Dr. Marcus H. Pendergrass

Interested in finding out what research in math or applied math is really all about?  I work with motivated students on problems in a variety of areas. Here are some examples of work students have done with me recently:

• Algorithmic Composition, Carlo Anselmo '18. Carlo developed algorithms for generating musical compositions based on the idea of a sequence recursion. A sequence recursion is a method for generating finite sequences iteratively. These sequences are then used to create sequences of pitches and rhythms for a musical composition. This research was two-pronged: Carlo explored the mathematical properties of sequence recursions, and then applied them to synthesize new musical compositions. 
• Realism in Messiaen's  Oiseaux Exotiques: A Correlation Analysis , Jahangir Iqbal '15.  The 20th century French composer Olivier Messiaen drew musical inspiration from bird song.  I fact, Messiaen believed that "it's probable that in the artistic hierarchy birds are the greatest musicians existing on our planet."  Is it possible to quantify the extent to which Messiaen's music derives from actual birdsong?  Jay addressed this question using statistical techniques.  Messiaen's composition Oiseaux Exotiques (Exotic Birds) references the songs of several specific birds.  By comparing the musical passages to recordings of these birdsongs, Jay was able to measure the similarity between the music and the birdsong that inspired it.  He presented his results at the National Conference on Undergraduate Research (NCUR) in the Spring of 2013.
• Mathematical Synthesis and Analysis of Sound, Ke Shang '13.  Ke studied two different problems in this work.  On the analysis side, he examined an alternative to the Fourier transform for frequency domain analysis, in particular the notion of the power spectrum with respect to instantaneous frequency.  On the synthesis side, Ke proved a theorem about the stability of phase modulated networks.
• A Survey Of Various Data Compression Techniques, Charles Alexander Smith '10. Alex looked at the underlying technology of lossy data compression, including MP3 audio compression and JPEG image compression as part of his Departmental Honors in Mathematics. Fourier analysis and wavelets were the main mathematical areas involved in his research. Alex's presentation of this work at the Spring 2009 regional meeting of the Mathematical Association of America took first prize in the poster session.
• Fast Multiplication Of Large Integers Using Fourier Techniques, Henry Clarke Skiba '10. Henry investigated the use of the Fast Fourier Transform (FFT) in carrying out the multiplication of very large integers (i.e. millions of digits). This work was part of the Applied Mathematics class taught in the Fall of 2008.
• A Frequency Domain Technique For Correcting Distortion Caused By Variable Recording Speed, Robert Hembree '09, Henry Clarke Skiba '10, Charles Alexander Smith '10. A variable sampling rate during recording (e.g. when the motor driving an audio tape machine fluctuates) will introduce distortions in the recorded signal. However, if a known reference signal is present, this distortion can be corrected. In the fall of 2008 our Applied Mathematics class developed algorithms to perform this correction, and tested the algorithms in various scenarios.

Probability, Mathematics and Music, Mathematics Pedagogy

• M. Leigh Lunsford, Marcus Pendergrass, Making Online Homework Work, PRIMUS, 26(6), 531-544, 2015.

• Marcus Pendergrass, Two Musical Orderings, Journal of Mathematics and Music, 8(3), 59-72, 2014.

• Marcus Pendergrass, A Path Guessing Game With Wagering, Probability in the Engineering and Informational Sciences, 24 (2010), 375-396.

• Robb Koether, Marcus Pendergrass, John Osoinach, The Lying Oracle With A Biased Coin, Advances in Applied Probability, Vol. 41, No. 4 (December 2009)

• M. Leigh Lunsford, Marcus Pendergrass, Phillip Poplin, David Shoenthal, The Naive Chain Rule, College Math Journal, 39 (2007), 142-145

• Andreas Molisch, Marcus Pendergrass, et al, UWB Propogation Channels, in UWB Communication Systems: A Comprehensive Overview, M. G. DiBenedetto, T. Kaiser, A. F. Molisch, I. Opermann, C. Politano, and D. Porcino (eds.), EURASIP Book Series on Signal Processing and Communications, Hindawi Publishing Company, 2006.

• Andreas F. Molisch, Jeffrey R. Foerster, and Marcus Pendergrass, "Channel Models for Ultrawideband Personal Area Networks", IEEE Wireless Communications, December 2003.

• Marcus Pendergrass and Joy Kelly, Proposal for Alternate Physical Layer for 802.15.3, IEEE 802.15.3 Document Archives, 03144r1P802-15_TG3a-TimeDomain-CFP-Document.pdf, March 2003.

• Marcus Pendergrass, Error Rates for a Class of Multiple Position Modulation Schemes, IEEE Vehicular Technology Conference, Spring 2002, May 6 - 9, 2002, Birmingham Alabama.

• Kyle Siegrist and Marcus Pendergrass, Generalizations of Bold Play in Red and Black, Stochastic Processes and Their Applications, 92 (2001), 163-180.

"The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit. Another claimant for this position is music." - Alfred North Whitehead,  Mathematics as an Element in the History of Thought

Ultimately, mathematics isn't about numbers, it's about patterns: understanding patterns, and understanding relationships between patterns.  Music is also about patterns, or rather, music is pattern: patterns of sound in time.  So it's not surprising that mathematics and music have had a close relationship, going all the way back to the ancient Greeks.

My recent research has focused on the mathematical structure of basic musical objects, such as scales and chords.  I have also explored how the patterns that appear in mathematics can be utilized to create new musical sounds and compositions.  Algorithmic composition is the field that applies techniques from mathematics and computer science to generate musical compositions. 

In my spare time I like to compose, arrange, play, and record music the old-fashioned way: by actually playing an instrument!  In my case, that means the piano mostly, with a little bit of guitar thrown in here and there.  Here are a few samples, all in MP3 format:

• Elephant Walk - an original
• Encomium - another original
• Alice In Wonderland - The Hilliard/Fain classic.  Hopefully fellow mathematician Charles Dodgson would approve...
• Where Or When - The Rogers/Hart classic.
• So Far - me again...