source file: mills2.txt Date: Sat, 1 Feb 1997 16:29:06 -0800 Subject: Meantone Explanation From: Gary Morrison <71670.2576@compuserve.com> Somebody asked me about the exact meaning of fifth-comma meantone. Those of you who are new to unusual tunings might find this useful information. ----------------------------------------------- The usual definition of an nth-comma meantone would be something like "a tuning based upon a circle of identical-sized fifths tuned flat of the 3:2 by 1/n commas, wrapping within an octave". Assuming that you use the syntonic comma in that definition, as opposed to the pythagorean comma, then quarter-comma meantone is devised to create a just 5:4 at the sacrifice of the accuracy of the 3:2. That is because a stack of four fifths lands you a major third above where you started: C G D A E. If those fifths were exactly just, then you'd arrive at an 81:64 ratio, which is a syntonic comma (81:80) sharp of the idea 5:4. So the quarter-comma meantone solution to that problem is to compromise the P5 down by 1/4 of a comma so that when you stack up 4 of them you end up 4/4 of a comma flat of 81:64, which is of course 5:4. Similarly, third-comma meantone is devised to temper the 27:16 major sixth down to the cleaner-sounding 5:3. As you've probably heard by now, 31TET is virtually identical with quarter-comma meantone, and 19TET, for all realistic purposes, *IS* third-comma meantone. If you stack up 31 quarter-comma flat fifths you land only 6 cents off from where you started, and if you stack up 19 third-comma flat fifths you land only about 1 cent off. So what about fifth-comma meantone? As with sixth-comma meantone, that is probably best characterized as well temperament. That particular size of fifth was probably chosen more to generally equalize tuning errors within a 12-toned framework, rather than to make any particular interval sound accurate. That in the sense that whether a 243:128 major seventh is really all that much more harmonically unpleasant than 15:8 major seventh (a comma flatter) or than any major seventh for that matter, is pretty marginal. Or relative to the tonic anyway; 15:8 is of course is mediant of the dominant, so perhaps one could argue it that way. Perhaps one could argue that fifth comma meantone was devised to make the third of V-chord just in the same sense that third-comma meantone makes the IV-chord just. That combined with the fact that the total error between the third of I (the mediant) and the third of V (the leading tone) is smaller that way. In the quarter-comma meantone case, the mediant's error is 0 and the leading tone's is 1/4 comma, whereas in fifth-comma meantone a total error of 1/4 comma, the mediant's is 1/5 comma, and the leading tone's is 0, a total of 1/5 comma. But I think that when you add to that formulation the error in the third of the IV-chord (the submediant), and also consider the effects upon minor keys, I suspect that the well-temperament explanation may be more realistic. Well anyway, fifth- and sixth-comma meantone definitely straddle the boundary between well temperaments and classic meantones. By the pure definition, they're certainly meantone tunings, but their motivation seems to match more closely that of well temperaments. Oh, by the way, I think that Switched-On Bach 2000 has some fifth-comma meantone on it. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sun, 2 Feb 1997 18:31 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA22655; Sun, 2 Feb 1997 18:31:02 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA22633 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id JAA05971; Sun, 2 Feb 1997 09:27:58 -0800 Date: Sun, 2 Feb 1997 09:27:58 -0800 Message-Id: <199702021224_MC2-10AB-B182@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu