Think I can figure out your question (but realize that my students drug me
to the bar after class ended tonight and I am not really responsible for
being logical and old grannies are not supposed to be bothering their
fragile brains with logs and exponents and such but......)
Marty's Magic number we will call A (could have been X but I liked A
You know that to get from Note 1 to a note a half tone higher you multiply
the frequency by A. We will use F1 to stand for the initial frequency.
to raise it an octave you must raise it 12 half tones or
F1*A*A*A*A*A*A*A*A*A*A*A In old FORTRAN notation A**12 indicating A to the
power 12... sorry I work in plain text and do not have exponents.
We know that the frequency of a note an octave higher is twice that of the
original note SO.... 2*F1 is equal to the note one octave higher
F1*A**12 = 2F1 (or F1 multiplied by A 12 times is equal to twice the
Dividing both sides of the equation by F1 you get rid of them and are left
A**12 = 2
To get Martys magic number, A, we have to handle that messy exponent and
that calls for the magic of logs.
Take the logarithm of each side..
ln(A**12) = ln(2)
12*lnA = ln(2)
ln(A) = ln(2)/12
taking the exponential of each side of the equation
exp(lnA) = exp(ln(2)/12)
But the left side of the equation is merely A
So you get A = exp(ln(2)/12) or
A = 1.059463094 or at least that is what I got when I typed
=exp(ln(2)/12) into an excel worksheet (couldn't find my slide rule)
Is that close enough to your number?
Note it does not matter what base you are working, base2, natural logs or
base10 etc just as long as your base is consistent throughout
Happy Halloween - I am headed for another chocolate bar then to get some
sleep - a three hour class of bird and butterfly identification has done me
in. Hope I managed to explain this clearly
From: Marty Michener [mailto:marty@...
Sent: October 31, 2001 1:18 PM
Subject: Re: [Nature Recordists] RE: Log base 2 and Halloween Sounds
>HI Barb: I love discussions of sound and music and numbers, esp.
>logs. All the sensory systems seem to do log input tricks.
Here is a question that is seldom asked or even wondered,
that I asked myself in high school:
What is the numerical music interval, the number by
which you multiply any music note by to get the
next (halftone) above it. Hint twelve of this operation
makes exactly an octave.
I use this number off the top of my head to at least
get music teachers attention and wonder some about math.
Ans: ca. 1.0595 - But how do you get it? - it is almost as
handy to know as pi, since you can always figure out the pitch
of any note starting with A=440 and then multiplying a lot. ;^)
my very best,
MIST Software Associates
75 Hannah Drive, Hollis, NH 03049
coming soon : EnjoyBirds bird identification software.
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