source file: mills2.txt Date: Tue, 19 Nov 1996 18:03:18 -0800 Subject: diatonic scale From: John Chalmers Re Hermann Pedtke's keyboard: The keyboard described by Pedtke in his Xenharmonikon 6 article on meantone (Summer 1977)is the two-manual Scalatron, not the generalized keyboard version. I think Brian is referring to Hans Luedtke's two patents for keyboards with hexagonal digitals, #2,061,364 of Nov. 17, 193 and # 2,003,894 of June 4, 1935. I am unaware of the Paris keyboard. RE the diatonic scale: I think our emphasis is a bit misplaced. As I understand European musical history, Pythagorean tuning and the rather artificial system of modes began to break down during the Middle Ages as Musica Ficta tended to erase the modal differences and the intonation softened towards just thirds and sixths, at least in English music, according to Walter of Odington. This tweaking went on for several centuries and eventually in the 18th century Rameau analysed the diatonic scale in terms of triads and theory caught up with practice. However Barbour showed that tuners in the 15th and 16th centuries, when Pythagorean tuning was being replaced by meantone, 12-TET and various irregular systems, devised complete chromatic octaves (12 notes) not merely the seven diatonic tones. Thus the diatonic scale was never viewed wholly in isolation from its modes and transpositions. No doubt by this time the diatonic scales were considered to be constructed of triads as all the chromatic tones were harmonized by triads, but most probably musicians thought in terms of five or six chords, not just three primary ones, though a specialist on early music could advise me here. I think it was Rameau's insight that the scale could be construed as the union of just three primary triads rather than 5 (including the conjugate ones on degrees 3 and 6) that led to the myth that it was deliberately constructed with this in mind. (I'm ignoring the supertonic triad deliberately.) It seems to me to be a coincidence that Ptolemy's Intense Diatonic in the Greek Lydian mode provides 4:5:6 triads on 1/1, 4/3 and 3/2. There is little evidence that the Greeks were aware of this fact and there were other diatonic tunings in favor without this property. The tuning does NOT have a mode with 3 minor (10:12:15) triads on these roots, however and neither does any other Greek tuning. It seems therefore, that the major-minor system presupposes some sort of tuning with acceptable thirds that ignores the syntonic comma. Meantone, ET and various irregular systems do just this. (Redfield's just scale does have this property as it consists of three minor triads on 1/1, 4/3 and 3/2. Every mode of this scale contains a note differing by a comma from a Ptolemaic mode. The generating tetrachord is 10/9 x 9/8 x16/15 rather than the Greek 16/15 x 9/8 x 10/9. ) I must say that I think Rameau's insight is very valuable for contemporary scale builders. I generated a number of analogous scales using various kinds of triads and mixtures in the 60's, but did very little with them save play a few on Partch's chromelodeon until I spoke with Max Mathews and saw David Lewin's article on Generalize Tonal Functions. As most scale builders soon find out, it is no easy task to find a harmonically generated scale with the melodic strength and logic of the diatonic scale. --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 20 Nov 1996 11:06 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA18711; Wed, 20 Nov 1996 11:07:54 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA15279 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id CAA04116; Wed, 20 Nov 1996 02:07:50 -0800 Date: Wed, 20 Nov 1996 02:07:50 -0800 Message-Id: <199611200504_MC1-C1D-A9BF@compuserve.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu