Re: Lilypond's internal pitch representation and microtonal notation

[Hans Aberg]

On 21 Sep 2010, at 11:46, Joseph Wakeling wrote:

> > A sharp is M-m and a flat m-M.
>
> If I understand right, this is a key "trick" of your system, since  
> such
> representations allow you to raise or lower the pitch without  
> affecting
> the degree.

Yes - accidentals do not affect the degree: they are of degree zero.  
One can add notes and intervals on this abstract level, and the  
degrees add as well. In mathematics, a function f is called a  
homomorphism (of abelian groups) when f(0) = 0, f(x + y) = f(x) +  
f(y), f(-x) = -f(x). The degree, which is the sum of the coefficients  
is a homomorphism.

> So by extension, if we say that q is a quarter-tone, to raise or lower
> by a quarter-tone would be to add (m-q) or (q-m); and to raise or  
> lower
> by 3/4-tone would be to add (M-q) or (q-M).

Typically, one might set the neutral second n between m and M, so  
there two choices: m + q and M - q. For an exact quarter-tone they  
would be the same when plugging in values, but algebraically they are  
different. So set n = m + q:

> .... but where/how in that system do we distinguish between for  
> example
> natural + 1/4 and sharp - 1/4 .... ?  Presumably the former is (m-q)
> whereas the latter is (M-m)+(q-m) ... ?

The whole pitch system is i m + j M + k n, where i, j, k are integers.  
If k = 1, then one needs two accidentals n-m and n-M: the amounts  
needed to go from the i m + j M to reach the n. If k = -1, then the  
other pair is needed: m-n and M-n. If k > 1, then one needs k of the  
first pair of accents, and if k < -1 one needs -k of the second pair  
of accents.

> > In the traditional typesetting, one has a minor second m and a
> > major second M, so it is all combinations p m + q M, where p, q
> > are integers, which can be identified with all pairs (p, q).
>
> ... so if we extend this vectorial representation to a 3D case for
> quarter-tones, (x_M, x_m, x_q), (NB my q is different from yours:-)

Yes, some mathematicians do that error, too, though in print, q as a  
variable might be typeset in italic, whereas as constant in upright  
type.

> we
> can think of natural+1/4 as being (0, 1, -1) while sharp-1/4 would be
> given by (1, -2, 1).

No: this is the tricky part. Introduce a neutral n = m + q. Then the  
two accidentals for k = 1 are n - m = q and n - M = n + q - M. In  
coordinates, this is (-1, 0, 1) and (0, -1, 1).

The reason is that you want to express a pitch i m + j M + n and  
notate it against a pitch of the staff system which is of the form i'  
m + j' M. So the difference has coefficient 1 on the n.

> Depending on the notational system desired, those could then be
> represented by the same accidental (the half-sharp symbol) or  
> different
> ones (natural-with-up-arrow and sharp-with-down-arrow).

If you are carful not writing any music which introduces the -n, you  
will actually only need two accidentals for raising and lowering the  
quarter-tone. I find the following example helpful for remembering:

In Persian music written (in one interpretation) in E53, one sets m =  
4, M = 9 (Pythagorean tuning), n = 6. The raising accidental is called  
sori and is n - m = 2, the lowering is called koron and is n - M = 6 -  
9 = -3. The point is that the koron lowers with more than what the  
sori raises. The algebraic system keeps track of this.

> Suppose now that I wanted to extend the quarter-tone system to one
> including eighth-tones (I'm thinking here of Ferneyhough's "La chûte
> d'Icare" which uses the "standard" quarter-tone accidentals  
> supported by
> Lilypond plus up- and down-arrows to indicate raising or lowering by  
> an
> approximate eighth-tone).  Would I have to add an extra dimension to  
> my
> vector space,
>
>   (x_M, x_m, x_q, x_e)
>
> ... or is there a cleverer way of dealing with this which lets you  
> keep
> only 3 dimensions?

The algebraic system allows you to do both are any number of extra  
pitches, and also change the scale of the staff system.

> The reason I ask is that I can't see a means of raising/lowering by an
> eighth-tone without altering the degree, unless I have the possibility
> to have forms (q-e) and (e-q).
>
> ... or did your term "neutral second" mean "something that does not
> alter the degree by definition"?

It works just like in the Persian music example above. If your eight  
tone is e, introduce a neutral n_2 = m + e. When the coefficient of  
n_2 is 1, one accidental raises 1/8 and the other lowers 3/8, and when  
the coefficient is -1, one accidental lowers 1/8 and the other raises  
3/8.

These are then algebraic relations. If you really want to have exact  
1/8 notes, then just introduce quarter-tone glyphs when the  
coefficient of n_2 is 2 or -2.

> Incidentally, if I understand right I think your system offers a way  
> of
> separating out the definitions of staff pitches and accidentals, since
> one can define the former merely in terms of degree values, while the
> latter can be defined independently in terms of different zero-degree
> "vectors"; ...

Yes, the values of the staff scale does not have anything to do with  
the accidentals: accidentals are only needed in order to express a  
difference between a note and the note value in the same degree on the  
staff.

And the accidental is a sum of pairs of seconds, which can be computed  
without any knowledge of available glyphs. The algorithm computes  
(relative an order of the seconds) a minimum of accidentals. So I  
think that if one wants to have a different representation, it would  
be easier to start off with the minimal one.

> ...although I wonder whether it might be better from a
> representational view to think of maps rather than vectors, e.g.
>
>    "Major second" -> 0
>    "minor second" -> 1
>    "quarter-tone" -> -1
>
> for quarter-sharp, since this kind of representation allows you to
> maintain different "neutral seconds" without risk of overlap.

Yes, this is how I wrote the Haskell code, as maps. I actually started  
with the vector representation, but when defining Just Intonation, one  
uses two major seconds: the Pythagorean M = 9/8 and the Just JM =  
10/9. One might want experimenting with them.

So therefore I made them maps from strings. This way, the system does  
not have any reference to any traditional scale or seconds m, M - they  
are chosen as special cases.

A drawback is that one must keep track of these seconds m, M when  
doing traditional typesetting. But one can write an interface hiding  
it away.

> (Forgive me if this kind of stuff is dealt with in your code; I  
> haven't
> looked, because my Haskell knowledge is extremely basic and these days
> are extremely busy in my day job...:-)

If you have more questions, just let me know.