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Path: utzoo!linus!philabs!sdcsvax!ajh
From: ajh@sdcsvax.UUCP (Alan Hu)
Newsgroups: net.math
Subject: Re: Scales again
Message-ID: <3623@sdcsvax.UUCP>
Date: Sat, 6-Aug-83 19:59:51 EDT
Article-I.D.: sdcsvax.3623
Posted: Sat Aug  6 19:59:51 1983
Date-Received: Mon, 8-Aug-83 23:56:44 EDT
References: <679@bronze.UUCP>
Organization: U.C. San Diego, CS Dept
Lines: 43

First of all, pure sounds do exhibit consonance and dissonance (I can't spell!).
Consider the following:  If you have one source playing sine waves at
440 hz (The A above middle C) and one playing at 441 hz, the sine waves
will drift in and out of phase resulting in beats, one per second.
As the frequencies drift apart, the number of beats increase.  Around
5 beats per second starts sounding really bad.  Now, if one source plays
440 hz and the other plays up one octave at 880 hz, there will be no drifting
in and out of phase; every wave from source 1 matches 2 waves from source 2.
There are no beats and the sound is pleasant.  Similar things happen with
perfect fifths and thirds.

On the subject of overtones, the actual sequence of notes if something like
(I don't have my handy Harvard Dictionary of Music with me.)
c' (I think that's middle C.) 		256 hz (I think that's pretty close.)
c'' (That's one octave up)		512 hz
g''					768 hz
c'''					1024 hz
e'''					1280 hz
g'''					1336 hz  and so on.
						(I don't remeber any more.)
Each frequency is an interger multiple of the original note.
All other notes can be derived from this.  If we let C be 1,
the G an ocatave above the fifth will be 3.  Therefore, the G
which is the fifth above C is 3/2.  Similarly, the E which is
a third above the C is 5/4.

All this garbage applies only to so called "just intonation".
A problem with this system is that intervals of the same size on
the keyboard have different sounds.  For example, C-G is consonant,
but D-A, isn't.  Most instruments are (de-)tuned to "well-tempered"
scales in which the ratio of succesive notes is exactly the same.
Even so, you can perform some simple experments on a piano to illustrate
some of these things.  Silently depress the C above middle C.
Stike middle C loudly.  You can here the upper C very clearly.
Silently hold g-c-e and strike the C two octaves below, loudly.
You can here the chord quite plainly.  You can do lots of other
(not very) interesting experiments like this.

I think I'm getting pedantic, and I'm probably boring you guys/gals/
people/others to death, so I'll stop.

					Alan J. Hu
					...sdcsvax!ajh