1. Theoretical framework

1.1 The intervals that create more harmonic tension are those of M3 and tritone.

Chords (or arpeggios) tend to «resolve» locally to some specific notes and not to others, and the harmonic tension of the chord is determined by the intervallic structures formed by its notes. This tension has a local context , but also —keeping the previous musical tensions in our memory— it may also have a global sphere; in this case, to the strictly-speaking local tension of the chord, the «tonal tension» will be added, which will have more or less influence according to the grade in which the attraction towards the tonic has been established. However, these tensions are of a different kind, and thus we will analyse them individually.

The melody also has its harmonic and tonal tension, but this tension is shared with other variables of a more complex nature, and, sometimes, it is difficult to analyse it just in one theory. For instance, when the melodic line reaches a tonic or tonicized note, it produces a relaxion that may contrast with a harmonic succession of tension in the same place. The melodic logic is very important, but in this work we will not study other aspects apart from the purely harmonic ones. It is also important to make clear that when I refer to harmonic tension I am not referring to the greater or lesser consonance or dissonance of the chords (when I refer to it I will use the term "tension of sonance "). For instance, the dominant seventh chord GBDF is more consonant than the major seventh chord GBDF{, but the first chord produces more harmonic tension in the sense that it has a stronger need to be resolved than the second chord, which has more sonance tension (it is more dissonant), but is nevertheless more stable, tonally speaking. Many compositions end up with this dissonant chord. Therefore, despite the fact that they are interrelated, we will distinguish between these three kinds of tensions: tonal, local harmonic and "sonance" tensions.

In a chord, arpeggio or melody (regardless of the tonal memory), the intervals that produce more harmonic tensions are those of M3 and tritone (and their inversions). Furthermore, we will see that these two intervals are fundamental in shaping and explaining the local sensations of tension or relaxion between chords, and also, over a longer period of time, tonal functions. We find the explanation in the great strength the third harmonic has for the auditory system accompanied by its multiple harmonics (6, 9, 12, 15...), which form what is known as the fifth of the fundamental (for further details on the influence of harmonics, see chapter 2). The ear hears the intervals that are very close to the perfect fifth as "imperfect fifths", "impure fifths" or "out of tune fifths", the sensation being that of resolution when

these "false fifths" match.1

The interval classes that are closer to the perfect fifth (P5/P4) are the M3 intervals (major third or minor sixth ) and the tritone (augmented third or diminished fifth). In the equal tempered tuning they are even closer.

The resolutions of these "quasi-fifths" intervals to perfect fifths (increasing or diminishing a semitone) are those that appear in figures 1 (M3) and 2 (tritone).

Fig. 1

Fig. 2

Fig. 3

As we can see in figure 1 in chord form, if we take the notes C-E as a sample of M3, E tends to resolve the «dissonance» to F, or C to resolve to B. With these two resolutions, the «quasi-fifth» is «tuned» as a perfect fifth. Later on we will see the harmonic consequences of it.

The case of the tritone is more complex, for, as we see in figure 2, it may resolve the «false fifths» in four different ways.2

1 The same could also be said for the second harmonic, the octave (see 2.3). But in this case we would be referring to the fact of "being out of tune" of the same note (the octave ) that resolves to itself. The m2 interval is an interval that is close to "auditory masking " (see page 33), which does not happen with M3 and tritone intervals.

2 Be that as it may, we must say that when the tritone is accompanied by more notes in a chord, the auditory system quickly positions the correct enharmony of the tritone notes. For instance, if we hear B-D-F, the auditory system hears the virtual fundamental G with its M3 B and its m7 F, but if we hear B-A¬-F, the ear captures the virtual fundamental D¬ with its M3 F and its m7 C¬ (not B).

If, as an example of tritone we take the notes E-B¬, the notes that resolve the «false fifth» to a perfect fifth (P5/P4) are F, B, A and E¬.

To sum up, of the two eventual "quasi-fifths" or "false fifths",3 one, the M3 interval (EC/CE), produces a harmonic tension that is resolved by the notes F or B, and the interval of the tritone (EB¬/A{E) creates a harmonic tension that is also resolved by the notes F or B (in addition to A and E¬), in all cases to adjust the fifth. We should note that the other two resolutions of the tritone (A and E¬/D{) are the major thirds of the main resolutions F and B; therefore, the simultaneous resolutions of F and A are compatible, for they are part of the F major chord (or, in other words, A is the 5th harmonic of F —as a "resonance"—). Something similar happens with B and E¬ (D{), for, in the tempered tuning, E¬may be heard as D{, although in this case the joint resolution does not seem so natural.4

If we consider the two intervals together, we obtain the resolutions of figure 3. These resolutions are boosted in the tempered tuning.

I call the local resolutions of these intervals homotonic resolutions. As we have already mentioned above, and in order to make it more comprehensible, everything that is said in this chapter about some representative notes can obviously be applied to all their transpositions.

From all that has been previously said, we can now state that:

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.

That is to say:

- The tone or the fundamental F resolves the tension of M3 (CE/EC) and the tension of the tritone (EB¬/B¬E). We will call it resolution of 1st order or htonal resolution (simplification of homotonic-tonal).

- The tone or the fundamental B resolves the tension of M3 (CE/EC) and the tension of the tritone (EB¬/A{E). We will call it resolution of 2nd order or Phrygian resolution.

We should observe that these two resolutive notes F and B are located on opposite poles in the circle of fifths and forms a tritone interval.

3 Formerly, the interval of diminished fifth was precisely known by this name: false fifth.

4 Scholastically, perhaps it would be possible to put A{ instead of B¬, but I prefer to write B¬ if C is in the bass, for although B¬ and A{ are at the same distance from E in the circle of fifths, the 5th and 7th harmonic of C are E and B¬ (not E and A{). And this is so despite the fact that we know that the harmonic B¬ is lower than the tempered B¬. Similarly, B¬ is closer to C than A{ in the circle of fifths.

I insist again on the fact that I take (C-E) and (E-B¬) as representative notes, but, obviously, the wording refers to every interval of M3 and tritone, and in any inversion. Generally speaking, we could also say that the M3 interval resolves to any one of its two "external" adjoining notes at a distance of a semitone when the interval is in the form of a third, or to any one of its two "internal" adjoining notes at a distance of a semitone when it is in the form of a sixth, and something similar for the tritone; however, it is a more complex wording than just putting specific notes as an example.

We should note that, in the two intervals that have the same resolution, a note appears twice, E; therefore, this note is of fundamental importance in harmonic tensions (provided that it is accompanied by the other two notes of the two intervals: C and B¬). We know that this note is known, in music theory, as the leading-note note of the tonality (in this case, F).

These resolutions of the "quasi-fifths" or "false-fifths" do not explain per se alone the phenomenon of tonality (tendency of a group of notes to resolve to some specific notes —more than to a specific scale—). To comprehensively explain tonality we should add to it the effect that the phenomenon of the harmonics of a sound provokes in the harmonic relations established by the auditory system.

The harmonic 2 (the octave, with its resonant "harmonics" 4, 6, 8, 10, 12, 14, 16, etc.) makes it possible for the auditory system to identify a sound with its octave, because this sound and its octave have many harmonics in common (50%), and, therefore, due to the effect of the virtual fundamental (see 2.2), the auditory system considers the octave as the "same" note, but "with a different `timbre', a more acute one". This is the reason why this second harmonic is essential in establishing the laws of harmony, and it really simplifies its principles (what the brain does whenever it can in order to save a great deal of information and energy), since, by matching the notes and their octaves, it very much reduces the "number" of possible notes.

The next most important harmonic is the 3rd one (the fifth, with its "harmonics" 6, 9, 12, 15, etc.). The fact that the fifth has a third of the harmonics of the fundamental is not enough any longer to identify fundamental with harmonic, but it is the note (different from the 8th) that best fits the harmonics of the fundamental. The auditory system does not consider it "the same note" but is certainly "the more familiar" with it. To put it in a more colloquial form: if C is the fundamental, as if G were the "fruit" of C and were generated by it . Therefore, if we play G and then C, the auditory system seems to recognise C as generating G, for it perfectly fits its harmonics and therefore produces a "resolution" effect. Furthermore, as we will see, the main harmonics of a tone actually form a kind of dominant seventh chord, which also makes its tendency towards the lower fifth easier (see details in chapter 2).

Therefore, of the two resolutive tones F and B of the intervals CE and EB¬, one of them, F, has an added relaxion because it is the lower fifth of C. C is the fundamental of CE (E is the 5th harmonic of C), C is the virtual fundamental of EB¬ (E and B¬ are the 5th and the 7th harmonic of C, see 2.2), and C keeps on being the fundamental of CEB¬. Therefore, the F tone, which generates C, has another kind of "hierarchy" with respect to the other resolutive note B, and has a relaxion of a different kind to that of the "quasi-fifth" tuning. It is a resolution of a "tonal" type, and this is the reason why I have called this local resolution htonal resolution. I do not call it "tonal relaxion", for it could be confused with a succession towards the tonic. A local htonal homotonic relaxion may happen —in fact it happens habitually— even though the fundamental to which it resolves may not be the tonic. The tendency to F of the structure made up by the notes CEB¬ is the basis of the tonality.

These two attractive forces that we have studied above (that of the resolution of "quasi-fifths" and that of the recognition of a harmonic generator ) are the principles that explain and generate tonality, the tendencies of a group of notes to resolve at each temporal moment to (one or several) specific notes.

1.3 The fifth of the fundamental is the less important note as regards the chord's harmonic function.
As we have seen, the note that has more harmonics in common with the fundamental is its octave. They have so many in common that the auditory system, due to the phenomenon of the missing fundamental (or virtual fundamental), regards it as being the "same note"; therefore, for a chord to have or not have the octave does not alter its harmonic or tonal function.

The following note that has more harmonics in common is the fifth, and, similarly, it is so strongly united with the fundamental that the fact of being or not being in the chord does not seriously alter its function (unless it forms new intervals of M3 or of tritone with other notes of the chord).

The meaning of the fifth as a powerful and consonant interval is often confused with the power to provide functional support to a fundamental. The intervals of octave and fifth are the more consonant and closer to the fundamental ones, and are therefore those which are less functionally important when they appear inside the chord. In fact, the fifths of the fundamentals are so powerful that they are heard as harmonics even when they are not in the chord. As we have seen and will see later on, the tendency to "tune" the perfect fifth builds up the tensions and relaxions between chords, but once we have obtained the perfect fifth, the tension disappears.

1.4 The 7M3 structure produces a tonal vector.

What I call 7M3 structure refers to the intervallic structure formed by the notes C, E and B¬ in any transposition and inversion. The 7M3 structure is the sum of the tensions of M3 and tritone (tensions which "want" to resolve to the notes F and B). Seen from the fundamental C, we have the intervals M3 and m7 (that accounts for the abbreviation 7M3).

The notes that form the 7M3 structure (CEB¬ in F) are known in music theory as dominant (C), subdominant (B¬) and leading-tone (E) of the tonality (in this case of F), and, as we have already seen, they have only been deduced from the common resolution of the "false fifths". We have not made use of any scale. The presence of these notes or of these intervallic relations in a melody or in a chord produces a tonal vector towards the tonic F (a tendency to resolve to this note); that is to say, towards the lower fifth of its fundamental C (in addition to the homotonic local Phrygian tendency towards B). We should note than when I refer to subdominant or dominant, I am not referring to the chords of subdominant or dominant, but only to THREE notes. Obviously, the dominant chords (in the F tone) contain CE, and those of the subdominant contain B¬, but the chords of this last tonal function may have the major or minor 3rd (of B ¬).

The M3 structure (C-E) alone also creates a tonal vector towards F, and the same happens with the tritone (E-B¬). When the two intervals appear at the same time, the vector is strengthened. The tonal vector of M3 is more powerful (because it contains the fifth of the tonic F, the dominant). In the case of a "competition" between two or more tonal vectors, the M3 of the last chord that can be heard always has the preference (always taking into account the Phrygian resolution of the 7M3 structure).

Fig. 4

In a short musical fragment we may easily find two or more 7M3 structures; for instance, a musical fragment containing the notes B¬-C-D-E-F{ has the structures CEB¬, DF{C and F{A{E. We have three vectors towards F, G and B (see figure 4). The "winning" vector will depend on the position of the notes, on whether the auditory system hears B¬ as B¬ or A{, and on the last chord or arpeggio. Usually, the last M3 heard is determinant. In these specific examples of figure 4 we have also added the melodic Phrygian cadence in favour of B (E-D-C-B); therefore, most probably, if we were to write the five notes randomly, the more resolutive note would be B. If we put all the notes together as a chord, the more resolutive note is also B.5

There is another structure that also creates a (less powerful) tonal vector: the intervallic relation made up of two m3 (EGB¬) due to the phenomenon of the virtual pitch (see 2.2). The reason for it is that this structure makes the ear hear C as a chord's missing fundamental , and, therefore, we are actually hearing the (C)EGB¬ structure; that is to say, even though the C is virtual, we have incorporated the 7M3 structure (with the fifth of the fundamental added), C being however virtual. So, EGB¬ also creates a tonal vector towards F.

