Re: Diatonic notation system

[Hans Aberg]

On 8 Dec 2008, at 12:28, Graham Breed wrote:

> > So it might be better to write an intermediate sound file with the  
> > diatonic
> > structure. Then it can be used to return the output without having to
> > recompile the typeset output.
>
> What's an "intermediate sound file"?

The idea is, instead of writing MIDI directly, LilyPond writes an  
other file with the diatonic information. Then this file is used to  
convert to MIDI or some other sound format, like Scala file, without  
bothering about the typeset output.

> > The question is the composer chooses pitches, or abstract  
> > intervals as in
> > Arab and Persian music. The latter is more powerful, because  
> > intervals can
> > be adjusted with the tuning.
>
> Yes, but *somebody* has to choose the MIDI output.

Yes, but the the one who wants to listen to the file may want a  
different tuning than the one who wrote it.

> > > > There is also a paper
> > > >  Tuning, Tonality, and Twenty-Two-Tone Temperament
> > > >  Paul Erlich
> > > > which constructs generalized 10-tone diatonic scales in E22.

> > I think he introduces special accidental symbols, which are  
> > unnecessary.
>
> Not necessary, but maybe useful.  I'm all for new symbols because it
> reduces confusion between different systems.  But others disagree and
> it isn't hugely important either way.

Something is needed to indicate the notation system. I think it would  
suffice to write it above or before the key signature. This way also  
specific tunings can be called for, if needed.

If people want to invent their own symbols, perhaps LilyPond should  
accommodate it. /but from the point of view of the m M model, it is  
just a change of symbols.

> > Also, I made keyboard map, which I have used in Scala playing in  
> > mainly E31
> > for the last couple of months:
> >   A#  B#  Cx  Dx  Ex
> >  A   B   C#  D#  E#  Fx  Gx  Ax  Bx
> >    Bb  C   D   E   F#  G#  A#  B#
> >      Cb  Db  Eb  F   G   A   B   C'# D'#
> >        Dbb Ebb Fb  Gb  Ab  Bb  C'  D'  E'
> > It will them work in any such generalized diatonic system.
>
> You mean you have a keyboard mapped like this, or you use a virtual
> keyboard in Scala?

It is just a key map for the computer typing key board, in the sense  
of a real music keyboard.

> Anyway, it'll only work in a system with a single spiral of fifths
> (singly positive or negative in Wilson's terminology).  No good for
> miracle or magic temperaments.

I am not sure what you mean here. This links
  http://en.wikipedia.org/wiki/Magic_temperament
says the latter keeps the octave, but chooses a different M - so it  
will work.

I cannot work out from this link
  http://en.wikipedia.org/wiki/Miracle_temperament
what scales they produce.

In terms of this link
  http://en.wikipedia.org/wiki/Regular_temperament
the key map works with any two generator system, if one can compute  
the equivalents of m an M.

It just displays the notes
    . ---> M
     \    /
      \  /  #, b
       v
        m
The letters then belong to the notation system, not the underlying  
diatonic system.

> > If you play standard music, you will find that the melodic  
> > development takes
> > place by moving between the diagonals /, where a change on such a  
> > diagonal
> > represents a scale alterations.
> >
> > In the p m + q M model, the number p + q is the scale degree.  
> > Altering with
> > accidentals does not change this scale degree. A similar thing  
> > happens in
> > Persian music, when adding the neutral second: the intermediate  
> > pitch ends
> > up on the / diagonal.
>
> By one interpretation of Persian music, maybe.

It is from Hormoz Farhat, "The Dastgah Concept in Persian Music".

The koron and sori actually derive from a E24 model invented by  
Vasiri, but the notation symbols are retained.

Then I worked through Farhat's book to check that this is how the  
symbols are used.

> > The diagram above makes it is easy to compute transpositions, as  
> > they are
> > merely translations.
>
> No transpositions in this system should break in Lilypond.  Do you
> have examples where they do?

I am not sure what you refer to here. The system I indicate will  
always work in with transpositions if one has two generators m and M.  
It does not work in a system where one uses in effect more generators  
than m M, but still uses the same notation system.

One example is Just intonation. The major seconds between C-D and D- 
E. But here the tradition is to just forget about it, temper the  
difference out. - I do not know about any specifically Just  
intonation music.

The accidentals are just symbols for passing between different  
combinations of p m + q M. So if one wants to notate Just intonation  
exactly, one has to introduce more accidentals, either as  
intermediate pitches using neutral seconds. Or if one wants to give D  
= 9/8, E = 5/4 relative C, introduce sharps and flats of different  
sizes. I haven't thought of the latter much.

> > > I think they can be made to work in Lilypond.  The problems are:
> > >
> > > - You have to override the standard octave numbers
> >
> > I suspect one has to scrap the current model, but if the change  
> > can be made,
> > it might be simpler. Only the core developers could know that.
>
> Maybe they can chime in.  But we can also search through the source
> code to find instances of the number 7.  As there isn't a constant
> defined for "number of notes to the octave" it won't be trivial to
> change.

Well, I do not know.

> As you're talking about systems with a single spiral of fifths --
> essentially meantone, Pythagorean, and schismatic -- the current model
> works very well.

I think that is what the Western notation system is designed for.  
Then oriental music split off, adding (an) intermediate pitch(es).

> For transposition to work you need to specify
> accidentals as a fraction of double the sharp alteration -- that is
> the difference between two notes 7 fifths apart, octave reduced.  Or
> the difference between C and C# rather than E and F.  I think that
> will work fine.  It won't sound right, of course.  So you should be
> asking for one more variable to control the size of the fifth (and
> another for the size of the octave).

Transpositions take place by adding a vector to the (p, q) pair in p  
m + q M. So even though the model originally derives from an  
iteration of 5ths, it is no longer important.

One may transpositions in fifths, but it is not needed:
  Fbb Cbb Gbb Dbb Abb Ebb Bbb
  Fb  Cb  Gb  Db  Ab  Eb  Bb
  F   C   G   D   A   E   B
  F#  C#  G#  D#  A#  E#  B#
  Fx  Cx  Gx  Dx  Ax  Ex  Bx
Think of it as a single line.

In Persian music, one then gets
  Fbp Cbp Gbp Dbp Abp Ebp Bbp
  Fp  Cp  Gp  Dp  Ap  Ep  Bp
  F>  C>  G>  D>  A>  E>  B>
  F#> C#> G#> D#> A#> E#> B#>
p = lowering koron, < = raisng sori.


> > But I think the simplest would be to use pairs (p, q) internally, and
> > compute octaves at need from that.
>
> Only for a rank 2 system.  It's no good for 5-limit just intonation
> and beyond.

Right, But I think that is the limitation of the Western notation  
system. And normally, I think one tempers it out, performing in a  
diatonic system.

> And there are plenty of cases where the same pitch can be
> written different ways for a rank 2 tuning.  Like a meantone notation
> with a new symbol for "diesis" shifts (1 step of 31, 50, 43, etc).
> You could write Db as the diesis above C# and you want it to stay like
> that.

Those are E31 enharmonic equivalents. A true meantone might use M =  
sqrt(5/4).

If you introduce enharmonic equivalents, or specific symbols for  
tonesteps, then it is tied to that tuning, and it cannot be retuned  
without lifting it to the diatonic structure.

> 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 will look at this and come back.

> > > - You need a different init file for MIDI and printing
> >
> > If one can write an intermediate diatonic file, as I described  
> > above, then
> > that could be used for creating MIDI-files at your favorite tuning.
>
> You didn't describe it above.  All you did was mention it.

It is just an idea. It is not possible to retune MIDI files, and not  
easy to extract from LilyPond innards to extract the right  
information to produce say Scala files. That was the situation that  
prompted the idea.

> > Then in the init file used for typesetting, the composer may have  
> > the option
> > to choose pitches or abstract intervals. But even if the composer  
> > chooses
> > pitches in a specific tuning, it may be best to let the  
> > intermediate file
> > still write it diatonically, if possible, so that retuning is  
> > possible.
>
> What would the "intermediate file" do that the original Lilypond  
> one doesn't?

Essentially be a sound oriented file, but retaining the diatonic  
structure needed for retuning.

> > > I think that's about as much as we can expect given how obscure they
> > > are.  Here's the shortlist of notation types we're trying to  
> > > support:
> > >
> > > http://groups.google.com/group/microtools/web/types-of-notation
> > >
> > > MicroABC can do some of them.  I think Lilypond should be able to do
> > > most of them with a bit of work.
> >
> > Thank you, it is a nice list, but unfortunately, most of the links  
> > lead to a
> > porn site :-).
>
> Did you report it as abuse?

No, I haven't time. Do it if you can. I do not know what caused it.

> I've cleaned up two pages, anyway.

Fine.

> > Several of the those systems are designed for a specific tuning,  
> > and the
> > problem with that is that is not how musical interpretation takes  
> > place.
>
> I think "several" is an exaggeration.  Maneri-Sims is obviously for
> one tuning.  Then there are two for just intonation.  And there's the
> AFMM, which is for any tuning but requires the tuning to be precise.
> In no case is an interpreter's freedom restricted.  They ignore
> precise metronome marks after all.

Anyway, there may be flux in the pitches.

> > This is something that has been criticized in the Turkish musical  
> > notation
> > systems. There are some papers:
> >  Ozan Yarman, "A Comparative Evaluation of Pitch Notations in  
> > Turkish Makam
> > Music"
> >  Ozan Yarman, "79-TONE TUNING & THEORY FOR TURKISH MAQAM MUSIC"
> >  Nail Yavuzoğlu, "Equal Temperament in Turkish Music"
> > (If you can't find them, I can send them to you.)
>
> I've read Ozan's PhD thesis.  I don't want to go through it in  
> detail again now.

I di not come that far, because I stopped believing in tying music  
notation to a specific tuning. I think Farhat's setup is right.

> > The first paper (I think) says that the E53 Arel-Ezgi-Uzdilek  
> > (AEU) notation
> > system was created by taking the 13th century Saffiaddin Ormavi's  
> > work, with
> > some errors. One such is setting the sharps # to the wrong value,  
> > as shown
> > here:
> >  http://en.wikipedia.org/wiki/Makam#Intervals
>
> He's trying to find a system that can handle all the complexity of
> Makams in practice.  He wants a multiple for 53 for reasons I wasn't
> clear about.

E53 is the Pythagorean tuning, which gives very good approximation of  
the perfect fifths 3/2. Multiples let you fine-tune intermediate  
pitches.

> There is a rank 2 tuning behind it but not one that
> would work on the keyboard layout you gave.

In the Turkish music notation, the symbol for raising 5 commas would  
have to be the internal sharp of my layout.

As for pitch fine-tuning, it tends to be heard as not new notes, but  
intonation. So tying that to notation seems me nad, though people  
experiment with that.

> > In any diatonic tuning, sharps and flats alter by the same amount  
> > M - m. In
> > E53, M = 9, m = 4, so these accidentals should alter with 5  
> > commas, not 4,
> > as in the link above.
>
> Yes, if sharps and flats are to follow the spiral of fifths.

Yes - this seems to be an error.

> > I think that the Turks want to change that, but it is difficult,  
> > because one
> > has to go back to the original model, and make a new translation.  
> > It is not
> > possible to do that from the current AEU notated music.
>
> I don't know that "the Turks" are of a single mind on this.  Last I
> heard Ozan was looking at 41-equal because the 79 from whatever system
> was too complicated.

It is a mess. So I think a system like Farhat's or the Arab would be  
best, and then indicate details as intonations or choice of tunings.

> > So that illustrates the problem of having notation tied to a specific
> > tuning.
>
> Does it?  What notation could possibly have handled these makams
> without tying itself to the tuning?

The Arab, Persian or the older system with letters, as here
  http://en.wikipedia.org/wiki/Makam#Intervals

> > But when playing this, I think two M's of different sizes in  
> > succession
> > sounds weird. So it does not bother me, too much. :-)
>
> It's a problem that's bound to arise with a 53 note system.

E53 does not have this problem.

> And I
> fail to see what it has to do with Lilypond.

It is only a problem if you want to notate Just intonation with  
transpositions.

  Hans