Re: [tuning] Bach: Pyth/5 or 5Synt/23

Dear Fred Reinagel,

Thank you for your highly interesting contribution and your version
to derive the "Bachgleichung" for the well tempered fifth Qw
"Werckmeister/Bach/wohltemperirt", being:

Qw^4 + 2Qw - 8 = 0.

I preferred to slightly modify your formulation concerning the
"sign" of the beats. The third is enlarged; the fifth reduced,
sign of identical expressions to be inverted; resulting in:

"Setting B(5) = -B(3) yields:
> 3*f(1) - 2*f(3) = 5*f(1)-4*f(2)".

The result, of course, remains unchanged.

But the most exciting feature of your communication was for me that
you reminded me of Alexander John Ellis' work, 1864. If I remember
correctly, the article I refer to is:

Ellis, Alexander John. "On the Temperament of Instruments with Fixed Tones",
Proceedings of the Royal Society vol. 13, London, 1864, pp. 404-422.

Therein Ellis defines - in terms of the syntonic(!) comma - a major
triad in which the beats of the third and the fifth occur at the
same rate! He calculates, as you did by our modern means, that
the temperament of the fifth in the triad that beats at unison with
the relevant third must amount to (about) S*5/23 (S=synt. comma).

If again I remember correctly, Ellis apparently did not realize that
the P/5 enters into play here!!! Because, approximately, P=23.5 cent and
S=21.5 cent. P/5=4.7 cent and S*5/23 = 4.67 cent. But as soon as
P/5 intervenes, then 7 perfect fifths and 5 such fifths, let's call
them unashamedly "wohltemperirt", will bridge 7 octaves.

But nearly a century before Murray Barbour, Ellis was not at all
conversant with irregular temperament. And missed to rediscover
Werckmeister's preferred temperament, later adopted by J.S. Bach for
Das wohltemperirte Clavier.

I learnt from the excellent book by my friend Hans Hanan Wellisch
(The Conversion of Scripts, Its Nature, History, and Utilization) on
transliteration of alphabets, that Alexander John Ellis had
attempted to catalogize all the existing phonems within all human
languages, an extraordinary undertaking.

There exists an amusing mnemnotics for "Werckmeister/Bachgleichung",
sort of little "joke":

For the perfect fifth there holds: 2Q - 3 = 0
For the meantone fifth there holds: Q^4 - 5 = 0
SUMMING UP (!) Q^4 + 2Q - 8 = 0

Isn't it logical that the fifth "wohltemperirt" lies somewhere between
the perfect fifth and the mean-tone fifth??
Nevertheless, this neat procedure, despite the argument above, is
mathematically at the verge of madness.

For my derivation of Bachgleichung, you may see:

Kellner, H.A.: Eine Rekonstruktion der wohltemperierten Stimmung von Johann
Sebastian Bach. Das Musikinstrument 26/1, Januar 1977, Seite 34-35.

As a P.S., you might be perhaps interested to see my best approximation
available to the solution of the quartic equation for Qw:

Qw = 1,4959535062432300212473083995637961...

FreinagelR@netscape.net schrieb:
> --- In tuning@y..., ha.kellner@t... wrote:
> ......
>
> >
> > Despite its pure third, the mesotonic triad will BEAT.
> To get > things "purer", shouldn't one attempt to reduce the beats?
> > These
> > triads beat: not because of the third that is pure, but due
> > to the fifth C-G.
> > This is one of the 4 fifths that has to be necessarily tempered
> > smaller in order to fit into its perfect third.
> > As the triad beats anyway, does it really make sense to insist
> > on a pure third? Obviously, NO! and the third may be relaxed,
> > up to the order of its beat-rate becoming comparable to the
> > rapidity of beats of the fifth. Enlarging the third somewhat
> > will slow down the beats of the fifth.
> > The ideal "meeting/compromise point" will be the situation
> > in which - within a tempered triad - both its principal constituent
> > intervals third and fifth beat at the UNISON. What sufficient
> > reason could
> > produce a BETTER COMPROMISE??
> > ******************
> ........
>
> The concept of the ideal "trias harmonica perfecta" being realized by
> a regular temperament where the _beat rate_ of the 5th of the major
> triad is identical to that of the major 3rd has compelling
> philosophical appeal. It should be appreciated that 1/5-(syntonic)
> comma meantone, which results in a 1/5-comma error in both of these
> intervals (5th narrowed, 3rd widened) does _not_ produce equal beat
> rates.

The 1/5 syntonic comma triad and the system has, in fact, the
appealing property that the fifth is reduced by the same amount as
the third is enlarged. But my ears are not that mathematical as to
perceive comma-fractions, but rather BEATS!

> The primary beat of the 5th is produced by the 3rd harmonic
> of the lower tone against the 2nd of the upper, whereas that of the
> major 3rd is the 5th harmonic of the lower against the 4th of the
> upper.
>
> Herein follows the derivation of the degree of temperament for this
> case:
>
> Let f(1), f(2) and f(3) be the frequencies of the triad tones in
> increasing order. Then:
>
> B(5) = 3*f(1) - 2*f(3)
>
> where B(5) is the beat rate of the 5th. Likewise:
>
> B(3) = 4*f(2) - 5*F(1)
>
> where B(3) is the beat rate of the 3rd. Setting B(5) = -B(3) yields:
>
> 3*f(1) - 2*f(3) = 5*f(1)-4*f(2).
>
> Collecting terms and transposing:
>
> 8*f(1) = 4*f(2) + 2*f(3).
>
> Dividing each term by f(1) yields:
>
> 8 = 4*R(3) + 2*R(5) [1]
>
> where R(3) equals the frequency ratio of the 3rd, f(2)/f(1); and R(5)
> that of the 5th, f(3)/f(1).
>
> Using the regular temperament definition for the major 3rd as four
> ascending 5ths reduced by two octaves, we have:
>
> R(3) = R(5)^4/4.
>
> Substituting into [1]:
>
> 8 = R(5)^4 + 2*R(5)
>
> which can be rewritten as the quartic polynomial equation:
>
> x^4 + 2*x - 8 = 0.
>
> An iterative numerical solution quickly converges to yield:
>
> x = 1.4959535....
*!*
>
> Converting to an equivalent portion of a syntonic comma:
0,217353263129005
> T = ln(1.5/1.49595535)/ln(81/80) = 0.2175428.... ,
*!*
*!*
0,217452477735437

> a value between 1/4 comma (0.25) and 1/5 comma (0.20). A very close
> fractional approximation to this value is 5/23.

Other rational approximations to T may be interesting, namely:

5 / 23
152 / 699
157 / 722
309 / 1421
7264 / 33405
7573 / 34826
14837 / 68231
22410 / 103057
104477 / 480459
>
> Respectfully,
> Fred Reinagel

Kind regards,
Herbert Anton Kelner
>
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