Sine Waves and Timbre

Updated 4/24/08

Timbre (pronounced tamber) is variously defined as 'tone color' and or the characteristic of sound that distinguishes one voice or musical instrument from another. It is determined by the harmonics of the sound and thus is distinguished from intensity and pitch.

To understand timbre we first need to make a side-trip into mathematics. Interestingly, many mathematicians are musicians, while few musicians are mathematicians. A greater percentage of mathematicians have a substantial understanding of, say, Bach and Mozart than musicians who are conversant with the Riemann Hypothesis or the Dirac delta function.

One of the trigonometric functions is called sine, which is often written as Sin to distinguish it from sin, which is a religious term. Sine and Sin are both pronounced 'sign'. In 1822 Jean Baptiste Joseph Fourier proved that any function of a variable, whether continuous or discontinuous, could be expanded into a series of multiples of that variable. Moreover, of course, the series now bears his name. He understood that when waves act independently of one another the displacement of any particle at a given time is simply the sum of the displacement that the individual waves alone would give it. It's what is now called 'superposition'. The importance of the superposition principle physically is that it makes it possible to analyze a complicated wave motion as a combination of simple waves. And the simple waves that Fourier specified are sine waves. In short, Fourier said all music is made from sine waves. In fact, almost all sounds are made of sine waves. The character of a sound's particular tone is directly related to the sine waves that compose it. Doesn't matter whether it's a tuba or a violin.

So, Fourier made it clear that musical sounds are simply combinations of sine waves (figure 3). And it turns out that the basic structure of one sine wave is identical to every other sine wave (figure 2). Timbre is the resultant sound envelope of the combination of various sine waves (figure 3).

As all sine waves are basically alike, how can we distinguish one from another? Sine waves can be varied in two ways: by varying length and/or height. Changing the length of a wave is identical to changing it's pitch and varying the height (amplitude) is identical to varying intensity (loudness, volume). The three major components of musical sound are pitch, intensity and timber and all relate directly to sine waves.

We now know what a sine wave looks like (figure 1) and how they can be changed—by varying length and amplitude. The next step is to introduce the concept of harmonics (partials). The fundamental (prime) wave is the longest that can fit into the tube. It's the absolute base note; the lowest pitch one can get from a particular tube. Harmonics are just divisions of that length (pitch).

So we've got the fundamental. Let's introduce the first harmonic. What's its length? It's 1/2 of the fundamental. And the second harmonic is 1/3, the third 1/4 and so on. But what this really means is that the first harmonic is an octave above the fundamental. The second harmonic is 1.5 octaves above the fundamental, etc. See figures 2 & 3 and notice that the length of each harmonic's wavelength is a whole-number fraction of the fundamental. Notice also that the amplitudes of the harmonics are arbitrarily (could be any value) graphed also as whole-number fractions of the fundamental's amplitude.

In a New Agey, feel-good-kind-of-way it can be said that any musical sound contains all octaves thereby all sounds. And it's probably true. But the higher harmonics are so small in amplitude as to be safely ignored. Since harmonics are higher and higher multiples of the fundamental we are out of the range of hearing pretty fast.

One last potentially confusing bit: The first harmonic is an octave above the fundamental, the second harmonic is musically called the Fifth. The third harmonic is two octaves above the fundamental. The fourth harmonic is called the Third and the fifth harmonic is a Fifth of the first harmonic. Unless you're familiar with musical nomenclature you can safely forget you ever read this paragraph.

Now you know the layout (lengths) of the harmonics, the pitch of each is just a whole number multiple of the fundamental. Most musical instruments work this way. Drums don't; with them the harmonics scale up in what's called an Eigen fashion, which for the terminally curious, is somewhat similar to the way quantum physics works.

But what about the amplitude (volume) of the harmonics? That's what the whole timbre business is all about. The harmonics have fixed pitches (in relation to the fundamental) but the amplitude of each is determined by the instrument, technique of the player, etc.. Violins, for example, have a large amplitude fifth harmonic—that's how we identify the sound. And in fact, the usual nomenclature for timbre uses the names and materials of instruments to refer to particular timbres. We say a timbre is 'brassy' not because brass sounds like that but because horns with very high aspect ratios are usually made of brass. They could just as well be made of cement. A timbre which is 'woody' is so called because it's identified with wooden instruments of lower aspect ratio which could have just as well be made of brass. Timbre is the resultant sound envelope of adding all the sine waves together--fundamental and harmonics (figure 3). The timbre of a tuning fork is close to the sound of a single sine wave--just the fundamental and few if any prominent harmonics so it should look like figure 1. Speaking or singing the vowel oo (as in too) comes close to producing a pure tone with few harmonics.

The distinctive timbre of the shakuhachi has a lot to do with the fact that it's end blown. The differences in timbre BETWEEN shakuhachi has to do with geometry of the bore and aspect ratio—mostly aspect ratio.

The biggest difference in timbre between shakuhachi and hochiku, for instance, is simply a matter of aspect ratio. High AR tubes favor the upper harmonics and lower AR tubes favor the lower harmonics. In high AR tubes the upper harmonics have greater amplitudes (relative to the fundamental) than do low AR tubes. What qualifies as upper and lower as far as harmonics are concerned? For the most part, the timbre of a shak is contained in the first dozen harmonics and its major elements in the first half dozen. So we can make an arbitrary distinction and place the dividing line between upper and lower as between the third and fourth harmonic--or there abouts. Whether particular harmionics are present is really just a question of their amplitude--a question of how strongly they are present. Adjusting the bass/treble knob on electronic equipment changes the amplitudes of respective harmonics. Your preference with the tone knob is a direct indication of the AR you'd be happy with in flutes.

Let's switch to looking at half-waves rather than full-waves for two reasons: the second half of a sine wave is just the inverse of the first half (figure 1) and the wavelength of a shakuhachi is twice the length of the tube. So looking at half-waves is representative of looking at the length of a shakuhachi bore.

Graph of the difference in the Sound Envelopes of lower and upper harmonic emphasis.

With a flute which plays second octave easily it shouldn't be surprising that the timbre is comprised of upper harmonics--it's a flute more favorable to higher pitched sound whether it be harmonics or fundamentals. All around, the flute just plays 'higher'. And the converse is true of a flute which strains to get second octave--it will be more comfortable with lower harmonics and fundamentals. It's mostly a case of aspect ratio.

One other thing. The area of the opening at the blowing edge does effect things in these regards. A smaller AREA (between lips and blowing edge) favors the upper harmonics, a larger opening--the lower harmonics. So you could have a tube with a low AR and by making the blowing notch shallow and narrow, boost the upper harmonics...some. Or go the other way and with a tight tube lower the harmonics by opening up the blowing area. That's really what meri-kari is about--changing the area of the opening. Not only does it change pitch but timbre as well by shifting the emphasis in the harmonic register.

If you want 'bright' timbre tighten up everything except the holes--they work the other way round. Big holes favor brighter timbre, small holes, darker. For that dark moody flute loosen up the bore, the blowing opening and tighten the holes.

The shakuhachi as a 'musical' instrument is built to boost the higher harmonics--thus it's particular bright timbre. The hochiku as a 'spiritual' instrument is designed to emphasis the lower harmonics and so it sounds darker. Apparently in the shakuhachi world a skinny tube is 'musical' and a fat tube 'spiritual'--it's a simple matter of timbre, which is largely a matter of aspect ratio. Play both and you've got it covered.

Bore and timbre

Hole size and timbre

Utaguchi and Timbre

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