source file: mills2.txt Date: Wed, 6 Nov 1996 15:31:05 -0800 Subject: Consonance coefficient From: PAULE Lorenzo, I don't think your method will lead you into any serious errors, although it seems like the various inversions of a given chord will often differ a lot less in consonance than fundamentally different chords. Also, I often prefer more dissonant chord voicings on guitar over more consonant ones, if the musical context befits them. Of course, for a six-string chord, you have 15 individual values for dissonance to add up. Since you are comparing intervals to their inversions and not to fundamentally different intervals, the results I posted for you were perfectly useless to you. Fortunately, my theory tells you that the certainty with which a tempered interval will be perceived as its most likely just interpretation is proportional to the denominator of that just interval (in lowest terms), and a factor, related to the degree of mistuning, which will be the same for all inversions of an interval, and thus ignorable for your purposes. So, interpreting this certainty as consonance, we can say that the perfect twelfth is twice as consonant as the perfect fifth and three times as consonant as the perfect fourth. The major tenth is twice as consonant as the major third and 2.5 times as consonant as the minor sixth. The major sixth is 1.666666667 times as consonant as the minor third or minor tenth. However, the increased consonance of wider intervals comes at the price of a thinner harmonic texture, which is not always desireable. For other intervals, the ratio-interpretation will vary according to inversion; contact me privately for more details. Also, if I'm treating the intervals as non-interacting and simply adding values for each interval in a chord, I'd feel better about adding dissonance levels than adding consonance levels. To be continued . . . -Paul Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 7 Nov 1996 01:12 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA06332; Thu, 7 Nov 1996 01:13:17 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA06319 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id QAA18287; Wed, 6 Nov 1996 16:13:13 -0800 Date: Wed, 6 Nov 1996 16:13:13 -0800 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu