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Re: Diatonic notation system

From: Graham Breed
Subject: Re: Diatonic notation system
Date: Tue, 9 Dec 2008 10:13:18 +0800

2008/12/8 Hans Aberg <address@hidden>:
> 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.

Lilypond code is already semantic markup for music.  Your intermediate
file would end up looking a lot like the original.

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

So they use a different init file, or change a few numbers in the
standard init file.  Lilypond caters for this by having init files and
allowing scheme expressions in them.

> 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.

When a musician's reading the notation, they're not looking above or
before the key signature.  They're looking at the notes.  So the notes
should look different to remind the musician that they should sound
different.  Otherwise it can get confusing as you switch between

> 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.

Right, and currently arbitrary symbols don't work.

> In terms of this link
> 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.

No, the key map needs to change for different choices of your p and q.
 For full generality you need to allow p and q to be specified, at
least one tuning parameter, the nominals in terms of p and q, and the
size of # and b in terms of p and q.  That would work but it's
obviously not what Lilypond does now.  But Lilypond would work fine
for meantone, etc., if there were a better way of setting the tuning
of the nominals.  I'm afraid changing the number of nominals is not
going to be easy.  But ignoring the octaves will work fine as a kludge
for the few of us who actually want more than 7 nominals.

>>> 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.

No, your system will not always work, as I've said, because diesis
shifts introduce more equivalences.  What example do you have where
Lilypond's transpositions don't work?

> 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.

You what????????? Elsie Hamilton, Harry Partch, Lou Harrison, Terry
Riley, La Monte Young, Kraig Grady, Toby Twining, Greg Schiemer, all
passed you by?  You don't know about the hexachords Willaert
supposedly used to get just intonation performances?  And yet you're
happy to tell the Lilypond developers how they should implement their
microtonal support!

> 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.

Yes.  Hence Sagittal, HEWM, Extended Helmholtz-Ellis, and Johnston's
notation.  They won't work with your system.  They will work for
printing with the current Lilypond system and third-party fonts.  They
would work for MIDI if Lilypond allowed the tuning of the nominals to
be specified ... and MIDI didn't suck quite so much.

>> 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

Okay, I know how to transpose.  What I'm asking is why it doesn't work
for you in Lilypond.  The way Lilypond does it looks fine to me.

>>> 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.

That's what Lilypond does.  You should be quite happy with Lilypond as it is.

>> 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).

They're equivalents in either E19, E31, E43, E50, and so on.  The
tuning doesn't matter.

> 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.

No.  In this case the equivalents (call them enharmonic if you like)
are not tied to the tuning.  The current Lilypond system allows them
to be distinguished but an abstract M m wouldn't.

>> 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 are other ways to get the pitches you want than starting with
53.  I think 58&72 looks good for the harmonies he looked at.  But he
specifically wanted a multiple of 53.  I think that can only be for
historical reasons -- some makams are specified in terms of 53 steps
to the octave and so the pattern needs to be preserved.

>> 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.

So, Lilypond being a notation program, you aren't worried about the
pitch fine-tuning?  It already does what you want.

>> 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.

Maybe the actual system they're notating is a mess when you try to fit
it into your theories, because it wasn't developed according to those

>>> 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

No, those specify a division of the whole tone into 9 commas.  They're
tied to the tuning of a comma.

>>> 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.

Of course it does!  C-D is 9 steps, D-E is 8 steps, if C-E is to be a
pure major third.


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