2010/9/22 Hans Aberg <haber...@telia.com>: > 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 _______________________________________________ lilypond-devel mailing list lilypond-devel@gnu.org http://lists.gnu.org/mailman/listinfo/lilypond-devel