From: sumrn@dssv01.crd.ge.com (Robert N. Sum)
Newsgroups: alt.guitar
Subject: Re: Note Frequencies needed
Keywords: frequency, note, hertz
Date: 17 Dec 92 14:25:27 GMT
Reply-To: sumrn@crd.ge.com
Organization: GE R&D, DICE Program

In article <102410@netnews.upenn.edu>, lginzb@ccat.sas.upenn.edu (Lionel
Ginzburg) writes:
|>In article <BzDu2M.Ao4@ecf.toronto.edu> ricegm@ecf.toronto.edu (RICE
George Matthew) writes:
|>>Hi all,
|>>
|>>I was wondering if anybody knew offhand what
|>>frequencies each of the strings vibrate at 
|>>(assuming that they're properly tuned of course :-?
|>>
|>Open E (6th String) 82.4hz
|>Open A (5th String) 110hz
|>Open D (4th String) 146.8hz
|>Open G(3rd String) 196hz
|>Open B (2nd String) 246.9hz
|>Open E (1st String) 329.6hz
|>A440 (1st String, 5th Fret) 440hz
|>
|>Note that if a note is an octave higher, frequency is doubled.  (And
|>likewise, if you go an octave down, frequency is halved.)
|>
|>>Also will any higher or lower harmonic frequencies also
|>>be obnoxiously present. The lower ones don't matter
|>>too much but the high ones ...ARRRGGHHHH.
|>
|>All notes sounded on a guitar (And indeed, any non-synthesized
|>instrument) will not be pure harmonically. (i.e. a sine wave)  All will
|>have overtones and undertones.  There is a pattern and an equation to
|>this, but I don't remember it offhand.
|>
|>>

  Just to add my $0.02:  The A above middle C being set to 440 Hz was
agreed upon near the beginning of this century as the usual reference pitch.
There are, however, groups that use somewhat different reference points.

There is some fairly simple math for calculating the frquencies, given certain
conditions such as the tuning system being used.  For example, if you are using
equal temperament (which is most common among electric instruments and those
with fixed tuning e.g., guitars, pianos, etc.), then the octave is split
into 12
equal parts logarithmicly.  (Logarithmically because that's how your
ears repond
to frequency.)  The octave is the only interval that retains its "pure"
relationship which, in terms of stringed instruments, results from using a
string exactly half (or twice) as long--everything else being equal.  Note: 
This is only a rough explanation, but basically the way the octave is
established.  See your basic physics book among other possibilities for
a deeper
explanation.

  So given our (arbitrary) reference point of A above middle C at 440 Hz and
the factor of 2, we can calculate the frequencies of all the As we want:


    27.5   55  110  220  440  880  1760 3520 7040 14080


(We note that our ears really don't work below about 20Hz or above
22kHz.  [This
limitation and a bit of information theory tells us why digital systems
like CD,
DAT can sample at around 44.1kHz or a little more and capture everything that
your ear can!])

  But, what about all the other notes?  Well, because our ear reacts
logarithmically and because western culture splits the octave into
twelve parts,
we split the factor of two into twelve parts via its 12th root i.e., the number
when raised to the 12th power gives us 2: approximately 1.059463.  For the
octave just above A440 we have

    A      A#=Bb  B      C      C#=Db  D      D#=Eb  E      F      F#=Gb  G
    440.00 466.16 493.88 523.25 554.37 587.33 622.25 659.25 698.46
739.99 783.99

    G#=Ab  A
    830.61 880.00

Note that it is a characteristic of equal temperament that the various #'s and
b's coincide.  There are tuning systems in which they do not.  Equal
temperament is a compromise that has this property.

  To compute the frequencies in other octaves, you have the choice of using the
above table and factors of 2 or computing by counting the distance from
A440 and
multiplying by the appropriate power of two.  For example, middle C is 9 notes
below A440 so

   440*(2^(-9/12)) = 261.63 = 523.25/2

where the - is used because we're going down and ^ indicates raising to power. 
(I've tried to keep things to only a couple decimal places.  Ear can't really
hear much more than that--depending upon ocatave, etc.  I ought to go
back to my
significant digit lectures from my science teachers.)

|>>
|>>Thnx in Advance,
|>>
|>>Matthew
|>
|>
|>-- 
|>Lionel Artom-Ginzburg
|>EMail to: Lionel@BWH.Harvard.edu
|>or, just autoreply.

  In any case, maybe some of you will find this interesting,

    -- Bob

==
  sumrn@crd.ge.com

  Robert Sum
  G.E. Corp. R. & D.
  P.O. Box 8, Rm. KW-C287
  Schenectady, NY 12301

  Tel. (518) 387-7696

Please note that the "Standard Disclaimer" applies here.