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Chessboard Canons

April 4, 2017

In 1946 P. J. de Bruyn brought back to light a curious form of musical puzzle canon employed in the 16th century by Ghiselin Danckerts, a Flemish musician. Danckerts was a composer, theorist and a chapel singer in the papal court from 1538 to his death in 1565. Danckerts' music was not well published, but was apparently well-regarded in his time. His unpublished treatise, known simply as Trattato di Ghiselino Dankerts is considered important in the history of music theory as contributing to conservative views of composition in the 16th century.


Danckerts' appointment to the papal chapel was undoubtedly supported by a curiosity, printed in 1535 in Naples, of a collection of musical fragments arranged on a chessboard, clearly, as evidenced by the text accompanying these fragments, a canon upon the familiar liturgical text beginning "Ave maris stella," or "Hail, star of the sea." 




One of the judges in the 1551 debate over the role of the ancient Greek 'chromatic' and 'enharmonic' tetrachords in sixteenth-century musical practice, Danckerts moderated between the composers Nicola Vincento and Vincente Lusitano. Vincento's treatise, is a source of canon. Danckerts was likely selected by virtue a the rare focus evident in his own treatise upon contemporary practice. This focus is largely reactionary, as Danckerts own views upon innovations in contemporary practice are of a conservative and quashing perspective, especially toward chromatic inflections (understood in the modern sense, not the ancient Greek sense), and some of the ficta that we now see as part of the gradual metamorphosis of music away from the eight modes and toward a major/minor understanding of tonality. As part of his rigorous musical life in the papal chapel, Danckerts would have learned the intricacies of employing ficta or 'accidenti' without harming the ecclesiastical modes in addition to training in canon. The implication that Danckerts composed without such innovations as Vincento simplifies the task of solving any puzzle canon by Danckerts; it is much easier to dismiss more questionable alternatives in the process of discovering the true solution.


It is all the more interesting that this curiosity of a chessboard puzzle canon should be presented by a composer who was a contemporary countryman of Mattijs de Castelein, the poet who was central to the Rederijkers (sp?).


J. P. Westgeest gave a solution to the canon in his 1986 article "Ghiselin Danckerts' 'Ave Maris Stella': The Riddle Canon Solved." Further parallels with the Rederijkers were drawn in his 1987 article 'Casteleins code gekraakt' [Castelein's code cracked]. Westgeest showed how Danckerts' canon is highly influenced by the Burgundian rhetorikeurs, poets, principally Matthaijs de Castelein (ca.1485-1550) of Oudenaarde near Ghent whose similarly cryptic chessboard yields 38 chauvanistic poems (ballades) through various motions from square to square, employing vertical, horizontal, diagonal and retrograde traversals. 




Much of what we have about Danckerts was rescued by Casimiri, and Bruyn published his 1946 paper in the wake of the death of Casimiri who died in 1943.


Danckerts' chessboard canon finds another contemporary precursor in Ludwig Senfl's similar canon 'Salve sancta parens' of 1520, this one on a 6x6 grid:


Published in 2010, Andrea Lindmayr-Brandl wrote a wonderful article on this Senfl canon: Magic Music in a Magic Square: Politics and Occultism in Ludwig Senfl's Riddle Canon Salve sancta parens  [Tijdschrift van de Koninklijke Vereniging voor Nederlandse Muziekgeschiedenis,Deel 60, No. 1/2 (2010), pp. 21-41], drawing parallels between the magic square and Senfl's canon and providing a solution and analysis.


A magic square is a game known amongst mathematicians, where a grid must be filled with numbers in such a way that each row, column and diagonal must add to the same sum. This is similar to, but not the same as a sudoku puzzle.



While Senfl's puzzle, composed entirely of breves, combines much like a magic square, resolving to block chords, Danckerts' is more rhythmically sophisticated. 





J. De Bruyn: ‘Ghisilinus Danckerts’, Tijdschrift voor nieuwe muziek, xvi/4 (1946), 217–52; xvii/2 (1949), 128–57.


Andrea Lindmayr-Brandl. "Magic music in a magic square: Politics and Occultism in Ludwig Senfl's Riddle Canon Salve sancta parens." Tijdschrift van de Koninklijke Vereniging voor Nederlandse Muziekgeschiedenis, 60, No. 1/2 (2010), 21-41.


Lewis Lockwood. "Danckerts, Ghiselin." Grove Music Online. Oxford Music Online. Oxford University Press, accessed April 14, 2017, http://www.oxfordmusiconline.com.colum.idm.oclc.org/subscriber/article/grove/music/07156.


Hans Westgeest. "Ghiselin Danckerts' 'Ave Maris Stella': The Riddle Canon Solved." Tijdschrift van de Vereniging voor Nederlandse Muziekgeschiedenis, Deel 36 (1986), 66-79.


Hans. Westgeest. "Castelein's code getraakt." Die Nieuwe Taalgids; 1987.


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