alves@aludra.usc.edu (William Alves) (11/10/89)
In article <1176@apctrc.UUCP> zcch0a@spock.UUCP (Chris Humphrey) writes: >The equal tempered western scale has an interesting >property: every interval (but 1) in the first 21 intervals >is a harmony, and nearly every possible harmony is >represented. By harmony do you mean "consonance"? >Many of the intervals are not >exact ratios, but all are within 1%, and there is no >smaller ratio that is closer. I'm afraid you're mistaken about that, assuming that pitch perception is non-linear; that is, when we look at cents many of these intervals you give are way off the mark: 5:6 = 315.641287 cents, thus the equal tempered m3rd is about 15 cents flat 5:4 = 386.313714 cents, the equal tempered M3rd is about 14 cents sharp Thirds sound awful in equal temperament to my ears, compared to these just ratios, and that is the big reason composers from about 1500 to 1750 rejected equal temperament, preferring to have their thirds sound good in the common keys at the expense of poor thirds in the less common keys. Despite what you say later, thirds were very important to composers of that period. 5:7 = 582.512193 cents, a beautiful interval, but not even close to the equal tempered tritone. 7:13 = 1071.70175 cents, even farther off from a major seventh. The old problem of finding a just equivalent of the tritone was studied with a computer in the Computer Music Journal a while back. There were several alternatives. >Each of these harmonies has its own distinct mood and feel, >some "positive", some "negative". Medieval music favored >2:3, in the Renaissance and Baroque they liked 3:4, and >romantic and pop music prefers 4:5. Where did you get this information? I'm afraid I don't think it is very meaningful or accurate. >It is possible to run interesting graphics off the interval >between successive or simultaneous notes. I wrote a program >for the Commodore-64 which makes very beautiful graphics >based on the harmonies. See John Whitney's "Digital Harmony: On the Complementarity of Music and Visual Art." Bill Alves USC School of Music / Center for Scholarly Technology