Vectorial Harmony

Preliminary note
The research explored in this article is part of the work I am currently doing at doctoral level at the Guildhall School of Music and Drama, London, under Professor Julian Anderson’s supervision.

Vectorial harmony is an idea derived from a morphological conception of musical harmony. It is obtained by choosing an initial chord and one or more sets of intervals to be applied to each of its notes. The ordered set is called an interval vector and is composed of numbers representing the intervals along with a sign: ‘+’ or ‘-’, meaning movement up or down. The word ‘vector’ is used to denote directionality of that movement.
The vector represents a defined shape for voice leading and is the very element that contains a defined morphology, controlling the transformation of the initial chord. In this way, vectorial harmony produces chord progressions from superposition of notes that move according to those shapes. As a composer, I use it mainly because – as will be made clear – it provides a very ordered and regular chord progression (a harmonic pattern) that one can extend, and direct up or down (depending on the sum of the intervals contained in the vector). The recurrence contained in this regularity stimulates auditory memory, and so there is a link between this technique and music cognition.

Vectorial harmony in my musical works
My recent work Shapes (2011) provides a good example of a stable case of vectorial harmony. Right at the beginning of the piece there is a chord of major ninths. The interval vector v = (+2, -11, +1, +7) was chosen. The vector is first applied to the lowest note of the chord. After that, the first rotation of the vector is applied to the second lowest note; then the second rotation of the vector to the third lowest note, and so forth. Because rotation is involved in the application of the vector to the chord’s pitches, this harmony is called rotational vectorial harmony. Figure 1 illustrates the technique at work.

Figure 1 - Rotational vectorial harmony along with the morphological depiction of the interval vector

Figure 1 – Rotational vectorial harmony along with the morphological depiction of the interval vector.

What is obtained is a sequence of four chords that is transposed down a semitone with each reiteration (-1 is the constant sum of the intervals of each rotation of the vector: 2 – 11 +1 +7 = -1). The intervallic structure of the resulting chords is as follows:

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where the prefix ‘c’ stands for a compound interval.

This realization of vectorial harmony is internally diversified in terms of intervallic content although, importantly, one can find some prevalence of ninths and seconds. Nevertheless, factors of good consistency are the recurrent strict patterns of voice movement — each voice moves according to a strict intervallic sequence — and the recurring transposed harmonic sequence of four chords. This last factor is very important as it immediately connects this technique with music cognition: auditory memory is stimulated since the same chordal intervallic structures are reiterated every four chords. Resemblance relations come into play as we compare two transpositions of those intervallic structures. They are similar, but not equal.

Going back in time a little more, I can see that my first real attempts at using this type of harmony lie in the experiments I was making some years ago while composing Aquecimento Global (in English: Global Warming, 2007, homage to A. Vivaldi). The chords underlying the first movement (I. Largo) are:

Figure 2 - Vectorial harmony in the underlying harmony of I. <em>Largo</em> of <em>Aquecimento Global</em> (2007)

Figure 2 – Vectorial harmony in the underlying harmony of I. Largo of Aquecimento Global (2007).

Sample 1 – Excerpt from the first movement of Aquecimento Global (2007).

The harmonic progression corresponds to the longer sustained chords. The first chord is heard transposed one octave down and the last chord is not featured in this sample.
Starting with the lowest pitch F3 of the first chord in Figure 2, the transposition intervals -10 and +7 (in semitones) are applied successively: we get G2, and then D3 on the first application, always the lowest pitch of the chords. The second lowest pitch of the initial chord – G3 – goes through a transposition of -5 and +2 semitones, generating D3, and then E3. These two interval sets are then applied in alternation from the lowest to the highest pitch. This creates parallelism between every other pitch: first, third, and fifth, and also between second and fourth. The sums can be computed as follows:

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The fact that the sums of transposition intervals applied to the pitches are the same for each and every pitch immediately implies that the entire chord is transposed after their application. We can see this by comparing the first and third chord in sample 1, for example. Later, I started calling these sets of intervals vectors, and this type of harmonic composing vectorial harmony. And so, in the present case, the vectors would be denoted by v1 = (-10, +7) and v2 = (-5, +2).
With this practice I was trying to control, on the one hand, the movement of voices and, on the other hand, the consistency of the progression by creating a resemblance relation: a chord recurs transposed after applying the vectors to it. These relationships are keys in stimulating auditory memory, contributing to coherent musical discourse. This concern has remained a major one in my music since then.
Coming back to the analysis, one could argue that the second chord is very different from the first, main differences being that there are no common notes, the intervals between adjacent notes are different (2-5-14-14 in semitones for the first and 7-0-19-9 for the second) and, also, the second chord has a more resonant bass part given by the perfect fifths and unison. The reason is that, in vectorial harmony, the movement of voices is more important than the intervallic consistency amongst chords. That is why the word vectorial is well suited. Nevertheless, and although one can try to find combinations that increase consistency (and they exist!), the recurrence of the chords involved in the application of the vector is always guaranteed through transpositions, as mentioned before.
To complete the analysis of sample 1 (Figure 2), one can note that from the third chord to the fourth, an internal rotation of intervals is applied. After that, the retrograde versions of the interval vectors are used. Internal rotation is applied again from chord 6 to 7. Finally, the last step of the progression involves a transposition of the resulting chord one octave up. These procedures show how vectorial harmony can be combined with other chord manipulations and itself subject to variation while being applied. They open this system to free interpretation.
Another instance of vectorial harmony is contained in the third movement of the same piece:

Figure 3 - Underlying harmony for III. Allegro of Aquecimento Global

Figure 3 – Underlying harmony for III. Allegro of Aquecimento Global.

Vectorial harmony is again alternated with internal rotation of chord intervals. The first couple of vectors feature the prime and retrograde form of the same vector, and so the designation retrograding vectorial harmony is used. The second couple of vectors used are different from each other, but their internal sums are still equal:

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Two final instances of the technique can be found in a later piece called Vectorial-modular, for orchestra (2011, first prize in the Póvoa de Varzim International Composition Competition, published by AvA Musical Editions). It constituted a very important step in my progress as a composer as I managed to use a good number of ‘formalized’ techniques to a satisfying musical result.
Vectorial harmony is used throughout the piece (but not exclusively). Let’s have a look at a first instance where it is combined with spectral enrichment. The harmonic progression was built by rotational vectorial harmony with vector v = (-14, +3, +3, +1) applied three times. Spectral enrichment with partials 9 to 11 was used from chord 6 onwards.

Figure 4 - Harmonic progression underlying the excerpt featured in Sample 2. Only sharps (including quarter-tone sharps) are used, lasting one chord and applying only to the notes on which they appear (IRCAM's OpenMusic accidental notation)

Figure 4 – Harmonic progression underlying the excerpt featured in Sample 2. Only sharps (including quarter-tone sharps) are used, lasting one chord and applying only to the notes on which they appear (IRCAM’s OpenMusic accidental notation).

Sample 2 – First excerpt from the piece Vectorial-modular containing the progression shown in Figure 4. The harmonic progression is present on the winds at first and then spreads to the whole orchestra towards the climax.

The score realization (as shown by the performance) shows yet another instance of liberty in the implementation of the technique: some chord-trills and ‘back-and-forth’ reading (that is, coming back to previously used chords) was used. Also, instrumental constraints had to be considered: some of the higher partials did not make their way into the final score.
In Figure 4 one can note a fair number of common notes between adjacent chords, contributing to harmonic linkage and coherence between them. This exemplifies how this technique can be ‘configured’ to provide very different types of progressions: number of common notes, pitch contour, intervallic consistency, etc.
Later in the piece, retrograding vectorial harmony is used in alternation with internal intervallic rotation of chord’s intervals. The vector va = (+3, -1, +6) is used two times, then the last chord is internally rotated followed by one application of vector vb = (+1, -6, +3), followed by rotation again, and finally vector vc = (+14, -6, -3) is applied one time:

Figure 5 - Harmonic progression based on retrogradational vectorial harmony underlying the music (IRCAM's OpenMusic accidental notation)

Figure 5 – Harmonic progression based on retrogradational vectorial harmony underlying the music (IRCAM’s OpenMusic accidental notation).

