Equal Beating Theorem

[Michael Wathen 556-9565]

Below is something that I found to be a mind opener.  I found that I
could teach my students how to figure out the appropriate test for any
interval in just a few short sessions.  Juxtapose this with two or
three years memorizing these same appropriate test after waiting for
someone to give you the answer.


The Equal-Beating Theorem                            Michael J. Wathen
                                                      February 3, 1993

Axioms

1).  The ratios of an interval can be represented by the frequencies of
the pitches involved.
           Example:  If A4 = 440 hertz and A3 = 220 hertz
                      then their ratio is:

				440/220


2).  Intervals are considered equivalent if the ratios of the
frequencies of each interval are equal.
           Example:  Taking the ratio above:


				440/220 is the same as 2/1

Definitions

An interval is just when the ratios of their frequencies can be
represented as the quotient of small positive whole numbers.  The
example from above is one of a just ratio.  This ratio may also
be regarded as the ratio of the partial numbers inverted.

Beats occur whenever one of the partial frequencies of one note of an
interval is in close range but  not equal to one of the partial
frequencies of the other note of the interval.  For example  m times f2
sounded simultaneously with n times f1  will produce beats, with m and
n being the partial numbers of the frequencies f1 and f2 such that the
difference of their products is small.  That is, |mf2-nf1|= á; where á
is some audible number of beats.


The Equal-Beating Theorem

If any two notes form a just interval, and from a third note sounded
simultaneously with one of the notes of the just interval beats are
produced, then the third note sounded simultaneously with the other
note of the just interval will produce an equal number of beats.


A theorem carries along with it a necessary sense of logic, its
conventions and its structure. A conditional statement is false only
when the consequent is false. This  means that to prove the statement
for the general case all that is necessary is to show that it is
impossible for the consequent to be false.

We have two conditions that we are to assume for the antecedent.  From
these two assumptions we will substitute the formal definitions and
rework them algebraically with the intent to deduce that the consequent
can indeed only be true.  The two assumptions of the antecedent
statement are: one, an interval is just, and two, a third note will
produce beats with one of the notes of the interval.  The mathematical
equivalent of these are listed below and come directly from the
definitions.

PROOF

Let the two notes of the interval be represented by the ratio of their
frequencies f1 and f2. Since the interval is just we must have:

			f2/f1 = n/m ......m,n are small whole numbers
Same as
			n*f1 = m*f2.


Let the third note be f3 with k its associated partial number that will
put its product in close range to, lets say, m*f2

			f3/f2 almost equal to  m/k
implies
 			k*f3  almost equal to  m*f2

and also by assumption:

			|k*f3 - m*f2| = x number of beats.

Now to deduce the consequent.  Simply put, because

			n*f1 = m*f2

we can substitute m*f2 for n*f1 into the second part of our assumption
to get:

		|k*f3 - n*f1| = |k*f3 - m*f2| = x number of beats.


The trick to proove an interval is just then is to find a good number k
such that the product of itself with the frequency of the third note
will be in close range to one of the other products.  If we have m and
n then we select a k to be a number close to m or n then this k forms a
ratio like:
		m/k

which looks like a just ratio but is in fact ratio of the partial
numbers of the one of the frequencies of the just interval and the
partial number of the third frequency.

Suppose I want to proof that some major third interval is in fact just.
Since I know beforehand that 5/4 is the ratio of a major third then to
find my third note or reference note I look for that k a small number
close to either 5 or 4.  How about 3?  Then I also have forehand
knowledge that 4/3 is the ratio of a fourth.  This tells me that I will
find my reference note a fourth above the higher note of my major third
because the fourth partial of that note is nearly matched to the third
partial of the reference note.  The Equal Beating Theorem guarantees us
that in this particular example that If the major third is just then
the fourth above the higher note will beat the same as the major sixth
above the lower note of the major third.


Michael J. Wathen
Tecnico de Pianos
College-Conservatory of Music
University of Cincinnati
Ohio EEUU  45221-0003
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