Verifying Harder's Conjecture for Classical and Siegel Modular Forms

A conjecture by Harder shows a surprising congruence between the coefficients of “classical” modular forms and the Hecke eigenvalues of corresponding Siegel modular forms, contigent upon “large primes” dividing the critical values of the given classical modular form. Harder’s Conjecture has already been verified for one-dimensional spaces of classical and Siegel modular forms (along with some two-dimensional cases), and for primes p 37. We verify the conjecture for higher-dimensional spaces, and up to a comparable prime p.

Effects of training and melodic features on mode perception

The two modes most widely used in Western music today convey opposite moods—a distinction that nonmusicians and even young children are able to make. However, the current studies provide evidence that, despite a strong link between mode and affect, mode perception is problematic. Nonmusicians found mode discrimination to be harder than discrimination of other melodic features, and they were not able to accurately classify major and minor melodies with these labels. Although nonmusicians were able to classify major and minor melodies using affective labels, they performed at chance in mode discrimination. Training, in the form of short lessons given to nonmusicians and the natural musical experience of musicians, improved performance, but not to ceiling levels. Tunes with high note density were classified as major, and tunes with low note density as minor, even though these features were actually unrelated in the experimental material. Although these findings provide support for the importance of mode in the perception of emotion, they clearly indicate that these mode perceptions are inaccurate, even in trained individuals, without the assistance of affective labeling.

Mental scanning in auditory imagery for songs

Four experiments examined how people operate on memory representations of familiar songs. The tasks were similar to those used in studies of visual imagery. In one task, subjects saw a one word lyric from a song and then saw a second lyric; then they had to say if the second lyric was from the same song as the first. In a second task, subjects mentally compared pitches of notes corresponding to song lyrics. In both tasks, reaction time increased as a function of the distance in beats between the two lyrics in the actual song, and in some conditions reaction time increased with the starting beat of the earlier lyric. Imagery instructions modified the main results somewhat in the first task, but not in the second, much harder task. The results suggest that song representations have temporal-like characteristics. (PsycINFO Database Record (c) 2012 APA, all rights reserved)