source file: mills2.txt Date: Tue, 17 Jun 1997 15:45:35 +0200 Subject: Tuning of Phrgian Mode From: John Chalmers As for the tuning of the ecclesiastical Dorian (Greek Phrygian) mode, I think it depends very much on how it is to be used.If one wishes to harmonize it as if it were what Ellis called a "trichordal", the best tuning would be 1/1 9/8 6/5 4/3 3/2 5/3 9/5 2/1 as this scale may be chorded with a major triad on 4/3 and minor triads on 1/1 and 3/2, in Ellis's nomenclature, ma.mi.mi. See pages 275 and 460 of Helmholtz for details. I personally do nŽind the 27/25 wide, epimeric "semitone" of 133 cents offensive in the upper tetrachord, but some might. One might also try the "dual" harmonies proposed by David Lewin, Blainville, etc. and build the triads downwards. In this case, the best tuning would be 1/1 10/9 6/5 4/3 3/2 5/3 16/9 2/1, with chords 2/1 5/3 4/3, 4/3 10/9 8/9 (16/9 in the lower octave) and 3/2 6/5 1/1. If the melodic pattern, T S T T T S T, is more important, then as Lydia stated, one may take it as a mode of the JI C major scale (which is the C mode of the E mode (Greek Dorian) of Ptolemy's Intense Diatonic genus 16/15 x 9/8 x 10/9). The scale is thus 1/1 10/9 32/27 4/3 40/27 5/3 16/9 2/1, whose tuning may be adjusted to 1/1 9/8 6/5 4/3 3/2 5/3 16/9 2/1. However, the scale may also be taken as a mode of the un-Greek natural minor scale, generated by the tetrachord 10/9 x 9/8 x 16/15. In this case the tuning is 1/1 9/8 6/5 27/20 3/2 27/16 9/5 2/1, the inversion of the preceeding, and may be retuned as before to 1/1 9/8 6/5 4/3 3/2 5/3 9/5 2/1. If the Didymos's diatonic 16/15 x 10/9 x 9/8 is employed, the scale as a mode of the Greek Dorian is 1/1 9/8 6/5 4/3 3/2 27/16 9/5 2/1. (There is less motivation to invert it as it is less harmonic than Ptolemy's form.) One could even use the other Greek and Islamic diatonic genera. Or one could follow Safiyu-d-Din and the other Islamic theorists and generate the scale from two identical tetrachords of the form T S T' (or T' S T) where T and T' are any whole tone-like intervals and S whatever completes the tetrachord. These Islamic theorists allowed all permutations of the tetrachord to form scales. Thus from Ptolemy's and Didymos's tunings we can get Phrygian-like scales with duplicated tetrachords such as 1/1 10/9 32/27 4/3 3/2 5/3 16/9 2/1 and 1/1 9/8 6/5 4/3 3/2 27/16 9/5 2/1. The "Pythagorean" tuning 1/1 9/8 32/27 4/3 3/2 27/16 16/9 2/1 has the advantage of being a "classical" mode generated by the 256/243 x 9/8 x 9/8 tetrachord. Other reduplicated tetrachords could be tried as well. Of course in ET's such as 12,17, 19, 22, and 31 that do not articulate the syntonic comma, the melodic pattern T S T T T S T is readily available. --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 17 Jun 1997 16:49 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA31421; Tue, 17 Jun 1997 16:49:15 +0200 Date: Tue, 17 Jun 1997 16:49:15 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA31406 Received: (qmail 9126 invoked from network); 17 Jun 1997 14:49:05 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 17 Jun 1997 14:49:05 -0000 Message-Id: <199706171047_MC2-18A6-4EA9@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu