Path: utzoo!attcan!uunet!apctrc!spock!zcch0a From: zcch0a@spock.uucp (Chris Humphrey) Newsgroups: comp.music Subject: Re: Eliminating the octave [Re: spectral composition] Keywords: Intonation systems, octaves, pianos Message-ID: <1176@apctrc.UUCP> Date: 8 Nov 89 20:47:01 GMT References: <6066@merlin.usc.edu> <3007@husc6.harvard.edu> Sender: news@apctrc.UUCP Reply-To: zcch0a@spock.UUCP (Chris Humphrey) Organization: Amoco Production Company, Tulsa Research Center Lines: 52 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. I have not made an exhaustive computer search for "lost chords", but there appear to be only a few, such as 6:7. The above statement is based on the empirical discovery by Pythagoras (not numerology) that what we hear as a harmony is a small rational number in the ratio of frequencies. Many of the intervals are not exact ratios, but all are within 1%, and there is no smaller ratio that is closer. This is not true for interval 1, i.e. two notes next to each other. Counting all the white and black keys, the correspondence between interval and small rational number is as follows: 2 - 8:9, 3 - 5:6, 4 - 4:5, 5 - 3:4, 6 - 5:7, 7 - 2:3, 8 - 5:8, 9 - 3:5, 10 - 5:9, 11 - 7:13, 12 - 1:2, 13 - 7:15, 14 - 4:9, 15 - 3:7, 16 - 2:5, 17 - 3:8, 18 - 5:14, 19 - 1:3, 20 - 4:13, 21 - 3:10. 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. What I think might be interesting (though possibly disconcerting because of unfamiliarity) is to dynamically alter the tuning of chords to make the harmonies exact. 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. Each individual figure is an ellipse where the major/minor axes ratio is the same as the harmony. Also the size in each dimension is inversely related to the size of the number. 2:3 is a large figure, 7:15 is small. To reduce the drawing time, I merely suggest the figure with radial lines. The number of lines is the same as the interval number. To further reduce drawing time, I draw the outermost point, 1 halfway to the center, 1 halfway from there to the endpoint, and one more halfway from there to the endpoint. Successive hits on the same interval cause it to cycle through 3 foreground colors and the background color (i.e. it will disappear). On the whole, however, the figures accumulate on the screen, and form complex interactions resembling a rose window. smail: 4502 E. 41st St Rm 325,Tulsa, OK 74135 C. Humphrey voice: (918) 660-4045 uucp: uunet!apctrc!zcch0a