source file: m1441.txt Date: Mon, 8 Jun 1998 21:36:19 -0400 Subject: Reply to Daniel Wolf From: "Paul H. Erlich" >Paul Erlich was describing the mean of two intervals while Graham Breed w= >as >describing the freshman sum. No, I only said "about halfway" and only in the case of 9:8 and 10:9. Anyway, the difference between the two means is 0.6 cents, which is too little to quibble over. >This does, however, raise an interesting set of problems. Given two ratio= >s >(a) will the freshman sum of the ratios (notated in specific octaves) >always yield the peak consonance? and (b) how would one determine the pea= >k >dissonance? = >I suspect that (a) is true as long as the two intervals are within a >certain magnitude (thus the freshman sums of 1/1 and 2/1, 3/2, or 2/2 and= >3/2, which is 5/4, are clearly peaks, but that between 81/80 and 2/1, >83/81, is not) while (b) is rather more complicated. Depending on registe= >r, >timbre, amount of mistuning and musical context I might find a slightly >mistuned perfect fifth to be more dissonant than a 12tet augmented fourth= >or minor second. Well, a VERY slightly mistuned interval will be more consonant that the freshman sum of the just interval with an equally simple one. >I don't have any elegant way of distinguishing these two= >kinds of dissonance. Any ideas? You've talked specifically about one kind of consonance and one kind of dissonance. What are the two kinds of dissonance you are thinking of? (I think I may be able to offer some answers once I'm sure what you mean).