source file: mills2.txt Date: Tue, 8 Jul 1997 19:23:15 +0200 Subject: Re: Octave invariance From: Daniel Wolf >From the discussion: >''>>If you allow for the octave equivalence of intervals, all triadic >>>inversions contain the same intervals. >>How so? The way I larnt it, a simple C major triad in root position consists >>of a minor third stacked on top of a major third. In first inversion, it >>consists of a perfect fourth stacked on top of the minor third. > Well, this premise is certainly clear you instead think of a root >position major triad as a P4 atop a m3 atop a M3, and first inversion anM3 >atop a P4 atop a m3, and so forth.'' The three inversions of a triad (that is, three tones without octave duplications) are intervallically distinct: (in 12tet interval classes:) 4,3,7; 3,5,8 ; 5,4,9. Throw in an additional tone, doubling the lowest pitch at the octave, and you get: 4,3,5,7,8,12; 3,5,4,8,9,12; 5,4,3,9,7,12, which are also distinct, shuffling the two-out-of-three combinations of (7,8,9). Triadic inversions are equivalent (a) in terms of pitch class content, or(b) under some octave modulus operation where inversions are equivalent. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 8 Jul 1997 19:25 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA20008; Tue, 8 Jul 1997 19:26:11 +0200 Date: Tue, 8 Jul 1997 19:26:11 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA20067 Received: (qmail 10644 invoked from network); 8 Jul 1997 17:24:35 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 8 Jul 1997 17:24:35 -0000 Message-Id: <199707081320_MC2-1A72-8173@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu