source file: mills2.txt Date: Mon, 16 Dec 1996 13:17:51 -0800 Subject: Genesis of a Music question From: Daniel Wolf <106232.3266@compuserve.com> Here is one approach to Partch: (1) Assume for the moment that he used stable and accurate instruments and that all intervals in the book expressed as ratios are tuned precisely. (In fact, he did a pretty good job under very difficult material circumstances). (2) Partch's _limit_ idea is based upon the series of odd whole numbers, and his particular expansion of tonal resources was based upon adding higher odd integers; some in the tuning community prefer to speak of _limit_ in terms of highest prime numbers, some forego the term limit altogether in favor of _genus_ (or set of generating factors). In Partch's case, it is sufficient to think of a limit as the defined by the highest odd number (x 2^n) used in either a numerator or denominator. The ratio 16/15, then, has to wait for a 15 limit to appear in a _diamond_ (Partch includes it as a _secondary ratio_, to both fill in the gaps in his scale and to allow some modulations, as well as an implied 15 limit tone (although he called it _inharmonic_). (3) As a music theory teacher, you may have probably noticed already that Partch's harmonic scheme has three principals: expansion through a kind of stacked third harmony, common tone chord modulations, and intervallic inversion. (To my ears, this puts him smack in the mainstream of harmonic practice and theory in the first half of this century.) In particular, the use of inversion will become very important to building his diamond, and it is critical that he identifies inverted harmonic structures by their _highest_ tone. (4) There are a couple of ways of looking at the diamond. Here's one without ratios: build a Major triad on c: c e g. Now invert these intervals and build a minor triad _down_ from c: c ab f. Next build similar minor triads down from e (e c a) and g (g eb c)and Major triads up from ab (ab c eb) and f (f a c). Note the shared tones between the triads, and the fact that all contain c, but in each chord in a different function. This arrangement is the diamond: G E Eb C C C Ab A F And transposed to G is the same as Partch's 5-limit diamond. A more generalized way of talking about the diamond is to think in terms of a set of intervals multiplied by its inversion (a technique familiar in the - albeit 12tet - music of Pierre Boulez), so we could map it onto twelve pitch classes as: 0 4 7 8 0 3 5 9 0 which starts to look like the corner of a twelve-tone row box (who says that Schoenberg and Partch didn't have anything in common?). In Partch's case, we are talking about Just intonation, however, so these pitches can be expressed as the ratios of small whole numbers in relation to a given 1/1, or tonic. In the five limit diamond, there are six triads, and each has a distinctive odd whole number (again times 2^n, to make it all fit in an octave) common to either the numerators of the minor triads (_utonalities_) or the denominators of the Major triads (_Otonalities_). For example, locate the minor triad with 5's in the numerator (5/4, 5/51, 5/3). Ignoring this common tone, now, the denominators will show the _function_ of each tone in the triad, which Partch called the _identity_: 5/4: 4 is 2^2, thus a 1 _identity; 5/5: 5 is the 5 _identity_; and 5/3: 3 is the 3 identity. Let me know if this was of any help... Daniel Wolf, Frankfurt Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 17 Dec 1996 03:07 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA06227; Tue, 17 Dec 1996 03:09:42 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA06181 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id SAA01865; Mon, 16 Dec 1996 18:09:35 -0800 Date: Mon, 16 Dec 1996 18:09:35 -0800 Message-Id: <199612162108_MC1-D44-3C23@compuserve.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu