Lilypond's internal pitch representation and microtonal notation

[Joseph Wakeling]

Hello all,

This email is an attempt to clarify some outstanding issues regarding
support for microtonal notation in Lilypond.  It's being written in
response to recent discussion with Graham Percival on Lilypond Issue
694: http://code.google.com/p/lilypond/issues/detail?id=694

To begin with, let's review how Lilypond represents pitch.  A pitch is
represented by a triple of numbers (octave, note, alteration) where
"octave" is an integer, "note" is an integer in the range 0 to 6
(corresponding to note names C D E F G A B) and "alteration" is a
continuous-valued variable whose unit is the whole tone.

With these three values Lilypond is able to represent any pitch in a
continuous frequency spectrum.

>From a notational perspective, the first two numbers are used to
calculate the vertical staff position of the notehead, while the value
of the alteration is used to determine the accidental: e.g. (1,1,-1/2)
corresponds to the D-flat a semitone above middle C.

*A key assumption of this approach is that there is a one-to-one
correspondence between accidental and alteration value.*  This clearly
holds for conventional Western 12-tone notation.  However, it does _not_
hold for many _microtonal_ notations.

For example, if we are using the very common 'arrow' notation for
quarter-tones, there are two distinct accidentals that can be used to
represent the alteration +1/4 (i.e. quarter-tone-sharp): the first is a
natural sign with an up arrow, the second is a sharp sign with a down
arrow.  There is currently no effective, well-defined way to indicate
which of the two is desired at any given moment.

The arrow quarter-tone notation is just one of a whole variety of
microtonal notations which operate not on the basis of single symbols
per alteration, but on the basis of asuperposition of a successive
hierarchy of symbols, each corresponding to smaller and smaller shadings
up or down of the pitch.  For example:

       sharp/flat  +  up/down arrow  +  plus/minus
        +/- 1/2          +/- 1/4         +/- 1/8

Lilypond's consideration of pitch alteration as a single number makes it
very difficult to adequately represent such hierarchical
pitch-alterations, and hence their corresponding notations.

Hence why I say that the issue of effective microtonal support still
requires action at the code level, and is not simply a matter of better
documentation ... :-(

Best wishes,

    -- Joe