source file: mills2.txt Date: Tue, 25 Feb 1997 13:19:30 -0800 Subject: 5-limit harmony, tritriadic scales From: John Chalmers Re 5-limit harmony: Five and seven limit scales apparently go back to Archytas (about 390 BCE) in Greece, though I'm not sure how "harmonic" their treatment was. Aristoxenos (circa 330) complained that the incomposite ditone in the enharmonic genus was being narrowed, a process called "sweetening" presumably from the 81/64 to the 5/4. This sounds very much like Archytas's 28/27 x 36/35 x 5/4 enharmonic tetrachord. However, in the diatonic genus, Archytas's tuning was septimal,( 28/27 x 8/7 x9/8) skipping 5 altogether, and Aristoxenos admits that this is a well known tuning (as 1/3 + 1 1/6 + 1 tones). Ptolemy, nearly 500 years later, implies that this is the most common diatonic tetrachord, though the Pythagorean 3-limit one is also in use. His own Syntonic Diatonic, 16/15 x 9/8 x 10/9 may be contrasted with the earlier Didymos's 16/15 x 10/9 x 9/8 tuning. This latter scale is rather non-harmonic in our sense. It, Pythagorean (Ditone Diatonic) and Archytas's diatonic (Ptolemy's Middle Soft Diatonic, Tonic Diatonic, etc.) all have many 32/27 and 81/64 thirds (the major scale has only 1). This suggests to me a possible avoidance of 5 limit thirds until Ptolemy's time. This is also true of his mixed modes which have predominantly septimal and pythagorean intervals. Traditionally, the development of 5 limit harmony is ascribed to late medieval English singers (fa-bourdon,etc.). I don't know of any data supporting 5 or 7 limit harmony in Greece though some late writers describe 5 and 7 limit intervals as "paraphonies" to distinguish them from consonances (symphonies) and dissonances (diaphonies). Paraphonies include, however, the tritone as well as thirds and sixths. Tritriadics: I would tend to agree that the derivation of the major and natural minor modes in JI from three triads to be somewhat artificial, despite Rameau and Ellis (i.e. his 56 tonal modes). However, it is still useful conceptually, even if comma tweaks be desired in the ii and VII (in minor) chords or the whole scale mapped into some convenient ET. The derivation is more plausible in Meantone, as these chords would be acceptable. Other "fixes" are possible: one might do as Blackwood recommended for some ET's, just avoid progressions with ii and VII. One might also make use of the 117 tones of Ellis's "Duodenarium," the 56 skhismatically related tones of his "unequally just" tuning or 53-tet. --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 25 Feb 1997 23:07 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA18952; Tue, 25 Feb 1997 23:07:21 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA18911 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id OAA04179; Tue, 25 Feb 1997 14:05:25 -0800 Date: Tue, 25 Feb 1997 14:05:25 -0800 Message-Id: <199702251701_MC2-11BC-C28E@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu