Greetings, My first calculation of the cents deviation from ET for the Wendell 2002 Well-Temperament: C= +6.8375 B= -0.9775 A#= +5.86 A= 0 G#= +2.9325 G= +5.86 F#= 0 F= +6.8375 E= -0.9775 D#= +4.8825 D= +2.9275 C#= +0.9775 I put these numbers in a spread sheet program and it looks like a pretty decent well temperament. Be sure to credit Robert Wendell. Paul Bailey Modesto CA ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ======================================================================== ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ P.S. I would be grateful for anything pointing people to the choir I founded 7 years ago, the Cantus Angelicus Choral Society at: http://www.cangelic.org We consciously cultivate just harmonies in singing the a cappella sacred music of the great composers. There is streaming audio and downloadable files are also available there. Sincerely, Bob Wendell (Forward from Tuning List) Message: 6 Date: Sun, 20 Jan 2002 02:39:06 -0000 From: "robert_wendell" <BobWendell@technet-inc.com> Subject: Tee-hee: Wendell 12-Tone Well-Temperament 2002 Wendell Well-Temperament 2002 Copyright Robert P. Wendell, January 2002 (Tuning group members may copy and use this as they wish as long as its authorship is recognized and respected.) p * C-E +5.9 * p p * ------------------- * p p * G-B +6.8 * p p * ----------------------- * p p * D-F# +10.8 * p p * --------------------------------- * p p * A-C# +14.7 * p p -*---------------------------------------*- p p * E-G# +17.8 * p p ------*---------------------------------------*------ p p * B-D# +19.6 * p p ---------*---------------------------------------*--------- p p * F#-A# +19.6 * p p ---------*---------------------------------------*--------- p p * C#-E# +19.6 * p p ---------*---------------------------------------*--------- p p * Ab-C +17.8 * p p ------*---------------------------------------*------ p p * Eb-G +14.7 * p p -*---------------------------------------*- p p * Bb-D +10.8 * p p * --------------------------------- * p p * F-A +6.8 * p p * ----------------------- * p p * C-E +5.9 * p p * ------------------- * p In the above diagram of the Wendell 12-Tone Well-Temperament 2002, the horizontal dashed lines represent the cycle of fifths from C back to C, the width corresponding to the sharpness in cents to just major thirds above them (4:5 frequency ratio). The vertical lines of asterisks represent the width corresponding to 12-tone equal temperament and those of p's, the thirds of so-called Pythagorean thirds (i.e., major thirds in 3-limit just intonation). The criteria for the development of this well temperament were: Keys closest to C more just than equal temperament and those remote less so. The justness of the keys most closely related to C traded off in such a way as to avoid the opposite extreme of so-called Pythagorean tuning as much as possible. Within this context, intonation as close to just as possible for the major triads in the key of C and those keys most closely related to it. A gradual, smooth, and consistent shift toward Pythagorean tuning as the keys move toward F#/Gb major, the most distantly related to C. Five of the 12 keys better, and the A/Eb axis in the circle not significantly worse than 12-tone equal temperament (thirds only one cent sharper in this realization). None of the keys quite as extreme as Pythagorean. These criteria were met by distributing the Pythagorean comma of +23.46 cents as follows: The first five elements of the Fibonacci sequence 0, 1, 1, 2, 3, 5... were added to total 12. The comma is distributed symmetrically from D in these proportions starting with 5 and proceeding sequentially to the lesser elements, so the Fibonacci elements represent multiples of 1/24-comma. Therefore five times 1/24-comma is subtracted from the fifth on each side of D, then 3 times 1/24-comma from the fifths on either side of those, and so on around the circle symmetrically so that the two zeros fall in the fifths between C# and D#/Eb and the total is 24/24- comma. The following characteristics result: The largest reduction of a perfect fifth is consequently 4.9 cents on either side of D and the fifths opposite D in the cycle are perfect. The maximimum deviations from 12-tone equal temperament are -7.8 for the most just thirds and +5.9 for the least just. The three major triads in the key of C are only 5.9 to 6.8 cents sharp. The sharpest thirds are 19.6 cents sharp and belong to the three major triads of F# major. These represent a somewhat softened "Pythagorean" tuning. The most just minor thirds are only 8.8 cents flat as opposed to 15.7 cents in 12-tone equal temperament. Five of them are better or equal to equal temperament. The two flattest minor thirds at -20.6 cents are still better than Pythagorean at 21.5 cents and the corresponding thirds are also two cents better than Pythagorean. At this extreme, tuning dissonance increases exponentially with pitch distance from just, so even small differences become significant enough to soften the dissonance.
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