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Re: Lilypond's internal pitch representation and microtonal notation


From: Hans Aberg
Subject: Re: Lilypond's internal pitch representation and microtonal notation
Date: Wed, 22 Sep 2010 10:29:04 +0200

On 22 Sep 2010, at 08:18, Benkő Pál wrote:

In algebraic terms, choose a neutral n between m and M. The total pitch system will be i m + j M + k n, where i, j, k are integers. But the staff system only has the pitches i' m + j' M. When taking the difference with
the
staff note, reducing the degree to 0, and taking away the sharps/ flat (a
multiple of M - m), there will result a multiple n - m or n - M.

a minor point: wouldn't it be clearer to use d (degree) and a (alteration) instead of M and m? d should be a second, i.e. M (or m), while a should
be the augmented prime, i.e. M - m.
then multiples of d gives the staff system (very roughly equivalent to the diatonic scale), linear combinations of d and a would give the usual chromatic system (all notes that can be notated with sharps and flats), and for microtonal and exotic music one should use (one or several) n.

I think this is what LilyPond has now, using d and M in E12 originally.

I think not.  I didn't mean to replace the whole of your system
by d and a, only M and m.  similarly to your P5-P8 example,
(1  0)(d) = (M)
(1 -1)(a)   (m)

This model with seconds only makes implementation easy. It easy to build an interface on top.

In the code I have, m and M are just seconds with no special standings. Joseph suggested independently such an approach. For example, in Just Intonation, one has two major thirds. One might want experiment with interchanging their roles.

Also, there is no requirements that one has at least two seconds, which could be used for E12 - 12 scale degrees.

But it becomes complicated when adding pitches. If one has seconds s_1, ..., s_k, then there is an accidental for each difference s_j - s_i and each s_i - s_j. With just m, M and n, one gets besides sharp M-m, flat m-M, also n-m, n-M, m-n, M-n. All four are used in Turkish music, but this system can
handle it algebraically by adding just one second n.

well, the two systems are equivalent, as M and m can be expressed by d and a: M = d, m = d-a (and n is common to both). I just think that d and a suits
better to classical music than M and n.

You can to use M and a instead when defining the tuning. It will lead to the matrix above.

In this system, d can always be computed. So it is not needed as a variable
to carry around.

I just meant it as another base in the modulus (or free Abelian group)
of intervals - I see your original proposal as a suggestion to replace
the current physical pitch based representation (which is essentially
a one dimensional vector space over the reals) to a theoretically
correct interval based representation (a modulus over the integers,
of dimension at least two, but incremented for microtonal purposes).

In mathematical terms, I look at the free abelian group generated by the intervals one wants to typeset. Since one can always compute the seconds, I use them as a basis in this group.

Another motivation is musical. One is typically not playing the accidental
but the neutral interval. So it is easier to describe the music using
seconds. An example of playing an accidental is major chord followed by a minor chord. But it is still more convenient to think of the minor chord built up by a minor and a major third rather than an alteration of the major chord. When playing the minor chord it has no relation to the major chord.

I'm not sure I got it - the minor third IS an alteration of the major third,
isn't it?

Yes, but one normally do not play it as such. It is the intervals of the scales that are important, not the intervals of the accidentals, which express the difference between the intervals you play. This is true also with ornaments: they should normally involve notes of different degrees, as they are notated in different positions on the staff.

anyway, I'm a big fan of using such a system: I've tried Pythagorean
and meantone MIDI-output by defining alterations, and MIDI was all
right, but the score had all the naturals which weren't defined to
be exactly zero (i.e. all except a); your system distinguishes nicely
between pitch systems and tuning (thoretical pitch and its physical
frequency).

Yes, this is another point. If creating music with these linear combinations of seconds, one can plug in values later, and it is easy to retune the piece. This is so because the staff system was created to admit different
tunings.

There is another part how to compute these seconds, which we have not yet come to. Traditionally, a system is defined by the pure fifth P5 = m +3M and
the octave P8 = 2m+5M.

yes; in my terms, P5 and P8 forms a base equivalent to M and m (or d and a);
in fact, for transposition purposes, this may be the best choice.

Transposition is just addition. To transpose from x to y add the note y - x.

Writing a matrix equation
 (1 3)(m) = (P5)
 (2 5)(M)   (P8)
the intervals of m and M can be computed by inverting the matrix on the left
hand side.

and we all know that m = 3P8 - 5P5, M = 2P5 - P8:
matrix inversion in musical terms!

I started out with a ChucK program where one had to compute them on your own, but that quickly became cumbersome. For example, meantone uses M3 = 2M which produces a factor 1/2 in the matrix inverse - a square root in the interval.

So it is easier to let the computer doing it.




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