Re: Byzantine chant and microtones

[Hans Aberg]

On 11 Nov 2010, at 18:47, Erich Enke wrote:

> > If you like, you might start using staff notation alone: there are  
> > similar
> > scales in oriental (Persian/Arab/Turkish) music. One then  
> > introduces some
> > microtonal accidentals.
> >
> > I have looked through some Byzantine scales, and can see more or  
> > less how to
> > do it. You might use E72 or E53, the latter if you learn Graham  
> > Breed's
> > implementation.
>
> Thank you for the pointers.  Since translation to Western notation
> remains a goal of mine, it would make sense to develop the staff
> notation.  The responses were encouraging enough that I'll begin
> looking into implementing this in lilypond.

LilyPond comes with a file makam.ly which you might try modifying. I  
think you should replace the the denominator 9 with 6 to get to E72.  
Turkish Arel-Ezgi-Uzdilek (AEU) notation has an error: the sharp  
symbol is assigned the wrong value, raising 4 commas (E53 tonesteps),  
instead of the 5 for correct transposition. You can see it here:
  http://en.wikipedia.org/wiki/Makam
So make sure to get the sharp the right value. This may not be a  
problem in E72, though.

You might choose another set of microtonal symbols, or not write them  
out at all, but they would be needed internally to get the right MIDI- 
files intonation.

When looking through the Byzantine definitions, I could not see the  
soft and hard chromatic scales are the same scale but different  
tunings, or different scales. If they are different scales, you might  
want to have special microtonal symbols for them, otherwise use the  
same set, but admit them change values.

I will show how to do a transposable theory for the hard chromatic  
scale, using the folloing values:

Hard Chromatic
  Pa  Vu  Ga  Di  Ke  Zo  Ni' Pa'
    m   P   n   M   m   P   n
  m + P + n = P4

1. Νεχεανές (Necheanes), 1881.

     Necheanes  E72
  m  256/243     6
  M    9/8      12
  P  243/200    20
  n   25/24      4

Here, the m and M are the minor and major seconds of Pythagorean  
tuning. So start off with a staff system for that. It is the system of  
all intervals p m + q M, where p, q are integers. Sharps and flats  
alter with the amount M - m. Then number p + q is a universal scale  
degree, which the accidentals do not change.

Now, we need to reach P and n, but one is unnecessary, acting just as  
a filler. So choose n. Then the system above is extended to all  
intervals of the form p m + q M + r n, where r = 0 or -1. In Turkish  
music, one can also have r = 1; in Persian and Arab music, one only  
needs r = 0, 1.

In Turkish music, this happens because the perfect fourth P4 is  
divided as
  n  P  n
So in order to down P, one needs to use both n and -n. In Persian and  
Arab music, one divides P4 as
  n  P  m
so only n is needed.

So compare this with the Byzantine soft chromatic scale, which in some  
versions is divided as in Turkish music.

This affects the number of accidentals needed. For each n and -n, one  
needs two accidentals to go from m resp. M to n or -n. So in Turkish  
music, one needs 4 microtonal accidentals, but in Arab and Persian  
music one only needs 2.

Now return to the hard chromatic scale above. Since n is smaller than  
m, one can use n' := M - n = 12 - 4 = 8 instead. Then one needs a  
microtonal symbol that goes from m to n', that is, raising 2 E72  
tonesteps, and another going from M to n', which lowers 4 E72 tonesteps.

Looking at the soft chromatic scale E72 values, assuming that the  
accidentals from the hard chromatic scale can be used, it might  
suffice to introduce another pair of accidentals: lowering 2 and  
raising 4.