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 (12^th), the main responsible for harmonic and tonal tensions. Nevertheless, curiously enough, this is in contrast

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. It shows that, in a perceptive way, harmonics may be reduced to those that are in prime position (that is to say, they are not harmonics of a prime harmonic ); more precisely, it studies the effect of the second, the third, the fifth and the seventh harmonic when establishing music laws (for they are the sole harmonics that have a "functional" influence). It develops the "harmonic" meaning of the major and minor scales, and their triads.

In the third chapter we carry out a study and functional classification of the chords according to their internal tensions, that is to say, according to the fundamentals that represent the "quasi-fifth" intervals. These fundamentals tend to resolve mainly (and locally) to another fundamental a lower fifth (or upper fourth),

or to another fundamental a lower minor second (or upper major seventh).

The fourth chapter analyses the secondary relaxions between chords and other successions of fundamentals, and presents a summary of homotonic relaxions.

The fifth chapter investigates the tonality and harmonic processes that originate a hierarchy among the notes (the tonal field). It analyses the cadences and modulations, as well as the tonal functions of the chords in line with the fundamentals representing them, which have been deduced in the previous chapters and that will provide us with much more information about the tonal vectors that are formed in a score.

In the sixth chapter we present chord progression examples with homotonic relaxions in weak tonal fields —that is to say, when there is no clear tonic, or when the latter changes constantly—, which happens when these local relaxions become more important, for, otherwise, the tonal tensions are predominant over the homotonic ones.

In the seventh chapter we may see some examples of homotonic and tonal analyses of fragments of musical works by several composers.

I have added four annexes that I already included in the work from 1994. The first one, which I have updated with new chord symbols for the most complex chords, carries out a functional analysis of every chord class that may be formed with the twelve-tone equal tempered tuning (that coincide with the "scale classes"). We prove that, altogether, there are 351 classes , and we select a representative for each one of them. In the second annex, we study and functionally classify all the modes that can be shaped with the eight main seven-note scales. In the third annex we organise the cyclic chords and cyclic modes , and in the fourth, the symmetric chords and symmetric modes .

Regarding the nomenclature that appears in the book, I often make use of the symbols that are used in English-speaking countries to specify the interval class:

m2: minor second / major seventh (also M7)

M2: major second / minor seventh (also m7)

m3: minor third / major sixth (also M6)

M3: major third / minor sixth (also m6)

P5: perfect fifth / perfect fourth (also P4)

Tritone: augmented fourth / diminished fifth

That is to say, if I use M2, I am both referring to the major second interval and to the minor seventh one (ex. C:D or D:C forms a M2 interval class).

When the chords appear in the middle of the text, the notes are written one after

the other, with their eventual accidentals; for instance, CEG{B (the accidental always refers to the previous note, not to the next one, that is to say, { applies to G, not to B).

Chapter 1