I. The Missing Fundamental
This demonstration explores the relation between the frequency content of a musical note and the pitch perceived by listeners. As we explore in Physics 160, musical notes are complex tones consisting of a fundamental frequency and higher harmonics (known as partials) that are integer multiples of the fundamental frequency. The particular mix of partials is part (but only part!) of what gives different musical instruments their individual character. The pitch of the note is related to the fundamental frequency of the complex tone. However, the pitch of the note remains unchanged even if this fundamental frequency is removed.
The accompanying .wav file consists of a complex tone made up of a fundamental and nine higher harmonics. The first tone heard has all the frequencies; the second tone has the fundamental removed but maintains all of the higher harmonics. Each successive tone sequentially removes the lowest harmonic. Notice that although the character of each note changes, the pitch remains the same.
Missing Fundamental demonstration.
II. Circularity in Pitch Judgment
This demonstration is my own personally-devised version of the circularity of pitch illusion first described by R. Shepard (J. Acoust. Soc. Am. 36, 2346-2353: ). This illusion is the audio analog of M. C. Escher's visual illusion The Ever Ascending Staircase (or is it descending?). If you listen to the accompanying audio file you will hear an ascending scale with the pitch rising from one note to the next; however, the scale can be played indefinitely with the pitch always rising from one note to the next without the notes becoming so high that the sound becomes inaudible. The file is in the .wav format and it consists of a series of notes that is one note short of 4 octaves. If you have a .wav player that is capable of looping to play the file repeatedly, the illusion will run for as long as you desire.
This aural illusion is used at the beginning and the end of my Physics 160 course to highlight progress in a number of the central topics covered in the course: complex tones, frequency spectra, logarithmic nature of pitch scales, loudness and the threshold of hearing. The explanation of the illusion given at the end of the semester draws on all these topics.
Circular Pitch Illusion
As I wrote above, this is my version, you can also check out a "professional" version given on the web page of the Acoustical Society of America. Theirs is a bit smoother--when I get a chance I'll revamp mine.
Sample Homework Question for the Physics of Music Course
Your group, The Basilar Membranes, after years of successfully switching to the latest trends in the music industry, is now desperately trying to decide the next wave in musical taste. Based on a combination of bad management, too much coffee, and a horribly flawed focus group study, your collective genius has decided that the next hot area will be a musical form based on dividing the octave into a nine note scale. As the only group member to survive with your numeracy intact after the group's ear-shattering experience with Gothic Speed Metal, you are given the task of calculating the frequency intervals between notes on this new scale. If all nine notes are equal intervals apart, what is the appropriate multiplicative factor to move from one note to the next (i.e. a number similar to that for the equal tempered twelve tone chromatic scale)?
If you don't know what a Basilar membrane is, or how to calculate the appropriate multiplicative factor for a nine tone scale, then perhaps Physics 160 is for you. I won't give the answer to this problem here but I will let you hear what a nine note scale sounds like. Enjoy.
Nine tone scale.
If you are interested you can also look at my web page concerning the Physics of the Didjeridu.