source file: mills2.txt Date: Fri, 27 Sep 1996 03:06:27 -0700 Subject: 88 cents approximations From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) Gary asked me to post the set of 88 cents tuning approximations that I had sent him. The differences are minute and therefore only possibly noticeable at larger intervals. They are least-squares approximations. More combinations are possible of course. The third column is the deviation of each interval in cents. 1: 3 2: 7 3: 8 4: 11 88.0484 cents, 13.6289/oct. Std. dev: 1.2803 cents 1: 7/6 2.7257 septimal minor third 2: 10/7 1.1490 septimal augmented fourth, Euler's tritone 3: 3/2 -2.4322 perfect fifth 4: 7/4 0.2935 harmonic seventh 1: 3 2: 4 3: 7 4: 8 5: 11 87.9745 cents, 13.6403/oct. Std. dev: 1.5049 cents 1: 7/6 2.9474 septimal minor third 2: 11/9 -4.4901 undecimal neutral third 3: 10/7 1.6663 septimal augmented fourth, Euler's tritone 4: 3/2 -1.8410 perfect fifth 5: 7/4 1.1064 harmonic seventh 1: 3 2: 7 3: 8 4: 11 5: 41 87.8356 cents, 13.6619/oct. Std. dev: 1.3071 cents 1: 7/6 3.3641 septimal minor third 2: 10/7 2.6386 septimal augmented fourth, Euler's tritone 3: 3/2 -0.7298 perfect fifth 4: 7/4 2.6343 harmonic seventh 5: 8/1 -1.2596 3 octaves 1: 7 2: 8 3: 11 88.0134 cents, 13.6343/oct. Std. dev: 1.3262 cents 1: 10/7 1.3940 septimal augmented fourth, Euler's tritone 2: 3/2 -2.1522 perfect fifth 3: 7/4 0.6785 harmonic seventh 1: 3 2: 4 3: 7 4: 8 5: 10 6: 11 88.1030 cents, 13.6204/oct. Std. dev: 1.4366 cents 1: 7/6 2.5619 septimal minor third 2: 11/9 -5.0041 undecimal neutral third 3: 10/7 0.7668 septimal augmented fourth, Euler's tritone 4: 3/2 -2.8690 perfect fifth 5: 5/3 3.3287 major sixth 6: 7/4 -0.3071 harmonic seventh 1: 3 2: 4 3: 8 4: 10 5: 11 88.0857 cents, 13.6231/oct. Std. dev: 1.7839 cents 1: 7/6 2.6138 septimal minor third 2: 11/9 -4.9349 undecimal neutral third 3: 3/2 -2.7306 perfect fifth 4: 5/3 3.5017 major sixth 5: 7/4 -0.1168 harmonic seventh 1: 3 2: 7 3: 8 88.0219 cents, 13.6330/oct. Std. dev: 1.9091 cents 1: 7/6 2.8052 septimal minor third 2: 10/7 1.3345 septimal augmented fourth, Euler's tritone 3: 3/2 -2.2202 perfect fifth 1: 4 2: 7 3: 8 87.8115 cents, 13.6656/oct. Std. dev: 2.3927 cents 1: 11/9 -3.8381 undecimal neutral third 2: 10/7 2.8073 septimal augmented fourth, Euler's tritone 3: 3/2 -0.5370 perfect fifth 1: 4 2: 7 3: 10 88.2160 cents, 13.6030/oct. Std. dev: 2.9412 cents 1: 11/9 -5.4561 undecimal neutral third 2: 10/7 -0.0242 septimal augmented fourth, Euler's tritone 3: 5/3 2.1987 major sixth 1: 3 2: 4 3: 10 88.2707 cents, 13.5946/oct. Std. dev: 3.1293 cents 1: 7/6 2.0588 septimal minor third 2: 11/9 -5.6749 undecimal neutral third 3: 5/3 1.6517 major sixth Manuel Op de Coul coul@ezh.nl Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 27 Sep 1996 20:54 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01082; Fri, 27 Sep 1996 20:54:32 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA01802 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id LAA27987; Fri, 27 Sep 1996 11:54:30 -0700 Date: Fri, 27 Sep 1996 11:54:30 -0700 Message-Id: <22960927185022/0005695065PK4EM@MCIMAIL.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu