Pythagoras and the Music of the Future Part II – Melody & Harmony

In the previous article, we examined the role played by the harmonic series in our perception of timbre; how the relative strength of the partials that constitute the ‘spectrum’ of a particular instrument determines its perceived tone colour. Timbre is not, however, the only element of music in which the harmonic series plays a determining role.  In fact, this microstructure of musical sound has a correlation with the macrostructures of pitch and rhythm too.  In the present article, we shall focus on the relationship between the frequency ratios present in the harmonic series and musical intervals.

We use intervals in two ways; melodically and harmonically: successions of intervals allied to rhythmic patterns form melodies while the simultaneous sounding of intervals forms chords; successions of chords form the structures we know as chord progressions. These musical structures are, in turn, based on underlying structures such as scales modes, ragas etc. all of which exhibit a clear relationship to the frequency ratios of the harmonic series, as we shall see.

First, a reminder of the structure of the harmonic series and the fact that its partials are in whole number ratio to the fundamental;

Harmonic Series

As we consider the harmonic series notated as musical pitches as above, we notice that successive pairs of partials form familiar intervals. When we look at the corresponding frequency ratios, we see that the frequency ratios forming these familiar intervals are as follows[1];

1:1          Unison

1:2          Octave

2:3          Perfect 5th

3:4          Perfect 4th

4:5          Major 3rd

5:6          Minor 3rd

8:9          Major 2nd [2]

15:16     Minor 2nd

Intervals such as major/minor sixths and sevenths are inversions of thirds and seconds respectively, and their frequency ratios, which do not appear between successive partials of the harmonic series, reflect this:  The ratio 5:6 (minor 3rd), with the second tone lowered by one octave, creates the ratio 3:5 (Major 6th); the ratio 8:9 (major 2nd), with the first tone raised by one octave produces the ratio 9:16 (minor 7th). The remaining intervals, with their corresponding ratios, are;

5:8          Minor 6th

8:15        Major 7th

Were we to hear the above example of a harmonic series played, we would hear the relationship between the harmonic series and familiar intervals. We would also hear that the interval of the octave (1:2) produces two tones which sound equivalent i.e. we perceive the upper tone to be the same note an octave higher. We even identify it as such by means of the same letter name. This is also the generally accepted reason why it is this interval that is universally divided into a number of smaller steps such as the semitone in the case of traditional western music. Various combinations of the above intervals formed the basis of scales, the ancient Greek modes (later to become the medieval Church Modes) and eventually, major and minor scales.  It is also noteworthy that even in non-western cultures; we find similar intervals, with, of course, similar relationships to the ratios of the harmonic series.

At least since the time of Pythagoras, musical intervals have been endowed with particular affective qualities. It is universally acknowledged that a melody can provoke a strong emotional response and this is partly attributable to the qualities that we ascribe to particular intervals, although contextual factors such as adjacent intervals, rhythm, repetition, tempo, dynamic and instrumental colour, all play their part in producing the overall effect.  Harmonic (i.e. vertical) intervals too, have qualities ascribed to them and here too, the effect of the intervals is emphasised or coloured by the context, with dynamics playing a particularly important role. Again, a correlation between the frequency ratios forming musical intervals and these widely accepted affective responses can be noted.

When we consider the moods or feelings that are customarily ascribed to intervals in relation to such frequency ratios, we notice that the simpler the ratios (lower numbers), the more stable we perceive the intervals to be.  Conversely, the more complex the frequency ratios of the intervals, the less stable we feel them to be.  We also notice, therefore, a correlation between the simpler ratios and the intervals that are generally considered to be consonant, with the less simple ratios creating intervals that are generally considered to be dissonant.

Of course, there is a lot of subjectivity involved in describing the affective qualities of intervals, but the following table outlines some widely acknowledged ones.  Unison here means repetition, or juxtaposition, of the same pitch;

Intervals Table

In medieval monastic music, all but the intervals with simplest ratios were considered consonant: In the 8th and 9th centuries, plainsong, Gregorian chant, was sung in unison. By the 10th century, organum where an original chant with second, parallel voice at the octave, fifth above had begun to develop. These additional intervals represent the first three partials of the harmonic series; unison; 1:1, octave 1:2, fifth; 2:3. It is unlikely that these intervals were chosen consciously. Rather, an awareness of their presence would have been enhanced by several things: firstly, and although this may be obvious, it is important, the music was performed by adult males, so the pitch of the notes would be low enough for their upper partials to be well within the audible frequency range. The octave and fifth would have been clearly audible in unison singing, partly because of their position in the harmonic series, and therefore their intensity, and partly because sum tones[3] would have sounded creating a very clear octave above the principal melody. This would, in effect, reinforce the first overtone of the series their voices were producing.  The interval of a compound fifth would also have been clearly audible as a partial, but when the chants were sung in parallel octaves, then sum tones would come into play again (f1 + f2 =  f3) and this 3rd partial would have been similarly reinforced. It would, therefore, have seemed quite natural for this tone to have been lowered by one octave to bring it within the vocal range of the singers. The fifth overtone, forming a compound major third with the note being sung, would also have been clearly audible. However, whilst it was indeed audible, and in that sense available, it was avoided at the time because it was considered dissonant: it had a ‘softening’ or destabilising effect on the intervals of the octave and the fifth.

During the Renaissance, the interval of the 3rd was fully established as a consonance and the triad, a structure containing intervals that correspond to the first five overtones of the harmonic series (where the root corresponds to the fundamental), were established as the basic building blocks of harmony.  Triads could be built on all degrees of the scale, but because of their relative strengths and their tendencies to ‘pull’ in a certain direction a hierarchy developed. The chord of the fifth degree of the scale, the dominant, because of the presence of the leading note[4] exerted a strong pull towards the tonic and is thus considered to be the strongest chord after the tonic. The ‘pulling’ tendencies of intervals (melodic or harmonic) and therefore the extent to which they would be deemed dissonant had existed for some time, indeed this is probably what led to intervals and indeed whole modes, having qualities attributed to them. More of this anon, but for now, we simply notice that the strongest chord – the one with the greatest ‘gravity’ – is that at a ratio of 2:3 from the fundamental.

After the tonic and dominant, the next ‘strongest’ chord in a tonal system is the subdominant which is at the same ratio of 2:3, but this time inverted: the fifth descends, but the strength of the relationship between the key-chord and the next chord is almost as clear.  The foundations of the hierarchical system of chordal relationships known as functional harmony thus demonstrate a close correlation to the ratios of the most prominent partials of the harmonic series.

Once modulation had become a frequent feature of music, the key most often modulated to was the dominant. It is difficult to escape the conclusion that this was again due to an awareness of the most prominent ratios of the harmonic series. In the case of modulation to the dominant however, the frequencies of all of the pitches of the home scale are multiplied in the ratio 2:3 or, put another way, by 1.5. Octave displacements of the resulting pitches would exhibit the equivalence discussed above. Again, this is a result of awareness and not a conscious, mathematical process.

The simplest means of modulating to more distant keys was to repeat the process of moving from the ‘home’ key to the key where the frequencies were multiplied by 1.5: i.e. via the cycle of fifths.  It is well-known that a ‘full’ cycle of 5ths eventually encompasses all the degrees of the chromatic scale[5], so using it as a modulating device, we can ‘go anywhere’. This is, in its simpler manifestations, a prominent and immediately recognisable feature of much Baroque music.

Hand in hand with the development of functional harmony came the acceptance of intervals representing higher ratios of the harmonic series; intervals which would, in earlier times, have been unthinkable. For example, the dominant 7th chord and the diminished triad, both commonly used in the Baroque era and beyond, contain the interval of the tritone – the Diabolus in Musica[6] as it was known in relation to medieval monastic music.  However, in the context of functional harmony, both have important, enriching roles: the minor seventh in the dominant 7th chord creates a strong need for resolution; the ascending tendency of the leading note being mirrored by the descending tendency of the minor seventh from the root. Likewise, the diminished chord has, as a principal characteristic, ambiguity and so creates a degree of flexibility in the choice of harmonic progressions and modulations[7].

So while, by the end of the Baroque era, dissonances – representing higher frequency ratios – were allowed, their functions did not change for centuries. Whilst the range of dissonances that provided routes to other keys, or which enhanced the feeling of resolution, were accepted there could be no question of these being allowed in anything but a functional harmonic context – until the twentieth century as we shall see.

In the Classical Period, sonata form became the standard model.  In the sonata, the symphony, the string quartet and many other forms of chamber music, a first movement in sonata form was de rigueur.  Here the notion of modulation is expanded; one key centre is not just visited briefly as in modulation, but replaced with another which itself might well incorporate shorter term modulations. A first subject, or subject group, in the tonic key, is followed by a second in the dominant.  Following a development section which would incorporate modulations to more distant keys, the music would return to state both the first and second subject groups in the tonic key. Again the workings of this large-scale and often complex structure are based on the tonic/dominant relationship and hence the frequency ratio of 1:1.5.

The table below illustrates (admittedly in rather simplistic terms) how this basic ratio and its corresponding harmonic relationship of tonic/dominant, together with its progression from microstructure to macrostructure, developed over time;

Tonic Dominant Table

So in the Classical period, sonata form became a medium by which composers could demonstrate degrees of ingenuity in leading the listener way from the tonic key into less familiar territory, surprise him with harmonic ‘twists and turns’ but eventually satisfy him when all was resolved back at the home key. Not unlike a refined and rather dignified roller coaster ride – great fun! In the hands of Beethoven however, sonata form was transformed from the beguiling mastery of harmonic and melodic development presented by Haydn and Mozart, for example, to a vehicle for extra-musical expression: modulating to more remote keys in the now greatly extended development section, whilst deploying  ever more daring, dissonant harmonies on the way, was  revealed to possess expressive, even dramatic power. It served not only to add a remarkable degree of emphasis to the eventual return to the tonic, but also it was symbolic of man’s struggle against, and eventual triumph over, adversity and/or oppression.

This was to have far-reaching consequences for musical expression and for functional harmony itself. The artists who shaped the 19th Century – the Romantics – are regarded by many as having been egotistical and introspective – preoccupied with discontent. Issues of unrequited love, political tension, or the burdens of existence itself became the ‘subject matter’ for expression via the medium of their art.  Composers harnessed and built upon Beethoven’s legacy, of expressing dramatic tension and eventual resolution by way of functional harmony, by developing it and stretching it further. The tensions created between the tonic key and those which were distant was reflected in a greater degree of chromaticism in the melodic writing and more dissonance in the harmony. But while dissonance was still there to be resolved initially, in the late 19th and early 20th centuries, again for purposes of extra-musical expression, the presence of clear key centres became more difficult, and at times impossible, to discern.  Famous examples include Liszt’s Faust Symphony (see below) and of course Wagner’s Tristan and Isolde.

Faust Symphony

In the above example from the opening of Liszt’s Faust Symphony, the harmony, consisting only of broken augmented triads, defies identification of a tonal centre.

The need to symbolise the ‘unresolved’ nature of so many aspects of human existence and indeed of nature, came to dominate musical thinking to the extent that some composers of the early 20th Century, most notably Arnold Schoenberg and Claude Debussy,  not only chose deliberately ambiguous harmonies and left dissonances unresolved, but often sought to avoid any reference to a key centre at all. This isn’t the place to go into detail about the workings of Schoenberg’s  twelve-tone system, or indeed of Debussy’s harmony, but as we will see shortly, the suggestion that music of ambiguous tonality and music without tonality are ‘divorced’ from the structures inherent in the harmonic series – as has often been suggested – is erroneous.  Consider, for example, the spectrum of a trumpet playing the note C4 ;

Trumpet Spectrum

Spectr Trumpet

The peaks in the spectrum could be notated as follows;

Notated Spectr Trumpet

What we see is, pure and simple, a C major chord. This illustrates very clearly the relationship between simple harmonies and the sounds of the instruments they are played on.  Now, let’s take a second look at the spectrum of the tam-tam which we first came across in the previous article whilst discussing inharmonic timbres;

Tam Tam Spectrum

Spectr Tam tam

If we were to notate the peaks in this graph we would find end up with a very different kind of chord;

Notated Spectr Tamtam

If we were then to create a simple passage using only the pitches of this chord (including octave displacements), we could get something like this;

atonal melody

If you have a piano or electronic keyboard to hand you can play these examples. If not, you can click below to play an audio example in which you will hear the ‘tam tam chord’ played and then, the above fragment;

Although this is not twelve-tone music (and there is a fleeting ‘octave’ relationship in the second bar!) it is not a million miles away from the sort of thing you might hear in the music of many earlier twentieth century composers.

From this then, we might draw the conclusion that while the structures of tonal music are closely related to the ratios to be found in the overtone series of natural sustaining instruments such as the trumpet or voice, the structures of atonal music are related to the frequency ratios of inharmonic timbres such as bells or gongs.  But this notion would be erroneous in the sense that it is incomplete.  Let us look at the spectrum of another natural instrument, the male voice;

Male voice spectrum

Voice Spectrum Boxed

Looking at the plot of the spectrum of a male voice, we see the now familiar peaks in simple whole number ratios corresponding to the intervals of the octave, perfect fifth, and third etc., but a closer look at the higher regions of the series reveals something interesting: the yellow-boxed partials, if brought within the same octave as the lower ones (and adjusted for pitch) would represent something very close to a chromatic scale! The pink-boxed partials would represent the pitches of microtonal intervals.

So within the very sound of the human voice – if not the earliest, then definitely among the earliest of instruments which broadly, have similar properties, we have the foundations of scales and of functional harmony, but we also have the basis – in terms of intervals – of twelve tone, and even microtonal, melodies and harmonies.

The only distinguishing factor is accessibility or perceptibility. The lower regions of the harmonic series are clearly audible and therefore easily accessible, consciously or in certain cases, unconsciously. The partials that create inharmonic relationships are found in harmonic sustaining instruments too, but in the higher reaches of the series and are, by virtue of their decreasing intensities, far less accessible. Nevertheless, the unconscious processes that create correspondence and unity between the elements of music based on ratios in the lower reaches of the series can, and do, do likewise for those in the upper.

In the case of inharmonic instruments such as the tam-tam, the partials that create such ‘atonal’ relationships are ‘brought down’ into the human auditory range – the same range as those that give rise to the structures of tonal music – and at a similar intensity. This is suggested as one explanation for the increasing use of inharmonic timbres during the mid-late 20th and early 21st centuries.

The use of inharmonic timbres is emphatically not simply about exoticism in tone colour. It is about the search for unity among the various musical parameters that serious composers have always instinctively undertaken.  The harmonic series is the source of such a unity, but now the frequency ratios that create inharmonic timbre provide the key to the integration of the musical elements of today and tomorrow.  It is noteworthy that the music of Pierre Boulez (b.1925) has demonstrated an increasing deployment of inharmonic timbres. We are told (not least by Boulez himself) that in Le Marteau Sans Maitre, there was a desire to expand the instrumental palette, by including instruments that were reminiscent of those of other cultures. But there is more to this than the simple expansion of the instrumental palette; it seems that in this work, there is a realisation of the relationship between atonal pitch structures and inharmonic timbres. Notable in this regard are the vibraphone which is a metallic instrument although fairly pure in timbre; the xylorimba which has inharmonic components to its spectrum, and the unpitched percussion and gongs which are wholly inharmonic in timbre. The work’s final gesture is a semitone trill on the alto flute accompanied only by the resonance of a suspended cymbal. This is prophetic. Since then, Boulez’s music has deployed inharmonic timbres increasingly prominently; Rituel, Repons and Sur Incise are notable examples as is the earlier Éclat which, in highlighting the resonance of instruments such as the piano and tubular bells, together with piano chords so staccato as to take on the aspect of ‘synthesised’, percussive timbres, generates an unmistakeably inharmonic soundscape[9].

Many younger contemporary composers have realised the richness of inharmonic timbre as a source of future musical development. Recent computer-assisted creation and manipulation of such timbres has resulted from research conducted at IRCAM[10] and other institutes such as Stanford University where John Chowning (b.1934) pioneered the technique of Frequency Modulation synthesis. Sound sources have been put ‘under the microscope’ so to speak and the term ‘spectral music’ has been coined to denote music conceived and structured in terms of the acoustical properties of such sound sources.

However, the contention here is that all music is – or should be – spectral music.  From this point of view, there is no essential difference between tonal and atonal music: they are merely founded upon structures from opposing regions of the continuum which is the harmonic series. Those who promulgate such a spurious distinction in order to justify the continued creation of works based on functional harmony and to exclude ‘atonal’ music on the premise that its structures are not implicit in the harmonic series, are therefore in error.

Now, the history of harmony’s ascent though the harmonic series has reached the higher echelons where the basis not only of tritones and minor seconds, but also of microtonal intervals are to be found. The following illustration summarises this.  The future may include the entire gamut of intervals and related harmonies, but if it is to do so, any suggestion of a return to functional harmony or anything that could be misconstrued as such, would need to be avoided.  A new language, stylistically and aesthetically unambiguous, and of its time, would need to be established.

Harmonics Timeline

Leonard Bernstein, in his famous series of lectures given at Harvard University in the 1970s, The Unanswered Question, claims that man has some kind of innate predisposition for tonal music. There are those who suggest that this is analogous to the human propensity for language acquisition as first propounded by Chomsky et al and thereby justify the abjuration of music not based upon functional harmony. Rather than invoke Chomsky, we should, perhaps, ponder Pavlov! For it is the conditioning of the last 800 years or so, together with the more recent ease and speed of communication, commercialism and the widespread relegation of music to the status of mere entertainment, that has created this illusion. Once there is sufficient familiarity to suspend conditioned responses to melodies, harmonic progressions, cadences and dissonances that tend toward resolution, then the sound worlds of contemporary music pose no challenge in terms of accessibility. The thing to do is to forget the familiar and focus on what is present; to suspend preconceptions and listen.


[1] Not to be confused with the accepted frequency ratios of Pythagorean tuning. The ratios presented are chosen to demonstrate only that ratios forming the intervals deployed in scales are present in the audible range of the harmonic series.

[2] The notated interval of the major 2nd occurs between the 7th and 8th partials of the overtone series. However, the 7th degree is naturally flat when compared to the intervals created in a variety if tuning systems including the Pythagorean. For this reason, it is not counted as the first naturally occurring instance of this interval.

[3] When two or more tones are sounded simultaneously, additional tones which are the sums of and the differences between their frequencies are produced.

[4] In the case of a single tone or unison, the 5th overtone of the dominant tone forming the interval of a compound major 7th with the tonic: in the case of a chord, a sounding tone forming a minor 2nd below, or major 7th above, the tonic. Therefore dissonance can also be said to operate in a sense ‘virtually’ that is to say by way of memory, irrespective of whether the tones involved are sounded as fundamentals or not.

[5] It is acknowledged that the cycle created by repeating the ratio of 1:1.5 is actually ‘off target’ in the sense that, as modulation became a common feature of music there was a need to adjust the ratio slightly creating equal temperament. To go into that here would be to stray off the point.

[6] ‘The Devil in Music’

[7] Jazz musicians refer to the dim. 7th as the ‘Clapham Junction’ of chords, creating a metaphor for the myriad branching railway lines to be found at the synonymous railway junction in London, UK.

[8] The term modulation is used to mean a short-term change of key as in a simple melody or simple form such as minuet & trio. The term key change is used by contrast to denote the wholesale replacement of one key centre with another as in the classical sonata

[9] There are many similar examples in the music of Boulez’s mentor, Olivier Messiaen. For example, successions of parallel chords in several of his works for piano, take on the aspect of timbre rather than, or at least as well as, harmony.

[10] Institut de Recherche et Cooordination Acoustique Musique, Centre George Pompidou, Paris

3 thoughts on “Pythagoras and the Music of the Future Part II – Melody & Harmony

  1. Nonlinearity is needed in order for sum and difference tones to appear. If the oscillating system is linear, or passed through linear amplification, there will be no real sum and difference tones. However, when talking about nonlinear standing waves or nonlinear amplification, sum and difference tones will form.

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    • The sum and difference tones I refer to are those that occur naturally when two tones are sounded simultaneously. This happens without any kind of amplification other than that provided by the bodies of instruments, nasal and chest cavities of singers etc. It is well known that string players play two adjacent strings, a fifth apart, when tuning. When the octave below the note produced by the lower string appears, the two strings are in tune: f-1.5f = .5f. A related process, this time using sum tones, is outlined in my short article Tuning Tips…

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  2. Interesting observations that become even more compelling when considering the changes in temperament over the last 200 years and the resultant affect on harmony and the emotional impact of music.

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