I recently picked up a copy of Arthur H. Benade’s, ‘Fundamentals of Musical Acoustics,’ and am enjoying the read. There’s some seriously trippy shit in there! Figuring some of you might enjoy this kind of thing, I plan to do little write-ups on the topics that pop out as fun and interesting. This article shows a neat example of how the timbre of an impulsive sound can affect the pitch you hear.
Part I: Impulse Frequency and Musical Tones
To get started, knock your knuckles on a hard surface.
When you do this, you are hearing what is called an impulsive sound. An impulsive sound is any short, impulse-like noise that is not sustained. We hear impulsive sounds all the time. They sound like raps, taps, claps, knocks, bonks and clonks.
Now, if you repeatedly tap your knuckles on the table – at a somewhat steady tempo – you create a rhythmic succession of impulsive sounds much like a metronome. Someone in the next room could ‘hear the beat.’
If you continue to speed up the frequency of those taps, cool things start to happen. At first, the distance between each impulse gets shorter and the tempo speeds up. However, at a certain point the impulses will be so close together that they produce a sustained sound. We perceive this sound as a buzz or ring – much like the sound of an alarm clock or telephone. This buzzing sound occurs at an impulse rate somewhere in the ballpark of 20 to 150 impulses per second.
By continuing to speed up the frequency of the individual impulses – beyond a rate of 100 impulses per second, or so – we begin to hear musical pitches. As the impulse frequency continues to increase, the pitch we hear gets higher and higher.
North American alarm clocks have an impulse repetition rate of almost exactly 120/second.
Part II: Successive Taps (In Unison) From Two Impulsive Sounds
This time, knock on the table with both hands at the same time. Since you’re hitting different surface areas – while using two different hands – the two impulsive sounds will slightly vary from one another in their timbre (the way they sound). If both sounds happen at nearly the same time we will either hear a new, homogenous sound – or – we may be able to differentiate between the unique timbres.
At this point, it does not make any difference. As we speed up the frequency of the simultaneous impulses – just as we did with the single table knock – the exact same effects occur. First, the tempo increases. Then, you hear a buzz. Finally, a musical pitch is produced that rises with the increasing impulse rate.
Part III: Successive Taps From Two Impulsive Sounds (Offset)
If you were to now knock on the table with both hands, but offset the striking pattern of each hand so as to flam*, things get interesting.
*In drumming rudiments, a flam is a basic rhythmic pattern where each note is preceded by a grace note.
Here’s a visual graph of what that flam might look like. Notice that the strike with ‘Hand I’ happens slightly before the strike with ‘Hand II.’ The distance between each strike remains constant.
When we do this, even with the offset striking pattern, a listener can still discern a tempo. Then, as we speed up the flams, we perceive the same effects as before – first an increase in tempo, that transitions to a buzz and finally ends up in musical pitch territory.
Interestingly, when we further separate the two impulses so that they lie directly in-between one another – much like an eighth-note subdivision – our mind still perceives the pitch based on the ‘slower’ quarter-note impulse rate. This is because the two impulsive sounds have slight timbral variances.
Impulses as eight-note subdivision.
The following visual shows both hands represented on the same line. The hash marks beneath the line show the distance from one impulsive sound to the next. This distance is half the distance from each successive ‘Hand I’ knock. Remember, however, that our mind perceives the musical pitch based on the distance from one individual impulsive sound (timbre) to its next reoccurrence. This is represented by the hash marks above the line.
This phenomena demonstrates the remarkable ability of the mind to differentiate between timbral differences – even when the impulse rate is so quick that it becomes impossible to hear the individual knocks. We know this is occurring because something amazing happens when we change the timbre of one impulsive sound to identically match the other. In doing so, we instantly perceive a pitch that is one octave higher than the original note. This change in pitch is due to the fact that we have effectively doubled the impulse rate per second of a single timbre.
Amazing!
Bonus Material: Harmonic Octaves Resulting From Successive Impulsive Sounds
Now, assuming that both impulsive sounds are identical, and we displace the hand taps ever so slightly – so that they are just off-center – we hear two pitches one octave apart. These harmonic octaves are perceived over a very narrow range of off-centeredness. And as we nudge the impulsive sound back toward an eight-note subdivision, the lower octave diminishes until it can no longer be perceived.
Pretty cool, huh?