source file: mills2.txt Date: Fri, 4 Jul 1997 08:37:16 +0200 Subject: Re: Definitions From: mr88cet@texas.net (Gary Morrison) >1. Tempering initially referred to how one deviated about the system of >Pythagorean tuning >Nowadays, however, temperament and tuning seem to be synonymous. At least among most tuning buffs, "temperament" is pretty much synonymous with "deviation from just intonation". And that is to say JI of whatever sort, although the Ptolemaic tuning of 1:1 9:8 5:4 4:3 3:2 5:3 15:8 2:1 is probably generally regarded as the most basic reference point for a major scale anyway. That is NOT to suggest, however, that Pythagorean tuning (1:1 9:8 81:64 4:3 3:2 27:16 243:128 2:1) is regarded as a temperament of that Ptolemaic tuning. Pythagorean is another JI point of reference. >3. Mean Temperament has two general meanings. First it means a system of >tuning by fifths such that four ascending fifths and two descending >octaves make up a perfect third. It also means any one of many slight >variations about this basic system. Meantone temperament is a tuning system based upon a consistent size of perfect fifth, that fifth being tempered downward to make other intervals Just, or at least a lot closer to just. Now in 99% of the cases they involved a "wolf fifth" at the point where the circle was broken, frequently between G# and Eb. That does not violate the single-size-of-fifth rule, because G# to Eb is clearly a diminished sixth instead of a P5. The system you described above is called "quarter-comma meantone", because its fifth is tempered flat by a quarter of a syntonic comma (81:80) so that four of them stacked atop on another, adjusting downward by octaves as needed, arrives at an exact, just 5:4. (Zikes! Please don't say "perfect third"; fifths can be perfect, but not thirds; that's what the word "just" is for.) Notice that I said in 99% of the cases it involved a wolf fifth. There are historically rare, but well-documented, cases of two meantone systems being completed by carrying them beyond 12 (or occasionally 13-14) tones per octave. The two systems that this works for a third-comma meantone, in which the fifth is flatted by a third of an 81:80 syntonic comma to produce a just 5:3 major sixth, and in quarter-comma meantone. When 19 third-comma flat fifths are stacked atop one another, you arrive at a pitch that is astoundingly close to exactly 11 octaves above where you started - only about 1 cent off! That means that "complete" third-comma meantone is so close to 19-tone-per-octave equal-temperament (called "19TET") that there's no real-world point in distinguishing between the two. And the analogous situation occurs with quarter-comma meantone and 31TET. Christian Huygens and others noticed this during the reign of quarter-comma meantone. >4. Bachís WTC refers to a mean tempered clavichord; in particular, >well-tempered means mean tempered. I am personally aware of no evidence of that. All evidence I've heard suggests that he wrote it for several of the Werkmeister well temperaments, which were used by the organs in the area. To the best of my knowledge "klavier" conceptually refers to any keyboard instrument, although it's astoundingly unlikely that it would be applied to the organ. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 4 Jul 1997 08:38 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04784; Fri, 4 Jul 1997 08:38:37 +0200 Date: Fri, 4 Jul 1997 08:38:37 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA04758 Received: (qmail 21446 invoked from network); 4 Jul 1997 06:37:25 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 4 Jul 1997 06:37:25 -0000 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu