source file: mills2.txt Date: Mon, 7 Jul 1997 18:39:02 +0200 Subject: Meantone, Partch From: John Chalmers Re Meantone: "Linear" is an historical term (Ellis?) contrasting with "cyclic" or closed. I agree that in general Meantone tone systems are 2-D, but under the common assumption of octave equivalence, the dimensionality may be reduced to 1. In which case, both ET's and meantone tunings are generated by one interval, the tempered fifth. Tempered systems are closed and have a finite number N of points on the line of fifths, while meantone are open and infinite. In this sense they are "linear" and open as opposed to closed and cyclic. BTW, Fokker used the idea of defining intervals in 2 and 3-D tones spaces in a somewhat different way. He chose sets of "unison vectors," which he set to 0, in 3x5 and 3x5x7 tone spaces to define temperaments as "periodicity blocks" in the space. For example, the diesis and the syntonic comma define a repeating block of 12 pitches as the area defined by the absolute value of the cross product of the vectors describing these intervals. (The box product is used for 3 intervals in 3-D space.) I believe the concept can be extended to higher dimensions by computing the absolute values of the determinants of the square matrices whose row vectors are the defining intervals. Paul Rapoport has also done much work in this area. Wu"rschmidt, I believe, was the first to identify "defining intervals" for tunings and to distinguish them from "constructing intervals," the successive intervals of the JI scales corresponding to each of the ET's as defined by this method (there are many possible sets of defining intervals for any given ET and there are many possible sets of 5 limit intervals interpretable as 12-tet.) Re Partch: HP's dual use of ratios to label scale degrees and to define functions can be somewhat confusing at first as Marion has pointed out. Partch assumed octave equivalence in deriving his theories, but did recognize that inversions have different sounds and inn his music used a variety of voicings. Hence, sometimes 1/1 appears in the middle of a chord where the voicing would indicate that the ratio should be 1/2, 2/1, 4/1 etc. As for odd numbers (or factors) in the denominators, they determine the roots of his otonal or harmonic series chords, though his term "numerary nexus," has not caught on. They also indicate the harmonic distances from the common tonic, the 1/1 (tempered G). When HP writes 8/5 1/1 6/5, the notation is a shorthand for 8/5 10/5 12/5 a major triad on the root 8/5, a minor sixth above his 1/1 G. This notation makes the harmonic relations clear, while somewhat obscuring the melodic movement and actual voicing. Usually, he's clear enough as to the actual inversion and voicing. Otherwise, he'd have to make the register explicit and increase the number of symbols, e.g., 1/3, 2/3, 4/3, 8/3 etc. for 4/3 (roughly C) in various registers.. Such schematic spellings are not unknown in conventional music theory. For the subharmonic (utonal) chords, the odd factors in numerator serve the same function. The Diamond diagram may clarify his meaning. One really has to read Partch's theoretical sections as an introduction and outline, not as a detailed composition manual. --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 8 Jul 1997 16:10 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA21599; Tue, 8 Jul 1997 16:11:01 +0200 Date: Tue, 8 Jul 1997 16:11:01 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA21507 Received: (qmail 24776 invoked from network); 8 Jul 1997 14:02:41 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 8 Jul 1997 14:02:41 -0000 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu