ordinary garage-band might use, (power chords!)
for
> what it's worth.
>
> It does show a little of the voice-leading potentialities, I guess.
> I don't know if "energy-states between two necklaces" really
> carries over from physics into music, that would be a question for
> a psychoacoustician. EIS A045612 only considers cases where
> the number of "positive and negative charges" are the same on the
> two necklaces so I am already stretching things a bit.
>
> -- PHj

Hi Paul . . . I'm listening.

Now, what is the matrix above? Easy as it may be, I'm afraid I don't
understand it.

-Paul

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > Below are the interval vectors for I,ii,iii,IV,V,vi against
> > I,ii,iii,IV,V and vi. (I am working on all white-key trichords
> > and also the full set of trichords (double-stacked necklaces))
> >
> > Here, there are only 8 unique ones. I've labelled the first
> appearance
> > of each one. (In this case it makes sense that, for example, I-IV
> and
> > I-V and ii-vi and iii-vi all have the same vectors. There are no
Z-
> > relations)
> >
> > 3,0,0,2,2,2,0 I-I
> > 0,1,4,1,0,3,0 I-ii
> > 2,1,0,2,2,2,0 I-iii
> > 0,1,4,1,0,2,1 ii-iii
> > 1,1,2,1,1,3,0 I-IV
> > 0,2,3,0,1,2,1 iii-IV
> > 1,0,3,2,0,2,1 ii-V
> > 2,0,1,2,2,2,0 I-vi
> >
> > This is so easy I could have put it on "tuning", but since I am
> doing
> > the others here I put it here also. So these are the energy states
> > between chords an ordinary garage-band might use, (power chords!)
> for
> > what it's worth.
> >
> > It does show a little of the voice-leading potentialities, I
guess.
> > I don't know if "energy-states between two necklaces" really
> > carries over from physics into music, that would be a question for
> > a psychoacoustician. EIS A045612 only considers cases where
> > the number of "positive and negative charges" are the same on the
> > two necklaces so I am already stretching things a bit.
> >
> > -- PHj
>
> Hi Paul . . . I'm listening.
>
> Now, what is the matrix above? Easy as it may be, I'm afraid I
don't
> understand it.
>
> -Paul

Each line is an interval vector of two stacked triads (necklaces).
Another way of saying it: The different sets of distances between two
stacked triads. For example, I-ii is all the distances between
C-E-G and D-F-A. There are 9. (C to D, C to F, C to A, E to D, E
to F, E to A, G to D, G to F and G to A) I'm afraid this is all
a little silly compared to all the advanced stuff that is done
on this group, but it keeps me busy. Not really a matrix, more
a set of "vectors."

-Paul Hj