source file: mills2.txt Date: Thu, 22 May 1997 19:29:04 +0200 Subject: RE:Equal Divisions -- of What? From: PErlich David, I agree that JI makes for the best harmonies in most cases. There are exceptions, such as a pentatonic scale played as a chord, where a meantone tuning changes the three thirds from Pythagorean to near-just, without giving up much consonance in the fifths. But yes, melody and harmony often have contradictory implications for tuning. >> Finally, consider the geometric series >> >> . .x^-9, x^-8, x^-7, x^-6, x^-5, x^-4, x^-3, x^-2, x^-1, x, x^2, x^3, >> x^4, x^5, x^6, x^7, x^8, x^9, . . . >> >> where x>0 and x does not equal 1. Whether the units of measurement are >> string length or wavelength or time period, as above, or frequency, as in >> the 800:900:1000 Hz example, the result is the same: equal temperament! >Alright. I figured out how to use the geometric series shown above to >acheive octave-repeating ET by supplying x with the nth root of two, where n >is the number of divisions/octave. But I don't understand what you mean by >putting a series of integrally-related frequencies such as 800:900:1000 Hz >into the equation. If you apply the geometric series to a _single_ frequency >you get ET. But I can't seem to get the 800:900:1000 Hz _series_ to make ET, >no matter what I do to it. All I was saying is that if you use frequency units as your units of measurement, the geometric series gives you equal temperament. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 22 May 1997 19:49 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA07883; Thu, 22 May 1997 19:49:02 +0200 Date: Thu, 22 May 1997 19:49:02 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA07871 Received: (qmail 13622 invoked from network); 22 May 1997 17:48:28 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 22 May 1997 17:48:28 -0000 Message-Id: <338434F1.7510@gold.tc.umn.edu> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu