Re: Diatonic notation system

[Hans Aberg]

On 9 Dec 2008, at 06:09, Graham Breed wrote:

> > > Are you on board with the regular mapping paradigm?  I may as well
> > > promote it while I'm here.
> > >
> > > http://x31eq.com/paradigm.html
> >
> > I looked a bit at it, the section "The Core Paradigm". The model I  
> > indicated
> > also chooses some generators, but in addition reflects the notion  
> > of scale
> > degrees that the Western musical notation system brings out, which  
> > in its
> > turn relies on an underlying empiric principle, also in the case when
> > augmented with intermediate pitches as in Persian and Arab music:
>
> Scale degrees work fine in the paradigm.  They're one possible
> generator.  I give the western scale in terms of those scale degrees.

It is necessary to have those scale degrees, and letters for pitch  
representatives of them for the system for the notation system I  
describe to work.

> What's "empiric"?

As far as I know, that is how the music happens to be structured, but  
I do not an underlying principle for it.

> > Given intervals x (resp. y) in a two (resp. three) generator model  
> > x = p m +
> > q M (resp. y = p m + q M + r n), define a scale degree deg x = p +  
> > q (resp.
> > deg y = p + q + r). Empirically, melodic development normally  
> > takes place
> > between different scale degrees, also on say chromatically altered
> > ornaments. This is also true in the Persian dastgahs, using Farhat's
> > description. If one alters scale degrees, melodic development  
> > still normally
> > takes place between different scale degrees, as on a parallel,  
> > altered
> > scale. In the Western notation system, one achieves this by simply
> > minimizing the amount of temporary accidentals, if the notes are  
> > not too
> > chromatically dense, which is very intuitive.
>
> That's partly because the harmony works that way.

Well, CPP harmony is a Western thing. There is some harmony based on  
4ths and 5ths.

> > Now, the construction on the link above seems to ignore those  
> > scale degrees.
> > If one defines a scale from say an octave P8 and a perfect fifth  
> > P5, then
> > scale degrees can be defined by setting deg P8 = 7, deg P5 = 4,  
> > and then
> > work it out for other combinations. This gives deg M = deg (2 P5 -  
> > P8) = 2
> > deg P5 - deg P8 = 1, deg m = deg (P4 - 2 M) = deg (P8 - P5 - 2M) =  
> > 7 - 4 - 2
> > = 1.
>
> How does it ignore them?  You saw the generators as part of the  
> core paradigm.

I just mean it is not enough to define generators: they must also  
have scale degrees. Different choice of scale degrees for the same  
choice of generators imply a different musical treatment, and will  
get a different musical notation.

> > Then, in addition, to get an (extended) Western notation system,  
> > one must
> > define pitches A B C D E F G ... of scale degrees 0, 1, .., (n-1),  
> > where n
> > is the number of scale degrees in what is called an "octave".
>
> Yes.


So then that is it. If one wants to have an intermediate pitch  
described a neutral n, then one needs one symbol to go from m to n  
which raises with the amount n - m, and a symbol to go from M to n  
which lowers with M - m. The sum of these intervals is M - m, the  
amount that sharps and flats alter.

For example, the Turkish AUE notation system
  http://en.wikipedia.org/wiki/Makam#Intervals
uses two neutrals of 5 and 8 commas. This will generate 2 pairs of  
symbols, the 4 that are used.

The Persian Farhat system, interpreted in E53, uses one neutral n of  
6 commas. One suggestion is that it is the rational interval 27/25.  
Since this interval is well approximated in E36, I made an  
interpetation in that, to work with E12 instruments.

  Hans