lilypond-devel
[Top][All Lists]

## Re: Lilypond's internal pitch representation and microtonal notation

 From: Benkő Pál Subject: Re: Lilypond's internal pitch representation and microtonal notation Date: Wed, 22 Sep 2010 08:18:53 +0200

```2010/9/22 Hans Aberg <address@hidden>:
> On 21 Sep 2010, at 21:31, 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)

> 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.

> 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).

> 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?

>> 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.

> 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!

p

```