source file: m1420.txt Date: Mon, 18 May 1998 19:22:47 -0400 Subject: Reply to Joe Monzo From: "Paul H. Erlich" >> I think the fact that the tritone in major nearly forms a 4:5:6:7 >> with the dominant, and that the tritones in minor nearly form >> an 8:10:12:14:17 with the dominant, were not inconsequential >> for the development of tonal harmony. >I don't think the fact that the tritones are close to septimal >consonances has anything to do with the development of >tonal harmony. Tonal harmony, as used during the so-called >"common practice" period 1600-1900, developed mainly out of >the fixing of pitches as absolute values (especially in notation), >the recognition of 5-limit ratios as consonances, and the intuitive >realization that because ratios have two terms, they are >related to other ratios in two different ways (Partch's >"Basic Monophonic Concept #2": "every ratio of a Monophonic >system is at least a dual identity"; "Genesis", p. 88), giving >rise to the major/ minor system. >In fact, assuming a tuning in 5-limit JI (or a meantone >which approximates 5-ratios well), the tritone could be one >of four ratios, all of which, while much less dissonant than >the Pythagorean tritones: >3^6 729/512 6.12 aug 4th >3^-6 1024/729 5.88 dim 5th >were pretty much at, if not beyond, the limit of what could >be considered consonant: >3^2 * 5^-2 36/25 6.31 dim 5th >3^-2 * 5^-1 64/45 6.10 dim 5th >3^2 * 5^1 45/32 5.90 aug 4th >3^-2 * 5^2 25/18 5.69 aug 4th >The 64/45 would normally be considered the tritone which appears >in the 5-limit Dominant 7th chord of 36:45:54:64. 4:5:6:7 is the >same as 36:45:54:63. This whole argument smacks of what I object to in the prime-limit theoreticians' approach. Harmonically speaking, an interval is never more likely to be interpreted as a higher-odd-limit, lower-prime-limit ratio than a higher-prime-limit, lower odd-limit ratio, holding the approximation error constant. I think the four 5-prime-limit ratios you listed have very little to do with the way a tritone is heard harmonically, even if tuned to exactly those ratios. Although I agree that the Pythagorean tritones, despite their low prime limit, are very dissonant, I don't think the JI tritones are "much" more consonant. In fact, they may even be more dissonant, since the Pythagoean tritones approximate simple 7-limit ratios better. Anyway, I maintain that the fact that the tritones reinforced the root of the dominant chord when combined with it harmonically helped to define major and minor as the "tonal modes", while other modes without such harmonic-melodic focus fell out of use.