Equal Beating Theorem

[Michael Wathen 556-9565]

Michael Wathen wrote:

 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.

in reply David Canright wrote:

Michael, your post was very interesting.  But I find the wording of the
statement above misleading.  Apparently what you actually mean is that the
third note must be chosen so that one of its overtones beats with the
*coincident overtones* of the two tones in the just interval.  Your wording
implies one could pick the third tone to be any that beats with one of the
others.

In reply:

David Canright,

It is as you say.  It is funny how long you can look at something and not
see it.  I should try to tighten up the wording to make the statement as simple
as possible.  Simplicity adds beauty.

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 slower
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 third
below 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