The current discussion of tonal drive and resolution in JI seems to me to miss an elementary point stressed by Harry Partch himself. JI increases the range of available consonance and dissonance however C and D are defined, see Tenney's book _A History of Consonance and Dissonance_. Hence, in some contexts the dominant seventh chord should be tuned 4:5:6:7 for maximum consonance, but in others one might wish to use 20:25:30:36 (1/1 5/4 3/2 6/5) or even 36:45:54:64 (1/1 5/4 3/2 16/9). One should read Chapter 11, "The Question of Resolution," in HP's _Genesis of a Music_, 2nd edition, pp 181-194. While HP wrote everything in the key of G (392 hz), in part because of material and technical limitations, in part for theoretical reasons (monophony is the harmonic expansion of a single tone, the 1/1, see the definition on page 71), he was very aware of resolution and chordal motion and wrote some extremely dissonant, yet powerful progressions. (Allen, do you still have your notes from his UCSD class?). Partch suggested several principles: Resolve voices by very narrow intervals, avoid 11 O or U identities (11 th harmonic or subharmonic of the tonic) in the bass, space the voices widely, and mix harmonic and subharmonic (O and U) chords in progressions. One does not have to use complete hexads (1 3 5 7 9 11 harmonics) and the remaining voices may be doubled as desired. Non-harmonic tones and tones outside the conventional 43-tone scale may also be used (HP sometimes did this on the canons and even changed reeds on the Chromelodea, i.e., 25/21 in Delusion.)
--John
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Pardon my ignorance but what is meant by "LCM" and can someone explain "minimum chords"?
thanx--
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Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 11 Nov 1995 23:38 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id NAA00386; Sat, 11 Nov 1995 13:37:59 -0800 Date: Sat, 11 Nov 1995 13:37:59 -0800 Message-Id: <951111213450_71670.2576_HHB6-4@CompuServe.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu
I believe that LCM is the usual mathematical abbreviation for "Least Common Multiple". Marion and probably others use that as an indicator of the overall harmonic complexity of a chord. The lower the LCM, the simpler the chord.
For example, the least common multiple of 4, 5, and 6 is 60. That's the smallest number that is an even multiple of all three (4x15=5x12=6x10=60). So that major triad voicing would be equally complex harmonically as a 10:12:15 minor triad (LCM is also 60), even though its numbers are somewhat higher on the whole.
Some on the list have disputed whether LCM is the best measure of harmonic complexity. Personally, it strikes me as a reasonable rule-of-thumb, although I haven't personally compared it with other possibilities.
Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sun, 12 Nov 1995 06:00 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id UAA02660; Sat, 11 Nov 1995 20:00:29 -0800 Date: Sat, 11 Nov 1995 20:00:29 -0800 Message-Id: <199511120358.WAA19511@freenet3.carleton.ca> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu