Pythagoras and the Music of the Future Part I – Timbre

Harmonic Wave 2This is the first in a series of articles in which we will explore the acoustical basis of the structural elements that together, form the phenomenon we call music. The title of the series is a tribute to the Greek mathematician Pythagoras, because much of what we understand to be the mathematical or physical basis of musical sound, and hence the remarkable insights that have been gained, is attributable to his discoveries and theories. There will be a minimal amount of mathematical content here, however, because the main aim of the series is to demonstrate that music – the art form – always has been, and always will be, bound up with principles that Pythagoras revealed.

Whatever the period, the style or the ethnic origin, all music has rhythm: two or more sounds in succession, within a not too long time interval, can form a perceived connection with one another and will mark the passing of time.  Much music has melody of a sort, by which is a succession of two or more pitches combined with rhythms so as to be perceived as a whole – a gestalt[1].

Some, not all, music has harmony; that is to say, at its very simplest, when two or more pitches sound together. Since all music requires there to be sound (even John Cage would probably have agreed!), there will inevitably be timbre since every sound has its own characteristics. Every sound also possesses the quality of intensity, or in musical terms dynamic.  However, although there is a clear acoustical basis for the perception of dynamics, we shall not dwell too much on it here: other than at a few isolated points in history, for example during the 1950s when integral serialism came to the fore, dynamics have not been a structural element of music in any sense analogous to rhythm, melody or harmony. Rather, they have been used as structural ‘highlights’ or for expressive purposes.

The same could be said for timbre which has, until relatively recently, been regarded as a colouristic rather than a structural element in music. However, it will be the first of the musical elements to be discussed for, as we shall see towards the end of the series, this is the element from which all other structural elements emanate and which has informed the course of western musical history since the Middle Ages. Whether this has been a conscious or unconscious process is an interesting question. I suspect a bit of both, but mostly the latter. Timbre occupies such a seminal position because most of the developments in western music since medieval times can be attributed to the fact that sustained sounds (notably the voice) were the primary materials in the early stages of the developments that, in succession, comprised the key developments in the history of western music.

The Harmonic Series

It is generally acknowledged that Pythagoras first propounded the principle that in the case of any musical sound, the pitch we perceive, known in acoustical terms as the fundamental, will be accompanied by a set of tones which have come to be called harmonics, overtones or partials, sounding simultaneously with it. The frequencies of these harmonics are in whole number proportion (ratio) to the fundamental (f) and, generally, their intensities diminish. The fact there is such a ‘fading’ of the partials might appear to be in contradiction to some of the observations later to be made, about the properties of individual instruments, but it will nevertheless be a very important factor in the discussion, in later articles,  relating to developments in ‘modern’ music and the future ‘direction’ of music.

The harmonic series is commonly represented thus, although it must be remembered that the overtones sound simultaneously;

Harmonic Series2

Overtones arise because a vibrating string or tube (these feature prominently in the sound production in all sustaining instruments, of course) vibrates not just along its whole length, but also, along shorter divisions of its length which are in whole number ratio to the overall length. These shorter vibrating lengths produce correspondingly higher frequencies. Since these shorter lengths are in whole number ratio to the overall length, the frequencies will also be in whole number ratio to the frequency of the whole string.

The following example separates and visualises these multiple vibrations and indicates the ratios in terms of string length and frequency ( f ).

Vibrating String Harmonics2

It will be noted that the higher integer divisions of the string vibrate with less energy and so possess progressively diminishing amplitudes, hence the diminishing intensities of the higher overtones.

The above representation of the harmonic series in musical notation is only partially accurate (excuse the pun), since the frequencies of the harmonic series do not match exactly the frequencies of the corresponding musical pitches of equal temperament. Indeed, the 6th overtone, notated here as B¨, and the 10th, notated as F#, would sound decidedly flat to the ‘well-tempered’ ear. The British composer Benjamin Britten made effective, expressive use of this phenomenon in his Serenade for Tenor, Horn and Strings in which the opening prelude makes use only of the horn’s natural harmonics. Click here to listen

The musically literate reader will notice (or indeed already know) that a clear intervallic relationship exists between the partials indicated  – transposing some of the higher partials down an octave or two makes this very clear. This will be explored in more detail, and expanded, later.

