source file: mills2.txt Date: Sun, 29 Jun 1997 18:33:20 +0200 Subject: RE: Partch Limit vs Prime Limit From: gbreed@cix.compulink.co.uk (Graham Breed) Marion wrote: >For me, the whole "lattice" approach is much less useful than >Aliquot Parts and Least Common Multiple for analyzing scales, >chords, and intervals. Aliquot Parts are unknown to me, so I can't comment on them. I have been a card carrying LCM-ist for quite a while, though, but I'm confused at this dichotomy between them and the lattices. I express the LCM in my lattice notation, but call it a region of harmonic space (ROHS) because an LCM should really be a number and not a matrix. An LCM of 60, then, corresponds to a ROHS of (2 1 1)H. In lattice terms, the ROHS is the size of the smallest n-box in n-dimensional ratio space that includes, on or within its boundary, all the notes under consideration. The distance between opposite corners of this n-box, measured using a city block metric with each dimension weighted according to the logarithm of the corresponding prime number, is the logarithm of the LCM of the same notes. Don't worry if you didn't get that last bit, folks, it isn't important! Anyway, my main point in writing is to draw a comparison between the odd-limit and the ROHS. Referring to my last message of a few days ago, the 7-limit clearly lies within the ROHS (x 1 1 1)H where x can be anything, to give octave equivalence. This is the same as the Euler genus 3.5.7. Not all the members of the genus 3.5.7 are within the 7-limit. (3 -1 1 0)H, or 15/8, for a start. For intervals within an octave, though, the ROHS (2 1 1 1)H contains all 7-limit intervals and only a few outside it -- (-1 1 1 -1)H or 15/14, etc. In otonal or utonal form, these exceptions disappear as odd primes cannot appear in both the numerator and denominator. If you reduce your chord so that all the intervals are within the first octave above the otonal root, the ROHS (2 1 1 1)H is identical to the 7-limit. The saturated chord being: (-2 0 0 1) (-2 0 1 0) (-1 1 0 0)H ( 0 0 0 0) The ROHS occupied by the largest interval within a chord will always be the same as the ROHS occupied by the whole chord. This is why the ROHS is superior to the odd limit from a number theoretic point of view. It is also why, IMHO, the ROHS is the superior way of classing scales. The equivalent utonal chord can be found by multiplying the chord above by -1. This is the same as arranging the notes in the octave below the utonal root. Where the otonal root will always be (0 0 0 0)H, the utonal root is equal to the ROHS with the octave term negated. Therefore, the octave reduced utonal (2 1 1 1)H chord is: (0 0 0 0) (-2 1 1 1) (1 -1 0 0) (-1 0 1 1) (2 0 -1 0)H (2 -1 -1 -1)H + ( 0 1 0 1)H (2 0 0 -1) ( 0 1 1 0) For consistency with the odd-limit idea, only otonal and utonal reductions are allowed. This rules out octave reduced (2 1 1 1)H, non 7-limit chords like the following: (-2 0 1 1) (-1 1 0 1)H ( 0 1 1 0) Hence o/utonal 7-limit chords are the same thing as o/utonal octave reduced ROHSs. Beyond the 7-limit, though, this equivalence no longer holds, just as the two kinds of odd-limit diverge. As the odd-limit idea is itself an extrapolation from conventional harmony, I would suggest that the ROHS method is the best way of extending it to higher primes. The ROHS is also a useful way of categorising chord changes. This is a very important subject, and one that is rarely mentioned on the list. However, I will not go into it today. I suggest that the ROHS should also be emancipated from octave reduction. Then the interval 15/1 comes out as less dissonant than 15/8. My ears tell me that there is some justification for this. I don't go with octave invariance, because different inversons of major triads definitely sound different. All the 'nice' versions of 4 note major triads fall within the ROHS (3 1 1)H. The ones that stay within (1 1 1)H also sound particularly nice. However, the (2 1 1)H ones are not neccessarily better than the (3 1 1)H ones. There seems to be a premium on having the root doubled. The idea of root doubling, though, implies the idea of octave invariance. I might some time get round to extending Kameoka and Kuriyagawa's dissonance algorithm to 4 note chords to see if it sheds any light on this. Graham Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sun, 29 Jun 1997 18:34 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA17021; Sun, 29 Jun 1997 18:34:36 +0200 Date: Sun, 29 Jun 1997 18:34:36 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA16996 Received: (qmail 2093 invoked from network); 29 Jun 1997 16:33:47 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 29 Jun 1997 16:33:47 -0000 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu