Adjacencies

If one takes all 924 possible hexachords, the counts for
both semitones (adjacencies)/major sevenths and perfect
fourths/perfect fifths are as follows:

0
60
600
1200
600
60
0

This is for frequency=0 through 6, and it is weighted. Unweighted
the counts would be 2, 60, 300, 400, 150, 12, and 0. "2" gets
obliterated.

These add to 2520. I don't know why they are balanced. I have been
looking at prime sets, such as triads in 7-tET and pentachords in 11t-
ET, these give the same counts for every interval. Sets such as
tetrachords in 9t-ET give the same count for every interval that is
relatively prime to 9, that is 1,2 and 4 (and 5,7 and 8).
interval-3 behaves differently.

So, I know that modes of limited transposition, symmetry,
complementability all have their influence on interval vector
value frequencies. Now it looks like the interval vector value
itself (if it is relatively prime to 12, say) determines something
about these same interval vector value frequencies.

Now you're all saying, Paul, what does this have to do with tuning?
Well, I don't know yet, but intervals are definitely based on
tuning, and of course, in 12t-ET, they are <1,2,3,4,5,6| or perhaps
<13,14,15,16,17,18| is a better mapping. (Is this how we map
intervals?)

I might have to rope in my Statistics-Ph.D. relative to push this
any further. Interval vectors-> possiblities-> quantum physics->
random Hermitian matrices->RZF? Doubt it.

Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> If one takes all 924 possible hexachords, the counts for
> both semitones (adjacencies)/major sevenths and perfect
> fourths/perfect fifths are as follows:
>
> 0
> 60
> 600
> 1200
> 600
> 60
> 0
>
> This is for frequency*=0 through 6, and it is weighted. Unweighted

* Should have said value=0 through 6, even though it is a frequency
of an interval. . .

> the counts would be 2, 60, 300, 400, 150, 12, and 0. "2" gets
> obliterated.
>
> These add to 2520. I don't know why they are balanced. I have been
> looking at prime sets, such as triads in 7-tET and pentachords in
11t-
> ET, these give the same counts for every interval. Sets such as
> tetrachords in 9t-ET give the same count for every interval that is
> relatively prime to 9, that is 1,2 and 4 (and 5,7 and 8).
> interval-3 behaves differently.
>
> So, I know that modes of limited transposition, symmetry,
> complementability all have their influence on interval vector
> value frequencies. Now it looks like the interval vector value**
> itself (if it is relatively prime to 12, say) determines something
> about these same interval vector value frequencies.

** Should be just "interval class", otherwise it looks like I am
talking about values relatively prime to 12...

>
> Now you're all saying, Paul, what does this have to do with tuning?
> Well, I don't know yet, but intervals are definitely based on
> tuning, and of course, in 12t-ET, they are <1,2,3,4,5,6| or perhaps
> <13,14,15,16,17,18| is a better mapping. (Is this how we map
> intervals?)
>
> I might have to rope in my Statistics-Ph.D. relative to push this
> any further. Interval vectors-> possiblities-> quantum physics->
> random Hermitian matrices->RZF? Doubt it.
>
> Paul Hj
>