88 cents approximations

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