Sample 3 – Second excerpt from the piece Vectorial-modular containing the progression shown in Figure 5.

Figure 6 - Score implementation of the harmony shown in Figure 5

Figure 6 – Score implementation of the harmony shown in Figure 5.

Relation with other techniques
A brief comparative analysis of my vectorial harmony technique and other pitch manipulations should provide a sense of its conceptual roots, context and novelty. Let’s consider two techniques: one by Boulez and one by Berio, two influences of my musical thinking.

In Dérive I (1984), Boulez obtains chords by transforming an initial pitch sequence (the Sacher sequence). The intervals are rotated and, from each rotation, pitch sets are obtained by starting always with the pitch E flat. These pitch sets are unordered and used to build chords by freely assigning their octaves. Therefore, the main result of the systematic procedure is to obtain pitch-class collections, irrespective of their locations in the register.

Figure 7 - The construction of harmony in Boulez's Derive I (1984). Only the first three chords are shown

Figure 7 – The construction of harmony in Boulez’s Derive I (1984). Only the first three chords are shown.

Whereas Boulez freely adjusts the octave of a given pitch that resulted from calculation, in vectorial harmony the expression of the vector’s shape is incompatible with octave equivalence. Intervals (defined by their magnitude and direction) are paramount to maintaining a characteristic (or pure) sonority of the process and its chords. As we know, and as Boulez was certainly aware, the spacing of a chord’s pitches affects its sonority (and that is why Boulez, in the end, carefully spaces the pitch collections to build the sonorities he wants, just like he did in Le Marteau sans Maître).

And so, in spite of the interval rotation procedure shared by the two techniques, the underlying conception is different. The absence of octave equivalence stems from acoustic considerations: as a given sonority is affected by the location of any of its constituent pitches in the register, so are manipulations. That is, one can speak of a characteristic sonority of a manipulation/transformation — much like a studio processing technique — in analogy with a characteristic sonority of a given chord (especially the ones derived from sound spectra). Concluding, the main result of vectorial harmony is a characteristic progression of sonorities — the aftermath of applying interval vectors to an initial chord.

Let us now turn to Berio’s use of pitch cycles. In O King (1967), a twenty-one-unit fixed-register pitch cycle is repeated throughout the piece. It contains only 7 different pitches, which occur 3 times each: 21/7=3:

Figure 8 - The 21-unit pitch cycle used by Berio in O King (1967)

Figure 8 – The 21-unit pitch cycle used by Berio in O King (1967)

It is interesting to note the use of recurrence to create consistency in the musical discourse. There are clearly three degrees of recurrence: the recurring individual pitches, the recurring melodic intervals between successive pitches, and, finally, the recurring pitch cycle. A more subtle degree can, perhaps, be found in the way the seven different pitches are obtained from the initial four (F, A, B, C#): the last three are shifted a semitone (and not by any larger interval). One could consider the altered pitches a distorted recurrence of the initial ones. Recurrence, use of fixed register and cycles are three aspects Berio’s technique shares with vectorial harmony. In the latter, the recurrence of chords after transposition and the recurrence of voice movement shapes are the key and have been discussed above.

Regarding the cycles themselves, one must be aware of a fundamental difference in purpose: Berio’s technique is predominantly melodic/horizontal, whereas vectorial harmony is predominantly harmonic/vertical. In spite of this, both techniques use cycles, the difference being that in vectorial harmony what is cycled is an interval set (the vector), instead of a pitch set. This interval set is strictly reiterated for each voice. Again, as the purpose is to create chord progressions, these interval cycles never become melodically autonomous (as are the pitch cycles in O King), but exist in tight and synchronized superposition with each other. Melodies can be constructed, but always by freely using pitches from each chord in turn.

A very important aspect of vectorial harmony worth stressing is that it is interval-based and not pitch-based. Instead of pitch cycles it creates chord cycles related by transposition: it creates cycles of intervallic vertical structures (sonorities). The focus on interval, and not pitch, has remained an important feature of my music.


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