The well-known dominant seventh chord already creates alone a powerful tonal vector, for it has these three notes available that play a leading role in the same chord, being the fifth of the fundamental optional or even alterable.

In figure 5 we may find some examples of (atypical) cadences due to the tonal strength of the 7M3 structure, in this case transposed to GBF. A flat between parentheses means that we may play this natural note or flat without affecting the resolutive sense of the progression. Therefore, each measure of the examples has between 4 (2 parentheses) and 32 (4 parentheses) possible cadential combinations, and the double if we consider resolution in major or minor.

Fig. 5

5 We have seen that the 7M3 structure also resolves "in a Phrygian way" to B. F{A{E also resolves in the Phrygian way to E{, but the ear does not identify the Phrygian resolution to E{ with the tonal resolution to F of CEB¬ due to the other note D, which provides support to F{ and not to G¬ (the ear recognises F{A{E and not G¬B¬F¬). If the note D were not there, CEB¬ and F{A{E would be two symmetric chords at distance of tritone, and then F and B, resolutively speaking, would be "tied".

In many combinations of figure 5, new 7M3 structures appear that may make the cadential sense «vibrate». For instance, if we have D¬ and E¬, we then also have the 7M3 E¬GD¬ structure, and we obtain a tonal vector added towards A¬ (figure 5c bis). or, also, in the cases of considering notes A and E¬, we then have the 7M3 FAE¬ structure with tonal vector towards B¬, which is «refuted» when hearing afterwards the B} of the dominant chord. We would even find more 7M3 structures hidden in these examples.

But if in the group of cadential notes where a new 7M3 structure is created we find its associated tonic augmented, we will note that this makes the power of the tonal vector diminish towards this new tonic; for instance, B} in FAE¬ or A} in E¬GD¬.

Furthermore, also in the examples of figure 5, the altered notes of the very GBF structure (G{, B¬, F{) could be added in the first chord, and even some in the second chord, although this would increase the possibilities of finding other stronger 7M3 structures (closer tonalities), and thus diminish or fully eliminate the cadential effect.

The only scales that contain a single 7M3 structure are the major diatonic scales and the minor scale with leading-tone (harmonic minor) (see 5.2). Probably, this was the reason why when all the modes were progressively incorporating the tritone into close notes (most probably since Glaureanus' dodecachordon) their decline began, because the incorporation of the 7M3 structure into the modes (in scarcely distant environments) entailed their amalgamation in the two aforementioned scales.

1.5 The htonal and Phrygian resolutions of M3 and tritone account for most cadences, secondary dominants and cadential harmonic progressions.

Besides the examples of (non-usual) cadences that appear in figure 5, many of the typical cadences and harmonic progressions that appear in the treatises on harmony also provide an explanation based on the resolutions of the tensions we have seen above. In figure 6 we may observe the most common cadential processes. We have used the notation (symbology) of chords and tonal functions that will be employed throughout the book and that will be explained in the following sections: 3.2-4 and 5.3. For the tonal functions of these chords, see chapter 5 devoted to tonality.
The authentic cadence is the htonal resolution of M3 (CE|F), with an increased effect when the fundamental has m7, because it also resolves the tension of the tritone that is formed with E (EB¬|F) (figure 6a).

The plagal cadence in the major mode does not take place due to the local resolution of a tension that creates a chord but (when a tonality is well-established) to the resolutive sense of resting on the tonic (chord), in this case from the

subdominant . However, the Locrian secondary homotonic relaxion (A¬-C) (see 4.1) does happen in the minor mode. Plagal cadence is less resolutive than the authentic cadence, and, in effect, as a succession of fundamentals, it is a local succession of tension that, contrasting with the tonal relaxation, gives this very characteristic colour to it. In fact, there is the so-called plagal half cadence (I-IV) (suspension in the IV degree instead of the V), which is actually an htonal resolution of M3.

Some of the half cadences, such as the so-called Phrygian one (figure 6b), are cadential progressions helped by Phrygian resolutions of M3 (CE|B). The half cadence that comes from the "dominant of the dominant" forms a htonal relaxion. Nevertheless, in a context in which tonality is well established, it is better to interpret a half cadence as a kind of suspension or "expectation" in the tension of a very familiar chord (the dominant chord), although the tension is not fully resolved. We could say that the ear is so much used to the authentic cadence that it regards it as being assimilated in the chord previous to "consummation" and allows the tension to remain suspended and not resolved, at least momentarily.

The deceptive cadence (figure 6c) is similar to half cadence in the sense that it is not a complete resolution of the tension, but the cadential progression is helped by the htonal resolution of M3 to two of the three notes of the 6th degree chord (in F major CE to (d)FA and in F minor CE to (D¬)FA¬). The deceptive cadence in the major mode is in fact an htonal succession of functional fundamentals C|F, if we consider F as being one of the two fundamentals that represent the D minor chord (in the following sections of this chapter we will explain the true functional fundamentals of the chords, as happens in the case of the minor chord).

The Neapolitan sixth chord is used before the dominant chord in a cadence, and the most frequent resolution consists in placing the ć cadential chord (the tonic chord in 2nd inversion) between the two chords (figure 6d). In the first case (without the cadential ć), we find two chords that, together, contain the cadential 7M3 structure (F{A{E), as we have seen in figure 5. In the second, more frequent, case, there is an explanation to be added: the tension of M3 of the Neapolitan sixth chord is resolved in a Phrygian way with the fundamental B of the tonic chord (it is a well-known fact that the cadential ć is perceived both as a tonic chord and a dominant chord with two appoggiaturas).6

There are also cases of chords with a structure of Neapolitan sixth that resolve directly on the tonic by taking benefit of the Phrygian resolutive power.

6 In the major mode if we consider the cadential ć as a pedal of the dominant in the bass (C in F major) and above the succession of chords (FA)-(EG); in this case note that (EG) is a Phrygian resolution of (FA).

Fig. 6

The chords of the lowered 2nd degree (as in the case of the Neapolitan sixth) may be regarded as having a subdominant function if they have the subdominant in the bass (sn) and precede the dominant, or they may also be considered Phrygian dominants (D'), especially if they have the fundamental in the bass and precede the tonic. See more details in chapters 5 and 7.

There are several chords of augmented sixth (figure 6e). The Italian augmented sixth chord and the German sixth are chords with a structure of dominant seventh (that is to say, they contain the 7M3 structure: CEB¬), but with B¬ enharmonised in A{ because it plays the role of a melodic leading-note. In effect, this chord benefits from the slight virtual fundamental that is located at a tritone distance of the fundamental of every dominant seventh chord due to the two virtual fundamentals that every interval of tritone has; for instance, (C)EB¬ and (F{)A{E.

The difference between the German and the Italian augmented sixth chords is found in the fact that the German has the fifth of the fundamental and the Italian does not (most certainly to avoid "Mozart's fifths"). Both make use of the Phrygian resolution of CEB¬ to resolve to the dominant (C|B in E) —or an htonal resolution from the point of view of the other hidden virtual fundamental (\F{|B).

The French augmented sixth chord makes the hidden virtual fundamental F{ become real and, therefore, it is a chord with two 7M3 structures at a tritone's distance. The resolution of the French augmented sixth to the dominant is a curious case of both Phrygian and htonal resolution at the same time. One of the two structures resolves in a Phrygian way (CEB¬(A{)|B) and the other does it htonally (F{A{E|B). The French augmented sixth chord also seems to be a solution to avoid "Mozart's fifths" by adding an anticipation of the fifth of the dominant.

In his Treatise on Harmony, Tchaikovsky regards augmented sixth chords as "dominant" chords that resolve to the tonic (from the lowered 2nd degree), not to the dominant (or else to the dominant as a local modulation); therefore, he considers Phrygian dominants (D') in the Italian and German cases, and a mixture of dominant Phrygian (D') and tonal (D) in the French case.

A dominant 7th chord may become another 7th dominant chord if we raise the fundamental half a tone, we lower the fifth half a tone and we leave the notes making up the tritone unchanged. This is what in jazz theory is called tritone substitution. This theory posits that this change of dominant sevenths can be made without it affecting the chord's harmonic function. Actually, it is similar to the use of the classic augmented sixth according to Tchaikovsky's views. That is to say, this substitution, which provides another fundamental at a tritone's distance, changes an htonal resolution in Phrygian resolution and a Phrygian resolution in htonal resolution (figure 6f).

All these examples of cadences and secondary dominants that appear in figure 6 only make use of the resolutions C|F, C|B and F{|B (as functional fundamentals of the M3 and tritone tensions), that is to say, htonal or Phrygian resolutions. Moreover, the theory of the tritone substitution proves that these two types of resolutions have a tonal function similar to that of the dominant (understanding the dominant as a chord that resolves its tensions to a tonic or tonicized chord ). That is to say, the major or dominant chords (see 3.1) whose fundamentals are a 5th above / 4th below or a semitone above the fundamental of the chord to be resolved can be used as a dominants chords (secondary or not). In other less complex words and making use of the terms employed in our research: a (secondary) dominant uses an htonal (D) or a Phrygian (D') resolution above the (passing) tonic chord. Most cadential successions used in tonal music may be summarised with these two kinds of relaxions.

Given the fact that M3 and tritone's intervals and their two main resolutions (htonal and Phrygian) are so important and help us understand a significant part of the tension resolutions of harmonic progressions in music analysis and in composition, would it not be possible to identify in a quick way these intervals within the chords, despite the degree of their complexity? The answer is "yes", and we will be able to find the solution thanks to a stunning coincidence.

1.6 A fundamental and its main harmonics also have a 7M3 structure.

In 2.1 we prove that the harmonics that are really important for the auditory system include at the most the first seven ones: 2, 3, 5 and 7 (4 and 6 are octaves of the first ones). See the discussion on the importance of each one of these harmonics when establishing the harmonic and tonal laws in section 2.1.
That is to say, based on a fundamental C, the main harmonics are those corresponding to the notes C (2), G (3), E (5) and B¬ (7 ) (!!) (this E natural being a bit low in respect to the tempered E or the Pythagorean E; and B¬, even lower in respect to the tempered B¬ or the Pythagorean B¬ [see 2.8]). In other words, the 7M3 (CE(G)B¬) structure is included (with the fifth of the fundamental added).

Does this mean that when we hear only one note in effect we are hearing a sort of hidden dominant seventh chord? Somehow this is true, and a proof of it is that we have a stronger sensation of resolution when after a note we hear the lower fifth than when we hear the lower eighth, although, according to the logic of the harmonics weight, the opposite should happen, since the 8th is the 2nd harmonic, and the 5th is the 3rd.
Therefore, the significant harmonics of a sound include the 7M3 structure, always taking into account that we are not referring to a real chord because the notes corresponding to the harmonics are sinusoidal waves (fluted timbre ) (see 2.1 and 2.2), and that there are important differences in tuning between the tempered minor seventh and the "natural" seventh (see 2.8), and also slight differences with the M3.

That said, we could ask ourselves: if the 7M3 structure is found in the harmonics and is therefore a natural phenomenon, why does such a natural structure, despite being real notes, create tension?

The main answer is that the harmonics that build up the musical "laws" in our brain are the octave and the fifth (see chapter 2). The harmonics 5 and 7 are too structurally weak, and not because, occasionally —timbrically in some instruments—, they can even be more strongly heard than the harmonics 2 and 3, but because the perception of the harmonics that the brain processes from childhood follows a formal and mathematical structure where the two first prime numbers have a fundamental importance (see Fourier transform and figure 14 in chapter 2). The harmonics 5 and 7 contribute to the «consonance» of the dominant seventh chord made up of real

notes but do not resolve the more powerful harmonic tension that is produced by the two real "quasi-fifths" they contain. We should add to it the supplementary tension that is produced by the equal-tempered tuning, which is unnatural.

This surprising fact allows us to elaborate a notation (actually a symbology) of the chords so that at first sight we know whose M3 intervals and whose tritone's ones are included in them, and, at the same time, show the authentic fundamentals of the chords according to the harmonics structure.

The method is very simple:

The CE or CEG chord will be simply notated as C (for E and G are its two main harmonics [after the octave]).

Optionally, if the fundamental has its fifth, we may write a dot above (~C).

We will notate the CEB¬ or CEGB¬ chord as C7 (since B¬ is the next harmonic —and the less— important one).7

We will notate the EB¬ or EGB¬ chord as \C7 (because C is the virtual fundamental of the EGB¬ harmonics). In the case of the tritone we have seen above that it has another virtual fundamental at a tritone's distance (F{) (very weak when the chord contains G).

When a fundamental only has the octave or the fifth (that is to say, neither has its M3 nor the tritone), we will spell it in a lower-case letter. For instance, the two-note chord CG will be notated as c or ~c.

As previously mentioned and if we prefer, in order to differentiate the chords with the fifth of the fundamental from those which do not have it, we may write a dot above the notation (or else write it in bold when we use some word processor). It might be useful in the analyses of some works with a very simple harmony; however, nothing happens if they are not differentiated in practice, for we have seen in 1.3 that the chord's harmonic function scarcely changes whether the fundamental's fifth is added or not; however, it is very important to note that this only happens as long as adding the fifth entails that no other M3 or tritone interval is formed (!!) (do not confuse harmonic function with sonance —chord's consonance/dissonance— because adding a fundamental's fifth or not may really alter its sonance significantly).

This notation of the chords' fundamental structures according to the harmonics' structure appears in figure 7.

7 In my book, La convergčncia harmňnica (1994), I used the symbol ş instead of 7, but I prefer to change it here due to the use made of symbol 7 to indicate the minor 7th, especially in jazz and modern music treatises.

Fig. 7

Therefore, to know if a chord has the tension of M3 or that of the tritone, we will simply have to make sure that there is some upper-case letter in the notation (if a 7 appears, we will know that it further has the tritone tension), whether crossed or not. We will also learn that its homotonic local resolutions will be —in the case of capital C— towards the fundamentals F or B. The resolution of C may be to F or B in upper-case letters, or to f or b in lower-case letters (!!), because here we are referring to the resolutive chord and, therefore, it may have or not have a new tension. We will call the fundamental, which represents the M3 and/or triton's tensions, spelled in capital letters, functional fundamental in order to differentiate it from the fundamental in lower case.

However, we see that in figure 7 only a small portion of the chords appears. What happens with the minor chord and all the others?

We will now deal with the next important point.

1.7 Most chords may be separated into one or two chords having the harmonic CEGB¬ complete or partial structure.

I call this harmonic CEGB¬ structure, whether complete or partial, convergent structure (its notes converge towards the fundamental C). That is to say, according to this formulation, most chords may be represented with one or two fundamentals, each one representing its part of the structure —therefore representing its tensions, as we have seen in figure 7.

This has been demonstrated in annex 1, where we may see a complete list of all possible chord classes represented by these fundamentals.
This is the reason for the title of the following section.

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.

Some examples of this separation or breaking down into CEGB¬ convergent structures of several chords may be seen in figure 8 (see 8.4 as well).

Fig. 8

The symbology of these chords will provide us with some information on the resolutive tendencies of the chords on a local level. For instance, if we find a capital G (even if it has been crossed), it means that the chord has a tendency to resolve locally or to become linked to other chords that have C or F{ as fundamentals (with capital o lower-case letters; the heightened sense of relaxion or resolution will depend on what the other notes of the chords and the other fundamentals in case they have more than just one will be, and, obviously, on the tonal context in which they are immersed: the tonal field). If, in addition, this G has a 7, the tendency to resolution —to the same notes— will be strengthened. In this case, since G is a capital letter, it is a functional fundamental. In principle, this symbology does not distinguish between inversions (to distinguish inversions, see 3.4).

When we find one M2 among the fundamentals, it means that we have a powerful tonal vector (the fundamental of the higher M2 must be functional, that is to say, notated with capital letters), for it will mean that, together, both chords contain the 7M3 structure (see figure 10). For instance, the fundamentals B¬/b¬ and C determine a tonal vector towards F; C must be spelled in capital letter (it contains the CE interval) and B¬ may be spelled in capital or in lower-case letter.

For further details on the meaning of the notation or symbology that we make use of, and the building and characteristics of chords' families, see chapter 3. We call this notation fundamental symbology, for it is made up of the chords' real or virtual fundamentals, according to the convergent separation that we have previously explained.

1.7.2 We can apply the fundamental symbology of chords to create or find local relaxed progressions of chords.

Regardless of the tonal forces that appear in a musical fragment we can use combinations of htonal and phrygian resolutions between fundamentals to link chords in a relaxed or fluid manner. In Figure 8bis we have some examples of this, using chords that have more than one fundamental. These examples are only local resolutions because if we are immersed in a tonal field we must always take into account the function of these fundamentals in the tonality. For example, in Figure 8bis (f) we have left-right homotonic relaxions; but we may notice that we have an M2 between fundamentals (B¬-C). As we have seen, this implies that with the notes of the two chords we have a 7M3 structure (CEB¬), which creates a tonic F, which is precisely the main fundamental of the first chord, that is, we also have a relaxion,

in this case tonal, in the opposite direction, from right to left. There is relaxion in the two directional senses, one with homotonic color and the other with tonal color.

