source file: mills2.txt Date: Wed, 4 Sep 1996 14:23:35 -0700 Subject: An interesting series of tunings From: PAULE I just happened to notice that in many of the tunings (equal temperaments, to be precise) that I work with, seven steps gives you an interval where the lower note sounds like the root. Let's see what happens if we formalize this idea -- we want the lower note of the interval to correspond to a harmonic that is a power of 2 (so it sounds like the root), and the upper note to be the next higher harmonic: Interesting tuning #n = round(7/(log((2^n+1)/2^n)/log(2))) According to this formula, interesting tuning #0 is 7-tET, which is the standard tuning of Thailand. Siamese compositions modulate, usually by fifths, much like those in the West. As the only other culture that relies on modulation, Thailand is the only other culture that uses equal temperament (though I have heard that this practice extends into Cambodia). Interesting tuning #1 is 12-tET. Apparently, interesting enough to satisfy the West for the last few centuries. Interesting tuning #2 is 22-tET. This is my main avenue of research; contact me for more info. This is also the first equal tuning with consistent representations of all harmonies through the 12th harmonic. See previous digests for discussions of consistency. Interesting tuning #3 is 41-tET. Recommended by Paul van Jenko around the turn of the century, this tuning is barely distinguishable from 9-limit just intonation. Try a "ninth chord" tuned (in cents) 0 380.5 702.4 965.9 1404.9. It also offers consistent representations through the 16th harmonic. Interesting tuning #4 is 80-tET. This is the first equal tuning with consistent representations through the 20th harmonic. That pretty much covers any harmonies I would consider musically useful, and then some. Interesting tuning #5 is 158-tET. Or it would be if there were anything interesing about it. Like most numerological speculations, such as Kornerup's and Yasser's, this one's luck runs out after a while. Then again, who am I to say that some great composer won't come along and demonstrate the wonderful properties of 158-tET? Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 5 Sep 1996 00:15 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA19911; Thu, 5 Sep 1996 00:17:17 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA19764 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id PAA15688; Wed, 4 Sep 1996 15:17:15 -0700 Date: Wed, 4 Sep 1996 15:17:15 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu