There seems to be some method in this construction after all, even when its\u00a0practical use is still unsure. And of course the use of numbers to create\u00a0structures belongs to the basics of composition. The architecture revealed\u00a0on the previous pages, however, is, as far as I know, the only example of its\u00a0kind. To compare it with the more usual style, we must have a look at the\u00a0works of the German composer Johann Sebastian Bach (1685 – 1750). Each\u00a0of them being a textbook example of mathematical figuration. Not just\u00a0because Bach\u2019s structures testify of his great craftsmanship, but mainly for\u00a0his abundant use of the figure \u201814\u2019.<\/p>\n
Who replaces letters by their alphabetical position, will soon discover that\u00a0according to the classic alphabet – to discern \u2018I\u2019 from \u2018J\u2019, and \u2018U\u2019 from \u2018V\u2019\u00a0results from more recent developments –\u00a0 \u201814\u2019 equals B + A + C + H:<\/p>\n
1 \u00a0 A \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 4 \u00a0 D \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 7 \u00a0 G \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 10 \u00a0 K\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span>13\u00a0\u00a0\u00a0<\/span>N\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span>16\u00a0\u00a0\u00a0<\/span>Q\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span>19\u00a0\u00a0\u00a0<\/span>T\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>22\u00a0\u00a0\u00a0<\/span>X<\/p>\n 2 \u00a0 B \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 5 \u00a0 E\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>8\u00a0\u00a0<\/span>\u00a0H\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>11\u00a0\u00a0\u00a0<\/span>L\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span>14\u00a0\u00a0\u00a0<\/span>O\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a017\u00a0\u00a0\u00a0<\/span>R\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span>\u00a020\u00a0\u00a0<\/span>\u00a0U=V\u00a0\u00a0 \u00a0 \u00a0\u00a0<\/span>23\u00a0\u00a0\u00a0<\/span>Y<\/p>\n 3\u00a0\u00a0<\/span>\u00a0C\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span>\u00a06\u00a0\u00a0\u00a0<\/span>F\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span>9\u00a0\u00a0\u00a0<\/span>I=J\u00a0\u00a0 \u00a0 \u00a0 \u00a0<\/span>12\u00a0\u00a0<\/span>\u00a0M\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>15\u00a0\u00a0<\/span>\u00a0P\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span>18\u00a0\u00a0\u00a0<\/span>S\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span>\u00a021\u00a0\u00a0\u00a0<\/span>W\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span>\u00a024\u00a0\u00a0<\/span>\u00a0Z<\/p>\n Fourteen chorales can be found in both Christmas Oratorio\u00a0and St. Matthew Passion, and \u00a0the motet BWV 227\u00a0Jesu meine Freude<\/i>\u00a0(\u2018Jesus my joy\u2019) is divided in fourteen sections.<\/p>\n And exactly halfway the\u00a0chorale-fughetta BWV 679\u00a0Dies sind die heil\u2019gen zehn Gebot\u2019<\/i>\u00a0(\u2018These are the\u00a0holy Ten Commandmends\u2019) the principal fuge theme is after eight\u00a0appearances out of ten, for a durance of 14 bars, replaced by a secundary\u00a0theme of 14 notes. But the number\u2019s use is not restricted to record\u00a0authorship only. Bach also uses his signature to turn following line from the\u00a0gospel of Saint Matthew, into his personal creed:<\/p>\n \u201cWahrlich; dieser ist Gottes Sohn gewesen.\u201d<\/em> 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\u2019s\u00a0lowest 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.<\/span><\/p>\n A very different and more complicated use of numbers is to be found in\u00a0the cycle of brilliant compositions Bach offered at 7 July 1747 to the king of\u00a0Prussia, a gifted flute player. Ten pieces in this \u2018Musical Offering\u2019 are canons,\u00a0a number that of course refers to the Ten Commandmends (Canon; Gr. a\u00a0rule that must be observed), as these canons observe some strict rules of\u00a0composition while exploring the possibilities of a rather challenging theme,\u00a0that was invented two months before by the Prussian ruler himself.