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The Foundations of Harmonic Tensions (The Fundamentals of Harmonic Tensions)

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The foundations of harmonic tensions (The fundamentals of harmonic tensions) Llorenç Balsach Provisional translation to english of the book Los fundamentos de las tensiones armónicas Primera edición. Octubre 2016 I.S.B.N: 978-84-89757-52-3 Dep. Leg: B-19535-2016 Coedición: Editorial Boileau y Edicions La mà de Guido Editorial Boileau Provença 287 - E-08037 Barcelona www.boileau-music.com Edicions La mà de guido Sant Pau, 54 - E-08201 Sabadell www.lamadeguido.com Per al Llorenç, el Roger, el Jan i la Sedes IndexIndexIndex Introduction 7 1. Theoretical framework 11 1.1 The intervals that create more harmonic tension are those of M3 and 11 tritone 13 1.2 The notes that form the M3 intervals (CE/EC) and/or tritone (EB¬/A{E) locally resolve to the notes F or B, or to the chords that have these fundamentals. 15 1.3 The fifth of the fundamental is the less important note as regards the chord's harmonic function. 16 1.4 The 7M3 structure produces a tonal vector. 18 1.5 The htonal and Phrygian resolutions of M3 and tritone account for most cadences, secondary dominants and cadential harmonic progressions. 1.6 A fundamental and its main harmonics also have a 7M3 structure. 22 1.7 Most chords may be separated into one or two chords having the harmonic CEGB¬ complete or partial structure. 24 1.7.1 We may summarise the harmonic tension of most chords with one or two fundamentals, which results in eight large families of chords. 25 1.7.2 We can apply the fundamental symbology of chords to create or find local relaxed progressions of chords. 1.8 The reduction of tonal functions to three (tonic, subdominant and 25 dominant) may be explained by means of the tensions of M3 and tritone intervals. 2. The harmonics of a sound as the basis of musical perception 29 2.1 Significant harmonics for the human auditory system 29 2.2 Virtual fundamental 35 2.3 Harmonic 2f 37 2.4 Harmonic 3f 39 2.5 Harmonic 5f 40 2.6 Formation of music scales 41 2.7 Minor chord and minor mode 53 2.8 Harmonic 7f 46 3. Functional study of chords 48 3.1 Classification of chords according to their harmonic tensions. The 48 fundamental symbology 3.1.1 Major chord family 50 3.1.2 Minor chord family 51 3.1.3 Dominant chords (unitonal) 52 3.1.4 Augmented chords 53 3.1.5 Symmetric dominant chords 54 3.1.6 Major-minor chords 55 3.1.7 Cluster chords 55 3.1.8 Suspended chords 56 3.1.9 Other chords 57 3.2 The functionality of chord families 57 3.3 Correspondence between the main known chords and the fundamental 59 symbology 3.4 Chord inversions and their optional symbology 61 4. Secondary relaxions and other successions of fundamentals 64 4.1 “Locrian” relaxion 64 4.2 “Dorian” relaxion 65 4.3 Successions of fundamentals without homotonic tension 66 4.4 Summary of homotonic tensions and relaxions 66 4.5 Homotonic relaxions and tonal axis theory 68 5. Tonality 71 5.1 The tonal field and its vectors 71 5.2 Tonality and the 7M3 structure 72 5.3 Tonal functions and their symbols (functional symbology) 73 5.4 Tonality and tonal axes 81 5.5 Cadences 83 5.6 Functional symbology in inversions 88 5.7 Modulation 89 5.8 Recapitulation 92 6. Examples of harmonic progressions with homotonic relaxions 93 6.1 Relaxions between chords with a single functional fundamental 93 6.1.1 Htonal relaxions following the circle of fifths 93 6.1.2 Phrygian relaxions following the 12 tones of the chromatic scale 95 6.1.3 Combinations of htonal and Phrygian relaxions 97 6.1.4 Using the full palette of chords 97 6.2 Relaxions with one or more functional fundamentals 99 6.2.1 Two functional fundamentals separated by a tritone (symmetrical 99 dominant chords) 6.2.2 Two functional fundamentals separated by a M2 (dominant 102 chords family) 6.2.3 Two functional fundamentals separated by a P5 (major chords 105 family) 6.2.4 Two functional fundamentals separated by a M3 (augmented 106 chords) 6.2.5 Other chords 109 6.3 Locrian homotonic relaxion 112 7. Examples of homotonic and tonal analyses 115 Annex 1 157 8. Morphogenesis of chords and scales 157 8.1 Equivalence level between chords 8.2 Equivalence level between scales 157 8.3 ‘Chords classes’ and ‘Scale classes’ tables 160 8.4 How to find the fundamentals or the fundamental symbology of the 163 chords 165 Chord/Scale class Tables 167 Annex 2 183 9. Modes of the first eight 7-note scale classes 183 Annex 3 187 10. Cyclical chord/scale classes 187 Annex 4 190 11. Symmetrical modes and chords 190 11.1 Symmetry between chords 190 11.2 Internal symmetries inside an octave 196 Bibliography 