Re: Diatonic notation system

[Hans Aberg]

On 7 Dec 2008, at 14:59, Graham Breed wrote:

> > 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.
>
> How is that a problem?

The rules for computing accidentals and transpositions will be  
different, and it will sound differently (see below).

> > The same problem is in meantone tunings, in E31, M = 5, m = 3, so  
> > that M/m =
> > 5/3.
>
> I still don't see a problem.

Same here: m is enlarged to about 116 cents, so that the major third  
closes into the rational interval 5/4. Then F# and Gb differ by a 31- 
ET tonestep (see below).

> > 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.
>
> Oh, this looks like a problem.

And this is what causes the problems. If done properly, the offsets  
should be computed properly, it should be done towards a scale that  
also involves m, as only in E12, there are six M in the octave.

> I don't see a way of defining the
> nominals after all.  The documentation for ly:make-pitch talks about
> "pitch C" and "pitch B" without saying what they're supposed to be
> tuned to.  So you need to define a Scheme function that takes a
> rational instead of an integer for the number of scale steps and sets
> "alter" accordingly.  And probably a Scheme function to set up 7
> definitions of each accidental to take account of the 7 different
> nominals they could be applied to.

I think that all is defined in term of relation to 6 M's in the octave.

> There's support for ancient
> notation so surely it must be easier than this to define historical
> tunings...?

Yes, and that is essentially what I am saying. The Western musical  
notation system is diatonic in the sense I described it. This is very  
clever, because it then work, regardless if it is Pythagorean, E12 or  
meantone tunings used: these latter are just differently intonations  
of the music.

So I think this system of defining an interval by a number and then  
looking up accidentals does not work. Instead, the pitches should be  
recorded as written, imposing any ET relations at need. But even  
then, it is common to stick to diatonic notation conventions, and  
only apply E12 conventions selectively.

Even chromatic runs may be notated in E12, but in a meantone tuning,  
that is selectively choosing flats and sharp to make up 12 in the  
octave. This is the difference between the 12-note "meantone", and  
the in general infinite "extended meantone" tunings.

> Yes, the intermediate pitches are specified as fractions of a whole
> tone.

For music without intermediate pithes, as integer pairs (p, q),  
referring to the abstract combination p m + q M.

For music with intermediate pitches, one extends this to sequences  
(p, q, r_1, ..., r_s), referring to the abstract interval combination  
p m q M + r_1 n_1 + r_s n_s.

> How else do you suggest that be done?  You can write a function
> that converts from your chosen units into fractions of a whole tone
> with whatever precision you choose.

Only if one knows m and M in advance, and it will generally destroy  
the musical structure.

> > 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.
>
> Could it?  Define the "half-flat" as a bit less than half a  
> semitone, then.

The problem is that one does not agree on what it should be, and the  
music may drawn towards E53 in some interpretations, so one is back  
to the problem of changing m. But the notation is still the same. So  
it would be better to keep underlying model preserving the musical  
intent.

> > 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.
>
> So change the init file.


There is the personal interaction problem: even though one agrees  
what the notation should be, one does not agree what the tuning  
should be. So as it is, set in E24, people may typeset a lot music  
that can be hard to retune.

A similar problem appear with music typeset in E12: if it should be  
retuned, one must first resolve enharmonic accidentals.

By contrast, properly notated music works in any tuning.

So the current system invites poor notation that cannot be easily  
retuned.

> > > 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.
>
> I made good progress during the summer.  It'll hopefully work as soon
> as someone looks at setting the relevant tables in the Sagittal font.
> A python function in Font Forge is how I'm told it's done.


I have a paper
  Sagittal A Microtonal Notation System
  by George D. Secor and David C. Keenan
by I haven't looked so much on it.

I developed the theory in the suggestion above while working a bit  
with a book by Hormoz Farhat on Persian dastgahs. In it, he describes  
them using the m, M and n above. I then it leads to a transposable  
theory in the abstract model.

Then the Sagittal paper above mentioned something that suggested a  
general tuning system. So that may then lead back the above  
suggestion - something to check.

  Hans