source file: mills2.txt Date: Tue, 25 Feb 1997 12:44:52 -0800 Subject: Re: History of the Diatonic Scale From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) From: PAULE I do agree that in most cases the diatonic scale existed without a standardized tuning. I believe that the creation of similar tetrachords was the only guiding principle in most cases, since I do not believe that 5-limit or higher ratios are relevant to purely melodic music. The 3-limit (and even 2-limit) approximations have a wide range of tolerance when harmony is not an issue, or when inharmonic instruments make the value of just intonation questionable. Many of the pentatonic scales of Southeast Asia and Africa exhibit two identical or nearly identical "trichords" each spanning a perfect fourth. A pentatonic scale with only two step sizes, such as in China or Thailand, obeys the principle of two identical trichords in every octave species. What is "mystical" about Ptolemy is that he insisted on superparticular ratios and a geocentric universe with purely circular motions as if these were some sort of revealed truths. They ended up requiring exceedingly complicated models to approach the fairly simple realities. It is just as fair to criticize his views as it is to criticize Aristotelian physics. Where would the world be if no one had ever criticized Aristotelian physics? If a schoolteacher were to teach Aristotelian physics today, would it not be fair to criticize them? If the Greeks had not their absolutist Pythagorean/Platonic insistence on ratios in explaining everything, to the extent that the discovery of irrational numbers was a forbidden secret, would they not have progressed farther in mathematics and science? In particular, would some mathematician not have interpreted Aristoxenus' ideas in terms of the twelfth root of two? I'm sorry if it's not politically correct to identify the intellectual stumbling blocks of past civilizations. Matt, you could be given a Mozart string piece which would have to descend by, say, seven commas from beginning to end, if it is to have all chords in just intonation and no comma shifts in sustained tones (this is actually a typical scenario). Even though the beginning and ending keys may be notated exactly, you would say that the piece ends in a distincly different key than it began. Would your analysis reflect the musical reality better than the traditional analysis? I think not! What if the piece was a keyboard piece? Forget it! Perhaps your platonic ideal is just intonation, and mine is a fixed diatonic scale. Perhaps we both need to loosen up a bit, and admit that both are sometimes compromised in favor of the other. The point of compromise can depend on such factors as the tempo of the piece. Slow tempi demand just intonation, while fast tempi demand symmetrical, digestible melodic patterns. The best musicians intuitively understand both ideals and how to compromise between them according to the musical situation. An analysis that purports to describe pitch in more accurate terms than what is actually notated would have to take these issues into account, or else it is nothing more than an exercise in Pythagorean mysticism. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 25 Feb 1997 22:22 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA09723; Tue, 25 Feb 1997 22:22:10 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA09711 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id NAA01065; Tue, 25 Feb 1997 13:19:30 -0800 Date: Tue, 25 Feb 1997 13:19:30 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu