Re: irrational meters

[H. S. Teoh]

On Wed, Jan 18, 2023 at 07:27:47PM -0500, David Zelinsky wrote:
> Silvain Dupertuis <silvain-dupertuis@bluewin.ch> writes:
> 
> > It is better not to confuse a /fraction/ (as an expression) and it's
> > /value/ (as a /number/) ­­— a number cannot have a numerator or a
> > denominator!
> 
> Well, a *rational* number does have a well-defined denominator:
> Because of unique factorization of the integers, there is a unique
> representation of a rational number as n/d where n and d are integers
> with no common factor and d is positive.

True, but time signatures do not have a 1-to-1 correspondence with
rational numbers: 6/8 and 3/4 are distinct as time signatures, even
though as fractions they signify the same rational number.


> Now if we wanted to think about time signatures N/D where N and D are
> elements of, say the integers with the square root of -5 adjoined,
> that's another matter.  Any takers, you composers out there?  :)
[...]

Adjoining square roots of some positive number may still be somewhat
imaginable, e.g., √5/4 means each measure consists of √5 (approximately
2.236...) quarter notes. Hard to perform precisely without computer
help, but still possible in theory.  But how is one to interpret √(-5)/4
??  How do you count up to an imaginary number of beats per bar, even in
theory?

//

Further thoughts about √5/4 as a time signature: due to the peculiarity
of standard musical notation, which only has symbols for note values
that are rational subdivisions of a whole note (and even that, not all
rational subdivisions are directly representable), any combination of
notes in a √5/4 bar will not fit exactly within the bar, √5 being
irrational. (This also applies to any other irrational fraction you may
choose for the top component of the time signature.) So that means the
last quarter note beat will be split across the bar line by some
irrational fraction, with the remaining duration overflowing into the
following bar.

Note that changing the notation may not help anything. Suppose we
introduce new notation for representing irrational note values. Then
either the new note values are commensurate with the (irrational) time
signature, or they're not.  If they are, then we've effectively reduced
the time signature to a traditional, rational signature: if we introduce
a note value of √5/6, for example, then 6 such notes would add up to 1
bar in a √5/4 signature, so we've effectively turned it into 6/4 and
it's no longer an irrational time signature. If the new note values are
not commensurate with the time signature, then we will still have notes
split across barlines even if we use the new note values, so nothing has
really changed.

Now if you have a long series of consecutive quarter notes, they will
span some number of bars, and every bar line will split a quarter note
in some unique, irrational fraction -- due to irrationality, no bar line
will fall exactly at the start/end of a quarter note, so excepting the
start of the first bar, everywhere else there will be notes split across
the barline. And each split note will be split in a unique, irrational
fraction.  If we individually move each barline so that it falls on the
closest start/end of a quarter note, then we end up with a series of
bars with regular (rational) time signatures, but with time signature
changes in an irregular pattern -- this is what I referred to in an
earlier post.  So, "rationalizing" an irrational time signature in this
way, we see the irrational number essentially serves as a source of
irregular bar lengths that never repeats.


T

-- 
The two rules of success: 1. Don't tell everything you know. -- YHL