For the system Bach/"wohltemperirt" we consider now
the circle of fifths:

Ab- Eb- Bb- F- C- G- D- A- E- B- F#- C#- Ab

From C upwards, the 5 fifths are tempered as follows,(cent):

-4.7, -4.7, -4.7, -4.7, 0(perfect between E and B), -4.7

The tempering-sum in this "closed" system is P, the
Pythagorean comma. Check: 5 *4.7 = 23.5 cent (=P).

In "wohltemperirt", within the C-major triad, the third C-E
beats  definition-wise at the same rate  as the fifth C-G.

It turns out that the intervening minor third E-G within
the basic triad of C-major, C-E-G, beats exactly FOUR times
at the fifth's rate. This represents a proportionally
beating triad - in very small numbers indeed.
Overall, omitting signs, 4,1,1; total sum 6.

In what follows, the relative beat-rates of a reduced
interval, i. e. smaller than perfect, gets a negative sign
(mathematically). An enlarged interval, the positive sign.

For the (MAJOR) C-triad "wohltemperirt" there holds:

minor third, major third and fifth beating at

    -4         1            -1.
Thus,           (-4, 1, -1)

Let - in a MINOR triad - the (upper) major third,
enlarged, beat at the same rate as the reduced fifth.
In this case, the not so well known relation holds:

minor third, major third and fifth beating at

    -3         1            -1.
Thus,           (-3, 1, -1)

Related to these obervations, a Belgian mathematician,
FRANK JANSSENS, discovered and published generalized
relations concerning beats in triads:

Janssens, Frank: On the Relations of Beats within Tempered
Triads. Acustica, S. Hirzel Verlag Stuttgart, Vol. 40/3,
1978, page 200-202.

His remarkable results, accessible in ACUSTICA, are
perhaps not generally known. I missed to suspect the
existence of such formulae and was unable to find them.
To recapitulate Janssen's results:

Let us designate the beats of a triad's intervals by
t,    T    and    F,     respectively, for beat rates of
minor third, major third and fifth.
(Beat rates carry mathematical signs - see above).

Then, for a MAJOR triad the first JANSSENS-formula holds:

   2t + 3T = 5Q

For the MINOR triad the second JANSSENS-formula holds:

    t +  T = 2Q

As an example for an application, let us now calculate
the beats of the minor third in B-major of "wohltemperirt".
The relative beat-rate of the enlarged major third therein
is 6, with the appropriate sign. The fifth then beats at -1.

Using these values, the first Janssens formula yields:

   2t  = 5Q -3T
   2t  = -5 -3*6 = -23
    t  = -23/2.

Here, multiplying by 2 removes the denominators:
(-23/2, 6, -1). Suppressing signs (absolute values), yields
(23, 12, 2).

This establishes the proportionality of beats in B-major
of "wohltemperirt", via the absolute values of the beats
(i.e. for beats that can be heard); overall sum 37.

A case much more simple and well known occurs e.g. within
a PERFECT octave. The tempered fourth at the top beats
twice as rapidly than the lower fifth. Exchanging these
intervals, fifth at the top, fourth at the bottom: both
intervals in this case will beat at the same rate.

Via the Janssen's formula above, all the further
proportionalities of beats within the  G, D, A triads
could also be derived. These formulae do apply as well,
of course, to the perfect fifth. Its beat-rate is zero.

The structure of "wohltemperirt" with respect to the
quality of major thirds, looks as follows:

1                    C

2                F       G     D

3           Bb                     A

4      Eb                              E     B

5  Ab                                           F#   C#

There are 5 platforms or levels. Descending by one step leads
to thirds tempered heavier by 4.7 cent.
C-major has the best third, only 2.8 cent off the pure third.
On the side with sharps, G- and D-major share one level,
likewise E- and B-major.

As concerns the side of the keys with flats, four descending
steps from C reach the Pythagorean third: Ab-C.
On the side of keys with sharps, four ascending steps from C
attain only E-major. This key is BETTER by one step
than Pythagorean: In this sense, the system "wohltemperirt"
gives preference and favours the sharpened keys.

Turning now to beat-ratios:

In C-major the beat ratio third/fifth is 1:1.
In B-major the beat ratio third/fifth is 6:1.

From C to the level E/B there descend 3 levels/steps. The
beat ratio between third and fifth increases thereby from
1 to 6, the difference being 5. Therefore, per step 5/3:

The        C-major ratio is                      =  1
The G- and D-major ratio will be   1+5/3         = 8/3
The        A-major ratio will be   1+5/3+5/3     = 13/3
The        B-major  ratio will be  1+5/3+5/3+5/3 =  6
(Note that the fifth on E is perfect; triad E-G#-B).

These beat ratios between thirds and fifths, upwards
from C-major are collected here :

C 1:1, G and D 8:3, A 13:3, and B 6:1.

It is according to the latter ratio 6:1 that the beats of
third and fifth within the the B-major triad must be adjusted
upon tempering.
The f# of the crucial fifth f#-B is reached on the keyboard
after the six descending perfect fifths downwards from C.
(Including octave-jumps upwards whereever necessary).

Thereafter, it only remains to bridge C-E by its 4 fifths
of equal size C-G-D-A-E in order to complete the bearings.

This tuning-procedure via the B-major triad divides the
Pythagorean comma by 5 and deducts this amount (4.7 cent)
from those five fifths that are indicated above as tempered.

A further example for a calculation:

What is in D-major the basic third's beat rate within
The fundament D lies by one tone above C, amounting to
approximately to 9/8. Within D-major, the beats of the
major third occur at 1+ 5/3 = 8/3 times the rate of the
fifth's beat-rate, as just shown. This must be multiplied
by 9/8, the factor of D. Thus 8/3 * 9/8 = 3.
As result, the D-major third D-F# on the keyboard beats THREE
times faster than the basic third C-E within "wohltemperirt".

This should be checked upon laying the bearings; one of
the checks suggested and recommended above.

HAK 22. 5. 2001