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Kirnberger's modulating canons through the fifths

Johann Philipp Kirnberger (1721-1783), who was a private student of J S Bach for two years (1739-41), published many examples of canon in his three-volume treatise (or five volumes, depending upon how you count), Die Kunst des reinen Satzes in der Musik [the art of pure composition in music] (1771-1779). Having written a little about modulating canons and canons per tonos, I thought it would be best not to overlook Kirnberger's two canons through the circle of fifths "durch Quinten", which can be found in Volume 2, part 3 pages 60 through 62.

These canons do not cadence or give any feel of repose in any particular tonality, instead rapidly modulating around the circle of fifths. They are emphatically perpetual canons, at least until the range extends beyond that of the instruments or practical performance. Although they are not fit for performance, they can be of pedagogical value for their sound and craft. And in that sense they are composers' canons, or theorists' canons at least. But they are most interesting theoretically.

The Enigmas

The way the puzzle's clefs are ordered is contrary to the practice that JS Bach followed in all but his earliest canons. While Bach would list the clefs in the opposite order of their use, leaving the leader's clef most proximate to its notes, Kirnberger has done the opposite with the first of these canons through the fifths. The initial voice, the dux or leader is meant to use the first clef, the bass clef with no key signature. This is evident after examination, since the canon is formed by rising fifths, and the keys must therefore increase by one additional sharp per entry. The numerical figures that appear above the first couple of lines (6 over the B, 9 over the C, 15 over the A) indicate by their metric positions the points of entry and by the number the interval above the given note made by the entry's first note. Thus the second voice enters a sixth above the note B with the note G; The third entry begins a ninth above the bass's C on the note D, and the final follower begins at the interval of a fifteenth (double octave) above the bass's A2 on the note A4.

Canon 2 follows the same plan. Here the numeral 2 above the bass's A indicates that the second entry begins on B at the interval of a second above that A, answering the leader's E up a fifth. The third entry begins on F#, as dictated both by its being a fifth above the previous entry and by the numeral #6 above the bass note A. The fourth voice begins, as expected, on C#, two octaves above the bass's C# as indicated by the numeral 15.

In both these canons, the clefs ascend by fifth as well, so that the answers can be read by means of these clefs and the key signatures given. Some judgment must be made as to the interpretation of accidentals, remembering that each voice represents a chromatic rather than a diatonic transposition.


Although these two four-voice canons are presented in the same way, their structure differs. Each is presented as a single melody to be imitated at the fifth until all four voices have entered. The canon then follows the successive tonalities of the entries as each claims a further step through the circle of fifths. Unlike most modulation canons, modulating part way through the octave and allowing subsequent iterations to complete the octave, Kirnberger's canons are notated all the way through the circle of fifths. In this regard, both canons appear to be long, with relatively close imitation (4 beats), but in fact the first canon is structurally shorter, consisting of three iterations of the same music. This convenience makes the journey through the fifths clear, and accomplishes the enharmonic shift in its notation, rather than leaving that task to the one solving the canon.

In the case of the first canon, four voices enter on C, G, D and A, each a fifth higher than the previous one, and the fifth entry in the key is made again by the leader, each of the followers doing likewise giving the key . Kirnberger could have notated only the first twenty beats and made an indication that the dux was to repeat a major third higher (on E).

A more concise but enigmatic puzzle might look like this:

fictitious alternate puzzle notation - Kirnberger #1

Regardless, the solution begins as such:

Of the two canons, the first is significantly weaker. Not only is the progression less compelling and less decisively anchoring each visited key, but of the twenty beats in each iteration, only three of these include all four voices.

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