[Smt-talk] Harmonics

[Greg Strohman]

I am primarily a trombonist, pianist and composer, but I have spent a great deal of time over the past eight years researching the acoustics and psychoacoustics of musical harmony and intonation and decided to comment on this thread.


> There is a great difference between overtones and chords as has been mentioned in this discussion before.

Fourier analysis reveals that there is no physical difference between overtones and chord tones. If one generates a chord which consists of five simultaneous sine waves tuned to [1/1, 2/1, 3/1, 4/1, 5/1] and compares it to a single pitch with a timbre that consists of only the first five harmonics of equal intensity, they produce identical sound pressure waves and are perceived as equivalent. Both produce a perfectly repetitive sound pressure wave with the same repetitive period.

Therefore, a single tone on an acoustic instrument is actually a complex harmony. A chord played by acoustic instruments is actually a poly-chord. By tuning that chord a particular way, we are also tuning the entire harmonic spectrum of that chord. A justly tuned [1/1, 5/4, 3/2] triad composed of pitches with harmonic timbres yields a justly tuned harmonic spectrum. Consequently, a justly tuned harmonic spectrum produces a sound pressure wave that repeats itself perfectly at some time interval.

Is there a psychoacoustic significance to perfectly repetitive sound pressure waves? When one considers sound as a medium for information transfer (music can certainly be regarded as some type of information), perfectly repetitive sound pressure waves can communicate their information perfectly within their repetitive period. A sound pressure wave which does not repeat requires an infinite amount of time to reveal that information. Humans likely evolved the ability to find perfect repetitions within sound pressure waves as a means of perceiving different vocal formants and distinguishing them from background noise. Our perception of harmonic stability and intonation might in the end be a by-product of our ability to perceive vowels and ultimately human language.


> The confusion is and potential for error is made even bigger in part because one can play for instance a dominant 7th chord as 1/1 5/4 3/2 7/4 in isolation without trouble and most ears will not hear it as out of tune instead only marvelling in it's highly synchronous sound. Yet once one starts to tune actual music this way, for instance in a piece by Beethoven or Bach or Chopin it will sound very out of tune to most ears.

I agree that performers should probably avoid tuning the music of Bach, Beethoven and Chopin like this. We often treat melodic and harmonic intervals as equivalent pedagogically, but physically they are not. When we perceive a melodic interval we perceive the "distance" between consecutive pitches. When we perceive a harmonic interval, we perceive the interference between the waveforms of the two pitches. In order to justly tune harmonic intervals we must usually distort melodic ones, and visa versa. If we try to tune the harmonies of Bach, Beethoven or Chopin to just intonation, we end up sacrificing the integrity and consistency of melodic intervals within voices. Intonation is often a tradeoff.

I would argue that equal tempered harmonies are often close enough to justly tuned harmonies to be perceived almost the same way. However, equal temperament has the advantage of preserving the consistency of melodic intervals. This is probably a good tradeoff for much Western instrumental music.


> The question then remains why is a 1/1 81/64 3/2 major triad perceived as "consonant" and as more consonant than a 1/1 9/8 3/2 suspension even though the latter has more simple ratios.

The Boston-based tuning theorist Paul Erlich has an elegant explanation for this called harmonic entropy. His model assumes that human perception of pitch, like all other human perceptions (and measurements of physical quantities), is uncertain. Therefore, we are capable of being convinced that the chord [1/1, 81/64, 3/2] is actually just a mistuning of [1/1, 5/4, 3/2].

A short summary of Mr. Erlich’s work can be found here:

http://www.tonalsoft.com/enc/h/harmonic-entropy.aspx


Greg Strohman
D.M.A. Temple University
www.gregstrohman.com