Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!linus!decvax!tektronix!tekmdp!bronze!philipl From: philipl@bronze.UUCP (Philip Lantz) Newsgroups: net.math Subject: Re: Scales again Message-ID: <679@bronze.UUCP> Date: Wed, 3-Aug-83 11:45:16 EDT Article-I.D.: bronze.679 Posted: Wed Aug 3 11:45:16 1983 Date-Received: Thu, 4-Aug-83 07:48:39 EDT Lines: 61 There is definitely something intrinsically beautiful about octaves, fifths, etc.; this is NOT an assumption! It is agreed to by all musicians, and is explained by physics (which I am about to do, if I can). There may be some minor things in here that are not 100% correct for the sake of simplicity. A sound made by a musical instrument is a mixture of different frequencies. The frequency we "hear" is called the fundamental, and is the lowest frequency. The rest of the frequencies are integer multiples of this one. This is caused by the way the sound is made in the instrument: If the sound is made by a string vibrating, for example, the string can vibrate with a wavelength of the length of the string, or a wavelength of half the length of the string, or one-third, etc. The ends of the string are both fixed, so it can't sustain a vibration at any frequency in between. Actually the string vibrates at all these frequencies at once, with amplitude of the vibration less at higher frequencies. The same effect occurs when the sound is made by air vibrating in a room. In this case, the ends are "fixed" by the essentially constant air pressure in the room (for a pipe with an open end), or by the constant volume available (for a pipe with a closed end). There are more complications dealing with open and closed ends, which I won't go into, unless I'm asked. When two notes whose fundamentals are an octave apart are sounded together, the frequency of the second harmonic of the lower note is the same as the frequency of the fundamental of the higher note. The fourth harmonic of the lower note is the same as the second harmonic of the higher note, and so on. There is a high degree of what musicians call consonance; the notes sound good together. Non-musicians can hear this, too; it's NOT training. (By the way, I'm not a musician, but I come from a family full of them.) When two notes are sounded together that are a perfect fifth apart, the third harmonic of the tonic is the same as the second harmonic of the fifth; the sixth harmonic of the tonic is the same as the fourth harmonic of the fifth. There is less consonance than there was with the octave, but still more than any other combination of two notes. The tone sounds good. When you play two notes a "seventh" apart, none of the harmonics match, until you get up to such a high frequency that the harmonics are so weak that they are barely or not audible. I put seventh in quotes because that term has no meaning except in connection with a equal-tempered scale; it doesn't correspond to any pleasing interval. By the way, I believe this does belong in net.math; music is a VERY mathematical thing. I believe net.music is concerned more with performers and recordings, whereas this discussion is about the mathematical and physical nature of music. I hope this helps anyone who is confused, but interested. If there are still things to be cleared up, let me know by mail, and I'll see if I can help. (Better yet, though, go to the library. It's hard to draw charts and diagrams on a terminal.) Philip Lantz tekmdp!bronze!philipl P.S. It seems to me there is an explanation of consonance that works for pure tones, also, (i.e., those with no harmonics present), but I couldn't remember it. Do pure tones exhibit consonance and dissonance, and if so, why?