The four-voice, posthumously published modulating canon "Mir lächelt kein Frühling" by Johannes Brahms is much like a round, but each new entry enters a melancholy semitone lower than the last. Once all the voices have entered and as each of these voices makes its way through the 16-measure melody, what we hear is a four-measure unit or iteration descending each time by semitone. By the time the dux repeats, it is a full ditone (or major third) lower than before. To the composer, aside from the task of accomplishing the modulation, the challenge of the canon is the same as that of a canon at the unison; invertible counterpoint is not a requirement.
Due to the composer's skill at modulation, the effect is however not of a blatant shift downward with each iteration, but a subtle one without any discernible point of downward chromatic shift.
Yes, each iteration is in a different key, but the fact that the successively descending iterations are otherwise identical also makes the canon easier to analyze.
What makes this canon most remarkable is the success of the modulation. That is to say, the modulation is convincingly deceptive, especially for such a brief one and such a distant one. Brahms uses the indispensable technique of subtle modulation using a pivot.
It can be seen from the graphic depiction above (download to enlarge) that once all voices have entered, each four-measure iteration of the canon is the same, keeping in mind that the canon modulates, so that each iteration is also a semitone lower than the previous iteration.
How does Brahms make the modulation convincing?
One subtle detail that helps the modulation is a chromatic line that uses the technique of hocket, where a melody is transferred from one voice to another as it progresses. In measure 15, the dux (voice 1) repeats the notes D-flat, C, then leaps down below the next chromatic note B-natural, singing instead the note F, while voice 4 sings the B, now a leading tone to C in a fully-diminished harmony.
When the ear listens to a diminished-seventh chord, it is the notes outside the chord that orient the listener as to the key or expected chord of resolution. In the vacillation between B and C those notes can be heard as leading-tone and tonic, but they can also be heard as dominant and minor submediant, and the latter interpretation is corroborated in measure 17 as F-sharp appears in the dux, soon followed by the new leading tone, D-sharp in voice 3. The modulation is handled craftily, as any sort of F is avoided following the downbeat of measure 16, where the lower tones E and G sound as the third and fifth of an inverted C-major triad—a tonic in the new key of C, but a weak submediant chord in the key of E minor. Brahms takes advantage of this weakness, the tendency for a first-inversion submediant triad to sound like a watered-down tonic triad, and simply makes it so, as the upper note C finally resolves down by step to B, the fifth of the E-minor triad.
The avoidance of either F or F-sharp (and any other note that might contradict either a tonality of C or of E, would spoil the pivot, which is maintained from measure 16, beat 2 through measure 17, beat 2. After this hiatus known as the pivot, the ear accepts the F-sharp and D-sharp and contextualizes the entire harmonic environment as the realm of E minor.
Moreover, in measure 17, the E-minor triad is approached in a way that sounds somewhat native to its key—at least after the downbeat of measure 16. Now all that remains is to confirm the new key of E minor with a clear and decisive dominant-to-tonic resolution. This happens twice over the course of measures 17 and 18, just precisely as it did in the key of F minor four measures earlier.
If you were not sure how this happened as you were hearing it, the four-measure iteration will be replaced by another, and although invertible counterpoint is not used to vary the iterations, the mere fact of their modulation does provide a sense of new flavor with each iteration.
A choral performance by Chor des Norddeutschen Rundfunks can be found on youtube.