In Search of the Missing Fundamental: by Richard K. Jones
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Frequency, Pitch and Colour

Complex tones, or composite waveforms (harmonic and non-harmonic), consist of many sine waves of different frequency added together. These individual sine waves are called frequencies components. What are consider to be musical sounds generally don’t have just one or two frequency components. Sounds that have only a few frequency components are not at all interesting or pleasing to listen to. They have no musical colour or timbre. Conversely, sounds that have too many frequency components, like the sound of a strong windstorm with rain, may be interesting and even pleasant to listen to but, these sounds don’t have a particular pitch so they usually aren’t considered musical “notes.”

When someone sings a note or plays a note on an instrument, a very particular set of frequencies is heard. Visualize each note that is sung or that is played on an instrument as a smooth mixture of many different pitches as shown above. These different pitches are called overtones or partials and are preferably harmonic as described above, but they can be either harmonic or non-harmonic. The human auditory system generally blends them together so well that you do not hear them as separate notes at all. Instead, the overtones or partials give the note its color or timbre. Notes that have many non-harmonic overtones are said to create inharmonicity.

In music, inharmonicity is the degree to which the frequencies of the overtones of a fundamental differ from whole-number (integer) multiples of the fundamental’s frequency. These inharmonic (non-harmonic) overtones are often distinguished from harmonic overtones (whole-number multiples) by calling them partials, though partial may also be used to refer to both. Whether we hear a sound as pitched or unpitched depends partly on the overtones of that sound. The more inharmonic a sound is, the less definite it becomes in pitch. Many percussion instruments such as cymbals, tam-tams, and drums create complex composite waveforms that are inharmonic sounds but yet they add a peculiar or colorful amount of inharmonicity or harmonic distortion to music, which is found to be palatable to the ear. Conversely, most modern professional-quality wind, brass and string instruments are designed to limit inharmonicity as much as possible in order to bring harmonicity to music.  In the hands of a master, timpani can bridge the gap and provide both.

If an oboe plays a middle C (C4 261.63 Hz) and then a clarinet plays the same note at the same loudness as the oboe, it is still easy to tell the two notes apart, because an oboe sounds different from a clarinet. This difference in the sound of the two instruments is the colour, or timbre, of the notes, which is based on each instrument’s own unique harmonic recipe. A note’s harmonic recipe is its number of overtones or partials (harmonic and non-harmonic) and their amplitude proportion relative to the fundamental. Along with their individual attack characteristics, it is these overtones (harmonic and non-harmonic) that give instruments their tone colour and without them, we could not tell one instrument from another.

The chart below graphs the first sixteen harmonic partials (with amplitude proportions) of a clarinet and oboe playing same pitch or frequency. The clarinet projects a strong fundamental and mostly odd harmonics while the oboe projects a strong fourth, fifth and sixth partial and a relatively weak or non-existent fundamental.

The oboe has almost no sound at the actual fundamental frequency, even though it is that pitch you hear when listening to it. The same is true of the bassoon sound. In fact, even if the fundamental frequency is omitted altogether, the ear still hears the same pitch as if the fundamental were. This missing fundamental effect is often referred to as virtual pitch or “reconstruction” of the fundamental. 2

A similar phenomenon also plays an important role in how we perceive timpani pitch. In the case of timpani, the pitch we perceive as being the “fundamental” is actually the second partial of a non-harmonic overtone series. It is the “construction” of a non-existent harmonic fundamental which is the objective when a timpanist tempers or “clears” timpani heads.

Chapter 2 of this book will introduce the theory of how circular vibrating membranes function and how the various modes of vibration contribute to the sound of timpani. However, as with any new discipline, there is usually a certain amount of new vocabulary, which needs to be learned and understood so the next section will be devoted to a short review of the concepts and terms introduced in Chapter 1. A stronger understanding of these terms and concepts may aid the reader in developing his/her own ability to listen to and better understand timpani tone.

Chapter 1 Review

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1) Chapter 1 Selected Bibliography and WWW Links