source file: mills2.txt Date: Tue, 17 Dec 1996 16:28:49 -0800 Subject: Re: Genesis of a Music question From: Matt Nathan > From: malkin@iwaynet.net (David Malkin) > Subject: Genesis of a Music question > > I am confused with Harry Partch's description of the 5 limit in Chapter 7 > of "Genesis of a Music." What is a 5 limit (or any "n" limit for that > matter)? Is it a scale that only includes ratios with a 5 in numerator or > denominator? Is the highest prime 5 that any number in numerator or > denominator can be divided into? > He uses a scale as follows: 1/1 6/5 5/4 4/3 3/2 8/5 5/3 2/1 > on page 109. > What about 16/15. This ratio's numerator and denominator are > divisible by primes that are 5 or less. Does he choose his scale based on > what sounded good, on how close to just he could get? I'm responding only on memory, not having Partch's "Genesis of a Music" here to refer to. I'm sure others will pipe in with refinements if needed. Partch used the idea of "identities" which are odd numbers in either the overnumber or undernumber position of the ratios which describe pitches ignoring octave transposition. A limit sets the highest identity he might use to generate a pitch set (ignoring octave transpositions). Identities are not factored together as in 3*5, so 16/15 is not in 5-limit, but requires 15-limit or higher. In my idiolect, a limit is different. A limit is the highest prime factor needed to analyse a given just pitch set. In this meaning, the interval 16/15 is a 5-limit interval, as is 243/125, etc. > On page 110 he speaks of the coexistence of Major and Minor and > draws the first Tonality diamond in the book. Why do these major and minor > sounds (ie Otonalites and Utonalities) result. It seems like they just > appear out of thin air. > What is a Numerary Nexus? His definition on page 72 is confusing to me. A Numerary Nexus is the number you keep the same in either the undernumber or over number while iterating the identities in the other position to generate a tonality. For instance, to generate the Otonality of 3, iterate the identities in the overnumber position while keeping 3 as the Numerary Nexus like so: 1/3, 3/3, 5/3, 7/3, etc. up to your limit. An Otonality is so called to match with "o"vernumber, since it is the set of overnumber identities which share a Numerary Nexus. All Otonalities follow the form 1/n, 3/n, 5/n, etc. (n being the numerary nexus). An Otonality is roughly analogous to major tonality, but may be closer to major chord quality, since we usually think of a major tonality as including the various diatonic chords available from a major scale, whereas Partch's tonalities move around with the individual chords, so to speak. The analogies drift apart rapidly with higher ratios. The septimal dominant (or whatever you want to call it) 4:5:6:7 would be a purely otonal chord, but would probably not be thought of by most as "major". Otonality is really an unique term which must be savored for its own theoretical functionality. Otonality is distinct from the overtone series, with which it is sometimes confused. The o in Otonality does not stand for overtone. An overtone series includes all positive integers in the numerator but does not include all octave transpositions of those pitches, while an Otonality includes only the odd overnumbers, but implicitely generates all octave transpositions of those pitches. An Utonality (u for "u"ndernumber) is the set of pitches you get from iterating the identities in the undernumber position, while keeping the same Numerary Nexus in the overnumber, of form n/1, n/3, n/5, etc. For example, the Utonality of 5 with 11 limit is: 5/1, 5/3, 5/5, 5/7, 5/9, 5/11. Utonality is roughly analogous to minor tonality or minor chord quality. Utonality is distinct from the undertone series. The pitch set you quoted 1/1 6/5 5/4 4/3 3/2 8/5 5/3 2/1 is what you get when you permute the identities in both the overnumber and the undernumber up to a limit of 5, remove the redundant instances, and express them in smallest ascending form (6/5 instead of 3/5 for instance), then throw in a 2/1 for the sake of "scaleness". The raw set would be: 1/1 3/1 5/1 1/3 3/3 5/3 1/5 3/5 5/5 The raw set may be what a tonality diamond is; I don't remember. Matt Nathan Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 18 Dec 1996 09:41 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01133; Wed, 18 Dec 1996 09:43:24 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA01131 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id AAA03779; Wed, 18 Dec 1996 00:43:21 -0800 Date: Wed, 18 Dec 1996 00:43:21 -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