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.

Well, the diagram was goofed up by the spaces getting ignored by
Yahoo. If the diagram is copied and pasted and the space reinserted
on the left and right ends of the dashed lines so the asterisks * and
p's line up vertically, it should look good and reveal graphically
the characteristics in a nice, intuitively comprehensible way.

Well, the diagram was goofed up by the spaces getting ignored by
Yahoo. If the diagram is copied and pasted and the space reinserted
on the left and right ends of the dashed lines so the asterisks * and
p's line up vertically, it should look good and reveal graphically
the characteristics in a nice, intuitively comprehensible way.

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:

/tuning/topicId_32946.html#32948

> Well, the diagram was goofed up by the spaces getting ignored by
> Yahoo. If the diagram is copied and pasted and the space reinserted
> on the left and right ends of the dashed lines so the asterisks *
and
> p's line up vertically, it should look good and reveal graphically
> the characteristics in a nice, intuitively comprehensible way.

Hi Bob!

Don't forget, if you're on the Web all you have to do is
go: "Message index" "Expand messages" and then everything is as you
intended it....

J. Pehrson

Thanks, Joe. I'll try again.