From: ted@nmsu.edu (Ted Dunning)
Subject: Re: Fundamental Frequencies of the Musical Notes
Reply-To: ted@nmsu.edu
Organization: Computing Research Lab
Date: Sat, 2 Jan 1993 00:38:24 GMT



i got a question on my last posting and reckon that my answer will be
of rather general interest.  i am sure that there are quite a number
of people out there who can correct me in the places that i go wrong.
please feel free to do so, if you know better than i.



To: williamb%ee.ubc.ca
In-reply-to: william burchill's message of Fri, 1 Jan 93 15:56:15 -0800 <9301012356.AA11807@fs0.ee.ubc.ca>
Subject: Fundamental Frequencies of the Musical Notes
Reply-to: ted@nmsu.edu
--text follows this line--

>   Could you tell me, is there agreement on the absoulute frequencies?

440 A and 256 C define slightly different scales that are used in
slightly different situations.  i think that most pianos are usually tuned
with 256 = C, while orchestras (in my limited experience) invariably
use 440 = A

>   Also, what is a "well tempered scale" and where can I read more on
>   this subject?

originally, scales were defined by going around the `circle of fifths'
(or the essentially equivalent circle of fourths).  this is done by
going up a perfect fifth from your base note (let's pick middle C).  a
perfect fifth is a factor of 1.5 in frequency and corresponds to 7
semi-tones.  from C, this takes us to G which would thus have a
frequency of 1.5 * 256 = 384 hertz.  doing this repeatedly gives us
the following frequencies:

tone	reduced	freq	note	
0	0	256	C
7	7	384.0	G
14	2	288.00	D
21	9	432.00	A
28	4	324.00	E
35	11	486.00	B
42	6	364.50	F+
49	1	273.37	C+
56	8	410.06	G+
63	3	307.54	D+
70	10	461.32	A+
77	5	345.99	F
84	0	518.98	C

sorting this by frequency gives us

tone	reduced	freq	ratio	note	
0	0	256		C
49	1	273.37	1.067	C#
14	2	288.00	1.053	D
63	3	307.54	1.067	D#
28	4	324.00	1.053	E
77	5	345.99	1.067	F
42	6	364.50	1.053	F#
7	7	384.0	1.053	G
56	8	410.06	1.067	G#
21	9	432.00	1.053	A
70	10	461.32	1.067	A#
35	11	486.00	1.053	B
84	0	518.98	1.067	C

there are several problems with this.  first, C an octave up isn't
right (it is 1.4% off, which is plenty enough to sound terrible).
secondly, chords other than the fifths used to construct the scale
sound off.  for example F# and the C above middle C have a frequency
ratio of 1.42 instead of the desired 1.5 (and if we use 512Hz instead
of 519Hz the ratio is 1.40 instead).  this error is absolutely gross
if you actually hear it.

to avoid this, many different tunings were developed other than the
one based on perfect fifths or fourths and great efforts were made to
avoid using the intervals that sounded bad in each tuning.  with
plucked or bowed string instruments, this problem isn't too terribly
bad since the natural, good-sounding chords are also the natural
fingerings for the most part.

just this problem of tuning was exactly the source of a profound
problem in pythagorean philosophy which was based entirely on the
apparent fact that various harmonious ratios seemed universal.
ultimately, it turned out that these ratios weren't all they were
cracked up to be, and irrationals had to be developed.

finally, bach popularized what he called `well tempered tuning'.  i
don't know the latin or german original phrase, but i am pretty sure
that bach didn't actually invent the tuning, but was merely the most
prominent composer to write prolifically for it.  this tuning allows
much greater flexibility in making chords by spreading out the errors
which occur in a natural tuning at the cost of making all the
pythagorean intervals such as fourths and fifths sound every so
slightly off.

this adjustment can be done numerically as was illustrated in my
previous posting.  i think that in practice, some slight adjustments
are still made so that common chords sounds slightly better than
others.  i know for certain that many synthetic music sources make
adjustments to the well tempered scale because there are limitations
on the frequencies which can be generated easily by frequency
division.

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