On Wednesday 08 February 2006 6:28 pm, Hudson Lacerda wrote:
> For an equal tempered scale, one can:
>
> - Define the tuning for the base pitches C D E F G A B (pitches C and c
> are equivalent mod N, but N doesn't have to be the 2/1 octave);
>
> - Define the number of steps included in a whole tone (C to D); the
> number of steps involved in a specific accidental can be represented by
> its ratio numerator.
Hi Hudson,
The above assumes a regular meantone-type ET (or EDO as they are sometimes
called). Sometimes, as in a non-meantone ET like 22-tet or 53-tet, there are
two sizes of major 2nd--one for the 9/8 approximation, the other, a smaller
major 2nd, representing 10/9.
So, the above only works for temperaments that temper out the 81/80 syntonic
comma (i.e. meantone-family temperaments)
Best,
Aaron.
> - Assing a numerical identifier to the scale: the denominator of
> accidental ratios.
>
> All operations above should be done inside %%pseudo-comments or
> I:commands (or [I:command]s).
>
> Then, we can obtain a notation like this, for 3 steps in a whole tone
> (e.g. 19ET), and using the ratio numerator for the steps:
>
> =C ^1/C ^2/C =D _1/ _2/D =C
>
> To use several scales simultaneously, one can use the denominator as a
> scale identifier (note that derivations of = have a meaning here):
>
> =/19C ^3/24C ^2/48C ^2/31C
>
> Therefore, a default scale has to be set for a section/voice of music.
> (BTW, abcm2ps default denominator is 2 in ^/ or ^3/ accidentals.)
>
> -----
>
> Problems
>
> - A problem with the suggestion above is that it is not compatible with
> occasional intonation changes, or re-divisions of the current scale,
> like in the current abc2midi implementation. They should be represented
> using floats in the numerator part (e.g ^0.5/19C), but I am fear this
> rather complex notation can be prone to syntax inconsistencies and
> ambiguities. (What do you think of that?)
>
> - Irregular scales are not well defined with that generic proposal. Each
> scale degree should be explicitly defined, e.g.: =C is 1/1, ^1/C is 60
> cents above, ^2/C is 126 cents above, =D is 188 cents above and so on.
>
> - The definition of the accidental glyphs is package dependent: abcm2ps
> uses special (user-defined) postscript operators, but what about other
> software pieces? How to use special fonts (for instance) in a portable way?
>
> - The suggestion is somewhat complex (perhaps too complicated?).
>
> Cheers,
> Hudson
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