The Harmonic Series and Timbre

The most well-known musical effect of the harmonic series is that of timbre or tone colour as it is sometimes called.  We all know that if you hear say, a succession of ‘Gs’ in the same octave, and of similar dynamic and duration, on the piano, the violin, clarinet, flute then trumpet, all the Gs will sound different: they will have markedly different tone colours. The reason for this is that the relative intensities of the overtones present, in the instrument’s spectrum, vary. The flute, for example, has very few accompanying partials and so the tone sounds quite pure. In the case of the clarinet sound, some partials are absent, and higher frequencies are prominent, lending the ‘buzzing’ 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 ‘pp’ sounds quite mellow or dull (in Stravinsky’s Symphony of Psalms it is used effectively in this way as the bass of a chord in which the other notes are played by three flutes!) whereas if  played ‘ff’ it is very bright or even strident in tone. The brain ‘averages’ these into the perception of a single entity, partly because the frequency ratios of the partials remain more or less constant. For the purposes of this article, the general principals being discussed should suffice.

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

a) Mixing together pure tones in predetermined frequency ratios and intensities to create -or literally synthesise – timbres; additive synthesis,

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

c) Creating additional partials thereby enriching a tone by rapidly modulating its amplitude or the frequency (amplitude modulation or frequency modulation synthesis).

More will be said about sound synthesis, and its increasingly important role in musical composition later, but for now, it illustrates the point that our perception of timbre is largely, though not exclusively,[2] depended on the overtone content of the tones we hear.

Although several subsets of timbres can be created, there are two basic categories. 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. Let us take a closer look at some.

The following illustrations were created using digital sound samples processed by software that can separate sound waves into their harmonic components (overtone or partials) using a complex (to me at least!) mathematical process called Fourier Analysis, after the 18th Century French mathematician.

You will note distinct ‘peaks’, denoting the most clearly audible partials. Each of the instruments analysed was playing the note G4 at similar a dynamic [3]. We will see that although the number and relative intensity of the partials varies (indicated by vertical axis on the graphs), the frequencies (indicated on the horizontal axis) are aligned and the whole number proportions can be seen.

 Flute Spectrum

Spctr Flute G4

.

Clarinet Spectrum

Spectr Clarinet

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 and more at the higher ratios.

Trumpet Spectrum

Spectr Trumpet

The trumpet spectrum exhibits very strong upper partials (compare to flute and clarinet) accounting, as mentioned above, for its rich, bright sound.

. Piano Spectrum

Spectr Piano

The piano spectrum was captured during the ‘decay’ phase of the sound, just after the attack, where the overtone content is clearer. There is just too much going on during the attack phase where the sound is very complex.

. Violin Spectrum

Spectr Violin

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. Once again; these are harmonic timbres. Other instruments, notably, bells, gongs and other metallophones have spectra in which the partials are not in whole number ratio. These spectra are termed inharmonic. Look at the following examples and compare with the above.

. Spectrum of Medium Gong

Spectr Gong

There is a lot of what would be termed ‘noise’ in the spectrum of a gong such as that sampled above, but there are distinct ‘peaks’ at certain frequencies, which we would perceive as having a distinct pitch. If we look at these frequencies, we see that they are in nothing like 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 simply with lowest partial. This is because the perception of a particular pitch or ‘note’ is much less certain. This much more pronounced in the case of the tam-tam where much more ‘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

Spectr Tam tam

We have seen then, that the harmonic series is the most significant factor in our perception of timbre and therefore the sonic characteristics of our musical instruments. We have also seen there is a difference in acoustical terms between harmonic and inharmonic timbres.

But the harmonic series has clear relationships to, and a governing influence over, much besides timbre. In fact, it permeates the whole gamut of music structures; intervallic, harmonic and rhythmic.  In future articles, we shall see that the most important developments in western music throughout history have been bound up with the influence of, and our perceptions of, the characteristics of the harmonic series. The distinction between harmonic and inharmonic timbres is central and will have a significant bearing on our future discussions of music in the 20th century and beyond. It will be very important to understand that it is not only the simple whole number ratios of the harmonic series but also the inharmonic ratios (still to found in the harmonic series – this will be afforded a dedicated discussion later) that have had, and increasingly will have, a profound influence on the development of musical thought and language.

I will even go so far as to suggest that the past trajectory of musical development can be projected forwards into the future and propose a new range of musical structures (what many would say comprise a ‘musical language’) which are radically different to those we are used to, but no less relevant regarding their relationships to the acoustical, natural properties of sound revealed to us by Pythagoras centuries ago.

The next article in the series discusses the harmonic series in relation to the development of melody & harmony and develops the ideas about its relationship to atonal music further. 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 perception of melody is to be found in my article Let’s Make Music (forget reading it for now)!

[2] 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.

8 thoughts on “Pythagoras and the Music of the Future Part I – Timbre

    • Hello Marco. I will be looking at the relationships between the harmonic series and intervals in the next article. I’m not too sure what you mean when you say the meanings of intervals, but I’d be very happy to explore it further with you. I’ll email you too. Many thanks for your contribution!

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  1. I add that tuning systems too are very dependent on the harmonic series. If you develop music on “harmonic” instruments you will probably end up using tuning systems like the Western one, if you develop music on “inharmonic” instruments (like in Bali, for example) you end up with completely different tuning systems.

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    • Thanks Carlo. I don’t know too much about tuning systems, but certainly, from what I have learned, you are right about tuning systems and the harmonic series. What may not be so clear (yet) is that I see the future of music involving both harmonic and inharmonic timbres, with much of the latter being produced digitally. So tunings might be well an issue in that context, but practicalities would also be a consideration; strings woodwind and brass can adapt nicely to various tunings, but traditional inharmonic instruments like tubular bells, gongs etc, and the piano (which I see as ‘cross-over’ instruments) I think would have to stay as they are for practical reasons. It’s an interesting question to which I don’t have an immediate answer.

      I hope I’ve understood your point correctly. Thanks for making it.

      By the way, thanks for liking the Facebook page! Please feel free to write something.

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  2. Inharmonic timbres and related tuning systems that can sound consonant with them can be easily created digitally. I suggest a great book by William A. Sethares called “Tuning, Timbre, Spectrum, Scale” on this matter. You can also find a few articles I wrote on the subject on my blog under “Tuning Theory” category (excuse me for the plug!)

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  3. Glad to see other people figuring out the Harmonic Series as the true source of all music — everywhere & anytime: http://whatmusicreallyis.com/research/source/

    One note about harmonic spectra: only the human voice has perfect harmonic overtones. Of all the instruments you mention, plucked and stroked strings are the most inharmonic, followed by bowed strings and blown pipes & reeds.

    Pianos for example exhibit inharmonicity, not as strong as metal idiophones, but still a noticeable departure from (or stretching of) natural harmonics of whole numbers, starting as early as the 6th harmonic and stretching so that piano overtone 27 corresponds to natural harmonic 28 for an 128Hz tuned acoustic string: http://imgur.com/D7OZpHZ

    “Relating Tuning and Timbre” is an excellent article that inspired the book Carlo talks about and summons it perfectly, written by the same author: http://sethares.engr.wisc.edu/consemi.html

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    • Many thanks for your comment. I had a look at your blog and it seems we are on the same wavelength – excuse any hint of a pun.
      I broadly agree with your points although I am not sure what you mean by ‘perfect’ overtone series.

      When I get round to writing the fourth and final article – ‘Pythagoras & the Music of the Future Part IV -Synthesis’, I think it will become clear that I feel the need for a correlation between the most prominent ratios occurring in inharmonic timbres and those in the domains of pitch and ‘macro’ musical time. My piece ‘Songs of the Aristos (2014)‘ embodies this principle. You’ll hear microtonal intervals in the electronic parts, together with temporal ratios that humans could not accurately realize, which sound totally ‘at home’ in the inharmonic context.

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  4. Pingback: Frequency, Pitch, Overtones, Harmonics and Timbre - Zeroes and Ones

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