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 the 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 also has a clear correlation with, and influence over, the macrostructures of pitch and rhythm.  Here, we explore the relationship between the frequency ratios present in the harmonic series and musical intervals and trace how the evolution of melody and harmony, over the course of musical history, is similarly related.

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. Both, in turn, are based on pitch systems such as scales, modes, ragas and so on, all of which comprise intervallic relationships similar to those that occur between partials of the harmonic series.

Harmonic Series

In the above figure, the harmonic series notated as musical pitches demonstrates how successive pairs of partials form familiar intervals. The frequency ratios that form these intervals are outlined below.[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 can also be regarded as 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 ratios of the remaining intervals are;

5:8          Minor 6th

8:15        Major 7th

Were the above notated harmonic series to be played, the relationships between the frequency ratios of the harmonic series and musical intervals would be easily discernible. It would also be evident that the interval of the octave (1:2) produces tones that sound equivalent i.e. we perceive the upper tone to be the same note an octave higher. We even identify it as such by assigning it the same letter name. This equivalence is also generally accepted as the reason why this particular interval is universally divided into a number of smaller ones, such as the semitone (minor 2nd) in the case of traditional western music. Various combinations of the intervals of the semitone, tone and, in certain cases, major and minor thirds, 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 in non-western cultures, we find similar intervals which, of course, also exhibit similar relationships to the ratios of the harmonic series.

At least since the time of Pythagoras, musical intervals have been endowed with quite specific affective qualities. There is universal awareness that melodies can provoke strong emotional responses and this is partly attributable to the qualities that we ascribe to 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. simultaneously sounded) intervals also have more or less consistent affective qualities ascribed to them and, as in the case of melody, their effect is conditioned or coloured by context, with dynamics and tone colour playing important roles. It can be concluded, therefore, that there exists a firm connection between the frequency ratios that give rise to musical intervals, presented melodically or harmonically, and the corresponding feeling responses.

It has long been noted that simpler frequency ratios produce intervals that are perceived as stable, unlike those produced by more complex ratios. This was proposed, notably by the nineteenth-century physicist Herman von Helmholtz (1821 – 1894), as the acoustical basis of the traditional musical concepts of consonance and dissonance, in which the simpler ratios created consonant intervals and vice versa.

Consonant and dissonant intervals, respectively considered stable or unstable, seem to trigger analogous emotional responses in listeners. Of course, a great deal of subjectivity will, inevitably, be involved in describing the responses produced by certain intervals, but the following table outlines some of the more widely acknowledged ones.  Unison here means repetition (when presented melodically), or juxtaposition (when presented harmonically), of the same pitch;

Intervals Table

Exactly which intervals are considered consonant or dissonant has, in fact, been subject to change over time.  In medieval monastic music, all but the intervals formed by the simplest ratios were regarded as consonant. Originally, in the 8th and 9th centuries, plainsong, Gregorian chant, was sung in unison but by the 10th century, organum, which consisted of an original chant plus a parallel voice at the octave or fifth above, had begun to develop. Therefore, the intervals deployed during this period corresponded to the first three partials of the harmonic series, with the frequency ratios 1:1, 1:2, and 2:3. It is unlikely that these intervals were chosen consciously. Rather, their adoption was probably a result of subliminal perception, assisted by several related acoustical phenomena: firstly, and although this may seem obvious, monastic music was performed by adult males. The pitch of their notes would have been low enough for the upper partials to be well within the audible frequency range. The lowest and most intense partials i.e. those representing the octave and the fifth would, therefore, have been heard clearly during unison singing. Furthermore, sum tones would have been produced, creating distinctly discernible intervals above the principal melody.[3] Two notes in unison produce a sum tone at the octave (f 1 + f 1f 2), in effect reinforcing the first overtone of the harmonic series the singers’ voices were producing.  The third partial, forming the interval of a compound perfect fifth, would have been similarly reinforced when the chants were sung in parallel octaves (f 1 + f 2f 3). It would not have been unnatural for this tone eventually to be lowered by an octave bringing it within the vocal range of the singers. The fifth overtone, which forms a compound major third with the fundamental, would also have been clearly audible. However, whilst it could indeed be heard, and in a sense available, it was avoided for some considerable time because it was, in fact, considered dissonant. It was regarded as having a softening or destabilising effect on the intervals of the octave and the fifth.

During the Renaissance, however, the interval of the major 3rd was fully established as a consonance and the triad, a structure containing intervals that correspond to the fundamental, plus the first five overtones of the harmonic series (which means that the minor third also was now included), was established as a basic building block of harmony. Triads could be based on all degrees of the scale, but due to the intervallic properties of those scales, triadic progressions exhibited an inclination to ‘pull’ in certain directions and by the 17th century, the hierarchical system known as functional harmony had developed.

The chord on the first degree of the scale, the tonic became in effect, the harmonic ‘home-base’ due to its strong feeling of stability and finality. The triad on the fifth degree of the scale, the dominant, whose root and third (the leading note of the underlying scale) exhibited marked tendencies to resolve and so came to be considered as the chord with the strongest ‘gravitational pull’ towards the tonic.[4] It is especially noteworthy that its constituent pitches are at a ratio of 2:3 from those of the tonic chord.  After the dominant, the chord in the hierarchy with the next most powerful attraction is that built on the fourth degree of the scale, the subdominant. Although the constituent frequencies are at the same ratio to the tonic inverted (3:2), creating a descending fifth, it can also be seen that the intervals forming the tonic to subdominant relationship are to be found between the third and fourth partials of the harmonic series. Whereas the dominant, with its powerful pull towards the tonic, features pitches with frequencies in the ratio 2:3 to those of the tonic, the subdominant, which exerts a less powerful attraction, embodies frequencies in the ratio 3:4. The principal relationships within the hierarchical structures of functional harmony thus demonstrate close correlations 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 too was due to an awareness of the most prominent ratios of the harmonic series since, a modulation to the dominant, essentially involves the frequencies of all of the pitches of the home scale being multiplied in the ratio 2:3 (or by 1.5). Octave displacements of the resulting pitches would exhibit the equivalence discussed above.  The simplest means of modulating to more distant keys was to repeat the process of moving from one key to another where the original frequencies were multiplied by 1.5 i.e. via the cycle of fifths.  It is well-known that a complete cycle of 5ths encompasses all the degrees of the chromatic scale so, by using this as a modulating device, it is possible to progress to more and more distant keys or, indeed, to come ‘full circle’.[5] This is, in its simpler manifestations, a prominent and immediately recognisable feature of much Baroque music.

It is implicit in the foregoing, therefore, that from the Middle Ages, a clear feature of harmonic development has been the gradual acceptance of intervals representing higher, increasingly irrational, ratios of the harmonic series; starting with the octave, then the fifth, then the third, leading to intervals which would, in earlier times, have been unthinkable. For example, the dominant 7th and diminished 7th chords, both commonly used in the Baroque era and beyond, contain the intervals of the minor 6th (5:8), the minor 7th (9:16), and tritone (7:10) – the Diabolus in Musica as it was known in relation to medieval monastic music.  However, in the context of functional harmony, these once unimaginably dissonant intervals assumed important, enriching roles: for example, the minor 7th and diminished 5th in the dominant 7th chord intensify the need for resolution, with the ascending tendency of the leading note being mirrored by the descending tendency of the minor 7th towards the 3rd of the tonic chord. Likewise, the diminished 7th chord which features two tritones, possesses the principal characteristic of ambiguity and so creates a high degree of flexibility as regards choices of harmonic progressions and modulations.[6] Although by the end of the Baroque era, such dissonances were commonplace, their functions did not change for centuries. Whilst increasing numbers of dissonances providing routes to other keys, or which enhanced the feeling of resolution, were accepted, there could be no question of their being allowed in any context other than that of functional harmony until the twentieth century.

During the Classical Period, sonata form became the standard model.  In the sonata, the symphony, the string quartet, and 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 merely visited briefly but is 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 were based on the tonic/dominant relationship and hence the frequency ratio of 2:3. The table below illustrates (admittedly in very simplified 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

Sonata form became a medium by means of which composers could demonstrate degrees of ingenuity in leading the listener away from the tonic key into less familiar territory, surprise them with harmonic ‘twists and turns’ but eventually give them satisfaction when all was resolved and the home key firmly re-established.  Gradually, however, composers became more aware of the potential of sonata form as a vehicle for extra-musical expression. In the hands of Beethoven (1770 -1827), the greatly extended development section, featuring modulations to more remote keys and deploying ever more daring, dissonant harmonies along the way, revealed sonata form as possessing extraordinary expressive, even dramatic potential. The harmonic conflict created during the development, followed by the eventual return to the tonic, became 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, ultimately, for functional harmony itself. The artists who shaped the 19th Century, the Romantics, are widely regarded as having been egotistical, introspective and somewhat preoccupied with discontent. Issues such as unrequited love, political tension, or the burdens of existence became the subjects of expression in all the Arts, and not least in the medium of music.  Composers harnessed, and built upon, Beethoven’s legacy of expressing dramatic tension, and eventual resolution by developing, and pushing the boundaries of functional harmony further. The tensions created between the tonic and distant keys were reflected in a greater degree of chromaticism in melodic writing and more dissonance in the harmony. But while dissonance was, initially at least, still to be resolved, in the late 19th and early 20th centuries, again for expressive purposes, the presence of clear key centres on which to resolve became more difficult, and at times impossible, to discern. Not only this, but in many cases, such tonal centres were avoided from the outset. Well-known examples include A Faust Symphony of Franz Liszt (1811 – 1886), composed in 1854, and the opening of Tristan and Isolde, the opera by Richard Wagner (1813 – 1883) written in 1859 and first produced in 1865. The example below illustrates how the harmony in the opening of the former, consisting only of broken augmented triads, defies identification of any tonal centre.

Faust Symphony

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 Claude Debussy (1862 – 1918) and Arnold Schoenberg (1874 – 1951),  not only chose deliberately ambiguous harmonies and deliberately left dissonances unresolved but, in the case of Schoenberg, eventually sought to avoid any kind of reference to a key centre at any time. This is not the context in which to discuss in any detail the workings of Debussy’s harmony or of Schoenberg’s twelve-tone system but, as is discussed below, the suggestion, notably by illustrious figures such as Leonard Bernstein (1918 – 1990), that music of ambiguous tonality and music without tonality are divorced from the structures inherent in the harmonic series is erroneous.

Justification of such a claim requires revisiting the harmonic series as represented in the spectra of certain musical instruments. Firstly, considering 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

The C major chord that results illustrates very clearly the relationship between simple harmonies and the sounds of the instruments they are played on.  Secondly, a re-examination of the spectrum of the tam-tam, featured in the previous article’s discussion of inharmonic timbres lead to a very different kind of chord;

Tam Tam Spectrum

Spectr Tam tam

Chord resulting from notating the peaks;

Notated Spectr Tamtam

The simple melodic passage in the following example uses only the pitches of this chord including some octave displacements;

atonal melody

In the accompanying audio example, first the chord is played then the melody. The similarity of this atonal melody to those found in the work of many earlier twentieth-century composers is immediately obvious.

This might lead to the conclusion that while the structures of tonal music are closely related to the frequency ratios present 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.  This would be true but slightly erroneous in the sense that it was incomplete.  Once again, the spectrum of another natural instrument, the male human voice;

Male voice spectrum

Voice Spectrum Boxed

It is noticeable that the spectrum displays the familiar peaks, in simple whole-number ratios, discussed above, that correspond to the intervals of the octave, perfect fifth, third and so on. However, closer inspection of the highest frequencies reveals that if those outlined by the yellow box were to be brought within the same octave as the lowest ones and re-ordered, the result would bear a remarkable resemblance to a chromatic scale. The pitches of the pink-boxed partials would create microtonal intervals.  Therefore, within the very sound of the human voice – if not the earliest, then definitely among the earliest of instruments which, have broadly similar properties, we find not only the foundations of scales and of functional harmony, but also the basis – in terms of intervals – of twelve-tone, and even microtonal, melodies and harmonies.

It could be concluded, as suggested previously, that the development of melody and harmony, from medieval times to the present, amounts to nothing less than an ascent through the ratios of the harmonic series. However, a major difficulty with this argument would be that of perceptibility. As has been discussed, the lower regions of the harmonic series generated by voices and harmonic sustaining instruments are clearly audible and able, via conscious or unconscious perception, to exert an influence. The partials creating irrational ratios, forming the bases of intervals considered dissonant, by contrast, occupy the higher reaches of the series and so are, by virtue of their diminished intensities, far less accessible. Nevertheless, it is suggested that the unconscious perceptual processes that create correspondences between, and unity of, the elements of music based on the lower, simpler, ratios of the series do likewise for the higher, irrational ones.

