There seems to be some method in this construction after all, even when its practical use is still unsure. And of course the use of numbers to create structures belongs to the basics of composition. The architecture revealed on the previous pages, however, is, as far as I know, the only example of its kind. To compare it with the more usual style, we must have a look at the works of the German composer Johann Sebastian Bach (1685 – 1750). Each of them being a textbook example of mathematical figuration. Not just because Bach’s structures testify of his great craftsmanship, but mainly for his abundant use of the figure ‘14’.
Who replaces letters by their alphabetical position, will soon discover that according to the classic alphabet – to discern ‘I’ from ‘J’, and ‘U’ from ‘V’ results from more recent developments – ‘14’ equals B + A + C + H:
1 A 4 D 7 G 10 K 13 N 16 Q 19 T 22 X
2 B 5 E 8 H 11 L 14 O 17 R 20 U=V 23 Y
3 C 6 F 9 I=J 12 M 15 P 18 S 21 W 24 Z
Fourteen chorales can be found in both Christmas Oratorio and St. Matthew Passion, and the motet BWV 227 Jesu meine Freude (‘Jesus my joy’) is divided in fourteen sections.
This total seems to be 11, but the first chorus is distinctively divided in three separate sections by new starts of the music in bars 37 and 53, while the centerpiece also restarts after an opening section of 36 bars.
And exactly halfway the chorale-fughetta BWV 679 Dies sind die heil’gen zehn Gebot’ (‘These are the holy Ten Commandmends’) the principal fuge theme is after eight appearances out of ten, for a durance of 14 bars, replaced by a secundary theme of 14 notes. But the number’s use is not restricted to record authorship only. Bach also uses his signature to turn following line from the gospel of Saint Matthew, into his personal creed:
“Wahrlich; dieser ist Gottes Sohn gewesen.”
‘Truly; this has been the son of God.’
The son of God is part of the Holy Trinity. Reason for this short chorus to occupy just a modest three bars. And the choir’s lowest voice subscribes the words by singing them on 14 notes. As a mucisian, Bach was a good bass singer, and he can be trusted to have sung this line while conducting.
At least this is what musicology makes of it. In fact the music covers two complete 4/4 bars plus the first quarter of the third one, while the base line enters the musical weaver by a quarter delay. As a result the signature under this creed on the second entity of the One God in Persons Three covers exactly two out of three bars.
A very different and more complicated use of numbers is to be found in the cycle of brilliant compositions Bach offered at 7 July 1747 to the king of Prussia, a gifted flute player. Ten pieces in this ‘Musical Offering’ are canons, a number that of course refers to the Ten Commandmends (Canon; Gr. a rule that must be observed), as these canons observe some strict rules of composition while exploring the possibilities of a rather challenging theme, that was invented two months before by the Prussian ruler himself.
A very strange feature of this magnificent gift, is the fact it was still under construction; five out of thirteen pieces, including a trio-sonata starring the king’s own instrument, arrived at court at a much later date. Which leaves us with the paradox of Bach to write some great music for a great king, and to deliver it, accompanied by a highly flattering dedication, in the most offensive way he could think of.
The respect he was due to a sovereign, demanded his gift to be perfect into the least detail. And delivery in terms does not agree with perfection at all. So, why did Bach permit himself such a liberty? To answer this question we must have a close look at a certain detail of his gift. The supplement includes a set of two canons for respectively two and four instrumental voices, of which Bach supplies no information on their performance; he just gives a melody without any sign at which intervals in time and pitch the different voices are supposed to tune in. And heading these musical puzzles is the challenge:
“Quarendo invenietis”
‘Seek, and ye shall find.’
Words again quoted from the gospel of Saint Matthew, this time from the Sermon on the Mount (Matt. 7; 7). A source indication enabling the preacher, as to be expected from a spiritual guide, to provide some good advise: follow your leader (the royal flute player) in the proper distance. Which in the four part canon is at pitch on seven bars in time for each next voice, and in the two part canon it is distance in space; the second voice tunes in after hearing a descending melodic seven: playing the tune in reversion and on the lower seventh. Because of the reversion this second voice is allowed to lead as well, playing an ascending seventh to mark the king’s entrance on the higher seventh. Before modern Bach research recognized the clue, these solutions had already been found by trial and error (in the case of the two part canon accompanied by another two playable options, derived from a sheer endless host of useless (dis)harmonies. Something to remember while plodding through the unavoidable pages with calculations lying ahead.
Seeking in the source indication, ye shall find Bach’s signature as well. But this time the reversion does not add the initials to it: 7 – 7 = 0. If this indicates these canons to count for zero, the cycle adds up to 14 pieces. Which is a little difficult to recognize, I admit, as we had thirteen pieces to begin with. But this official amount depends on a trio sonata to be counted for one, in spite of the fact that each of its four movements makes a separate piece. Apart from that 7 – 7 = the date on the dedication. Obviously this latter interpretation was to Bach worth a sacrifice. Nobody would have noticed the intended connection between challenge and dating, if the gift had been presented in the recommended state of perfection.
The seventh of July being the date, Bach’s best option when he realised the scope of his project, was to postpone delivery by twelve months. Yet, he decided otherwise. And the reason for this blunt haste is only discovered very recently, when some genius – I do apologize for not having been able to trace his name – had a good look at the host of all solutions, possible and impossible, for the puzzle canons:
They amount to a score of 1747 bars.
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