source file: mills2.txt Date: Thu, 27 Feb 1997 21:15:47 -0800 Subject: Aristoxenus's + Ptolemy's enharmonics From: kollos@cavehill.dnet.co.uk (Jonathan Walker) John Chalmers wrote: > As for the use of 5/4. Eratosthenes, who may be responsible for > the linear misinterpretation of Aristoxenos's parts, used 19/15 as > his enharmonic ditone. > Frankly, I'm somewhat suspicious of Ptolemy's enharmonic and > chromatic genera as they have a rough 1:2 division of the pyknon > (Barbera), even if it meant that he had to reorder the intervals > resulting from "katapyknosis" to obtain usable superparticular > ratios. But, I do agree that the kithara and lyra tunings probably . > represent practice I'll deal with Eratosthenes first, then Ptolemy. I was discounting Eratosthenes precisely because of the general view that he tried to interpret Aristoxenus's tetrachordal figures in terms of string-length proportions. Ptolemy provides us with a version of the enharmonic tetrachord from five different theorists, including himself (Bk.II, sect.14). Of the four who give ratio descriptions, Archytas, Didymus and Ptolemy all agree on 5/4 for the large interval; only Eratosthenes differs by giving 19/15 instead. This leaves only Aristoxenus, who gives us a large interval of 24 parts (out of 30 for the 4/3). How are we to interpret Aristoxenus in ratio terms? If we call our Aristoxenean measure "A", we can use the formula exp(Aln(4/3)/30) and then find the first few convergents of the continued fraction expression of the result. In the case of A 24, the third convergent is 5/4 (the fourth is 34/27). If this is indeed Aristoxenus's interval, then the remaining pyknon of 16/15 is divided equally into 3 +3 parts. This would yield a tetrachord that is perceptually indistinguishable from the enharmonic of Didymus, who is clearly aiming for near-equal division of the pyknon (31/30 and 32/31 descending), while Archytas and Ptolemy both choose to divide the pyknon unequally. We have good reason then to identify the enharmonic of Aristoxenus with that of Didymus. What of Eratosthenes then? Let's take him seriously in his own right for a moment, rather than as a mere misinterpreter of Aristoxenus. As it happens, his ratio of 19/15 is in fact one cent closer to 24 Aristoxenean units (if they are interpreted as a 30-fold equal division of the 4/3). 24 parts is about 398 cents; 5/4 is about 12 cents smaller, while 19/15 is about 11 cents higher. I have no idea how Eratosthenes made his calculation, but clearly he didn't use logarithms and convergents, nor could he extract the 30th root of 4/3. Nevertheless, 19/15 would seem considerably harder to arrive at for lyra and kithara players, and it never reappears except under Eratosthenes' name. Given that three out of our four sources for the enharmonic agree on 5/4, I would suggest that the balance is in favour of this ratio, rather than 5/4. The only further option is to take 81/64 -- the ditone -- as Aristoxenus's intended interval. At this point, I'll confess that I've been disingenuous in the above two paragraphs, since I don't believe for a moment that Aristoxenus's 30 parts of a 4/3 allow anything like such precision. The ditone is in fact Aristoxenus's own _stated_ preference for the large interval of the enharmonic tetrachord, and he laments that the trend in his time was in favour of a more intense tuning, which "sweetened" the ditone (clearly the 5/4 of the other theorists -- see El. Harm. 23, 12ff.). For what it's worth, then, how close does the ditone come to the cents measurement of 24 parts of (4/3)^(1/30) each? The improvement on 19/15 is only one cent more. This should give us a good idea of the margins of vagueness and precision within which we should Aristoxenus's figures. Archytas was a near predecessor of Aristoxenus, or an older contemporary, and he, as we have seen, was already content to give 5/4 as the width of the large interval, so the trend was already well under way some time before Aristoxenus wrote; we might even speculate that Aristoxenus was stating his preference for a practice that had died out by the time he wrote. Whatever the case may be, Barker argues persuasively that neither Aristoxenus nor Archytas convinced themselves of the relative sizes of ditone and 5/4 by means of the monochord (a method which would hardly have claimed much of Aristoxenus's time!) nor by listening alone, but rather by observation: they would have seen players tune their string by fourths and fifths to arrive at the ditone, after which they would have noticed the players tighten the string until the "sweetness" of the interval between that string and its higher neighbour satisfied them. > Frankly, I'm somewhat suspicious of Ptolemy's enharmonic and > chromatic genera as they have a rough 1:2 division of the pyknon > (Barbera), even if it meant that he had to reorder the intervals > resulting from "katapyknosis" to obtain usable superparticular > ratios. But, I do agree that the kithara and lyra tunings probably . > represent practice I don't see that there is any more reason to doubt Ptolemy's pyknomatic tetrachordal divisions than there is to doubt his kithara and lyra tunings. Of course players would not have been able to intuit the sizes of Ptolemy's ratios within the pyknon, but nor would they have been able to intuit the ratios of any of the other theorists either. No, what leads me to believe that Ptolemy's enharmonic and chromatic divisions were trustworthy as a description of contemporary practice (though perhaps not earlier) is the peculiar and awkard method he devises for calculating how the pyknon was to be divided. This has nothing to do with the constraint of finding superparticular ratios, since the tunings Ptolemy takes issue with -- the enharmonics of Archytas, Eratosthenes and Didymus, and the chromatics of Eratosthenes and Didymus -- all remain within these constraints (only Archytas's chromatic contains an epimeric ratio). Rather, he has devised the method to arrive at a result which gives a higher interval that is noticeably larger than the lower, indeed, as John Chalmers says, about twice as large as the lower interval. He criticises Archytas for reversing the order of large and small in both enharmonic and chromatic, and Didymus for the same offence in his chromatic. The only plausible explanation would seem to be that Ptolemy heard such divisions of the pyknon practised by players; his task, based on this perception, was to find the method of generating superparticular ratios that best reflected this perception. -- 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; Fri, 28 Feb 1997 09:38 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA32745; Fri, 28 Feb 1997 09:38:49 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA32732 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id AAA11631; Fri, 28 Feb 1997 00:37:01 -0800 Date: Fri, 28 Feb 1997 00:37:01 -0800 Message-Id: <33209164.125761749@kcbbs.gen.nz> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu