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ECE Professor's Research makes Music more Harmonious

Electrical and computer engineering associate professor William A. Sethares is finding ways to make music more harmonious. He is making a wide variety of sounds work better for both conventional and unusual musical scales.

Sethares bases these improvements on the idea that the tonal quality of a sound influences listeners' perceptions of the consonance (or smoothness) and dissonance (or roughness) of notes of any two pitches (referred to as musical intervals). He has developed technical definitions of "consonance" and "dissonance" that lets him sculpt sounds and intervals that precisely control the amount of consonance and dissonance in musical passages.

Any single sound can be thought of as constructed of many waveforms called or partials. The number and arrangement of partials determine pitch--say "C" or "B flat"--and give a sound its character. A pure-sounding note, say from an instrument such as a concert flute, will contain few partials while a rich-sounding note, such as >from a bowed violin, will contain many. It is the interrelationships of the partials of the notes that determines consonance or dissonance of an interval, Sethares says. When the important partials of two notes either line up perfectly or are far apart, the notes will sound consonant. But if they're close, yet not close enough, the note will sound dissonant. So to increase or decrease consonance, Sethares moves partials by changing their frequency.

William A. Sethares

William A. Sethares (large image)

Currently, Sethares can't move the partials of tones from live instruments as they're played (although increases in computer power may soon make such a real-time harmonic-shifting box possible). Instead, he converts (or "samples") the tones into stored digital versions, creates computer maps of the digital tones, and programs a computer to gently adjust the partials' frequency. "You're manipulating the sound at the level of the partials in such a way as to make this dissonance as small as possible," he explains. And the changes in the harmonic structure are so subtle as to barely change the note's normal character, he says. "If you don't know what you're listening for, it's pretty hard to tell the difference."

The new sounds can then be played back directly from the computer or with an electronic keyboard, as is common in popular music. The scales used in popular music, however, aren't the most interesting applications for this technique, Sethares says. That's because virtually all modern Western music uses an octave that is divided into 12 equal parts. In this system, the common intervals are close enough to consonant. But if you divide the octave into fewer or more than 12 notes, the intervals can become quite dissonant (although some divisions, such as seven or 19 notes, result in may intervals that are consonant). The 10-note octave, for example, is notorious among composers because it results in so many difficult-to-listen-to intervals. "It's hard to find any pair that sounds good," he explains.

To prove a point, Sethares has constructed 10- and 13-note-per-octave electronic-sample sets that may eventually be released commercially. But, he says, the market for such odd-ball scales is currently limited and is likely to remain so, unless either prominent composers adopts his technology or non-Western music--some of which relies on specialized intervals--becomes popular.

Using a similar technique, Sethares has written software to "reverse engineer" tones: Instead of adjusting the partials to match a predetermined scale, it can analyze a tone and generate the most consonant-possible scale. Sethares has, for example, created scales to match bells and then written compositions based on these scales. "The effects are strikingly unusual and at the same time not dissonant," he says. Examples of this and other techniques will appear on a compact disk that will accompany a book--titled "Spectrum and Scale: A Perceptual Synthesis"--that Sethares will publish this year.

Sethares is also applying his high-tech methods to a centuries-old problem: making intervals sound as pure as possible in a 12-note octave. Before the mid-eighteenth century--the time of Johann Sebastian Bach--the octave used in Western music was divided into 12 uneven intervals. In "just intonation" and later the "meantone" system, the specific intervals instruments were tuned to varied depending on the key of the composition to be performed. These intervals sounded very consonant in the original key, or those closely related to it. But if the performer changed--or modulated--to a distant key, many of the common intervals would become dissonant. As composers became more interested in frequent and radical modulations, the octave was divided evenly, sacrificing very pure intervals for the ability to change keys freely.

To get the best of both systems, Sethares has developed software that analyzes music as it is performed on an electronic keyboard, and subtlety adjusts the notes' pitch (rather than the partials) so that the intervals are as consonant as possible. The system isn't perfect. In some simple music, these real-time adjustments can result in a noticeable drift of some notes. But for complex compositions--such as a Bach choral--these effects tend to average out. "You really can play in just intonation," Sethares says, "and it doesn't matter how much you change keys."