Philip Taylor escreveu:
[...]
> For what it's worth, BarFly uses a single signed short to represent
> pitchbend, with a resolution of 1/256 semitone, and it's mapped log
> (if I understand that correctly).
I mean: the linear mapping of integer numerators, for example, from -255
to 255 fractions of semitone, corresponds to a logarithmic scale of
frequencies. If one uses the given non-quantised fractions -- abc2midi
supports (1 to 4096) / (1 to 4096) -- the resolution is not linear, but
based on sets of rational numbers (1/3 == 3/9, but 1/3 != 33/100 and 1/3
!= 85/256).
> I use this to set global
> intonation, and don't yet support microtonal accidentals, although I
> probably will in the future once a consensus appears. 1/256 semitone
> is the basic resolution of Quicktime Music Architecture, which is
> what I use for playing, and seems sufficient resolution to me, since
> it's generally reckoned that one cent is the best resolution of the
> human ear.
True, one cent can be taken as a limit for pitch distinction, but in an
harmonic context, such a difference is not negligible: for example, just
intonation intends to be beating-free; for high pitches, one cent error
can give rise to beatings. Moreover, if the quantisation error for a
pitch can be one cent, the interval between two quantized pitches can
sum 2 cents of error.
But I think the final resolution is implementation specific. We can try
instead to define precisely the meaning of ``NUM'' in Remo/Keenan's BNF:
<note> ::= [<accidental>] <pitch> [<octave>] [<duration>]
<accidental> ::= "^"+ | "_"+ | "=" | ("^" | "_") [ (<fraction> |
<quarters>) ]
<octave> ::= "'"+ | ","+
<duration> ::= <fraction>
<quarters> ::= "'" | "," | "/"
<fraction> ::= NUM [ "/" NUM ]
Is ``NUM'' an integer? A float? What is its valid range?
We could accept the 14bit MIDI range as reference for ``NUM'' range:
2^14 = 16384 values for +2/-2 semitones, then:
one semitone = 2^14 / 4 = 4096.
Therefore, we can have 4096 different pitchbend values inside a
semitone. Is it a too big value? Perhaps, but this allows a full control
of MIDI pitch bends by using directly the values (1 to 4096)/4096.
Unfortunately, ``NUM'' from 1 to 4096 will call for 24 bits to store the
fraction. To store both numerator and denominator in a 16 bit variable
instead, the range must be 1 to 256.
Someone with a good knowledge on mathematics could tell us how many
groups from this sequence are needed to represent all (1 to 4096)/4096
values (after quantisation, of course):
{{1/1}, {1/2, 2/2}, {1/3, 2/3, 3/3}, {1/4, 2/4, 3/4, 4/4}...}
Perhaps that number could be used as the limit for resolution.
Other software (like the so far postponed abc2sco) can use a very higher
precision to handle fractions like 1/3, 1/11, 1/13, without MIDI
resolution restrictions.
Hudson
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