source file: mills2.txt Date: Mon, 23 Jun 1997 22:59:35 +0200 Subject: Re: the 6th chord and odd-limit theory (Paul E) From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) From: "Paul H. Erlich" >> (i.e., chords in >> which each interval is within the 7-limit but the chord as a whole is in a >> higher limit). >I know what the first bit means, but I don't know any other definition of >the limit of a chord. Could someone explain this to me so that I can set >about (dis)proving it? I know this whole business can be very confusing, but let me try to explain this step by step. First of all, as the title of this message indicated, I am talking about odd-limit theory, as in Partch's original definition. The limit is defined as the highest number needed in the representation of the interval or chord after all factors of two are discarded (octave equivalence is the justification for discarding them). This also agrees with Ralph D. Hill's observation that every new odd number adds a new quality, as well as a higher degree of tension, to a chord. I used to believe (though I'm not sure in light of my recent obsevation here) that a given style of music established a certain odd limit as the norm; higher-limit chords will sound dissonant and need to resolve, while lower-limit chords will sound incomplete. This is certainly applicable to Common Practice and the 5-limit; 3-limit pentatonic examples are easily constructed, and I believe I've taken a first step toward realizing the 7-limit case (see my upcoming Xenharmonikon paper). Now let us state what we mean by "representation." For an interval, this is just a lowest terms ratio p/q; we can define this to be the frequency ratio, in which case the period (or wavelength, string length, . . . ) ratio in lowest terms is just q/p. Either reprentation implies the same odd limit for the interval. Following Partch, we will define integer multiples from 1 to n of a frequency to be an "n-otonality," and integer multiples from 1 to n of a period (or wavelength, string length, . . .) to be an "n-utonality." So we can define the otonality-limit as the lowest n for which a chord (with all factors of 2 removed) is a subset of an n-otonality, and similarly for the utonality-limit. For a chord of 2 notes, the limit of the 1 interval in the chord is the same as the utonality-limit and the otonality-limit of the chord. For more than two notes, the otonal or utonal representations may differ in complexity, in which case the simpler one is often chosen and the chord is deemed as appropriate. For example, 10:12:15 is a "5-limit utonality" and 4:5:6:7 is a "7-limit otonality." As Paul Hahn has pointed out, the 3-limit, 5-limit, and 7-limit are particularly easy cases to deal with, since if all intervals of a chord belong to one of these limits, the chord itself will belong to that limit either as an otonality or as a utonality. However, the 6th chord 12:15:18:20 has an otonality-limit of 15, a utonality-limit of 15, but no interval is beyond the 9-limit. Kami Rousseau's 6th chord 14:18:21:24 has an otonality-limit of 21, a utonality-limit of 21, but again no interval is beyond the 9-limit. Paul Hahn's example 18:22:24:33 has an otonality-limit of 33, a utonality-limit of 33, but no interval is beyond the 11-limit. It appears then that there is a new class of "saturated" chord beyond the otonal and utonal chords that Partch envisioned. By saturated I mean that no new notes can be added without increasing the odd-interval-limit of the chord. I think this finding merits investigation as a natural extension of Partch's work. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 23 Jun 1997 23:01 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA05735; Mon, 23 Jun 1997 23:01:23 +0200 Date: Mon, 23 Jun 1997 23:01:23 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA05728 Received: (qmail 26044 invoked from network); 23 Jun 1997 20:59:26 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 23 Jun 1997 20:59:26 -0000 Message-Id: <009B63B0DDB21546.5CEC@vbv40.ezh.nl> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu