4. Secondary relaxions and other successions of fundamentals

We have seen that the two local (homotonic) main resolutions or relaxions of the functional fundamentals are (in the next chord) the lower fifth (or upper fourth) fundamentals (htonal resolution) or the lower minor second (or upper major seventh) fundamentals (Phrygian resolution).

This was the result of the tendency of the auditory system to «adjust» the «false-fifths» or «quasi-fifths» which are formed with the M3 and tritone intervals.

4.1 "Locrian" relaxion

There is another succession of fundamentals that produces significant relaxion, although is little used in tonal music because it does not fit within the diatonic scales, except in few cases, as in some half cadences in the minor mode.

Consists of the passage from a functional fundamental of a chord to another fundamental at M3 distance. For example, ~C-~E (CEG-EG{B chords). I call it «Locrian» distension.

The acoustic demonstration of this relaxion would also be related to the resolution of the "quasi-fifth" of the M3, but this time only considering the fundamentals of the chords or intervals (figure 36).

Fig. 36

In this relaxion it is important that in the «resting» (second) chord the fundamental has its fifth since it is precisely the note of resolution. If the E is a functional fundamental (has its M3 [figure 36b]), the relaxion that occurs is clear despite the chromatism involved (G-G{). If the fundamental of arrival is not functional, for example, the fundamental (lowercase) of a minor chord, then the relaxion is not so clear because there is a sequence of tension between fundamentals in the opposite direction (C-G) (figure 36c). But we must distinguish between successions of tension and successions in which there is relaxion in the two directions.

There is also relaxion in both directions if the first chord is from the minor family (C_a) because we also have tension in the opposite direction (a-E) (figure 36d). These two chords (C_a-E) could be continuously linked as a kind of perpetuum

mobile, and either could be the final chord because there is homotonic relaxion in both directions (Locrian and htonal) (see also example 6-32 and figure 39). Therefore this Locrian relaxion or resolution establishes a relaxation between chords with certain conditions. The Locrian relaxion «wins» to the htonal one in the other direction if C_a chord is in second inversion, because C and A sound like appoggiaturas of the chord with fundamental E. Scholastically speaking we would be in the case of a cadential æ in the minor mode (for example, see chapter 7 example 7-15 bars 78-79).

Melodically, without chords, the two notes that form an M3 interval can be «resting» notes; in this case the tonal field is determinant to establish the direction of relaxion (if we have strongly established, for example, a tonic C, then we have melodic relaxion in the two cases: E|C o A¬|C).

This resolution becomes much clearer if the fundamental of the first chord has the minor seventh since then (for example, C⁷|E/e) there is a double resolution of «quasi-fifths» EC and EB¬ that resolve in the perfect fifth EB. B¬ is usually enharmonized to A{.

In fact, Locrian relaxion is a variation of the Phrygian one since what causes relaxion is the Phrygian melodic resolution of the interval of M3 by means of a descending step of m2 of one of the notes.

4.2 "Dorian" relaxion

There is another succession of fundamentals that also produces some local harmonic "relaxation", but in this case I would no longer use the name «resolution» since it is very weak. It is the ascending major second succession of fundamentals (figure 37). This harmonic progression is used continuously in tonal music. Much more than its inverse sequence, the succession of fundamentals by a descending major second. As examples we could include the typical tonic-supertonic (I-ii /i-ii), subdominant-dominant (IV-V), deceptive cadence (V-vi) in the major mode, ii-V, an so on. I call it «Dorian» relaxion. We cite it only to take it into account, but it has little weight compared to the other three homotonic relaxions. The most powerful Dorian succession is that of genre C_a⁽⁷⁾-D if in the bass there is a jump of P5 (a|D).

Fig. 37

4.3 Successions of fundamentals without homotonic tension

Of the twelve intervals (six interval classes) that can be formed with the twelve notes palette of the equal-tempered chromatic scale we have established so far —once these two secondary relaxions have been added —eight intervals (four interval classes) between fundamentals, which determine successions of tension or relaxion. They are the fundamental (lower/upper) jumps of P5/P4 (htonal), m2/M7 (Phrygian), m6/M3 (Locrian) and m7/M2 (Dorian). We would only have pending to study the intervals of unison, m3 and tritone. These intervals form a tonal axis (according to Bart?k/Lendvai's theory of tonal axes).

The sequence of chords whose fundamentals form one of these intervals (they are in the same tonal axis) do not establish, locally, nor succession of tension or homotonic relaxion (would have tonal tension if they were established in a tonal field). They are unusual chord successions in tonal music because they do not fit within the palette of notes of the major and minor scales. In figure 38 we have some examples, they are sequences of chords that seem to be floating, not knowing exactly where they are going.

Fig. 38 (Listen on youtube)

4.4 Summary of homotonic tensions and relaxions
In Table 1 we have all local (homotonic) tensions and relaxions between the functional fundamentals of two chords, that is to say, independently of a possible tonal memory (tonal field) in which they could be immersed (remember that a capital letter represents an M3, for Example C = CE, having its fifth or not).

In case these chord progressions occur in a musical neighbourhood where the tonic is very established, the tonal tension/relaxion should be added (not to be confused with the homotonic htonal relaxion, which is simply a certain local jump between fundamentals). For example, the plagal cadence: the tonal relaxion (resting in the tonic chord) has more force than the homotonic local tension between fundamentals (the fundamental jumps an upper fifth and therefore is a local succession of tension). In general, to know the overall tension between two chords in a piece of music, we have to take into account three factors: tonal tension (if we have an established tonic), homotonic local tension and sonance tension. The sonance tension simply means that the passage from a dissonant chord to a consonant one produces relaxion, tension in the opposite case.