Hand in hand with the inclusion of intervals representing more complex ratios of the harmonic series has been the deployment, particularly by composers of the mid-late 20th and early 21st centuries, of an increasing variety of percussion instruments which are, on the whole, inharmonic in timbre. Such use of inharmonic timbres cannot simply be attributed to a desire for novelty or exoticism in tone colour. It is more to do with the search for unity among the various musical parameters that serious composers have always undertaken intuitively.  While the harmonic series has always been the source of such unity, now, the frequency ratios that create inharmonic timbres provide the key to the integration of the musical elements today and, possibly, tomorrow.  It is noteworthy that the work of Pierre Boulez (1925 – 2016) afforded increasing prominence to inharmonic timbres. We are told (not least by Boulez himself) that in Le Marteau Sans Maître (1952 – 1955), there was a desire to expand the instrumental palette by including instruments reminiscent of those of other, particularly Asian and African, cultures. However, it is emphasised that there is more to this than simply an enlargement of the range of instrumental colour. In fact, this work creates the strong impression that a realisation of the relationship between atonal pitch structures and inharmonic timbres has taken place. Articulating its post-serial pitch structures are the vibraphone, which is a metallic instrument although fairly pure in timbre; the xylorimba which has inharmonic components to its spectrum, as well as gongs of various sizes and unpitched percussion instruments all of which are wholly inharmonic in timbre. Since then, Boulez’s music has deployed inharmonic timbres increasingly prominently. Notable examples are Rituel in memoriam Bruno Maderna (1974 -1975) Repons (1981 -1984) which includes electronically generated inharmonic timbres, and Sur Incise (1996 – 1998). The earlier work, Éclat (1965) 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.[7]

Of course, countless other composers have created such sonic environments and many younger contemporary composers have realised the richness of inharmonic timbre as a source of musical development. Relatively recent computer-assisted creation and manipulation of such timbres has resulted from research conducted at IRCAM and other institutes such as Stanford University where John Chowning (b.1934) pioneered the technique of Frequency Modulation synthesis which he used to extraordinary effect in his Stria (1977).[8] Here, inharmonic timbres are liberated from the uncontrollable ‘attack – decay’ envelopes of traditional instruments such as bells and gongs.

Elsewhere, notably at IRCAM, sound sources have repeatedly been put ‘under the microscope’ by composers such as Gérard Grisey (1946 – 1998) and Tristan Murail (b. 1947) and the term spectral music coined to denote music conceived and structured in terms of the acoustical properties of such sound sources. However, the contention here is that music has, essentially, always been – and should continue to be – spectral music.  Viewed from this perspective, there is no intrinsic difference between tonal and atonal music: they are merely founded upon frequency relationships from opposite regions of the continuum which is the harmonic series. Those composers and commentators who promulgate such a spurious distinction in order to justify the continued creation of works based on, or at least bearing a superficial resemblance to, functional harmony and to exclude atonal music on the premise that its structures are not implicit in the harmonic series, are therefore in error.

As alluded to earlier, Leonard Bernstein, in his famous series of lectures given at Harvard University in the 1970s, entitled The Unanswered Question, claimed that man possesses an innate predisposition for tonal music. This could be seen as analogous to the human propensity for language acquisition as first propounded by Chomsky et al and thereby used as justification for the abjuration of music not based upon functional harmony.[9] However, rather than invoke Chomsky, we should, perhaps, ponder Pavlov for it is the conditioning of the last 800 years or so, reinforced by the ease and speed of modern communication, by commercialism and the widespread relegation of music to the status of mere entertainment, that has created this illusion. Once the listener is able to suspend such conditioned expectations with regard to melody, harmonic progression and dissonance that must be resolved, then the sound worlds of contemporary music pose no challenge in terms of accessibility. Of course, the composer must avoid courting such conditioned expectations and ways of accomplishing this are proposed in the fourth and final article in this series, Synthesis.

Related articles


[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 of 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 inaccurate in the sense that, as modulation became a common feature of music there was a need to adjust the ratio slightly to create equal temperament.

[6] Jazz musicians sometimes 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.

[7] 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 harmony. See Quartet for the End of Time.

[8] Institut de Recherche et Coordination Acoustique Musique, Centre George Pompidou, Paris.

[9] Noam Chomsky (b.1928), American linguist and philosopher.

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