philipl@bronze.UUCP (Philip Lantz) (08/03/83)
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?
brucec@orca.UUCP (Bruce Cohen) (08/04/83)
I believe (not positive, bu I don't currently have access to any of my acoustics or sensory psychology/physiology books), that the reason the pure tones can be consonant or dissonant is that the ear is not a linear transducer. In converting the sound waves to mechanical movements of the sensory fibers in the inner ear a degree of non-linear mixing takes place, which generates harmonics (and sum and difference frequencies) which did not exist in the original sounds. Bruce Cohen UUCP: ...!teklabs!tekecs!brucec CSNET: tekecs!brucec@tektronix ARPA: tekecs!brucec.tektronix@rand-relay
ajh@sdcsvax.UUCP (Alan Hu) (08/07/83)
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