Re: Diatonic notation system

[Graham Breed]

2008/12/8 Hans Aberg <address@hidden>:
>
> On 8 Dec 2008, at 04:53, Graham Breed wrote:

>> Right.  But the actual fifth has to be specified so you need an init
>> file to do that.  The exact meaning of the alterations also have to be
>> specified in an init file.  So there has to be a different init file
>> for each Sagittal context if you want any reasonable MIDI output.
>
> 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 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.

>>> There is also a paper
>>>  Tuning, Tonality, and Twenty-Two-Tone Temperament
>>>  Paul Erlich
>>> which constructs generalized 10-tone diatonic scales in E22.
>>>
>>> He then does not seem to realize that standard Western music notation
>>> will
>>> work, if one only alters the number of notes per octave.
>>
>> He does so!  There are examples on the last page.
>
> 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.

>> I am looking at other cases like this.  There's a list to discuss
>> microtonal tools, which started out very optimistically.  What you're
>> talking about are, indeed, "diatonic notations":
>>
>> http://groups.google.com/group/microtools/web/diatonic-notations
>
> This might be the same idea: to get the Western music notation working in
> this generalization, one needs really only abstract m and M, and an octave
> plus names A B C D E F G ... for specific combinations of them. To get the
> G- C- or F- clef working in this generality, those note names needs to be
> defined.

The pitches need to be defined, in the given system.  But mostly you
can define it as a maximally even scale, so only 2 numbers.  For
regular staff notation it'd be 7 from 12, for example.  (The 12
needn't be equally tempered.)

> 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?

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.

> 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.

> 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 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.

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

> 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.  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.

Are you on board with the regular mapping paradigm?  I may as well
promote it while I'm here.

http://x31eq.com/paradigm.html

>> - 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.

> 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?

>> 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?  I've cleaned up two pages, anyway.

> 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.

> 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.

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

> 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.

> 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.

> 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?

> A thing that cannot be handled in a transposable way by the Western notation
> with strict pitch interpretation is strict Just intonation, as it uses two
> M's of different sizes. this can be illustrated in E72 (or E53) where (in C
> major), the E should be lowered one tonestep). One then gets an diatonic
> intermediate pitch.

It can't be handled explicitly with or without transposition.  You can
always use schismatic notation, so that the E becomes Fb, if you like.

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


                               Graham