Posted by John Swensen on November 12, 2001 at 13:19:15:
In Reply to: can-o-worms posted by js on November 11, 2001 at 23:30:52:
Joe's observations are entirely consistent with the established theory on intonation.
Most of us hear intonation problems through the beating of two notes together. For example, suppose we play two sine waves, matched to 1% accuracy, one at 110 Hz (cycles per second), the other at 111 Hz. The tuning is 1 cent off, and the two notes will beat with a frequency of one per second. If we are playing anything faster than a quarter note at 60 MM, we will never hear the beating. If we repeat the experiment at 880 Hz and 888 Hz (still 1 cent difference), the beats will be 9 per second, and we will hear beating if the notes are slower than 32nd notes at 60 MM. The higher the frequecies of the notes, the easier it is to hear the beats (this is why the easiest way to get two piccolos in tune it to prevent one of them from playing).
We don't, typically, play sine waves in music, and the harmonics of the notes contribute to the beating, as well. Our 110 Hz fundamental will have harmonics at 220 Hz, 330 Hz, 440 Hz, 550 Hz, etc., and our 111 Hz fundamental will have harmonics at 222 Hz, 333 Hz, 444 Hz, 555 Hz, etc., with beat frequencies of 2 Hz, 3 Hz, 4 Hz, 5 Hz, etc., respectively. If we play a 111 Hz fundamental, our 444 Hz harmonic will beat with the oboist's 440 Hz fundamental, provided there is enough energy in that harmonic to be heard.
A big, 6/4 tuba with a fundamental-rich, dark sound, will have much less energy in the higher harmonics than a smaller, brighter tuba, so beating due to intonation problems will be much harder to hear on the big horn as on the smaller horn.
Intonation, as well as many other, issues are discussed in Arthur Benade's excellent book,
Fundamentals of Musical Acoustics.