Among the pieces in Bach's Musical Offering, dedicated to Frederick the Great, is a puzzle canon with the epigram: "Ascendenteque Modulatione ascendat Gloria Regis" (As the modulation ascends, so may the king's glory ascend).
This is a canon 'Per Tonos" or "by a tone," as described by Giovanni Maria Bononcini (1642-1678), quoted in translation by Alfred Mann in his book The Study of Fugue:
A canon that rises a whole-tone step above its beginning with each new return
Bononcini uses the Italian word Tuono, and as of today I do not know whether Bach was the first to call his canon by the category 'per tonos." However, it seems to have stuck.
Notice that the direct (squiggle) at the end of the alto line indicates that the repetition is to be a step higher (on D) than the first note (C). While the given voice in the lower staff begins on G (tenor clef), it repeats beginning on A. The bass voice begins on C and repeats on D.
The canon per tonos is a variety of modulating canon. Since voices can easily and quickly modulate beyond their range, composers tend to favor modulating canons that either rise or fall by step. They sometimes traverse the circle of fifths by clever imitation, so that each voice enters a whole step higher or lower anyway.
In fact, Johannes Brahms' appropriately mournful canon for four female voices "Mir lächelt kein Frühling" (I don't smile in spring) falls a semitone with each new entry. In the published version (1881), the first voice enters on A-flat, the second on G, the third on F-sharp, the fourth on F-natural, and the first then repeats on E, and so on. (In enigmatic manuscript it is notated a half step lower, beginning on G.). With this canon, although each entry is a semitone lower than the previous one, any single voice repeats a major third (a ditone) lower each time; the canon could then technically be called a 'canon per ditonos' if we were to stretch the nomenclature a bit too far.
This only points out with more immediacy the conundrum that all modulating canons face: the need for enharmonic notation as the modulations drift relentlessly ever more flatwise or sharpwise.
In addition to Bach and Brahms, many composers wrote modulating canons: Hilton, Morley, Kirnberger, and Dowland to name a few. I will be featuring some of these in upcoming posts.