7. What are wave shapes and spectral content?

The shape of a wave is directly related to its spectral content, or the particular frequencies, amplitudes and phases of its components. Spectral content is the primary factor in our perception of timbre or tone color. We are familiar with the fact that white light, when properly refracted, can be broken down into component colors, as in the rainbow. So too with a complex sound wave, which is the composite shape of multiple frequencies.

So far, we have made several references to sine waves, so called because they follow the plotted shape of the mathematical sine function. A perfect sine wave or its cosine cousin will produce a single frequency known as the fundamental. Once any deviation is introduced into the sinus shape (but not its basic period), other frequencies, known as harmonic partials are produced.

Partials are any additional frequencies but are not necessarily harmonic. Harmonics or harmonic partials are integer (whole number) multiples of the fundamental frequency (f) (1f, 2f, 3f, 4f…). Overtones are the harmonics above the fundamental. For convention’s sake, we usually refer to the fundamental as partial #1. The first few harmonic partials are the fundamental frequency, 8ve above, perfect fifth, 2 8ves above, 2 8ves + major 3rd, 2 8ves + major 5th as pictured below for the pitch 'A.' After the 8th partial, the pitches begin to grow ever closer and do not necessarily correspond to equal-tempered pitches, as shown in the chart. In fact, even the fifths and thirds are slightly off their equal-tempered frequencies. You may note that the first few pitches correspond to the harmonic nodes of a violin (or any vibrating) string.

Most acoustic instruments not in the ‘noise’ category produce some combination of fundamental and harmonic partials. Bells are a category of sound that produces inharmonic partials, so called, because they do not correlate to the harmonic partials above.

The collection of frequencies produced by a waveform are prime contributors to its spectrum (pl. spectra). For a periodic waveform, the shape of the wave determines not only the frequency of the partials, but also their relative strength, another important factor of timbre.

In the 18th Century, a mathematician named John Baptiste Fourier determined that:

1) all complex periodic waves may be expressed as the sum of a series of sinusoidal waves
2) that these waves are all harmonics of the fundamental and
3) that each harmonic has its own amplitude and phase (which we have not discussed yet).

With the advent of computers and digitized audio data, it has become routine to perform Fourier analysis on existing sound to break it down into its component frequencies along with their specific amplitudes. It has also become commonplace to 'reverse engineer' this process and create complex timbres by carefully mixing sine waves in a process called additive synthesis, practiced masterfully by composer Jean Claude Risset amongst many others.

Early on, researchers were able to quickly visualize real-world sound in two important ways—either take a snapshot of the instantaneous amplitudes of a sound’s component frequencies at a single given point in time (called an FFT or Fast Fourier Transform) or create a sequential series of snapshots to give a picture of how a particular sound evolves over time. This form of FFT gives tremendous insight into how the various partials of an instrument evolve in the first few milliseconds of its attack.

Pictured below are two FFT's. The first plots the frequencies and strengths of an instantaneous moment of an oboe note in two dimensions. The second analysis demonstrated the attack phase of an oboe note in 3 dimensions, frequency, amplitude plus time. It can be viewed as a collection of snapshots put together to show how the note spectrally evolves.

Notice in the 'snapshot' FFT that the 2nd partial has a greater amplitude than the fundamental and also that partials #9 and #13 are missing from the spectrum. Notice in the second FTT how the various partials develop over time and change relationships.

Composers have been able to get inside these analyses and manipulate the pitch, timing and evolving amplitudes of individual partials with very interesting results (made easy by programs such as Audiosculpt (IRCAM) and Spektral Delay (Native Instruments). A related process called phase vocoding allows these successive analysis snapshots to be “resynthesised” by creating a sine wave for each partial (or band) which follows the original analysis—the beauty is that a sound can be “reconstituted” at any desired speed, forward or backward, without altering its pitch, as would be the case if a tape were sped up or slowed down. This technique exemplifies a type of synthesis by analysis.

It is beyond the scope of this article to discuss the specific spectra created by synthetic waveforms (such as triangle waves, square waves, etc.), but please visit Basic Synthesis Facts for further information).

In the real world, many different frequencies dynamically combine, either from the same source or from different sources to create our impression of timbre. It should be noted that the summation of waveforms creates a series of complex interactions over time. However, at any instantaneous moment in time, only one sound pressure level is acting upon our eardrum and only one voltage is being applied to a loudspeaker, even if you are listening to or reproducing an orchestra of 80 instruments. We perceive the mixture of pitches through the complexities of a varying pressure stream and also a bit of psychoacoustic extrapolation on our part (the 19th Century “father of acoustics” Helmhotz believed that our perceiving the musical pitch of a sound was dependent on the presence of its fundamental sine wave — this has now been proven to not always be the case).

For further study, see Hyperphysics->Waveforms

An Acoustics Primer, Chapter 7
URL: www.indiana.edu/~emusic/acoustics/wave_shape.htm
Copyright 2003 Prof. Jeffrey Hass
Center for Electronic and Computer Music, School of Music
Indiana University, Bloomington, Indiana