5. Tonality

5.1 The tonal field and its vectors

According to Carl Dalhaus (1967) and the New Grove encyclopedia, the term tonal or tonality was introduced by Castil-Blaze in 1821 —although this term had already appeared before, at least in the dictionary of Alexander Choron (1810)— to give relevance to three notes or three intervalic relations: the tonic, its 4th and its 5th (calling them cordes tonales in contraposition to the cordes mélodiques), that is, what would later be generalized as subdominant (4th) and dominant (5th) of the tonic.

Now instead the term tonality is closely intertwined with hierarchical relationships of notes within musical scales and more specifically of two scales or modes: the major mode and the minor mode.

Thus, for example, according to how the notes of a musical piece are fitted with the structures of certain major or minor scales, we say that that piece or fragment is in the «key» of D major, or is in the key of E minor, etc.; in practice the term key disappears and we simply say that the fragment is in C major and then modulates to G major, A minor, etc. But a representative major or minor scale is usually in mind.

Our concept of tonality goes some way back to its origins, we consider the major and minor modes of the same tone as two modes that have a different color, but which are subject to the same laws of tension and resolution. The two have the same note in which the chords tend to rest and have four identical significant notes: tonic, subdominant, dominant and leading-tone (although these notes may occasionally be altered in some musical fragment without losing «the key of the work»); the other notes simply add color to the musical discourse. Therefore, it would also be correct to say that we are in the «key of», without specifying whether it is major or minor; actually tonality is independent of the use of a scale, which does not necessarily have to consist of 7 notes.

Continually, in the course of any musical piece, a hierarchy is established between the 12 notes of the musical palette so that there are some that the auditory system gives them the power to be more resolutive or more stable than others; to say it in a simple way: there is a note (or several) that are the best candidates to finish a melody or to be root of an ending chord since they provide the highest degree of «resolution» or rest. This note would be, according to our way of seeing the tonality, the tonic, regardless of the scale or mode that can be associated. Sometimes I like to use the term tonal vector when, in a certain melodic or harmonic transition or in a transient modulation, some note is tonicized (I usually employ the term: tonal vector «towards note x»). There may be several vectors at a time, and with different lengths (their

strength). Our graphic vision of the «tonality» of a piece of music consists of this living organism formed of arrows (tonal vectors) which, in the course of a work, are lengthened and shortened. This `tonal field' can be very static for pieces in which the tonic is always the same or very dynamic for chromatic pieces that are continually changing their tonic. In pure twelve-tone works this tonal field would once again be static and would be formed by an organism of 12 almost equal vectors without one of them predominating over the others. This is our vision of tonality.

Therefore, our work in analyzing tonally complex works will not be to find, at any moment, a scale of reference, but rather the tonic (or tonics) that appear in the course of the work. In more diatonic pieces, we will simply be in a major mode if the third of this tonic is usually major and we will be in a minor mode if the third one is usually minor, although they may eventually vary, for example, using the Picardy third (major 3rd) in the minor mode.

5.2 Tonality and the 7M3 structure

We have seen in the previous chapters that there is a very powerful structure that creates a strong tonal vector: the structure 7M3. This structure creates a tonic (transient or not, even if the tonic note, to which it tends, is not present in the musical fragment) and is formed by the fifth, the fourth and the leading-tone of this tonic (as notes, not as chords). That is, in conventional theory, they would be the dominant, the subdominant and the leading-tone of the key. It varies slightly from the term introduced by Castil-Blaze in 1821, since he quoted the structure formed by the tonic, the fourth and the fifth, although if we consider this structure formed of functions or fundamentals of chords instead of notes, it comes to be a similar concept.

Finding the 7M3 structures in a musical work will therefore be very important in order to establish its tonal vectors.

We will see below that from the 7M3 structure we deduce the major and minor scales (with leading-tone). They are the only two scales of seven distinct notes that contain a single 7M3 structure and therefore have no conflict of tonics.

Let us take, for instance, the key of C:

The fixed notes of the 7M3 structure in this key are G, B, and F. Apparently, in order to maintain C as resting tone, the other notes may be any, but we will rule out certain combinations.

D¬: This note creates another 7M3 structure with the fixed notes: D¬-F-B (C¬), that is, a tonal vector towards G¬. If there is also E¬, another is formed: E¬-G-D¬ with a vector towards A¬. If, in addition, A is natural, we find a hidden one: A-D¬(C{)-G with vector towards D. This note (D¬) is discarded if we want a single tonal vector in the scale.

E¬: If A is natural, it also creates another 7M3 structure: F-A-E¬ and a tonal vector towards B¬. In addition there would be a second 7M3 structure with the note B: B-E¬(D{)-A and a tonal vector towards E. Therefore, E¬ can only be accompanied with A¬ if we only want a single clear tonic.

A¬: If E is natural, it creates another 7M3 structure with D: E-A¬(G{)-D and a tonal vector towards A.

Therefore, the only scales with a single 7M3 structure are the major and harmonic minor (with leading-tone) scales (figure 41).

In practice, these note combinations that we have discarded are used continuously in tonal music, even in the simplest, but we have tried to show that by applying only the tonal force of the 7M3 structure the two most common forms of major and minor scales are deduced.

Fig. 41

As we have seen in 2.7, in the minor mode, in order to avoid the exotic interval A¬-B¬, the notes A}-B} are sometimes used locally in the ascending tunes or B¬-A¬ in the descent ones. In the latter case, what is lost in tonal vectorial power is gained in internal scalar coherence (fifths F-C-G and fifths A¬-E¬-B¬) and we get an ascending melodic tonal resolution A}-B}-C and a descending melodic Phrygian resolution to the dominant (B¬-A¬-G), known as basso di lamento.

5.3 Tonal functions and their symbols (functional symbology)

Before continuing with the 7M3 structure we will make a parenthesis and apply what we have learned in the previous chapters about the tensions that appear in the chords of the diatonic scales and will see the coherence of the true chord fundamentals with the reduction of the tonal functions to three: the tonic, dominant and subdominant functions (theory officially consolidated by Hugo Riemann).

The compendium of tonality in three «chords» or main functions was already present in J. P. Rameau and especially in J. F. Daube. Rameau, in his Démonstration du principe d'harmonie (1750), deduces the major scale as a result of considering the fundamentals C, F and G and their first five harmonics. Daube, four years later, in his work General-Bass in drey accorden (1754), argued that any accompaniment can be made with three basic chords: the tonic chord (CEG), the subdominant chord with an added sixth (FACD) and the dominant seventh chord (GBDF). De facto, matching inversions, with Daube we find a succession of degrees I, II, V, which is

also the foundation of the functional theory of jazz. Also William Jones in his book A Treatise on the Art of Music (1784, p. 13) says: "Therefore these three chords [C, G, F] comprehend all the native harmony of the octave; and the three notes C, G, F, are the fundamental notes, because they carry all the degrees of the octave in their accompaniments". Recall here figure 18 of Chapter 2:

We will introduce our functional symbology (not to be confused with the fundamental symbology of the chords), making some changes with respect to the Riemann school's symbology, applying everything we have seen in the previous chapters.

In Figure 42 we have the triadic chords of the major diatonic scale. Above the pentagram we have our separation of chords into fundamentals (fundamental symbology, seen in 3.1) and below the functional symbology used by Riemann and his followers. These symbols show the importance of the M3 and tritone intervals to establish the tonal tensions and therefore their functions. In the major diatonic scale, C, F, and G are the only notes that have an upper major third. And the only tritone found is B-F that determines its virtual fundamental G. Functionally these are the main fundamentals of the scale. Note that the uppercase (functional) fundamentals are only three (C, F, G) and match the T, S, and D of the Riemann school. In the minor mode, tonic and subdominant may be minor chords, but the dominant G is always in major (otherwise, we would be in eolian mode). That is, as dominant, always has the note B, the leading-tone. Having a subdominant chord (either major or minor) and a dominant chord automatically implies having the 7M3 structure, since, as we say, the leading-tone is always included in the dominant chord.

Fig. 42

The minor mode is much more complicated. Riemann was dualist, that is, he considered the minor chord as an inverse reflection of the major chord. As we have seen that the diatonic major scale can be deduced in various ways from harmonics (see 2.6), the dualist theory believes that the minor chord and the minor mode (although there is no single minor scale) can be deduced from the symmetrical inversion of the major chord and the major scale, that is, as if the musical notes also had a kind of subharmonics (fact that is not an acoustic reality).

Following this theory Riemann designed symbols for the minor mode that complicated, and in our opinion distorted, his great success, which was precisely to simplify the tonal functions to three. Some followers of their symbols, such as Wilhem Maler and Diether de la Motte, were also aware of this and changed their symbols, among others, that of the minor tonic chord, that of the minor subdominant and that of the «minor dominant», putting simply the symbols t, s and d. This is what we have also done, though for other more complex chords our functional symbols are different in order to be consistent with the fundamental symbology we have seen in Chapter 3.

As can be deduced from what we have been explaining about our concept of tonality, our theory is not dualist, we consider the major and minor modes of the same tonic, as we have said, two modes with different character and color, but ultimately submitted to the same tonal laws towards the same tonic.

In figure 43 we have triadic chords of notes within the orbit of the different scales of C minor (we could still place more). Above the pentagram we have again the chord symbols according to its fundamentals and below our tonal functional symbology of the chords with respect to tonic C.

Fig. 43

What is the meaning of T¶, S¶, D¶ ?

T¶, S¶¶ and D¶ are the tonic, the subdominant and the dominant of the relative major (key) of the minor, key to which always tends the minor mode when the 7th degree is not the leading-tone (they have the same key signature). If we have a musical fragment in minor mode with the functions T¶, S¶, D¶, these symbols will give us information that in that part we can have a transient modulation to the

relative major. For example, in figure 44. We could use either of the two symbols below, using T¶, S¶, D¶ or the brackets: when we have a small transient modulation, we can put the new functions in the new key inside brackets (depending on the length of the modulation).

The inversions of the chords are not specified in the functional tonal symbols, except in a few cases (see 5.6).

Fig. 44

We could do something similar to the major and its relative minor, as, for example, the fragment of Figure 45 (C major / A minor / C major)

Fig. 45

or with its "relative major" (C major / A major / C major) (figure 46).  

Fig. 46

In the latter two examples, the new symbol D§ appears as the dominant function of the relative minor, and in figure 46 the symbols S§, D§, T§ as subdominant, dominant and tonic functions of A major (within a context of C major). When we see the symbol D§, it means that we have a tonal vector in the direction of the relative minor of the major mode (or the relative in major if T§ appears).

I prefer to leave aside if we are in a major or minor mode and I like to talk about the upper relative or the lower relative of a tone (they belong to the same tonal axis). The symbols T¶, S¶ y D¶ give us information on a trend to the upper relative and the symbols S§, D§ y T§ information of a tendency to the lower relative of the tonality in which we are established, characterized by the symbols S, D, T (or s, D, t).

Symbols are different if viewed from the perspective of one tone (key) or another. For example, in figure 47 we have the functional symbology of the triads in A minor from the perspective of A (in a tonal context of A minor) or from the perspective of C (in a tonal context of C major). See also 1.8 with figures 9 and 10.

As we have said, the symbols T¶_p, S¶_p y D¶_p will be simplified by t, s y d (T¶_p = t, S¶_p = s, D¶_p = d) most of the time. When a transient modulation occurs towards C from A (C is the tonic of the upper relative [T¶ ] of A) these symbols can be used since A ceases to be momentarily the tonic.

Fig. 47 

Note that the chord's fundamentals of the functions T¶ , T y T§ form minor third intervals (also the fundamentals of S¶, S, S§ and those of D¶, D, D§ ). That is, they are on the same tonal axis, as seen in 4.5.

Only three tonally functional fundamentals would be missing to complete the circle of fifths with all 12 possible fundamentals. These three fundamental would form a tonal vector towards the tonality which, in conventional theory, is considered the most distant one: that which is at a distance of tritone (a fact debatable in our theory and that of Lendvai). We will put the symbols S', D' and T' to the functions that represent these fundamentals. The function D', in fact, is not a strange function with respect to T, since it is its «Phrygian dominant» and is used continuously in tonal music. For example, in C, the function D' would be a chord with fundamental D¬, which resolves, with Phrygian relaxion, in C. The D' function can share the subdominant function with respect to the original tone if the subdominant is in the bass (would form the well-known Neapolitan sixth chord) and is followed by the classical tonal dominant. Into the Andalusian mode used in flamenco music, this chord (transposed to F) is also called dominant, in its classical version resolving in E (major) (C_a-G-F-E). This function (D') is also usually used as the «(Phrygian) secondary dominant» of the dominant, often in the form of a augmented sixth chord (see 1.5).

Fig. 48

In figure 48 we have an example, within a tonal context of C, where we see transient modulations following the tonal axes (C, a, f{, d{), and the respective dominants are D, D§, D' y D¶. The symbols appearing in the second lower line are alternative notations (to which we will give preference in the analyzes to provide greater simplicity and clarity).

sⁿ (in bar 5) is the Neapolitan sixth chord (acting as subdominant, such as the postriemannian school symbolizes it) and Dæ the cadential æ chord (as appoggiatura of the real dominant). They are the only two chords that —depending on its inversion and the function of the subsequent chords— can have a double tonal functionality. And this because of the strength of the note that is in the bass which, in these cases,

acquires a greater functional protagonism. In figure 49 we have examples of D¬

major chord acting as Phrygian dominant (D') or as subdominant (sⁿ) and the second inversion of C major chord acting as tonic (T) or as dominant (cadential æ) (Dæ).

Fig. 49 (Listen on youtube)

Taking a tonic as reference, for example C, any fundamental (or fundamentals) of a chord, however strange it is to the tonality of C, is functionally classified using these symbols, which are summarized in the circle of fifths of figure 50. The symbols are representative of the tonal tensions of M3 in a tonal context of C.

Fig. 50

It should be made clear that the only symbols that determine the classical concepts of tonic, subdominant and dominant functionality, as they are known since Rameau and Riemann, are the symbols T, S and D (and their variants in the secondary chords T_p, S_p, D_p, \D, but without the ¶ , § , or ' additions).

Most harmony books speak of tonic, subdominant and dominant chords, but, apart from saying that they correspond to the chords of degrees I, IV and V (with leading-tone) of the scale and to give rules of progressions between them, their meaning with respect to musical perception is somewhat ambiguous. 

There are coincidences in saying that the tonic is the function of rest, of end, of resolution, of «returning home». The dominant is defined as the function of tension, which is resolved satisfactorily if it is followed by the tonic function. The function of subdominant is darker, it is spoken of «expansion», passage chord towards the dominant… It is also said that the tonic function is in the center, that of dominant towards one direction (an upper 5th) and the one of subdominant towards the opposite direction (a lower 5th) and that then the subdominant and the dominant define the key and the tonic is in the middle and is its center of gravity. Riemann said: «Thesis is tonic, antithesis is subdominant, and synthesis is dominant».¹

Objectively, from a harmonic (and from the harmonics) point of view, we can only say that T is a htonal and tonal resolution of D, S a htonal relaxion of T, D a doric relaxion of S in the major mode and a Phrygian relaxion of S in the minor mode and that, certainly, S and D define a key or tonality (a tone of rest, a tonic) since (together) they contain the 7M3 structure (when I speak of T, S, D, I also include —in a key— the chords T_p, S_p, D_p, \D, D⁷ but note, not T⁷ and S⁷, which, by themselves, already include 7M3 structures strange to the original tone). These homotonic relaxions have been the cause that they are the most used progressions in classic tonal music: T_(p)-S_(p) more than S_(p)-T_(p), S_(p)-D_(p) more than D_(p)-S_(p) and D-T as paradigmatic final resolution.

As we say, T, S and D are the basic tonal functions and the symbols T¶, S¶, D¶, S§, D§, T§, T', S', D' basically indicate tonal vectors towards the tonic relatives. However, if we extend the concept of dominant as any chord that discharges its tension in the next chord or we extend the concept of subdominant as a chord that links well with the dominant and produces local (homotonic) relaxion when it sounds after the tonic, then some of these symbols can also be considered somehow tonic, subdominant or dominant with respect to the original tone. Specially, as we shall see, T', S' and D' since the D'-T sequence gives us a Phrygian resolution (D', Phrygian dominant), just like the sequence T-S', which is also a Phrygian homotonic resolution. In fact D'⁷ has two of the three notes of the 7M3 structure included within the dominant seventh chord Dµ. They have the same tritone (F-B) and, therefore, share their tension and resolve it similarly towards T.

In a much less pronounced way we also find homotonic resolutions in the sequences D¶ -T (especially D¶_pµ-T, which also contains two notes of 7M3: G and F) and T-S§, which form two weak Doric homotonic resolutions. There are other combinations with Phrygian resolutions like D§ -T¶ , D¶ -T§ , D-T', T§ -S¶ , T'-S, T¶ -S§ and we obviously have the htonal relaxions D§ -T§, D'-T', D¶ -T¶, which, as we have already seen, indicate a small transient modulation to the relative tones.

¹ Riemann "Musikalsche Logik" (extracted from Harrison [1994], p. 267).

Due to this large coincidence of behaviors (functionalities) similar to the classic functions of T, S and D we consider that can be convenient to use these new symbols eventually and occasionally in strange (and isolated) chords within a tonality. But when there is a clear transient modulation, we will prefer to use the symbols T, S, D of the new key (between brackets). We will also prefer to put the secondary dominants (htonal or phrygian) in parentheses before the tonicized chord. For example, (D)|D (dominant of the dominant) is the same as S§ |D; and (D')|D (Phrygian dominant of the dominant) is the same as S¶ |D; they are dominant chords with respect to the dominant, but with respect to the tonic are subdominants that link with the dominant. In Chapter 7 we can see examples.

5.4 Tonality and tonal axes

In figure 50 we have the tonal axes (variations of T, S and D) in a similar drawing to Ernö Lendvai's theory of tonal axes. As we have seen in 4.5, this theory roughly says that one could substitute chords of the same axis (tonic, subdominant or dominant axes) with one another and this is what, according Lendvai, Béla Bartók did in many of his works. For example, replace S with S§ or S¶, or D with D' or D§, etc. I do NOT say that this can be done, except for a few exceptions, if we want to keep the attraction of a SINGLE tonic. Indiscriminately substituting families of T, S or D of the different axes with each other soon loses the attraction of a single tonic and we enter in a multitonal field, which can be perfectly a system of composition (as can be the twelve-tone of Schoenberg), but moves away from the classic tonality concept with a main tonic, which is what we are studying in this chapter.

We have presented all these symbols now because, sporadically, it is possible to put strange chords to a tonality without being considered modulation and this symbology helps us to locate the function of any chord in reference to the main tonality.

Classifying functionally (within a key) a foreign chord has been a problem in all schools when you want to take a scale (major or minor) as reference. In the Riemannian school many strange chords can not be functionally symbolized. Schoenberg himself, in his book Structural functions of harmony (1948), simply cross out the degree of the root when it is a chord that does not fit within the scale.

That said, however, as we have advanced before, some chords can be substituted, without losing the main key tonal vector, when in the passage from subdominant to dominant or dominant to tonic or tonic to subdominant we use homotonic (htonal or Phrygian) relaxions.

For example, in figure 51 we have substitutions of D for D'. In the case of D'µ we find the familiar tritone substitution used in jazz (two notes of the chord are

changed but the tritone [F-B] is conserved and therefore also its tension). It converts an authentic cadence (tonal dominant D) into a Phrygian one (Phrygian dominant D'). I have put the tonic chord in major, but it could be in minor.

Fig. 51 (Listen)

In figure 52 we have substitutions of S for S¶ or S¶_p. The passage from subdominant to dominant then becomes a Phrygian resolution or, if it is to say otherwise, the subdominant makes the role of phrygian dominant of the dominant. If we want to see this way, we may put D' in parentheses. When S¶ is a seventh chord, if G¬ (melodically) functions as F{ (resolves in G) and in the bass there is the fundamental, we are in the case of the scholastic use of the Italian and German augmented sixths (case [c], in the case of the German one the «Mozart fifths» appear). As before, the tonic chord may be major or minor. In case of major mode, these substitutions imply giving it a color of minor.

If the tonic is well established, S can also be replaced by S§ (then S§ makes the role of dominant of the dominant, i.e., the htonal resolution to the dominant), but it would take more than four chords if we do not want the tonal vector towards the dominant exceeds that of the tonic.

Fig. 52

Of course we can also make these substitutions in a consecutive way, as in figure 53, or even putting two fundamentals of the same tonal axis within a same chord

(Figure 54); in this case, if the fundamentals are separated by a tritone, we have (two) seventh dominant chords with a lowered fifth (in the same chord: symmetrical chord). Case (b) is the known succession of the French augmented sixth.

As before, the tonic chords of figures 52, 53 and 54 could also be minor chords.

Fig. 53 (Listen)

Fig. 54 (Listen)

5.5 Cadences

We could divide the cadences into conclusive ones —which serve to finalize a work or a musical fragment (such as the authentic cadence or the plagal cadence)— and «suspended» cadences which do not fully discharge the tension but allow a certain rest or local relaxation (such as the half cadence or the deceptive cadence).

In the concluding ones, and in view of the tonal music literature from the end of the 19th century until today, we would extend the cadential progressions simply by saying that we found a cadence when there is a rest in the tonic, that is to say, mainly a resolution in a chord of the family of the major or minor chords (may include major sevenths and other intervals, see definition in 3.1.1 and 3.1.2) normally

with the tonic as fundamental when it (the tonic) has been well established. See for example the previous figures 49, 51, 53 or 54 although there are many more possibilities.

If the tonic has been well established, any chord (degree) can go before the final tonic chord, without losing conclusive meaning. Hovewer, if in the previous chords we hear the 7M3 structure (the one that defines the tonic), this conclusive cadential sense will be more intense and definitive.

The tonal cadential skeleton par excellence would be that shown in figure 55, being much more used progressions (a) and (b) than (c).

Fig. 55

In the skeleton of figure 55 other notes could be added and if these notes do not form new 7M3 structures they do not lose the conclusive cadential sense. But even if new 7M3 structures appear, the resulting new progression can give us a satisfactory cadence, as shown in the examples in figure 5 of chapter 1 (1.4). In fact we could include again all part 1.4 of Chapter 1 here, so I suggest to the reader his rereading.

By making a more systematic and traditional arrangement of the different possibilities of filling the conclusive cadential skeleton we would obtain the examples of figure 56 (we only give two chords previous to the not shown tonic chord). In these examples the cadential æ may be interleaved, especially when the sub-dominant has also dominant character (the last four) (Figure 56b).

Fig. 56

In the dominant part one could also add E or E¬ forming the chords EGB or E¬GB; then the dominant chord is somewhat «tonicized» and at the same time E or E¬ can be heard as appoggiaturas of D (figure 57), although D does not have to sound in the cadence. Riemann considered the E minor chord to have a dominant function (Dominant-Parallele), but many harmony treatises consider it with function of tonic. Indeed, the E minor chord (EGB) has two notes of the tonic triad chord (CEG) and two notes of the dominant triad chord (GBD), but I believe that the dominant part prevails over the tonic part due to the tension of the M3 (GB) (dominant + leading-tone) and if we separate the chord into fundamentals according to the harmonic structure (see 1.7, 2.7 and 3.1), the chord has the functional fundamental G, which is the dominant. But one has to take into account the tonic component of this chord, especially if we place E in the bass. As is often the case in analysis, there is no need to opt for one thing or another, but at certain times the two functions can be shared.

Fig. 57

In these examples (figures 56-59) we are placing the subdominant and the dominant in the bass, but it would also work by putting the chords in any inversion, although the resolution effect, with tension release, of the cadence, can diminish according to the note that is on the bass. Recall also that the figures are cadential skeletons, they are not examples of voices conduction.

Fig. 58

The subdominant may also be included in a seventh chord and also, obviously, the dominant and the leading-tone (figure 58). In example (a) we should have the tonic previously defined because the chord Fµ (contains the 7M3 structure of another tone) defines another tonic, although after listening to GB (with the previous F) we have C tonicized. Something similar happens in (g), but here, even if the first chord defines the tonic G¬, this chord is at the same time Phrygian dominant (D'µ) of the final tonic C.

In all the examples of figures 56 and 58, resolution in the tonic chord (major or minor) is not shown. This resting tonic chord of the cadence can also have the major seventh (B) and even the ninth (D), which gives a slight dominant feeling to the tonic chord and a half cadence air to the cadence, but it is fully conclusive, widely used in jazz. The ear accepts this dissonant chord as final because it has the main harmonics of the main harmonic of C, which is G (figure 59). You can also use the sixth (A) as this note does not create a new functional fundamental. If the sixth goes along with the ninth (D), it (the 6th) loses meaning as root of a possible chord based on A minor since the auditory system accepts A as harmonic (fifth) of D. Located within the funcional harmonic limits one could even add F{ to complete the D chord, forming a global chord C^GD by fifths (of fundamentals). Even all these added notes (BDF{A) may be used together in the final chord.

With the minor chord you can also use the major seventh, but we have an even more dissonant ending (since B does not have now the support of E). We could also put the sixth (A) and the ninth (D) together (because they support each other forming a fifth), but putting them appart would be more conflicting due to the tension with the E¬, although neither can be ruled out as final chord.

In all these cases, we have sonance tension in the final chord but at the same time a tonal resolution (figure 59).

Fig. 59

The two classic non-conclusive cadences are half cadence and deceptive (or interrupted) cadence.

The half cadence, as its own name indicates is an «incomplete» cadence. Creates a suspension in the dominant chord and therefore does not resolve the tonal tension. We could say that the ear is so accustomed to authentic cadences that when the dominant chord sounds, it already has the tonic chord in mind before it actually sounds, creating a sort of suspensive expectation if this familiar resolution does not occur.

However we can found chords, previous to the dominant chord, that help half cadence to acquire a more «cadential» character. They are the chords resoving homotonically towards the dominant using a Phrygian or htonal relaxion; and, in fact, are the most used progressions in classic half cadences.

In C, for the Phrygian resolution to the dominant, we would need A¬ (S¶ ) as functional fundamental and for the htonal resolution to the dominant we would need D (S§ ) as functional fundamental. The htonal resolution is more conclusive because in fact it consists of a small local modulation (see figure 60: the (¬) in parentheses means that E can be natural or flat).

Other half cadences come from the subdominant F (S) as functional fundamental, producing also a weak Doric homotonic resolution (see 4.2).

Fig. 60

The deceptive cadence, like the half cadence, does not completely resolve tonal tension. It also leads us to a sort of temporary suspension which calls for clarification.

The 6th degree chord to which the dominant chord is resolved is not a tonic chord but is familiar to it because it has two of the three notes of the tonic triad. In the major mode the functional fundamental of the 6th degree chord is precisely the tonic and taking the functional fundamentals of the 5th and 6th degrees we obtain a homotonic htonal distension.

In figure 61 we have a deceptive cadence in major and minor modes. In both cases, if we only considered the notes of the upper staff, that is, all but the bass, we would have an authentic cadence. The bass, in the final chord, converts the authentic cadence into an unexpected progression, but at the same time the ear hears the familiar authentic cadence of the other notes, producing this characteristic and sweet pseudo-cadence.

Fig. 61

In section 5.3 we already mentioned the Neapolitan sixth chord when using the cadential skeleton. But the most usual progression of the Neapolitan cadence is placing —before the dominant chord— the æ cadential chord. We already know that the æ cadential chord is the second inversion of the tonic chord. Considering it as a tonic chord this sequence gives us a more fluid progression since between the Neapolitan sixth chord and the tonic chord we have a Phrygian homotonic resolution (figure 62, see also the example of figure 6d).

Fig. 62

5.6 Functional symbology in inversions
In general, with the functional symbols we have introduced (the functional symbology), we do not distinguish between a chord and its inversions since in most cases the inversion does not significantly vary the tonal function of the chord. But, as we have seen, there are exceptions. Depending on the note in the bass and depending on the tonal progression that follows, a chord can acquire different tonal functions or, in fact, share the two at the same time. Therefore, for some inversions and progressions the functional symbology used may be different for the same chord. They are relatively few cases. In figure 63 we have the only chords with possible dual tonal functionality and therefore with two different functional symbols depending on the tonal situation in which they are immersed.

We also add in this figure, as a summary of the different symbologies for the same chord, the two possible functional symbology of some of the minor mode chords that we have already seen in 5.3 (t, s and d).

Fig. 63

5.7 Modulation
We have already explained in 5.1 our vision of the tonal field as a living set of dynamic tonal vectors appearing during a musical work and that signalize new tonics or tonicization processes.

When this process of tonicization takes on a certain consistency, it is referred to as "transient" modulation and when the tonic is firmly established, then we speak of true modulation. With the algebraic simile we use we would draw the case of one of these new vectors clearly highlighting the others. Also, for example, the use of secondary dominants would create very small vectors as a tonicization of the fundamental to which it resolves.

Let us remember that these vectors signal tones, that is to say, concrete notes and secondarily modes, therefore the changes in a musical fragment based on a major scale that lowers its third degree (and optionally the sixth degree) and becomes minor (or vice versa) for us it is not exactly a modulation. A modulation implies a change of tone.

Although the term modulation is often used when passing from major to minor (or vice versa) of the same tone, theorists like Piston and Schoenberg also seem to suggest something similar: «At first glance it might appear that the tonalities of C major and C minor are quite distant, since there is a difference of three flats in the key signature. But because of the similarity of their harmonic functions, these two tonalities can be considered the same in many aspects, since they have the same tonal degrees and actually differ only in the third degree» (Walter Piston, Harmony, p. 223 in the Spanish version) and «In classical music the minor and major modes are frequently exchanged without so much formality, in the sense that a passage in major mode can be followed by a passage in minor mode without a harmonic help and vice versa» (Arnold Schoenberg, Structural functions of harmony, p. 90 in the Italian version).

Picardy third is well-known to us: when a work in minor mode ends with the major triad chord. Although it has its origin in modal times, its use has remained

relatively alive in the baroque and in classicism and also later composers like Chopin have made use of it (see for example 7-22).

We could also quote as examples Beethoven's Symphony No. 5 in C minor that ends in C major, not only because of the last chord, but because it uses this mode in most of the last moviment; or, for exemple, the Rondo of his Sonata No. 21, Waldstein, where, from bar 153, major and minor chords alternate each other; or also the start of Also sprach Zarathustra by Richard Strauss. We would find numerous musical examples of the direct alteration of the third of the tonic converting passages from major to minor and vice versa, thus producing a change in color but not a change in tone, that is to say, without changing the note that does tonic function, the note that our auditory system assigns as the best note of rest or resolution.

For there to be a change of tone (key), transient or not, again the 7M3 structure is fundamental. This structure creates a powerful tonal vector (a well-established tonality has a unique 7M3 structure associated to it). If in a musical passage (except modal ones) the 7M3 structure is unique, we can say that «we are» in the tone (key) associated with this 7M3 structure and the (major/minor) key will depend on whether the third of this tone is, in this passage, mostly major or minor.

Hence a modulation will simply consist of moving from one 7M3 structure to another. Let us keep in mind that a musical fragment can share one or more 7M3 structures at a time, in this case the power of the tonal vectors will depend on the harmonic progressions that are used.

