Re: Diatonic notation system

[Hans Aberg]

On 7 Dec 2008, at 12:31, Graham Breed wrote:

> > > The only 12-ness is in the tuning of MIDI files, which I think  
> > > you can
> > > change to get extended meantone.
> >
> > The implementations in Turkish music seemed to force a multiple of  
> > 12,
> > whereas it should be E53.
>
> Ah.  Well, that's one configuration file, which you can change.  The
> tunings are expressed as a fraction of a whole tone but I don't think
> there's a limit on the precision.

That is not the problem , but that whole tone is not the double of  
the half tone. M = 9, m = 4, so that there are 5*M+2*m = 53 tonesteps  
or commas in an octave.

The same problem is in meantone tunings, in E31, M = 5, m = 3, so  
that M/m = 5/3.

> > > What do you mean by "a diatonic
> > > notation system"?
> >
> > It is what yo get by adjoining to the notes A B C D E F G,  
> > octaves, sharps
> > and flats. When reducing in octaves, this is a finite system  
> > exactly when it
> > is an n-equal temperament. The underlying mathematical structure  
> > is simple:
> >
> > It is the set of pitches that is generated by p m + q M, where p  
> > and q are
> > integers, and m (resp. M) is a minor (resp. major) (abstract)  
> > second. Sharps
> > (resp. flats) raise (resp. lower) with the amount M - m.
>
> I think you can do that in the configuration files.  I haven't tried
> it yet, mind.  I think you have to specify every note you might use in
> advance. At least that's the way I've always seen it.  Check that
> Turkish example file.  I don't think the nominals have to start in
> equal temperament.

When I looked at it, it computes accidentals as an offset from the  
whole-tones, which are six to the octave. That forces E12, and for  
intermediate pitches a multiple of 12.

> It's all Scheme code.

And particular part was not alterable by Scheme code.

> There would be a problem in using a notation system with more than 7
> nominals to the octave.  That's hard-wired.

This, too. Though rare, one is experimenting with more the 7-tones to  
the octave, and the model I gave applies to that as well.

> (I'm talking about the development branch here.  There's a different
> system for accidentals in the stable release.  But you can't be
> looking at that because it doesn't have the Turkish accidentals,
> right?)

Right, and I looked at it at the beginning o this year.

> > When choosing values for m and M, and a tuning frequency, one gets  
> > a tuning
> > of the notated music; one of them suffices if the octave is  
> > assumed. For
> > example, one extended meantone tuning popular during the  
> > Renaissance uses M
> > = sqrt(5/4).
> >
> > One can also describe intermediate pitches on top of this. It  
> > works with
> > oriental music, I think this is what the Sagittal microtonal  
> > notation system
> > notation does.
>
> Sagittal is a big set of accidentals that can have different meanings
> in different contexts.

Yes, that would be the very point with my suggestion.

> I expect there'd need to be a different
> configuration file for each of those contexts when it eventually
> works.

Yes, like one would have to define M7m = 2 to get E12.

> As Sagittal isn't working we are a bit short of sharp symbols.  There
> are some arrowed accidentals on the way, though.

This would be the long haul.

A more immediate concern would be fixing Arab music. The guy who didi  
it fixed it at E24, because they use such symbols, but everyone  
agrees that the intermediate pitch is not an exact quarter-tone.

Now that fixation is not only a problem for MIDI files, but may cause  
transposition problems, as a half-flat may be erroneously altered to  
a half-sharp on the semi-tone below.

  Hans