Goldbach
Variationen
There are some claims about Bach having used the
golden ratio in his music.
The golden ratio, or Phi, is supposed to be the
“Divine proportion”, the best ratio between two related items; it approximates
to 1.618. As music deals with integers, I will only consider here integer
relations; so 162, for example, will admit an only golden section, 100, and the
minor golden section will be 62.
While golden ratio is mostly searched within a single
piece, I had the idea to study a well-known binary form, Prelude & Fugue,
inside the “48”, the two volumes of the Well-tempered clavier. When a Prelude
has x bars and the associated Fugue y bars, are there cases where x and
y are the minor and major golden sections of x+y?
If we consider executed music, i.e. with the repeats
of the preludes, there are two solutions, and these are the two Prelude
& Fugue in F sharp minor (fis-moll) which rank 14 in volume 1
(1722) as well as in volume 2 (1744).
The first step in Bach numerology is the “signature
14”, with 14 being the sum of the B-A-C-H letters in a numeric alphabet of 24
letters which was used at this time. If there is not any explicit proof of this
signature 14, it seems to have convinced most scholars, although some extreme
sophistications are quite difficult to accept.
So BWV 859 has 64 bars, of which minor and major
sections are 24 and 40, and BWV 883 has 113 bars, of which minor and major
sections are 43 and 70, and Preludes and Fugues 14 equate these numbers (a
board for both volumes is given here).
It’s often possible to find “meaningful” words having
wanted values (like CREDO=43 and IESUS=70), but not always clearly relevant.
Here stands a fantastic opportunity. Bach chose to name his work Das
Wohltemperirte Clavier (in his own write), a title which has 24 letters for
an exploration of the 24 tonalities… Bach is likely the first to have used this
word Wohltemperirte which has 14 letters… and its value in that
24-letters alphabet is 177, i.e. the number of bars of pieces ranking 14 in
both volumes, starting with a prelude in 24 bars (24+40+43+70=177).
I’ll come back to this, but I have much “curiouser”
now.
My study of the “48” gave two other interesting cases.
The first volume ends on the only Prelude &
Fugue with repeats for the Prelude. Without the repeats, these pieces of 47
and 76 bars give a very good approximation of Phi, and the Fugue ends with the
famous letters SDG, acronym for Soli Deo Gloria, but its value of 29 is
often taken for an other signature of JSB himself; 29 is the golden section of
47.
There is a strange crossing between volumes 1 and 2
for the pieces in C sharp minor (cis-moll): P1 (39 bars) is the golden
section of P2 (62) while F2 (71) is the golden section of F1 (115).
The sum for these pieces in cis is
39+115+62+71=287, now it happens that 177, the sum for pieces in fis, is
the golden section of 287. There are only in both volumes two sets of four
pieces giving perfect golden relations, and a perfect golden ratio appears
between these two sets.
It should be noticed too that cis is the fifth
of fis, the fifth being obviously the musical “golden ratio” (much more
obviously than the real golden ratio should be THE factor of beauty).
There would be many other little things to add, but I
come back to this number 287 which already appeared in Bach numerology.
A strange thesis held by Van Houten and Kasbergen in
their Bach en het Getal (1985) is that Bach lived in such a perfect
harmony with the Cosmos that he knew very soon the exact day of his death on
the 28th day of the 7th month of 1750. Hence Bach’s music
is full of allusions to this date, mostly under the forms 287 or 209 (7/28 is
the 209th day of the year).
That’s what THEY say. Now, finding this number 287
implied in a deep golden relation, I had the curiosity to look for the little
and big golden sections of 209, which are 80 and 129, and this is quite funny
because Bach was born on March the 21st, which was the 80th
day of the year 1685. Actually there is at least one example (in B minor
Mass) where VH&K consider the sum of bars 129+80=209 as signifying
Bach’s birth and death.
It just lacked the Golden Ratio to be fully Cosmic,
now it’s done.
Still being curious, I looked for possibilities of
finding 80 as well as 129 or 209 bars in consecutive pieces of the “48”. The
three numbers appear only one time each, together, in a kind of ideal pattern
which escaped to Messrs VH&K. That’s in first volume, where Preludes
& Fugues in f-moll, Fis-dur, fis-moll give the sum 209 not only with
their bars, but too with the values of these tonalities in the numeric alphabet!
This board gives the main relations yet encountered:
|
set |
tonality |
=287 |
|
P1 |
F1 |
|
P2 |
F2 |
|
|
4 |
cis-moll |
78 |
|
39 |
115 |
|
62 |
71 |
287 |
|
|
|
+ |
|
|
|
|
|
|
|
|
12 |
f-moll |
54 |
|
22 |
58 |
|
(70) |
(85) |
|
|
13 |
Fis-dur |
74 |
|
30 |
35 |
|
(75) |
(84) |
|
|
14 |
fis-moll |
81 |
|
24 |
40 |
|
43 |
70 |
177 |
|
|
|
209 |
= |
76 + |
133 |
|
|
|
|
Before going any further I feel it necessary to
clearly state I’m not at all attracted by the ideas of VH&K. Then:
The pattern ends with set 14 which was at the
beginning of this investigation.
