Re: Microtonal accidentals

[Hans Aberg]

On 7 Nov 2013, at 21:47, Keith OHara <address@hidden> 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.