Re: Microtonality

[Hans Aberg]

On 27 Sep 2013, at 08:45, David Kastrup <address@hidden> wrote:

> Hans Aberg <address@hidden> writes:
> 
>> On 26 Sep 2013, at 17:16, Phil Holmes <address@hidden> wrote:
>> 
>>>> The section originates with me but I got diverted into trying to
>>>> create a more elegant solution for how to rewrite accidentals in
>>>> transposed music. It was all related to the need for an effective
>>>> chromatic transposition solution that also worked well with
>>>> arbitrary microtonal accidentals.

>>> I think it's waiting for someone to propose how it could be
>>> represented in LilyPond.
>> 
>> For one microtonal accidental, one needs, in addition to the
>> minor/major seconds m and M, a neutral second n. For a pitch x = r*m +
>> s*M + t*n, compute its degree deg(x) := r + s + t, which is its staff
>> position, and subtract the staff pitch.
>> 
>> There remains a new pitch, which I also call x, but now with r + s + t
>> = 0. As sharps/flats alter with a multiple of r - s, reduce using them
>> so that only one of r, s is non-zero.
>> 
>> Assume first that t = 1, i.e., one n. Then it must be either n - M or n - m.
>> 
>> We have six microtonal symbols, sharp/natural/flat with up/down
>> arrows, but it will, as we shall see, suffice with four. One way to
>> make a choice is to conceptualize n as below or above (m + M)/2: if it
>> is a small or large neutral. This choice is purely formal at this
>> point, but will be of importance when plugging in values.
> 
> [...]
> 
>> If the absolute value |t| of t is larger than 1, then one needs as
>> many arrows as |t|: up if t is positive, and down if t is negative.
>> 
>> Two symbols where not used: sharp with up arrow and flat with down
>> arrow. But they conceptually fall without the region of raising a
>> sharp M - m or lowering with a flat -(M - m), and can in fact be
>> reduced using a natural with up/down arrow plus a sharp/flat. So here,
>> one would need notation simplification algorithm.
> 
> Well, today's xkcd, at the surface more being about LilyPond's choice of
> extension language, still seems somewhat on-topic here:
> 
> <URL:http://xkcd.com/1270/> (mark the mouse-over text)

Indeed.

> Now I appreciate that you are no longer expounding on Abelian groups
> here, but this still is not a text you'll find in a musician's handbook
> (not even if he's called Arnold).  

You must be mixing me up with some microtonalist - they are well aware of this 
thing, though they did not use the "A" word.

> If you are interested in getting your
> ideas conceptualized in a manner that will make both musicians and
> LilyPond programmers understand them to a degree where they can work
> with them and actually want to do so, you need to diverge further from
> the abstract.
> 
> I remember that my initial (and it turns out terminal) reaction to your
> initial group theoretic treatise a year ago or two was "I'll read this
> some other time".  If you take into account that I'm the sort of guy who
> chose to do a treatise on number-theoretic transforms for convolutions
> as an undergraduate term paper in an engineering course, that should
> raise a lot of warning flags.  So how do we stop this from putting a
> terminal halt on the discussion this time?

What I wrote is more or less equivalent to engraving microtonal symbols 
(leaving that as an exercise :-) ). 

So if you, with mathematical training, for some reason find it intractable, how 
do expect others here to deal with?

One way would be that somebody asks questions: "hey, how does this work?".

Hans