Hello. For chords such as 3:7:10, and 4:6:10, where the difference
tones land on the chord tones (ie, 10-7=3, 10-6-4; Helmholtz commented
on these, and I like them too!)...I had a question. This is probably
axiomatic to the math adept (but an exciting mystery to me!):what math
principle is it that accounts for the fact that if you take 3 adjacent
numbers in a sequence (eg, 4,5,6...or 3,5,7) and double the middle one
(to get 4,6,10...and 3,7,10), that you get 10-7=3, 10-6=4? (Don't
laugh...)

A similiar question, is: take the chord 12:15:20. You have 5/4 x 3 =
15:12, and 4/3 x 5 = 20/15. Then, 15-12 equals the difference tone 3,
and 20/15 equals the differnece tone 5. It just seems neat that 5/4 x
3 = 15/12 (the position of the interval in that particular chord), and
that the difference between the two numbers (15 and 12) is the same
number with which the lower fraction (5/4) was multiplied. So why are
the differnce tone and the 'interval height' the same?

thanks, Kelly