source file: mills2.txt Date: Thu, 27 Feb 1997 15:03:01 -0800 Subject: More Meantones From: John Chalmers Least Square Tunings: With somewhat simpler assumptions than Paul Erlich, I generated some linear temperaments designed to minimize the errors of the Fifth, Major Third, and Harmonic 7th (C-A#). It seems to me that the 5/3 and 6/5 are not independent of the 5/4 and 3/2 and not really necessary, though I believe Thomas Young included all the simple 5 limit consonances in 3 or 4 octaves to generate his temperaments (5/18, 11/40 comma, approximately 50-tet). Therefore, I have used only functions of the cent values of the intervals and X (as the temperament fifth) to construct error functions Thus (F-X) is the error of Fifth, (T-4*X-2*oct)the error of the Major Third, and (S-10*X+5*oct) that of the natural 7th. These error functions may be squared, summed, differentiated and set equal to zero as Paul described. Analogous functions may be written for positive systems where X is greater than 700 cents: (F-X), (T+8*X-5*oct), and (S+14*X-9*oct). Finally, the terms may be weighted inversely as the magnitude of their primes. Thus the (F-X) term may be divided by 3, the Major Third term by 5, etc. after squaring. Negative Systems: 1/4-comma Meantone, 5/4 Just696.5784 LST35696.8947 LST37696.9328 LST57 696.8406 LST357 696.8843 Just 7/4 NEG696.8826 Weighted LST35 697.0857 Weighted LST37 696.9982 Weighted LST57 696.8269 Weighted LST357 696.9229 LucyTuning 695.4930 Wilson's Meta-Meantone 695.6304 Wilson's Meta-1/5-Comma Meantone 697.0676 3 1/3-Comma, 6/5 Just 694.7862 3op5, 1/5-Comma, 15/8 Just 697.6537 Positive Systems: Pythagorean , 3/2 Just 701.9550 6/5 Just, 1/9-SKHISMA, 3 Sabat-Garibaldi 701.7379 15/8 Just, 1/7-SKHISMA, 3op5 701.6759 5/4 Just, 1/8-SKHISMA, Helmholtzian 701.7108 7/4 Just 702.2267 LST35 701.71454 LST357 702.0992 The unweighted negative Lst Square systems and those yielding certain ratios appeared in Xenharmonikon 2, 1974 along with a number of other tunings at the 11 limit not shown here. --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 28 Feb 1997 00:06 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA32330; Fri, 28 Feb 1997 00:06:15 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA32447 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id PAA11720; Thu, 27 Feb 1997 15:04:20 -0800 Date: Thu, 27 Feb 1997 15:04:20 -0800 Message-Id: <970227170944_1382852835@emout19.mail.aol.com> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu