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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.
- Re: Lilypond's internal pitch representation and microtonal notation, (continued)
- Re: Lilypond's internal pitch representation and microtonal notation, Carl Sorensen, 2010/09/21
- Re: Lilypond's internal pitch representation and microtonal notation, Graham Percival, 2010/09/21
- Re: Lilypond's internal pitch representation and microtonal notation, Hans Aberg, 2010/09/21
- Re: Lilypond's internal pitch representation and microtonal notation, Benkő Pál, 2010/09/21
- Re: Lilypond's internal pitch representation and microtonal notation, Hans Aberg, 2010/09/21
- Re: Lilypond's internal pitch representation and microtonal notation, Benkő Pál, 2010/09/22
- Re: Lilypond's internal pitch representation and microtonal notation,
Hans Aberg <=
- Re: Lilypond's internal pitch representation and microtonal notation, Hans Aberg, 2010/09/22
- Re: Lilypond's internal pitch representation and microtonal notation, Hans Aberg, 2010/09/21
Re: Lilypond's internal pitch representation and microtonal notation, Graham Percival, 2010/09/20