jgk@osc.COM (Joe Keane) (07/11/90)
In article <1278.2694ccd9@gp.govt.nz> philip@gp.govt.nz (Philip Dorrell) writes: >A rather popular chord sequence is - > >C major --------------> G major --------------> D major ------------> C major > (again) >C ----(* 3/2)--------> G --( * 3/4)------> D (*3/2) --->A (*3/5)---> C > >total - > > C -----------------------------( * 81/80)-------------------------> C The A in the key of C major is defined to be a major sixth (5/3) above C. The A in the key of D major is defined to be a perfect fifth (3/2) above D. If we say D is a major tone (9/8) higher than C, the first A is lower than the second one by the ratio 81/80. The interval 81/80 is called a comma and pops up all over the place. This kind thing of thing has been studied since Pythagoras. There is actually a rich theory dealing with somewhat subtle differences, where you talk about different kinds of commas, chromatic versus diatonic semitones, major versus minor tones, and so on. One conclusion is that the second note in a major scale is somewhat flexible. Unfortunately, the result of equal temperament has been to pave over this theory. If you ask a piano player what is the difference between G# and Ab, he will say there is none. This ignores the fact that some organs had keys for each, or that modern electronic instruments can easily play them as different notes. Also, if you ask people what are CX or Ebb (excuse my ASCII) you're likely to get blank looks. This is interesting, but not math; followups are directed to `comp.music'.
gsmith@ronzoni.berkeley.edu (Gene Ward Smith) (07/20/90)
In article <1269.26909383@gp.govt.nz> philip@gp.govt.nz (Philip Dorrell) writes: >>>All [...] intervals in music can be expressed in terms of integral powers of >>>2, 3 and 5. Taking logarithms, this gives a 3-dimensional vector space with >>>basis vectors log 2, log 3 and log 5. What you should say is that music based on 2,3 and 5 forms a subgroup (free of rank 3) of Q*, the multiplicative group of positive rationals, and that the log map takes this in a natural way to a lattice in R3. If we assume a mean-tone type system, the kernal of the homomorphism from this rank 3 abelian group to the rank 2 group which covers mean-tone systems is generated by 81/80, which is the well-known diatonic comma: the difference (9/8)/(10/9) between the two just major tones of the just diatonic scale. >It seems that some approximation error always occurs somewhere, and this is in >fact a consequence of my theory - according to it, every tune contains a 'proof' >that 80 = 81. For example : suppose there existed some tune played on an exactly >tuned scale that did not somewhere come up against the problem that an interval >in the tune was out by a factor of 81/80 from what you wanted it to be - >according to my theory, such a tune would not be a tune. Your theory is wrong. Lots of music can be made on just the diatonic major scale. Make everything triadic, and so long as the minor supertonic (ie d minor over C major) is avoided, you are home free. ucbvax!brahms!gsmith Gene Ward Smith/Brahms Gang/Berkeley CA 94720 Fifty flippant frogs / Walked by on flippered feet And with their slime they made the time / Unnaturally fleet.