source file: m1486.txt Date: Mon, 27 Jul 1998 16:12:59 -0400 Subject: RE: Nano-temperament From: "Paul H. Erlich" >The method of nano-temperament seems useful in a range of musical >contexts, for example in approximating various shades of meantone. >Since a meantone major third is derived from four fifths, it is >possible by varying the size of _some_ of these fifths by one tuning >unit to achieve shades of _average_ temperament only 1/4 tuning unit >apart. The major thirds will vary by an amount four times this size: a >single tuning unit. The principle Margo delineates for realizing Pythagorean tuning is one I recently used to realize 1/4-comma meantone, the most important tuning of 1500-1700, on my Ensoniq synthesizer. Though I recently posted the basic idea in this, I will repeat myself for the benefit of this discussion. In 1/4-comma meantone temperament, the major thirds are tuned pure. The Ensoniq VFX-SD, despite its cents-based tuning tables, has a true internal tuning resolution of 512 notes per octave (unbeknowest to most). The best approximation to the major third is 165 steps of the tuning, or 386.7188, as compared with the just value of 386.3137 cents. A major third played in the middle register with the 165/512 oct approximation would beat once every two seconds, too slow to be noticeable in most music. The major third is constructed from four fifths, which would have to be approximated by an average of (165+512+512)/4, or 297.25 units. So tuning every fourth fifth to 298 units (698.4375 cents) and the rest to 297 units (696.0938 cents) makes all 8 major thirds in the tuning virtually just. This is my 1/4-comma meantone setting on my Ensoniq. Just as in Margo's Pythagorean nano-temperament, the minor thirds here vary in size. Any tuner during the period when meantone was predominant would be able to tune the four augmented triads (Eb G B, Bb D F#, F A C#, C E G#) almost perfectly by ear by eliminating beats in the constituent major thirds, but tuning them to one another using approximately identically tempered fifths was an imprecise endeavor. So in practice the tuning could rarely have been much better than the approximation I have described. ------------------------------ End of TUNING Digest 1486 *************************