<\/p>\n A very strange feature of this magnificent gift, is the fact it was still under\u00a0construction; five out of thirteen pieces, including a trio-sonata starring the\u00a0king\u2019s own instrument, arrived at court at a much later date. Which leaves\u00a0us with the paradox of Bach to write some great music for a great king, and\u00a0to deliver it, accompanied by a highly flattering dedication, in the most\u00a0offensive way he could think of.<\/p>\n The respect he was due to a sovereign, demanded his gift to be perfect\u00a0into the least detail. And delivery in terms does not agree with perfection at\u00a0all. So, why did Bach permit himself such a liberty? To answer this question\u00a0we must have a close look at a certain detail of his gift. The supplement\u00a0includes a set of two canons for respectively two and four instrumental\u00a0voices, of which Bach supplies no information on their performance; he just\u00a0gives a melody without any sign at which intervals in time and pitch the\u00a0different voices are supposed to tune in. And heading these musical puzzles\u00a0is the challenge:<\/p>\n \u201cQuarendo invenietis\u201d<\/em> Words again quoted from the gospel of Saint Matthew, this time from the\u00a0Sermon on the Mount (Matt. 7; 7). A source indication enabling the preacher,\u00a0as to be expected from a spiritual guide, to provide some good advise:\u00a0follow your leader (the royal flute player) in the proper distance. Which in\u00a0the four part canon is at pitch on seven bars in time for each next voice, and\u00a0in the two part canon it is distance in space; the second voice tunes in after\u00a0hearing a descending melodic seven: playing the tune in reversion and on\u00a0the lower seventh. Because of the reversion this second voice is allowed to\u00a0lead as well, playing an ascending seventh to mark the king\u2019s entrance on\u00a0the higher seventh. Before modern Bach research recognized the clue,\u00a0these solutions had already been found by trial and error (in the case of the\u00a0two part canon accompanied by another two playable options, derived from\u00a0a sheer endless host of useless (dis)harmonies. Something to remember\u00a0while plodding through the unavoidable pages with calculations lying ahead.<\/p>\n Seeking in the source indication, ye shall find Bach\u2019s signature as well. But\u00a0this time the reversion does not add the initials to it: 7 – 7 = 0. If this\u00a0indicates these canons to count for zero, the cycle adds up to 14 pieces.\u00a0Which 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\u00a0for 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\u00a0latter interpretation was to Bach worth a sacrifice. Nobody would have\u00a0noticed the intended connection between challenge and dating, if the\u00a0gift had been presented in the recommended state of perfection.<\/p>\n The seventh of July being the date, Bach\u2019s best option when he realised\u00a0the scope of his project, was to postpone delivery by twelve months. Yet, he\u00a0decided otherwise. And the reason for this blunt haste is only discovered\u00a0very recently, when some genius – I do apologize for not having been able\u00a0to trace his name – had a good look at the host of all solutions, possible and\u00a0impossible, for the puzzle canons:<\/p>\n They amount to a score of 1747 bars.<\/p>\n —<\/span><\/p>\n —<\/span><\/p>\n go to next chapter<\/a>\u00a0– – – – – – – – – – – – – – – – – – – – –<\/span>\u00a0back to the previous chapter<\/a><\/p>\n There seems to be some method in this construction after all, even when its\u00a0practical use is still unsure. And of course the use of numbers to create\u00a0structures belongs to the basics of composition. The architecture revealed\u00a0on the previous pages, however, … Continue reading \n
This total seems to be 11, but the first chorus is distinctively divided in three separate\u00a0sections by new starts of the music in bars 37 and 53, while the centerpiece also restarts after an opening section of 36 bars.<\/span><\/em><\/h5>\n<\/blockquote>\n
\n<\/i>\u2018Truly; this has been the son of God.\u2019
\n<\/i><\/span><\/p>\n\n
At least this\u00a0is what musicology makes of it. In fact the music covers two complete 4\/4\u00a0bars plus the first quarter of the third one, while the base line enters the\u00a0musical weaver by a quarter delay. As a result the signature under this creed\u00a0on the second entity of the One God in Persons Three covers exactly two\u00a0out of three bars.<\/span><\/em><\/h5>\n<\/blockquote>\n
\n<\/i>\u2018Seek, and ye shall find.\u2019
\n<\/i><\/span><\/p>\n