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

"wohltemperirt"?

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