source file: mills2.txt Date: Thu, 27 Feb 1997 10:43:19 -0800 Subject: Re: 5-limit harmony, tritriadic scales From: kollos@cavehill.dnet.co.uk (Jonathan Walker) John Chalmers wrote: > > 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. Ptolemy rejects the 81/64 ditone as a possible melodic (i.e. an interval between two adjacent degrees) in his "rational systems", since it is epimeric (i.e. not superparticular). The only superparticular melodic approaching this magnitude is the 5/4. However, as I've said before, Ptolemy distinguishes between the generic pattern arising in singing, and that which players of kithara and lyra prefer to follow: namely, the intense diatonic (10/9, 9/8, 16/15 downwards) in the former case, and the ditonic diatonic (9/8, 9/8, 256/243) in the latter. However, the 5/4 is only a composite interval made up of two melodics in the intense diatonic; it is only available as a melodic in the context of the enharmonic tetrachord, and of the few sources we have, there appears to be a consensus in favour of 5/4 as the leading ratio, which is endorsed by Archytas, Didymus and Ptolemy, although they differ in their division of the remaining pyknon. One of the grounds on which Ptolemy objected to Archytas's divisions was the retention of 28/27 as the following (i.e. lowest) interval in all three genera, a feature which was contrary to practice (and thus unacceptable, in the context of Ptolemy, because it is evidence of reasoning which has not been submitted to empirical testing). Ptolemy also provides us with important evidence for a trend in favour of the more tense tetrachords ("more tense" meaning that the mutable inner degrees of the tetrachord were tuned higher in relation to the fixed outer degrees), since he tells us that his enharmonic and soft chromatic, though constructed in accordance with reason, were unfamiliar to experience (by his time), and so the intense chromatic is the softest of all the tetrachords featured in Ptolemy's careful description of the tunings that were actually used in his time (it appears in the tropoi kithara tuning, mixed, of course, with the even diatonic). -- Jonathan Walker Queen's University Belfast mailto:kollos@cavehill.dnet.co.uk http://www.music.qub.ac.uk/~walker/ Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 27 Feb 1997 19:47 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA32181; Thu, 27 Feb 1997 19:47:47 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA31649 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id KAA19356; Thu, 27 Feb 1997 10:44:22 -0800 Date: Thu, 27 Feb 1997 10:44:22 -0800 Message-Id: <3315C527.58C9@cavehill.dnet.co.uk> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu