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 with that same partial
that is coincident in the just interval, 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 prove 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.

In the world of aural tuning The Equal Beating Theorem is the most powerful tool we have available. It can be used to demonstrate whether an interval that is not quite just is either wide or narrow of just. This makes it indispensable in setting an equal temperament. For example, suppose I tune a 3/2 fifth according to a beat chart calculated for equal temperament and suppose that chart gives a beat rate of 3 beats in five seconds as a narrow of just interval. It is difficult to tune an interval with a slow beat pattern. Generally the slow beat the harder it is to verify that it actually exists. We have the numbers 3 and 2, pick another number close by. How about 5? Well if the 3rd partial of the bottom note of the fifth is matched with the 2nd partial of the top note of the fifth then 5 represents the fifth partial of the reference note. I know going into this that 5/3 represents the ratio of a major sixth. I can put the beat rate of this sixth in a comfortable place, say four beats a second larger than just. Now suppose I find that this reference note played with the upper note of the fifth beats three times a second then this tells me that the fifth is most likely narrow. I say most likely. The converse of the Equal Beating Theorem is not always true. In this case, however, if the 5/2 interval was beating three times a second on the narrow side you can bet that the fifth could not be mistaken for a nearly just interval.

The Equal Beating Theorem also proves invaluable for dealing with and verifying aurally inharmonicity in instruments such as a piano. Piano tuners generally refer to octave types when talking about tuning of octaves. For example, I were tuning A3 to A4 I might want to tune this as a 4/2 just octave. Then I would look at the 4 partial of the lower note and match it in frequency with the 2 partial of the higher note. To prove that it is in fact just I would need a reference note which also has this frequency or near it as one of its partials. We have 4 and 2. How about 5? I know going into this that 5/4 is the ratio of a major third. So my reference note can be found a major thirdbelow the bottom note.Next I look at the 6/3 coincidence for this same octave. Reference note? How about 5 again? I'm willing to bet that your A3 - A4 octave tuned just at the 4/2 level will not be just at the 6/3 level.

Michael Wathen