The 80 bars of set 12 part in 22 and 58, 22 being the
little golden section of 58, as 80 is the little section of 209; this is again
the only case within the two volumes.
The 65 bars of set 13 fit with the age of Bach at his
death, 65 years… and 129 days given by adding the following 64 of set 14.
There are some striking resemblances between fugues 12
and 14 (marking Bach’s birth and death following VH&K!):
– they have both 4 voices, entering both T-A-B-S
– much more peculiarly the exposition gives tonic-fifth-tonic-tonic;
I’m not aware of any other example of this within Bach
– the notes BACH, without order, appear in the first
tonic theme; the only other example within volume 1 is in the last fugue, where
the theme is very chromatic (and it’s too the other golden set of volume 1).
Here are the details for calculating the values of the
tonalities:
f-moll = 54 (6 + 12 + 14 + 11 + 11)
Fis-dur = 74 (6+9+18+4+20+17)
fis-moll = 81 (6+9+18+12+14+11+11)
total = 209 !!!
No other pattern of
successive tonalities equates 209. It was the 287 bars of set 4 in cis-moll
that lead me to this pattern 209, now cis-moll = 78, and 209+78 = 287
!!!
So in this pattern appear
Bach’s dates, 209 and 287 for the death, and 80 for the birth. I wondered if lacking
213 (3/21) could be found, and the values of the 3 minor tonalities add to 213
(78+54+81). And the golden partition of 213 in 132 and 81 is actualized by
tonality 14.
It’s often claimed that
anything can be deduced from any range of numbers, but this set of four numbers
78-54-74-81 is the only one allowing to find by adding up 287 as well as 209,
213 as well as 132 (one could even think of 80 as eis-moll!!). It
is quite independently from names and values of tonalities that I came to
distinguish these sets 4-12-13-14, first from the golden ratio which is often
referred to Bach, then from numbers 287 and 209 that I didn’t invent either.
Actually this pattern seems so perfect that there would have been many entries
to discover it, and I have to forget here other properties.
The problem is that it seems
too perfect to be fully intentional; Bach could not decide these numbers
78-54-74-81 would be values of tonalities! It cannot be considered as simple
chance either, as there are many examples of such perfect numeric patterns, so
what? I have no easy answer, I’ll just suggest to look towards Jung’s
synchronicity for instance, where the idea of non-causal effects is not at all
uncommon.
Nevertheless the pattern
shows the limits of logical approach. Jung studied astrological data not
because he believed in astrology but because the conjunction of some planets
had a mythical and psychological meaning, and found there some strange results.
As Bach’s music became mythical as well, could this interfere with the composition
of the music itself?
I won’t go any further here
in interpretation, my English being too poor to explain my thinking. I’ll end
with another pattern which has nothing to do with VH&K.
If the 24-letters title fits
with the 24 tonalities, then the 14-letters word Wohltemperirte fits
with the 14 tonalities from cis-moll to As-dur. The Prelude
& Fugue 8 of volume 1 has the peculiarity to show the Prelude in
es-moll and the Fugue in dis-moll. Maybe there was a
“well-tempered” reason for that, but it allows an other extraordinary
gematrical coincidence.
The golden section for 14 is
9.
The golden section for the
value of Wohltemperirte is 110, and that is the value of the 9 first
letters.
The es-dis trick makes
it necessary to count two times the other tonalities, this leading to a sum of
1664 for the tonalities of all 14 Preludes and 14 Fugues. The exact golden
section 1028 is found for the first 9 of them.
|
|
|
tonality |
|
P |
F |
P+F |
|
W |
|
cis-moll ”” |
|
39 |
115 |
|
|
O |
|
D-dur ”” |
|
35 |
27 |
|
|
H |
|
d-moll ”” |
|
26 |
44 |
|
|
L |
|
Es-dur ”” |
|
70 |
37 |
|
|
T |
110 |
es-moll dis-moll |
1028 |
40 |
87 |
|
|
E |
|
E-dur ”” |
|
24 |
29 |
|
|
M |
|
e-moll ”” |
|
41 |
42 |
|
|
P |
|
F-dur ”” |
|
18 |
72 |
|
|
E |
|
f-moll ””
2 |
|
(22) |
(58) |
80 |
|
R |
|
Fis-dur ”” 0 |
|
(30) |
(35) |
|
|
I |
|
fis-moll ””
9 |
|
(24) |
(40) |
129 |
|
R |
67 |
G-dur ”” |
636 |
19 |
86 |
|
|
T |
|
g-moll ”” |
|
19 |
34 |
|
|
E |
|
As-dur ”” |
|
44 |
35 |
|
|
|
177 |
|
1664 |
375 |
608 |
209 |
These 14 sets add up to 1192 bars, the golden
section of which being 737. It does not fit with the first 9 (826), nor with
the cutting between Preludes and Fugues (451 and 741, not far), but there is a
way to obtain a perfect relation by remembering the perfect pattern of 209,
with the golden section falling precisely at this step 9. By isolating these 3
sets, the remaining 11 sets offer the perfect cutting of 1183 in 375 and 608.
Vertically, these relations cut the 14 sets in
8-1-2-3, while these pieces stand right amidst the tonalities of C and ABH.
Rémi Schulz