source file: mills2.txt Date: Thu, 27 Feb 1997 01:51:16 -0800 Subject: Re: Reply to Matt Nathan From: Matt Nathan > From: PAULE > > > > ...Deriving the > > > diatonic scale from the tonic, subdominant, and dominant > > > triads is utterly inaccurate, in my opinion. There are too > > > many pieces in major that focus on I-ii-I progressions, > > > and pieces in minor (aeolian, to be more academic) > > > that focus on i-VII-i progressions, for this to be an > > > accurate explanation, for not even the tuning of the chords > > > will be correctly specified by such an explanation. > > > Here would be cases, like I-ii-I, which I would say are > > implying more than 7 pitches, > > A severe violation of Occam's razor! Haha, yes it involves only 6 pitches! This is amusing but a misunderstanding. The diatonic symbol IV pretty assuredly reflects the rational structure 4/n 5/n 6/n where n is 3. The diatonic symbol ii is a case of confounded rational structures however. There are at least two distinct rational structures which suffer lossy compression in a diatonic assumption and get lumped under "ii". You can sometimes tell by context which of these is being implied. The ii chord can function in context as either the relative minor of IV, which comes from the rational structure n/4 n/5 n/6 where n is 5/3, or as the secondary v (minor) of V, which comes from the rational structure n/4 n/5 n/6 where n is 27/16. A diatonic piece which uses both I-(v of V)-I and I-IV-I (no I didn't mean I-V-I) implies at least the 8-pitch set 1/1 9/8 5/4 4/3 27/20 3/2 5/3 27/16. In vector notation, this is (using fixed-width font): cell: 5 | 1 - 3 pitch set (C tonic): A,- E | | F - C - G - D - A | F' Matt Nathan Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 27 Feb 1997 11:56 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA29740; Thu, 27 Feb 1997 11:56:14 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA29759 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id CAA15800; Thu, 27 Feb 1997 02:53:33 -0800 Date: Thu, 27 Feb 1997 02:53:33 -0800 Message-Id: <331566DF.682C@ix.netcom.com> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu