Math 5: Resources - FALL 2011

Theory: General references

• On reserve at Baker library (the Paddock ones have been moved to Baker):
  • Gareth Loy, Musimathics, volume 1. My list of ERRATA for first printing (2006) of this book. The second printing I have been promised corrects more of these. Author's website.
  • Arthur Benade, Fundamentals of Musical Acoustics, (Dover edition, 1990), is a beautiful, non-mathematical introduction at about our level (in fact with less math than we'll do). Benade was a renowned researcher of the workings of instruments.
  • Rossing, Moore and Wheeler, The Science of Sound, 3rd Ed. (2002). (I think 2nd Ed. is also on reserve). This massive book introduces many of the concepts of our course, at our level, with a engineering/physics rather than math spin. It tends to be encyclopedic, with a lot of sometimes tedious or incomprehensible data and graphs on particular instruments, while missing the elegant universal themes.
  • Fauvel, J., Flood, R., and Wilson, R., eds. Music and Mathematics : from Pythagoras to fractals (2003). Lovely collection of (unrelated) articles connecting music and math. Great source of project ideas, esp. for those who prefer historical and music analysis projects.
  • Charles Taylor, Exploring Music: The Science and Technology of Tones and Tunes (1992). Very intuitive, has lots of engineering type demos, pictures of waves, etc. Not mathematical.
  • Johnston, I. H., Measured tones : the interplay of physics and music (2002). As with the above, a nice, intuitive book with concepts explained via text rather than equations. Good explanations of tuning systems, pipe and string modes.
• Music: A Mathematical Offering by David J. Benson (Cambridge University Press, 2007), a freely-available 10MB PDF file. This is a wonderful study, with beautiful and deep musical-math connections, and almost-obsessive detail about e.g. tuning systems, Fourier methods, world-music instruments, symmetry, etc. It is at the advanced undergrad math major level, but don't be scared to dig in!
• Other books:
  • Fletcher, The Physics of Musical Instruments (1991), wonderful investigations into details of instrument acoustics, is more mathematically sophisticated than our course. Fletcher is known for his original research in this area.
• College of Santa Fe Auditory Theory is a nice online textbook, with demos too.
• Basic Physics of Sound course by Rick Van Kooten at Indiana.
• Longitudinal and transverse wave animations from Kettering.
• Physics of Music course by Christopher Munroe at UMD.
• Victor Stanionis book on Computer Music, reviews digital recording and synthesis, some physics of waves.
• ECHOES Newsletters from the Acoustical Society of America. Interesting research and news. Great starting point for projects and aural postings.
• Vibration Data collection of online newsletters with interesting acoustics applications, good for project ideas.
• Internet Searching Guide: links on assessing web sources and searching, by our library staff.
• General notes on exam and study technique (geared mainly towards final exams in intermediate-level math courses).

Mathematics

• Math review with some tools you'll need, and a very few practise questions.
• Three pre-calc sets of practise problems: trig, exp and log, algebra, from Bethel U.
• Trigonometry shown with nice interactive circular motion applets.
• Addition formula for sin derived with rotating vectors.
• Function plotter applet with ok mouse zoom and pan. Another from University of Vienna; another one with mouse pan and axis labels but no mouse zoom; and one that explores a family of graphs by allowing you to slide parameters (eg sin(k*x) for various k), no mouse zoom.
• Fourier series applet finds harmonic content of a waveform you draw by hand.

Instruments

• University of New South Wales (Australia) musical acoustics research group has many pages about acoustics and various instruments.
• MIMO database of musical instruments in museum collections, includes audio recordings of over 900 instruments, and photos of 50000 more.
• Tuning of bells, a wealth of information and recordings on their site.
• Acoustical properties and construction of the tabla, compared to other drums. Some cool re-tuning of resonant frequencies using a non-uniform mass distribution on the tabla surface. tabla Chladni figures
• violin string motion during bowing from +Plus magazine, and guitar string motion waveforms captured with iPhone 4 video (fun!)
• My summary of what affects string timbre for this course, fabulous plucked string applet
• Scientific American article on Tuvan throat singing, by our own Theodore Levin and co-author.
• 5 lectures on the acoustics of the piano from KTH, Sweden.
• Why can't we make a perfectly tuned trumpet? by Thomas Moore (2006).
• 1-page review of physical models for instruments by Christopher Munroe at UMD.
• Java applet for variable vocal-tract formant generation, shows spectral filter and modes for each peak! Instructions for making formant machines (great project idea!)

Music Theory and Perception

• Tuning harpsicord by ear to Bach's temperament, by Bradley Lehman.
• Auditory Demonstrations from Houtsma, Rossing, Wagenaars.
• Consonance and Dissonance, online graduate level music course notes from Ohio State.
• Ernst Terhardt page with lots of info on pitch perception, consonance, and other scientific auditory research.
• `Just' intonation resources including sound files to compare to equal-tempered tuning.
• Applet to explore tuning systems (including playing music excerpts and intervals), with excellent explanation by Pierre Lewis.
• Applet to play the 23-note Hindu scale (needs you to install JSyn first via your browser)
• Dissonance curves and explanation of Helmholtz theory.
• My matlab code helm_diss which counts numbers of dissonant partials for various intervals.
• Circularity in pitch judgement with Shepard tones (continuously ascending scale); also see applet at Exploratorium and this applet.
• 15th century woodcut showing Pythagoreans with strings of different tensions (note the frequency ratios are given by the square-root of tension, so would not form a just ratio).
• Essay by Bill Hammel on mathematics in music composition (skip to section on group theary).

Acoustics

• Falstad's Fourier synthesis applet.
• History of the wave theory of sound, excerpt from book by A. D. Pierce (ASA).
• Falstad's ripple-tank simulator, wonderful sandbox for exploring time-domain waves [applet]
• Information on DTMF telephone codes with example recordings, from EE at Notre Dame.
• My summary of what you need to know about the Doppler effect. Also a Doppler effect applet 1, and applet 2, both showing waves spreading in 2D.
• Nice intro to Decibels (note we call I_0 instead I_r), and What is a decibel? (shows Flectcher-Munson phon curves)
• Binaural recording such as the Virtual barbershop
• List of interesting articles on human hearing and acoustics research.
• How to generate and write out your own mathematical functions as WAV files using MATLAB or octave software. Simple code here.

Music/sound sources

• How to generate and decode DTMF (telephone touch tone) signals: here
• DRAM academic music collection (requires Quicktime plugin to listen via web streaming; can't download unless you record your own computer's audio using audacity...)
• Primate vocalizations from Hauser lab at Harvard.
• Binaural beats examples.