Not So Wild A Dream

If one counts the septachords in a 31 symbol system (not neccessarily
31-tET) you get 32,000 different scales. This is based on having one
C, one D, etc.

Therefore,

4 X 5 X 4 X 4 X 5 X 5 X 4 = 32,000.

Now if one wishes to reduce for transposition, you need this grid.
The LH side shows how far one can go in a chain of fifhs, negative,
and the right side how far one can go in a chain of fifths,
positive. The rows are bb, b, natural, #, ##. Some are not allowed,
such as Cbb, Fbb, E## and B## (not one of the 31 symbols). One needs
to add the values on the left, plus those on the right, plus one
for "staying put" to get the number of transpositions that must be
divided out. The function is one row chosen for each column, that is,
cannot have Bb and B together.

Here is the grid

B E A D G C F | B E A D G C F
bb 0 ,1, 2, 3,4 *,* 26,27,28,29,30,*,*
b 5,6,7,8,9,10,11 19,20,21,22,23,24,25
nat 12,13,14,15,16,17,18 12,13,14,15,16,17,18
# 19,20,21,22,23,24,25 5,6,7,8,9,10,11
## * ,*, 26,27,28,29,30 *,*, 0,1,2,3,4

The normal scale, CDEFGAB, has 12 + 12 + 1 transpositions (25) which
is the highest. The lowest would be 1 transposition.

Question: How to program this to find the total number of scales,
reducing for transposition (key signature) in a 31 tone system. I
expect about 3,200 scales.

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> If one counts the septachords in a 31 symbol system (not
neccessarily
> 31-tET) you get 32,000 different scales. This is based on having
one
> C, one D, etc.
>
> Therefore,
>
> 4 X 5 X 4 X 4 X 5 X 5 X 4 = 32,000.
>
> Now if one wishes to reduce for transposition, you need this grid.
> The LH side shows how far one can go in a chain of fifhs, negative,
> and the right side how far one can go in a chain of fifths,
> positive. The rows are bb, b, natural, #, ##. Some are not
allowed,
> such as Cbb, Fbb, E## and B## (not one of the 31 symbols). One
needs
> to add the values on the left, plus those on the right, plus one
> for "staying put" to get the number of transpositions that must be
> divided out. The function is one row chosen for each column, that
is,
> cannot have Bb and B together.
>
> Here is the grid
>
>
> B E A D G C F | B E A D G C F
> bb 0 ,1, 2, 3,4 *,* 26,27,28,29,30,*,*
> b 5,6,7,8,9,10,11 19,20,21,22,23,24,25
> nat 12,13,14,15,16,17,18 12,13,14,15,16,17,18
> # 19,20,21,22,23,24,25 5,6,7,8,9,10,11
> ## * ,*, 26,27,28,29,30 *,*, 0,1,2,3,4
>
> The normal scale, CDEFGAB, has 12 + 12 + 1 transpositions (25)
which
> is the highest. The lowest would be 1 transposition.
>
> Question: How to program this to find the total number of scales,
> reducing for transposition (key signature) in a 31 tone system. I
> expect about 3,200 scales.
>
> PGH

Forgot to mention that you can only transpose the lowest value on
the LHS plus the lowest value on the RHS, plus one. For example,
all nat would be 12 (LHS) 12 (RHS) plus one.

PGH