Pythagoras and the Music of the Future Part I – Timbre

This is the first in a series of articles in which we explore the acoustical basis of the elements that together, form the phenomenon we know as music. The title of the series pays homage to the Greek mathematician and philosopher, Pythagoras, because the theories that inform our analyses and explanations of musical sound, in terms of physics and mathematics, are largely attributable to him. There is, however, minimal mathematical content in what follows since the principal aim of this series is to demonstrate that the acoustical structures of music – the art form –  have always been, and always will be, inseparable from the principles that Pythagoras propounded.

Musical works generally, although not in all cases, consist of combinations and juxtapositions of rhythm, melody, harmony, dynamics and tone colour.  Whatever the historical period, the style or ethnic origin, all music has rhythm; when two or more sounds in succession, of similar or different duration, played within not too extended a time period, ‘connect’ with one another to create patterns in the mind of the perceiver.  When such patterns are combined with successions of pitches, they give rise to the phenomenon we know as melody. A melody, like a rhythmic pattern, is conceived of as an ‘object ‘ – a whole which is greater than the sum of its parts or, in other words, a gestalt.[1]  Some, but not all, music features harmony; at its very simplest, when two or more pitches sound simultaneously. There will inevitably be timbre in any music since every sound possesses particular properties that give rise to the perception of tone colour – the attribute that differentiates the sound of a piano from that of an oboe, for example. All music consists of sounds of constant or varying intensity, dynamic in musical terms.  Although there is a clear acoustical basis for the perception of dynamics, this is not addressed in any depth in what follows. Other than at a few isolated points in history, for example during the 1950s when integral serialism came to the fore, gradations of loudness or softness has not been seen as an essential structural component of music in any sense analogous to rhythm, melody or harmony. Rather, dynamics have been used, in the main, as ‘markers’ or ‘highlights’ or for expressive purposes.

It is generally acknowledged that Pythagoras discovered the mathematical relationship between the length of a vibrating body, such as a string or a tube, and the pitch it produces. Both strings and tubes, of course, feature prominently in sustaining instruments, including the human voice if one considers the vocal cords, in broader terms, as strings. He knew, of course, that a string, when excited by plucking, vibrates at a certain frequency and emits a sound of particular pitch. The note that we hear and identify as possessing a definite pitch, is known in acoustical terms as the fundamental and it is the result of the vibration of the entire length of the string.  Pythagoras later found, however, that a string vibrates not only along its whole length but also, at the same time, along shorter divisions of its length in whole-number ratio to the overall length. These shorter sections produce correspondingly higher frequencies (pitches) which are also in whole number ratio to that of the fundamental and to each other.

The following example illustrates these multiple vibrations separately and indicates the ratios in terms of string length and frequency with f  indicating the frequency of the fundamental.

The simultaneously sounding set of tones of higher pitch are referred to as harmonics, overtones or partials. It will be noted that the higher divisions of the string vibrate with less energy and so possess progressively diminishing amplitudes, hence the diminishing intensities of the higher overtones. The intensities of the upper partials are not as uniformly related as the diagram would suggest. There are, in fact, considerable variations in the relative intensities of overtones, mainly due to the design of instruments and the materials from which they are made.  The relative strengths of the harmonics are what give the sound of any instrument, or voice, its particular tone-colour or timbre.

In common with dynamics, timbre has, until relatively recently, been regarded as a merely colouristic, even decorative, feature of music rather than an essential one.  However, it is timbre, due the inherent frequency ratios of the partials that create it, which is the element most responsible for analogous relationships in terms of rhythm, melody and harmony, and which has underpinned the course of western musical history since the Middle Ages.

The Harmonic Series

The totality of partials in integer proportion to the fundamental, continuing to infinity, is known as the harmonic series. It is commonly represented in musical notation as follows, with f once again representing the fundamental.  Although the fundamental is indicated here as C2 this is merely for convenience – the principal under discussion applies irrespective of whichever note happens to be sounding;

The above notation of the harmonic series is not entirely accurate since the frequencies of the pitches indicated do not match exactly those of the corresponding pitches of equal temperament. Indeed, the 6th overtone, notated here as Bb, and the 10th, notated as F#, would sound decidedly flat to the ‘well-tempered’ ear (the British composer Benjamin Britten (1913 – 1976) made effective, expressive, use of this in his Serenade for Tenor, Horn and Strings in which the opening prelude makes use only of the horn’s natural harmonics).  The musically literate reader will notice (or indeed already know) that clear intervallic relationships exist between the partials indicated in the above example. This is explored in, considerable detail, in the second article in this series.

The Harmonic Series and Timbre

As has been touched upon, the most well-known musical effect of the harmonic series is that of timbre.  Everyone knows that a succession of ‘Gs’ played, in the same octave and of similar dynamic and duration, on the piano, the violin, clarinet, flute then trumpet, will all sound different in that they will have markedly different tone colours. This is because each instrument emits a sound spectrum in which the intensities of the overtones present differ to those of the others.[2] The flute, for example, has very few accompanying partials so the tone has a certain purity. In the case of the clarinet, certain partials are absent, and higher frequencies are prominent, lending it its characteristic sound quality.  The trumpet, on the other hand, has a rich spectrum with prominent upper partials, lending the sound a quality of brightness. This is, in some ways, an over-simplification since the relative intensities of the partials vary not only from instrument to instrument but also within the same instrument at different phases of a note, at different pitches and particularly, at different dynamic levels. For example, the trombone played pianissimo produces a quite mellow, even dull, sound whereas when played fortissimo it is very bright or strident in tone.  The brain ‘averages’ such differences into the perception of a single entity, partly because the frequency ratios of the stronger partials remain more or less constant. For the purposes of this article, the general principles relating to the relationship between the relative intensities of overtones and instrumental timbre will suffice.

One practical application of our knowledge of the correlation between harmonic content and timbre is the electronic synthesiser. Synthesisers work to create new timbres, or reproduce existing ones, in one of three basic ways;

  1. a) Mixing together pure tones in predetermined frequency ratios and intensities to create synthesise timbres, a technique known as additive synthesis,
  2. b) Starting with a rich timbre and filtering out certain bands of frequencies – including in dynamic ways according to the settings of envelope generators (subtractive synthesis), and
  3. c) Creating partials from pure tones, thereby enriching the timbre, by rapidly modulating its amplitude or its frequency. These are termed amplitude modulation and frequency modulation synthesis, respectively.

There will be a considerable amount of discussion of sound synthesis, and its increasingly important role in musical composition in the final article in this series, but here, its discussion illustrates the point that our perception of timbre is largely, though not exclusively, dependent on the overtone content of the notes that are heard.[3]

Although several subsets of timbres are acknowledged, there are two basic categories. The sounds of most string, woodwind and brass instruments, together with the piano (most of the time) and the human voice, exhibit spectra in which the basic harmonic series, with overtones in whole-number ratio, are present. These are termed harmonic timbres.

In the following illustrations which were created using digital sound samples processed by software that separates sound waves into their harmonic components (overtone or partials), using a mathematical process known as Fourier Analysis after the 18th Century French mathematician, distinct ‘peaks’, denoting the most clearly audible partials are notable. Each of the instruments analysed was playing the note G4 (392.00 Hz.) at a similar dynamic.  It will be seen that, although the number and relative intensities of the partials (indicated by vertical axis on the graphs) varies, the frequencies (indicated on the horizontal axis) are aligned and their whole number ratios can be discerned.

Flute Spectrum

Clarinet Spectrum

Most listeners would probably describe the sound of the clarinet as being richer and brighter than that of the flute. Studying the above graphs of the frequency content of both sounds demonstrates clearly why this is: the clarinet timbre is much richer in partials with more at the higher ratios.

Trumpet Spectrum

The trumpet spectrum exhibits strong upper partials compared to the flute and clarinet accounting, as mentioned above, for its rich, bright sound.

Piano Spectrum

The piano spectrum was captured during the ‘decay’ phase of the sound, just after the attack, where the overtone content is clearer. The harmonic content is extremely complex during the attack phase and exhibits ‘inharmonic’ qualities which are discussed below.

Violin Spectrum

In all of the above examples, we see variations in the numbers and intensities of harmonics, but no variation in the ratios of their frequencies. It is this that gives rise to harmonic timbres. Other, non-sustaining, instruments such as bells, gongs and other metallophones, exhibit spectra in which the partials are not in whole-number ratio. These timbres are termed inharmonic.

. Spectrum of Medium Gong

There is much of what is known as ‘noise’ in the spectrum of a gong such as the one sampled above, but there are distinct ‘peaks’ at certain frequencies, which we would perceive as distinct pitches. Close examination of these frequencies makes it clear that they are not in whole-number ratio.

Frequency (Hz) 131 242 315 424 559 672 863
Ratio to Lowest* 1.00 1.85 2.40 3.24 4.27 5.13 6.59

* In this example the term fundamental is replaced with lowest partial due to the fact that the perception of a specific pitch is much less certain. This is even more pronounced in the case of the tam-tam, where a great deal of ‘noise’ is present and a great many more partials are to be perceived. Again, there is nothing resembling a whole number relationship between the partials.

 Spectrum of Tam-tam

It has been seen that the relative intensity of simultaneously sounding overtones is the most significant factor in our perception of timbre and, therefore, our appreciation of the sonic characteristics of musical instruments. We have also seen that there is a difference in acoustical terms between harmonic and inharmonic timbres. However, the harmonic series has clear relationships to, and a governing influence over, much besides timbre. In fact, it permeates all aspects of music; melodic, harmonic, and rhythmic.  In subsequent articles, it is proposed that the most significant developments in western music throughout its history have been due to the influence of, and perception (conscious or unconscious) of, the characteristics of the harmonic series. The distinction between harmonic and inharmonic timbres is central to the arguments that ensue and has a significant bearing on the discussion of music from the Romantic period to the 20th century and beyond. It will be vital to understand that it is not only the simple whole-number ratios of the harmonic series but also the inharmonic ratios – which are, in fact, also contained within the harmonic series – that have had, and increasingly will have, a profound influence on the development of musical thought and language.

It is even suggested that the past trajectory of musical development can be projected into the future and that a new range of musical structures, radically different to those we are accustomed to, but no less relevant in terms of their relationships to the natural, acoustical properties of sound revealed to us by Pythagoras, can be proposed.

The next article in the series discusses the harmonic series in relation to the development of melody & harmony in the contexts of both tonal and atonal music. Please click here to read;  Pythagoras and the Music of the Future Part II – Melody & Harmony.

 

 

[1] Further discussion of gestalt psychology as applied to the perception of melody is to be found elsewhere on this site in the article, Let’s Make Music (forget reading it for now)!

[2] The term spectrum is applied in relation to sound in a manner analogous to its application in terms of light. Just as white light consists of a spectrum of colours at different wavelengths, sound incorporates a spectrum of harmonics also with varying wavelengths.

[3] The other most significant factor affecting our perception of timbre, and therefore the characteristics of musical instruments, is the attack phase of the sound – put simply, the speed of the onset of the sound.