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!")

A Pitch-Time Analogy in Carl Vine's First Piano Sonata

Australian composer Carl Vine's Piano Sonata from 1990 begins with the pianist silently depressing A0, C1, E1, G1, and B1 and keeping them undamped with the sostenuto pedal for the first 12 measures. In a very short preface, the composer tells the performer that the tempo markings are "not suggestions but indications of absolute speed." The first 103 measures of the work, before a "poco allargando" in m. 104 and "Poco meno" tempo change involving a factor of 11 in m. 105, establish various isochronous pulses with wavelengths longer than 250 ms. These pulses group into exactly five "tempo classes" by equivalence of 2:1 ratios. If the first "tempo class" in the work is represented as 1, then the other four tempo classes can be represented as 1.2, 1.5, 1.8, and 2.25.  If the frequency of the pitch class A is represented as 1, then the other four pitch-class frequencies in the opening "silent" chord can be represented in just intonation as 1.2, 1.5, 1.8, and 2.25. (Such an analogy could also be demonstrated using the method David Lewin used in his 1987 book Generalized Musical Intervals and Transformations to analyze a passage from Carter's First String Quintet.)