The most effective way of modifying a 7M3 structure (and to avoid duplication of 7M3 structures) is precisely by varying the notes that shape the 7M3 structure. Doing this way will get the tonalities that are considered more «close» to the original or starting tonality.

For example, let's stand in the tonality of C (C major and C minor scales). The 7M3 structure consists of notes G, B and F (GBF).

If we change G for G{² , we obtain, in C major, the new 7M3 structure EG{D creating a tonal vector towards A (minor mode, since we have C natural).

If we change B for B¬, we obtain, in C major, the new 7M3 structure CEB¬ creating a tonal vector towards F (major) and, in C minor, the new 7M3 structure B¬DA¬ creating a tonal vector towards E¬ (major).

If we change F for F{³ we obtain, in major and minor,⁴ the new 7M3 structure DF{C creating a tonal vector towards G (major).

² If we change G for G¬, the ear actually hears G¬ as F{ since F{ is the P5 (harmonic 3) of the 7th degree B and the M3 (harmonic 5) of the 2nd degree D, so we will consider it as a variation of F, case we will see next.

³ If we change F for F¬, the ear hears it as E since E is the P5 (3rd harmonic) of the 7th degree B and the M3 (5th harmonic) of the 1st degree C.

That is, by changing the notes of the 7M3 structure of C we obtain the tonal vectors towards A (relative of C major), Mi¬ (relative of C minor), F (lower fifth of C) and G (upper fifth of C), which are the tones (keys) considered the closest to C.

But we can work out modulations to other more «distant» tones by simply showing the 7M3 structure of the new tone we want to go to, and if we want them more consistent, we can apply one or more cadential processes, discharging the tensions of the 7M3 structure in a tonicized chord, as we have seen in 5.5.

Fig 64 (Listen)

One procedure that can help us to modulate in a fluid way would be to apply htonal and Phrygian homotonic relaxions between the chord progressions responsible for the modulation. In figure 64 we have some examples. The arrows indicate htonal or Phrygian homotonic relaxions. They are fast modulations with few chords, illustrative purposes. Surely modulations too fast from a classical and romantic point of view, where key changes occur in a more gradual way and in closer tones. In example (d) we modulate very fast from C minor to F{ minor using sharps; in example (e) we modulate to the same tone (enharmonic)(G¬ minor), also with few chords, but using flats and a symmetrical chord with dual function of htonal dominant and Phrygian dominant introduced by a double homotonic resolution from the previous chord ì_f7. In the two examples we start from the same C minor chord but advance in opposite directions, in (d) we apply a Phrygian resolution and in (e) a htonal one, resolutions which, as we know, are at tritone distance (their resolution fundamentals).

⁴ In C minor we would also get enharmonically the structure A¬CG¬, but as we have said in note 2, the ear hears G¬ as F{, creating a conflicting 7M3 structure, but we have to keep in mind the fact that this tendency towards D (major) would also exist.

5.8 Recapitulation

During these chapters we have seen that the phenomenon of harmonics shapes the mechanisms of musical apprehensions. In particular, the third harmonic (the fifth), the perfect consonance, is responsible for creating the «laws» that govern our perception of harmonic tensions.

Harmonics create the palette of available sounds and intervals, and the second harmonic (the octave) is the cause that all intervals between notes can be reduced (apart from unison) to six: m2 (M7), M2 (m7), m3 (M6), M3 (m6), P4 (P5) and tritone. Of these intervals there are two very close to the perfect fifth consonance (half-tone difference), are the «quasi-fifth» intervals of M3 and tritone, which discharge their tension if one of their two notes is directed toward the note (or fundamental) that would «resolve the tuning of the fifth». These are the resolutions we have called htonal and Phrygian, very important interrelationship between chords for the establishment of harmonic and tonal tensions and relaxions.

If we take these two intervals together, so that we match the resolution notes (which are separated by a tritone), we obtain the 7M3 structure, formed by an M3 and a tritone (for example CEB¬), structure that reinforces and increases the resolutive tendency towards these two notes (F and B} for CEB¬).

This structure (7M3 = M3 + tritone) together with the tendency of tones to resolve a lower fifth (due to the harmonics structure) is what generates tonality (the notes that form the 7M3 structure are known as dominant, leading-tone and subdominant of the tone in which they want to discharge their tension).

These principles (we might also call them musical forces) are therefore very simple and elementary (in fact everything could be reduced simply to the tendency to tune the interval of fifth and to rest in the lower fifth), but, as in nature, combinations of elemental principles (or forces) can give very complex perceptual structures.

In the next two chapters we will give examples of how we can apply these principles in composition or musical analysis.

Chapter 6