Tabla 1

When the chords have more than one fundamental (functional or not) the issue is somewhat complicated, but as a general rule we can say that if the fundamentals form homotonic distension in both directions, the sequence will not have as much tension or will have relaxion in the two directions. This will depend on each specific case.

For example, Table 2 is the result of applying Table 1 to all link possibilities between major and minor chords. An arrow in both directions means that there is homotonic relaxion in both directions, dashes without arrows mean that there is neither tension nor relaxion. To these local harmonic tensions the slight tension of sonance, also local, should be added when passing from a major chord to a minor one (since the minor chord is less consonant than the major chord) or relaxion in the opposite sense. In the case of double arrow, the notes in the bass (and also in the soprano) can be decisive.

Tabla 2

It is not necessary to pay much attention to the apparent complexity of Table 2, as we say is simply the result of applying Table 1, which is the one really important; and, within it, the two main relaxions stand out: the htonal and the Phrygian ones.

As we have been saying, the tonal memory also influences (and much). For example, the authentic cadence in the minor mode actually has homotonic relaxions in the two directions; in F minor (C-A¬_f) we have htonal and tonal resolution (C|f), but in the opposite direction we would obtain the Locrian resolution (A¬|C); if the tonic is F, we have more rest in the F minor chord, but if F is not a clear tonic, if we only play these two chords —detached from a tonal field— we can rest in either.

When we have this equilibrium, a variation of sonance can decant the balance. For example, by placing the F minor chord in second inversion (figure 39).

We will insist in saying that the global tension between two chords is always the sum of the homotonic, sonance and tonal tensions. Now we are dealing only with the homotonics, that acquire more force in weak or indeterminate tonal fields. The tonal tensions will be studied in the next chapter.

Fig. 39

4.5 Homotonic relaxions and tonal axis theory

If the Locrian homotonic resolution was in the opposite direction, we would have obtained a perfect demonstration of the tonal axis theory proposed by Ern Lendvai (1955 and 1993), which, according to this author, was often used by Béla Bartók in his compositions.
But we shall see that, even so, the formulation of these homotonic relaxions gives sufficient consistency and reason to this theory, or, if one means otherwise, the two agree on many points.

Lendvai's theory, in short, comes to say that we can divide the 12 notes of the equal-tempered chromatic scale into three groups (axes) of notes (which can also be understood as chords roots or tonal regions) and that between them we found similar relations to those of tonic, subdominant and dominant.

The notes of each group are separated by a m3 or a tritone or, in other words, each axis would form a diminished seventh chord (although the theory goes beyond the diminished seventh chord).

For example (that the group is tonic, subdominant or dominant will depend on the compositional discourse):

- tonic axis or group: C, A, F{, E¬

- subdominant axis or group: F, D, B, A¬

- dominant axis or group: G, E, D¬, B¬

If we join these notes with a line in the circle of fifths, these groups will form 3 perfect squares.

If we understand a «dominant» as a note or a chord resolving in a «tonic», according to tonal axis theory, the notes or chords of the «dominant» axis «resolve» in notes or chords of the «tonic» axis (and the same with the tonic-subdominant and subdominant-dominant axes). Well, our homotonic theory largely coincides with the tonal axes one.

That is, according to tonal axes theory, we have the following resolutions or relaxions: G|C, G|A, G|F{, G|E¬, E|C, E|A, E|F{, E|E¬, etc. If we look at Table 1, we will see that they coincide with the htonal, Phrygian, Dorian resolutions and fail with Locrian. However if we look at Table 2, we can see that if we take minor chords, the «Locrian sequences» are in equilibrium (due to the htonal relaxion that occurs in the opposite direction). In addition, Lendvai's theory divides each axis into two tritones, if we only take the resolutions between the «primary» tritone of each axis, we get our main htonal and Phrygian resolutions.

Theories also coincide in considering that between notes or chords of each axis there is neither (local) tension nor relaxion.

The coincidence between relaxions of the two theories is reflected in figure 40. If we apply the htonal, Phrygian and Dorian relaxions to points of a square, we obtain points of the following square. If we only take into account the htonal resolution, the squares (tonal axes) tend to rotate and resolve counterclockwise —towards the square (axis)— a lower fifth. As we have said, it only fails with the Locrian resolution since, if it is applied to the points of the same square above, it gives points of another square, which is not drawn in the figure (the GED¬B¬ axis).

Fig. 40

The similarity between these two theories will help us, when we study tonality, to identify tonal chordal functions in complex tonalities or in remote but transient modulations.

Going back to our theory, if chords have several functional fundamentals, as a rule, the homotonic local tension-relaxion logic between chords will follow the logic of tensions-relaxion between fundamentals, always keeping in mind the tonal field that is being created, which can quickly change the overall relaxion between chords. The utility of local homotonic relaxions is to find a fluid sequence of chords (rather than to look for cadences), especially in chromatic fragments where the tonal vectors change rapidly; in these cases the tonal field weakens and the homotonic and sonance tensions gain weight. Of course, if that is what we want in our composition, since we might be interested in the opposite, namely, to obtain sequences of chords always in tension, both tonal and homotonic or sonance; in this case we will continue applying the same principles, but in the opposite sense.

Chapter 5