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From: Silvain Dupertuis <silvain-dupertuis@bluewin.ch>
To: lilypond-user@gnu.org
Cc:
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Date: Wed, 18 Jan 2023 19:20:35 +0100
Subject: Re: irrational meters
To make things clear, a fraction is a mathematical
_expression_ which is a specific representation of a
number (or of a formal _expression_), comprising a numerator, a
denominator, and one kind of division sign (which can be written
in different ways, as there are different ways to write the
operator).
A ratio of two numbers is a number resulting from the
division of those two. It can therefore very naturally expressed
as a fraction.
So I have no problem considering a musical metric as a
fraction!
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!
But whenever we write a fraction in a mathematical _expression_,
it is the value which is implied, not the fraction
itself.
So we do confuse them very commonly, like we commonly confuse a
word and it's meaning in common language (and we survive quite
well with this confusion).
In case we need to avoid these confusions, one can use
quotation marks... It can be useful or essential when teaching
mathematics or when working in mathematical logic or in
linguistics.
Hope this clarify the matter.
Silvain
Le 18.01.23 à 18:51, Shane Brandes a
écrit :
Because terminology amuses me here. Years ago, I learned
that time signatures were decidedly not fractions but ratios
from a one Richard Hoffman. But even before that I learned
ratios consisted of antecedents and consequents, which also
seems to overlap musical structural terminology in a weird way
making that also fairly useless as a nomenclature.
Shane
--
Silvain Dupertuis
Thanks, that was a wonderful discourse.
It made me wonder about the caution to not mix up the fractional representation with the number it evaluates to.
Made me wonder, what does the value represent?
I guess it is obvious once you think about it, but it represents the number of whole notes in the bar.
So, 4/4, 2/2, 1/1, and 8/8 all have a value of 1 and represent a whole note in total duration.
Also makes me wonder: do we have a word for time signatures that have duration other than a whole note?
In this sense, evaluating the duration of non-dyadic signatures is just as transparent as those that are not a whole note in duration.
For example, understanding the length of 3/4 is straightforward, and evaluating the length of 6/8 and 4/6 are about the same, since you have to reduce the fraction to 3/4 or 2/3.
However, from a philosophical perspective, I have a difficult time thinking of the time signature itself as a fraction. I feel like it is rather the combination of a multiplier and a fraction.
The bottom number tells you what note gets the beat, which we represent as fractions of a whole note, with an implied numerator of 1. Then the top number tells you how many beats are in the bar.
Combining a multiplier with a fraction of course yields another fraction.
But the meaning of the resulting fraction is not as obvious, or I guess is not typically directly musically relevant. So musically, I don't think we tend to think of time signatures as fractions.
In fact, even the fractional part of this, which is baked into the concept of which note type gets the beat, I would argue, is not typically experienced as a fraction. In fact, it seems like what time signatures are doing is more integer-ization of a fraction: it says that this thing that was previously defined as a fraction, that thing should now be considered 1 unit beat. (Ignoring compound time conventions).
So, I guess I come to the conclusion that the only reason we think of the time signature as a fraction is because we define our note values as fractions of a whole note.
Elaine Alt
415 . 341 .4954 "Confusion is highly underrated"
elaine@flaminghakama.com
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