Re: meantoning of non-meantone temperments

> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
> Subject: Reply to Robert C Valentine
>
I had written :
>
> > "given an equal tempterment, can I write a program that will come up
> > with a reasonable, quasi-meantone treatment".
>
> >For instance, type in "17", and get out
>
> > major scale : 3 3 1 3 3 3 1
>
>

To which Paul replied :

> Yuck! Have you heard the triads this produces?
>

Nope, but in my spreadsheet, it looks like the major triad will have a
good deal of 'snarl'. The point was really in the following paragraph.

"""""
The idea being that transposition and alterations used in 12 TET would
have a mathematically consistant mapping in the new temperment (i.e.,
sharping the 'F' will produce a G scale that is a transposed version of
the C scale).
"""""

The algorithm works great for quasi-meantone temperments, 19, 26, 31 etc...
These were the ones I was most interested in, and I wanted to make sure that
the same common music text file would get the correct notes in the new
temperment.

You are correct that applying these rules to other temperments
will produce some 'interesting' cases (hold your nose AND your ears).

FOr instance, the algorithm I adopted, given a request for 21 tone equal,
determines that

whole step = 5
half step = -2
sharp = 7 flat = -7

and C D E F G A B (C) = 0 5 10 8 13 18 23 (21)

Add an F# (8->15) and sure enough, starting on G produces a transposed
sequence.

Along these lines, I attempted the same thing with scales of the type

l m s l m l s (for instance 53TET 9 8 5 9 8 9 5, or 22Tet 4 3 2 4 3 4 2,
or 17TET 3 2 2 3 2 3 2).

This immediately showed some of the problems with "non-meantone" tunings.
The same goal (consistent trasposition) immediately falls apart. What the
algorithm WANTS to do is create a vector for the sharps and flats. For
instance, F# in 22TET wants to be (0, 0, 2, 0, 1, 0, 0) to move key of C
to key of G. In two or three transpositions, the original root starts to
drift. Non-key alterations produce bizarre results (for instance, minor
tonality seems to want a different system).

Bob Valentine