197 Introduction Why when we hear a dominant seventh chord, or simply a major third, or else a tritone, are there notes that resolve the chord’s tension or the interval’s one? Why do we also notice relaxion* when we hear the so-called Phrygian cadence? Why does a specific succession of notes establish a (or several) tonic, and hearing this tonic (or tonic chord) produces a relaxation irrespective of the previous chord? Which tension do the chords of the Neapolitan sixth and those of the augmented sixth establish? Why is a specific succession of chords fluent and released and another produces tension? In the following pages we will try to provide answers to these and other questions, always trying to look for the reasons accounting for musical tensions and relaxions . Furthermore, we will inquire into the nature of sound and our harmonic perception. We are aware of the fact that many of these questions could be answered in official theory by taking as a basis a scale and its associated concepts: degrees, tonal functions, cadences, etc.; nevertheless, there is usually no acoustic-harmonic explanation provided for the tensions and relaxions that are produced. In my book La convergència harmònica (Balsach, 1994) and in the article “Applications of virtual pitch theory in music analysis” (Balsach, 1997), I already introduced many of the concepts and explanations that appear in the following pages. However, this time I have tried to present the main ideas in a more ordered and structured way, besides including new contributions and suggestions, mainly in the field of tonality. This is the reason why the first chapter has been devoted to presenting in an abridged and ordered form the main and new ideas and conclusions of the book . In fact, if anyone wanted to get an idea of the book’s contents, they could achieve it by just reading the first chapter. I proceed on the basis —and this is not new— that the continuous perception of the harmonics phenomenon of a sound —from birth— is what shapes in the brain the sensations of harmonic tensions and relaxions when we listen to music. And, according to our theory, we will see that only the first seven harmonics play a role in it —at the most—, being the third, the interval of fifth (12th), the main responsible for harmonic and tonal tensions. Nevertheless, curiously enough, this is in contrast 8 Los fundamentos de las tensiones armónicas with the fact that, apart from the octave, the fifth of a fundamental is the least important note in a chord; and this is so precisely because it does not produce tension in it, as it happens with the octave (the second harmonic), which does not produce tension in it either. Conversely, the intervals that are closer to the fifth (a difference of a semitone) produce tension, such as the tritone intervals (augmented fourth/diminished fifth) and the minor sixth/major third ones. As we will see, this is a fundamental point in the theory, for the situation of these intervals in the chords will provide us with much information on the resolutive “preferences” of the chord to free this tension of the “quasi -fifth “ interval that the auditory system perceives as “something is wrong here” in the chord (as a kind of “dissonance”). We will see that the tension of these “quasi-fifth” intervals and the tendency of the notes to their lower fifth are also responsible for the tonality (we will see that a major third and a tritone combined in a specific way create the most powerful tonal vector that may appear in a score). The formations of the major and minor scales with leading-tone will be the result of such tensions, but not the cause. That is to say, we do not base ourselves on the degrees of a scale to explain the harmonic or tonal discourse. Regarding tonal functions, we could say that our theory is a neo-Riemannian theory that also agrees with some aspects of Ernö Lendvai’s tonal axes theory. Over the years, I have reached the conclusion that we may separately study three kinds of harmonic tensions: the purely local tensions or relaxions between two chords or arpeggios, irrespective of the tonal memory, which I call homotonic relaxions, the tonal tensions, and the chord tensions taking into account their dissonance or consonance (which I call sonance tensions). To these three harmonic tensions we should add the melodic tensions that, despite being intertwined with the harmonic tensions, have their own laws, among which the second’s descending movements producing relaxion, stand out. The second chapter analyses thoroughly the phenomenon of the harmonics and the virtual pitch (missing fundamental) theory.
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