newsham@wiliki.eng.hawaii.edu (Timothy Newsham) (07/20/90)
In article <1307@fs1.ee.ubc.ca> jthornto@fs1.ee.ubc.ca (THORNTON JOHAN A) writes: The major scale is indeed based on harmonics. In the key of C (what else?) : Note Decimal frequency fraction C 1 1 D 1.125 9/8 E 1.25 5/4 > F 1.333333... 4/3 ^^^^ G 1.5 3/2 A 1.666666... 5/3 ^^^^^ B 1.875 15/8 C 2 2 This is of course the true tempered scale. Johan Thornton EE UBC okay, some more stupid questions (hey, i'm gettting good at this) i did find out what 4/3 came from (inversion of 3/4 or 3/2, the perfect fifth, so its a fifth below the tonic, i hope i got that right), but i am still baffled about the 5/3. inverting it u get 3/5. either way u dont get a power of 2 on the top or bottom. how do u get this note? also, if u are accepting 4/3 what about 8/5? ...etc? anyone ever exerimented with those? -Tim
maverick@fir.berkeley.edu (Vance Maverick) (07/20/90)
> i did find out what 4/3 came from (inversion of 3/4 or 3/2, the perfect > fifth, so its a fifth below the tonic, i hope i got that right), but i am > still baffled about the 5/3. inverting it u get 3/5. either way u dont > get a power of 2 on the top or bottom. how do u get this note? > also, if u are accepting 4/3 what about 8/5? ...etc? anyone ever > exerimented with those? > -Tim People have experimented with them all, and continue. In a tonal context, I think of 5/3 as "a major third above the fourth degree", i.e. 5/4 * 4/3. This is not the only tonal "meaning" for the sixth degree, though -- how about "a perfect fifth above the second degree", i.e. 3/2 * 9/8 = 27/16. This is a major tone (9/8) above the fifth degree, not a minor tone (10/9). Any interval can be "decomposed" in this fashion; 8/5, for example, is "a minor third above the fourth degree" (6/5 * 4/3) or "a major third below the octave" (2 / (5/4)), which as you see comes to the same thing. The just intonation people eat, sleep and breathe this kind of arithmetic. I'm working on a system which I hope will enable ratio-based interval selection independent of explicit scale-building. If you're in a position to experiment, check out the ratios involving 7 -- 7/4, for example, is the first candidate for a "flatted seventh degree" one can draw directly from the harmonic series built on the root, yet it sounds pretty strange used melodically in a tonal context. To my ear, 9/5 ("a minor third above the fifth degree") sounds more normal, which is hardly to say better. There's a neat HyperCard stack by Robert Rich (JI Calc, shareware from Soundscape Productions, PO Box 8891. Stanford, CA 94309) which allows you to twiddle ratios to your heart's content, building scales, playing them over the Mac speaker, or dumping them to a MIDI synth. Because of the MIDI orientation, it assumes octave equivalence and twelve notes per octave, but this is reasonable for most people's music. Gerald Balzano wrote an article in Music Perception (spring? 1986) in which he derived the rudiments of standard tonality from group-theory properties of twelve-tone equal temperament. Pretty implausible historically, but I think he was being provocative to make a point -- that the degrees of the scale do a lot more than make pretty intervals together, and that there are a lot more factors influencing the construction of scales than the availability of perfect triads.
ROGER@pucc.Princeton.EDU (Roger Lustig) (07/20/90)
In article <8667@uhccux.uhcc.Hawaii.Edu>, newsham@wiliki.eng.hawaii.edu (Timothy Newsham) writes: > >In article <1307@fs1.ee.ubc.ca> jthornto@fs1.ee.ubc.ca (THORNTON JOHAN A) writes: > C 1 1 > D 1.125 9/8 > E 1.25 5/4 >> F 1.333333... 4/3 > ^^^^ > G 1.5 3/2 > A 1.666666... 5/3 > ^^^^^ > B 1.875 15/8 > C 2 2 > Johan Thornton >okay, some more stupid questions (hey, i'm gettting good at this) >i did find out what 4/3 came from (inversion of 3/4 or 3/2, the perfect >fifth, so its a fifth below the tonic, i hope i got that right), but i am >still baffled about the 5/3. inverting it u get 3/5. either way u dont >get a power of 2 on the top or bottom. how do u get this note? >also, if u are accepting 4/3 what about 8/5? ...etc? anyone ever >exerimented with those? Think of these as ratios or operations. The approach given above is based on 3 operations: doubling, tripling, quintupling. (x4 is twice x2, so it's not necessary to mention it.) What we're tupling is the frequency of a fundamental. What we've done here is to take the three operations, apply them to the fundamental, and get a bunch of intervals. Starting with low F, x3 is a 12th above: c. x5 is a 17th: a'. The interval c-a' is a major 6th. As I pointed out before, the C major scale above is based on a fundamental F, which is one of the sticky bits in the theory. Since all notes are calculated wrt C, you'd expect a common denominator that gives C primacy. Ratios above have also been reduced to lowest terms. But to get the ones you see here all over a common denominator, you need an F at the bottom. 8/5 is a minor 6th, such as e' to c". And 9/8, the major whole tone, is simply a fifth on top of a fifth. (The minor whole-tone is 10/9.) Roger Lustig (ROGER@PUCC.BITNET roger@pucc.princeton.edu) Disclaimer: I thought it was a costume party!