On 7 Nov 2013, at 21:47, Keith OHara <[email protected]> wrote: > Hans Aberg <haberg-1 <at> telia.com> writes: > >> I have just defined pitch names for E53 note c with accidentals, using > Graham’s file regular.ly: >> cff cffu cffuu cfdd | cfd cf cfu cfuu | cdd cd c cu | >> cuu csdd csd cs | csu csuu cssdd cssd | css r2. | >> Here, d (resp. u) is down (resp. up) one E53 comma (tonestep). I just > checked with midicomp that the output is >> correct. >> > > Hans, I am late, but can I persuade you to try this with sharp and flat > representing 4 tone-steps rather than 5 ?
No, since in regular E53, the minor second m = 4, and the major second M = 9, so sharps and flats alter with M - m = 5 E53 tonesteps. So the minor second in E53 is 4*1200/53 = 90.566 cents, which is very close to the Pythagorean m = 256/243 which 1200*log_2(256/243) = 90.225 cents. The sharp alter with 5*1200/53 = 113.208 cents. > Then a sharp is 91 cents, near the 92-cent alteration corresponding to > the ratio 5.3.3.3/128 that is often represented by a sharp -- the shift > from F that is the IV of C-major to the F-sharp in a D-major chord that > we see when we modulate to G-major. > > By comparison the Helmholtz notation uses sharp to mean 114 cents, > taking F to the F-sharp from a sequence of 7 perfect fifths as in > Pythagorean tuning, but in music this sharp is most often seen > lowered by a 22-cent comma. The normal is to depart from the Pythagorean tuning, and adding approximations for JI. > Ben Johnston's notation uses flat to mean -70 cents (a ratio 24/25) > but this is most often seen on notes B E and A that Johnston has lowered > already from the cycle of fifths by a 22-cent comma. > > The notation looks simpler and more familiar if we define sharp and flat > such that C-sharp is below D-flat <http://k-ohara.oco.net/Lilypond/> So you need to choose your m and M first, but for regular ETs that is done by merely seeking the best approximation of the perfect fifth 3/2. Quarter-comma meantone, which sets the major third exact, is approximated by E31, which has m = 3, M = 5, and 1/3 comma meantone, which sets the minor third to 6/5, is approximated by E19. These would give narrower sharps alterations. _______________________________________________ lilypond-devel mailing list [email protected] https://lists.gnu.org/mailman/listinfo/lilypond-devel
