<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=Content-Type content="text/html; =
charset=iso-8859-1">
<META content="MSHTML 6.00.2800.1264" name=GENERATOR>
<STYLE></STYLE>
</HEAD>
<BODY bgColor=#ffffff>
<DIV><FONT face=Arial size=2>Hello list</FONT></DIV>
<DIV><FONT face=Arial size=2>Like all of you, I imagine, I was first =
made aware
of these beats when listening to a piano (in my case pipe organ) being =
tuned. My
curiosity led me to analyse the reason and I turned to simple =
mathematics for
the answer. Enter my "small equation" where the fundamental (1st.Harm.) =
is
multiplied by the 12th. root of 2 to derive the next fundamental in the
ascending equal tempered chromatic scale. (Simply devide if you =
want to go
down chromatically). In every case the resultant becomes a Fundamental =
Note - or
1st.Harmonic. I know this is all "old hat" but there's probably some =
meat in it
to be chewed over just the same. Now taking the Fundamental and =
multiplying by
"Two" one gets the second harmonic. Multiplying by "Three" gets the =
third
harmonic ad. inf. The really interesting point now comes... Take your =
M10 based
on A# (which computes at 466.16376.....)The 5th. harmonic of that A# = =
2796.9826...... the M10 is F(which computes at 1396.9129.......) The =
2nd.
harmonic of that =
F =
=2793.8259......
There we have a difference of 3.1567..... Hz which is the beat you hear. =
The
same applies, of course, to any and all intervals in the equal tempered =
scale,
be they M3,M4,M5 - whatever.</FONT></DIV>
<DIV><FONT face=Arial size=2>It is interesting to apply that "small =
formula" to
log all the notes of the equal tempered scale on a mathematical basis =
and
thereby to extract the "Beat-rate" information whole-sale. To me this
information is "an invisible tool" - and I use it - as do you. But it's =
nice to
<EM>see</EM> an invisible friend sometimes! </FONT></DIV>
<DIV><FONT face=Arial size=2>Of course! There are many pitfalls! =
Many pianos do
not produce the exact mathematical frequencies according to my table. =
Those
times when the first covered (wrapped) Bass string are played with their =
M10 -
and the beat rate does not conform to the mathematical progression of =
the tables
as produced in the "metals" section of the piano. Well..., That's life =
:-)
I find the "scientific calculator on the PC best as it produces so many =
decimal
places! In using such a calculator you can go right up the scale =
and land
up, an 8ve later with exactly twice the number you started out with. Now =
that's
Magic! </FONT></DIV>
<DIV><FONT face=Arial size=2>Regards</FONT></DIV>
<DIV><FONT face=Arial size=2>Michael G =
(UK)</FONT></DIV></BODY></HTML>