Sunday, September 9, 2012

Musical Ciphers

The fancy name for this is gematria from the Hebrew, but it could also be derived from the Greek geometria. In either case it refers to assigning numerical values to letters or phrases. Here is the Wikipedia article. In music, it is the practice of associating musical notes with letters so the music, or even the notation, can contain a message. Another method used in music was number symbolism in which the number of notes can be symbolic. Guillaume Dufay offers a couple of examples of this. In his motet Vasilissa ergo gaude written to celebrate the marriage of the daughter of an Italian nobleman, Cleofa Malatesta, to the son of the Byzantine emperor Manuel II the total number of notes in the motet are 800 which in Medieval numerical symbolism means 'eternity'. The actual symbolic number is 8, but you can't write much of a piece with only eight notes! Here is the motet:

Dufay would also sometimes sign his name in the score by writing "Du" followed by the note "fa". But the great master of number and note symbolism was J. S. Bach and the most famous instance was his signing his name to the last piece in his Art of Fugue. In German, the way the notes are pronounced is different than in English. For historical reasons, Germans pronounce B flat as "B" and B natural as "H". Therefore Bach could write his name in notes: B flat, A, C, H. Here it is as it appears in the last (unfinished) piece in the Art of Fugue:

And here is that fugue, Contrapunctus XIV:

It ends abruptly, incomplete... The BACH cipher appears at the 9:38 mark in the recording. This is a triple fugue. It starts with a subject derived, as are all the subjects in the Art of Fugue, from the original theme. Then, after a substantial fugue on that theme, a new theme in eighth notes is introduced, and in this second part, the theme from the first part returns. Then, after a full cadence, the new theme, BACH is introduced and a third fugue begins. The plan would have been to then reintroduce both of the other themes, but it remains incomplete. With most pieces by great composers left incomplete, someone always steps in to complete it. The most famous example is the Requiem of Mozart, completed by his student Süssmayr. Yes, there are completions for this piece too, but the suspicion always is that no-one alive is quite qualified to complete it. So it is often, as above, left incomplete... Here is the Wikipedia article on the Art of Fugue. There are two entries of the BACH cipher that are backwards, 'cancrizans' in musical parlance. Here is one of them:
This is quite similar to another famous cipher used by Dmitri Shostakovich. Taking advantage of the fact that in German E flat is pronounced "es", he writes his own name in notes this way: D S C H. Here it is in notes:
Those intervals differ from H C A B by only one note, the last one. This theme permeates all of Shostakovich's String Quartet No. 8 and was also, appropriately, used on his tombstone:

Click to enlarge

Here are the first and second movements of the string quartet. That four note theme is presented as a fugue in the very beginning:

Going back to Bach to close, he also used number symbolism as Dufay did. On his own printed copy of the Goldberg Variations he jotted down fourteen canons based on the bass of the theme. Why fourteen? It is the number of his name: B = 2, A = 1, C = 3 and H = 8. Total = 14. This was only discovered in the 1970s! You might also recall that the Art of Fugue also has fourteen fugues? And there may be more symbolism that we haven't noticed yet. One scholar has pointed out that the Mass in B minor seems to have some interesting numbers in it. The only music in the mass that is repeated is the chorus Gratias agimus tibi from the Gloria which returns as the Dona nobis pacem to end the piece. The theme of these two sections begins with the notes D, E, F#, G, or, numbering the notes, not in the key of the movement (D major), but as if they were in C major, we get 2 3 4 5. Now this may be of no significance whatsoever, but, as a matter of fact, there are precisely 2345 measures in the mass as a whole. Hmmm. Here is that Dona nobis pacem:

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