Equal Beating, what does it mean?

[Richard Moody]

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> From: Clark <caccola@net1plus.com>
> To: pianotech@ptg.org
> Subject: Re: Equal Beating, what does it mean?
> Date: Sunday, October 03, 1999 7:05 PM
> 
> 
> 
> Jorgensen's _Tuning_ being right next to me,> 
> "Equal-beating temperament: any temperament that contains two or more intervals
> that beat exactly the same speed. In this case, the intervals can never be the
> same size." (glossary, p771)

 But what does "size" mean?   is it the ratio of the interval or the beat rate? It
is possible to have two thirds in an octave beating the same.  OK, their ratio's
might not be exactly the same mathematicaly, but musically they are  heard as
thirds. And musically this might not lead to a "proper" temperament, but I bet I
could  tune such a temp and palm it off as an ht. 
> 
> A search of the pianotech archives leads me to believe that this definition is
> more strict than what may be accepted, since I would interpret the above as
> meaning different intervals (ie. octaves and fifths, fourths and fifths, etc.).

Yes.  In certain temps SOME of the Thirds and Sixths from the same root beat the
same.  In ET within the octave, the fourth on the bottom beats the same as the fifth
on top, but the fourth on top beats twice as fast as the fifth on the bottom. 
Indeed equal beating can be used to prove an octave. A minor third on bottom equals
a major sixth on top.(when you can hear it) (gives 6:3 including inharmonicity)  > 
> 
> This may be useful as far as defining specific instances of EB intervals, not
> accounting for inharmonicity. For EB {M3, M6} relative to a common root {P1};
> interval names mean the freq. of the upper note in hz.
> 
> M6=ABS({4(M3)-[(5+5)*P1]}/-3)
> P4=M6-M3
> P5=2(M3)-M6
> m3=2(P1)-M6

If you could supply some numbers and work it through...it looks like it could be
simplified, as (5+5) looks like it could be (10). The formulas I am familiar use
intervals presented as ratios.  ie  M6 = 5/3.  Which also gives the beat rate,  the
5 and 3 signify the coincident harmonics, the difference of which gives the beat per
second. M3 = 5/4. Thus if M6=M3 then 5/3 = 5/4.  Or the fourth harmonic of the M3 is
equal to the third harmonic of the M6, giving an equal beat rate between M3 and M6..
I suppose that means a pure fourth between the upper note of the 3rd and upper note
of the 6th.....? ? 
 
> The limitation of this formula is that it will not automatically generate the
> remaining notes in the scale. I'm interested to see how you are determining the
other
> intervals.

In most temperaments (12 tone) the intervals are determined by the multiplication of
a constant fifth 11 or 12 times. OK, in at least three temps this is the case.   ie
in  Pythagorean there are 11 pure fifths at 1.5.  In Meantone,there are 11 fifths at
1.495.  In  ET there are 12  fifths at 1.498.  Thus if you take a starting  note of
any freq and multply it by 1.498307 twelve times and divide by 2 when ever necessary
you will get a 12 tone ET scale ending at double the starting freq. 
	Now there some temperaments (again 12 tone) that use fifths of TWO diff sizes. One
is these is the Werckmeister III  recently posted.  Here 6 pure Fifths are tuned
down from MC (middle C) to and including Gb.  Next 3 meantone fifths are tuned up
from MC.  then two pure fifths, which gets us to B.  Now from B to F# is supposed to
be another meantone fifth, but remember that F# has been already  tuned as Gb, so we
cannot this Fifth.  This is the intrique of this temp, is the resulting B--F#/Gb
Fifth really a Meantone Fifth??. 

Below are freqs of f# and B from a spread sheet I constructed to create beat rates
for a Werck III from data in the earlier book by  Owen Jorgesen.  

367.497 / 246.037 = 1.493665586883     a meantone 5th is 1.49534.   The cents diff
is 
1.97cents.    That is close enough for Werckmiester, but how did he figure out that
the untuned fifth in his scheme would be a Meantone fifth or very close?

BUT the super intrigue of this Werck III is the fact that only three notes need be
tempered, and they are tempered according to equal beating combinations. 	You can''t
miss.After the 6 fifths are tuned pure down from c,   d is tuned to beat equal with
Bb and f# .(Bb=d=f#) Here are two equal beating thirds, that got mentioned
earlier...: )...Then G is tuned to beat equal with c an d,
(G--c = G--d) and F is tuned to beat equal with A  and A--d. Now this is supposed to
result in 8 pure fifths and four meantone fifths. Does it really?  The spread sheet
shows very close.  The G--d G--c is off by half a beat per second.  

MY question is did Werckmiester realize only three notes need be tempered, or is
this Jorgensen's conclusion?  If either it is certainly a brilliant deduction. If it
is that easy then it should have been very popular.   And being so simple I can't
see how it would have been referred to as "well tempered" ---ric