A Chromatic Striving in Mahler's Second Symphony

In a solemn brass chorale from the last movement of Mahler's Second Symphony "Resurrection," starting at "Etwas energischer in tempo," the first six chords are Ab+, Eb-, C+, Bb-, Ab-, and F+ (+ for major, - for minor). During these chords, the bass line moves continuously downward, putting all the chords in root position, and the soprano line moves continuously upward: Ab, Bb, C, Db, Eb, F. The bass and soprano lines are entirely diatonic, either within the notated key signature of D-flat major or the secondary key of A-flat major. Apart from the fourth chord, which has a minor third between soprano and bass, the soprano and bass notes in the other chords span perfect intervals, which means the quality of the triad could be either major or minor. Therefore, in fixing only the outer voices, there are 2^5 or 32 possible progressions of consonant triads.

There are multiple ways to measure the distance between two consonant triads. One way—call it Tonnetz distance—is to count the minimum number of L, P, and R transformations needed to change one triad into the other. Among the possible 32 progressions, the only completely diatonic progression (Ab+, Eb+, C-, Bb-, Ab+, and F-) minimizes Tonnetz distance for every triadic adjacency, and is the only one whose five triadic-adjacency distances sum to the smallest possible value of 11. Among the possible 32 progressions, Mahler's progression is the only one that maximizes Tonnetz distance for every triadic adjacency, and the only one whose five triadic-adjacency distances sum to the largest possible value of 18.

Later in the movement, one hears these words: "Mit Flügeln, die ich mir errungen, In heißem Liebesstreben, Werd'ich entschweben Zum Licht, zu dem kein Aug' gedrungen!" ("With wings which I have won for myself, In love’s fierce striving, I shall soar upwards To the light which no eye has penetrated!")