Fig. 8bis


We have devoted chapter 6 to see some examples of these local homotonic relaxions.

1.8 The reduction of tonal functions to three (tonic, subdominant and dominant) may be explained by means of the tensions of M3 and tritone intervals.

It is a well-known fact that at the end of the 19th century, Hugo Riemann reduced the chords' tonal functions (at least within the diatonic scales) to three different «types»: the functions of tonic, subdominant and dominant (most probably, as an evolution of Rameau, Daube and Jones' previous works). Riemann regarded the major triads of tonic, subdominant and dominant as the primary «harmonies», and maintained that every other secondary chord in major or minor modes also (and only) adopted a tonic, subdominant or dominant, «meaning». This theory was rather well accepted, especially in Germany, so that many treatises on harmony, in the tonal analyses, replaced the Roman numerals of the theory of scale degrees (I, II, III, IV, V, VI, VII) with the symbols based on T, S and D (Tp, Sp, Dp, \D7, etc.).

We will see that this is a straight consequence of considering the really «functional» intervals: those of M3 and tritone.
In effect, if we take the M3 and tritone intervals that appear in the major mode, we will obtain figure 9 (a). In it, above the pentagrams we have written the fundamental symbology, as we have seen in 1.7, and below, the symbols used by Riemann and his adherents. Note that only three functional fundamentals (C, F y G) appear, those corresponding to the tonic, the subdominant and the dominant. If in (a) we complete the triad on the upper part, that is to say, if we add the 5th in the first three chords and the minor 3rd in the fourth, we obtain the chords that appear in (b), which have the same function as those appearing in (a) because, as we have seen above, the 5th of the fundamental does not have a great influence on the variation

of the chord's "tension" and therefore on its function. The symbols representing the tonal function (T, S and D) keep on being the same.

Fig. 9

On the other hand, if we complete the triad in its lower part (we add a lower minor 3rd to the first three chords), the situation changes: the major chords become minor but, by keeping the M3 interval, they still keep its tension, and therefore keep a large part of their function; Riemann's school writes the symbols that appear in the lower part in (c), keeping the symbols of T, S and D, and adding below a lower-case p (in the upper part of the pentagram we find again the separation of the chords, in this case, the first three ones are minor chords that break down into two fundamentals, although the functional fundamental is still the same.8

In the minor mode, the symbology used by Riemann and his adherents (Grabner, Maler, Motte) is much more complicated and has actually undergone continuous revisions, its global result being a labyrinth of symbols; however, the base is still the same: the use of the T, S and D in capital or lower-case letters (D§, şDp, şSp, Sg, Dl, Tg, dP, sP, dL, tG…) to represent the tonic, dominant or subdominant functions.

In this book, we will write the symbols of the tonal functions based on Riemann's notation; however, in order to maintain the coherence with the fundamental symbology that we have seen in 1.7, a new revision of the Riemannian symbology will be unavoidable for the more complex and chromatic chords (together with the contributions of Ernö Lendvai's tonal axes theory). As a sample in figure 10 we show the skeleton of a typical T-S-D-T cadence both in the major and minor modes.

8 Let us observe that an m3 interval also has a very weak virtual fundamental in the note corresponding to the lower m3, that is to say, (C)EG(B), (F)AC(E). Riemann, who even changed the whole symbology when these chords adopted this function in some harmonic progression, also accepted this.

Fig. 10

In the case of the minor mode, in figure 10 we write the functional symbols first from the point of view of C major (local modulation in A minor), and below, the symbols in a context in A minor: h means htonal resolution and f the Phrygian resolution. The chords based on S and D contain a 7M3 structure and establish the key (tonality) of C. Likewise, s and D (also Sp y D§ ) contain a 7M3 structure, in this case of A, creating a tonal vector towards A minor.

Obviously, figures 9 and 10 are only schemes of intervals and chords , and not an example of voice conduction.

The M3 interval thus defines the primary structures of T, S and D, and the tritone, by means of its virtual fundamental, the structure of \D virtual dominant.

For further details on tonal functionalities, see chapter 5 devoted to tonality; for a summarised general overview of the whole theory, see the recapitulation in 5.8.


Chapter 2