6. Examples of harmonic progressions with homotonic relaxions

In this chapter we will see examples of (local) homotonic relaxions between fundamentals in weak tonal fields, that is, sequences of chords that create a continuously fluctuating tonality so tonal tensions do not influence much because they are weak. It is in this situation that homotonic relaxions acquire greater meaning, in spite of the chromatism of the chord links.

An arrow will indicate a htonal or Phrygian relaxion between fundamentals.

The examples can be heard in a playlist on youtube (https://www.youtube.com/user/llbape/playlists).

6.1 Relaxions between chords with a single functional fundamental

In this section we will give examples of htonal relaxions using major, minor and dominant chords with a single functional fundamental, that is, only containing an interval of M3 or tritone (or both).

6.1.1 Htonal relaxions following the circle of fifths

In Example 6-1 we can see htonal relaxions between major triads following the circle of fifths.

Ex. 6-1 (Listen on youtube)

In Example 6-2 we have htonal relaxions between minor triads following the circle of fifths. The chords are in first inversion but with the functional fundamental in the bass. If other inversions were used, the effect of homotonic relaxion between chords would continue to occur in the same way, perhaps adding possible sonance tensions or relaxions if the inversions of the chords were different between them.

Ex. 6-2 (Listen)

Example 6-3 shows htonal relaxions between dominant seventh chords following the circle of fifths.

Ex. 6-3 (Listen)

In Example 6-4 we have htonal relaxions between minor seventh chords (or, in other words, major triads with the sixth) following the circle of fifths. They are also in first inversion with the functional fundamental in the bass.

Ex. 6-4 (Listen)

In these four previous examples we have seen sequences of relaxion between chords with the same structure and therefore do not intervene other tensions like those of sonance (the tonal tensions act very slightly, since their vectors are changing continuously). But any combination of intervalic structures in these chords, using any inversion, would be possible without losing the relaxed sense produced by the homotonic progression. As, for example, the progressions in Example 6-5.

Ex. 6-5 (Listen)

The consecutive htonal successions are very abundant in the musical literature. In the following chapter we will see many fragments, like Examples 7-1, 7-2, 7-3 and 7-5 of Beethoven and Brahms.

6.1.2 Phrygian relaxions following the 12 tones of the chromatic scale

In this section we will make examples of Phrygian relaxions using major, minor and dominant chords with a single functional fundamental.

In Example 6-6 we can see a progression of major triads (some in first inversion to avoid parallel fifths or octaves, although in Phrygian successions the parallel fifths do not sound bad, because they are chromatic) whose fundamentals are traversing the 12 tones Phrygianly.

Ex. 6-6 (Listen)

In Example 6-7 we hace done Phrygian successions using minor triads (in first inversion).

Ex. 6-7 (Listen)

Between the chords of Example 6-7 a small local tonal tension is continuously produced —or a (tonal) relaxion if the sequence were in the opposite direction— since any two contiguous chords contain notes corresponding to a 7M3 structure (we have a M2 interval between fundamentals, see 1.7.1). For example, between the first two chords we find AC{G and therefore the two chords create a tonal vector toward D, which is the dominant tone of the G minor chord (the first one), so there would be a micro-tonal rest in this chord if we were playing these first two chords repeatedly.

In Example 6-8 we can see Phrygian successions between dominant chords, the second one with a virtual fundamental.

Ex. 6-8 (Listen)

In Example 6-9 I use minor seventh chords, alternating root positions with first inversions. We also have slight tonal tensions (due to the tonal relaxions that are observed in the opposite direction).

Ex. 6-9 (Listen)

As before, any combination in the chords structure (and in any inversion) is possible without varying the sensation of Phrygian relaxion, as long as the step between fundamental is Phrygian; as in Example 6-10. To the relaxions appearing in it a sonance relaxion should be added at the beginning of each bar, since we have a more consonant chord.

Ex. 6-10 (Listen)

These Phrygian relaxions are relatively abundant in music literature. Similar sequences can be seen in Brahms symphonies (Examples 7-6 and 7-18) or in Chopin (Example 7-19).

6.1.3 Combinations of htonal and Phrygian relaxions

Fluid chromatic chord sequences can also be obtained using combinations of htonal and Phrygian relaxions, as in Example 6-11, in this case using only major triads (with inversions in the 2nd bar).

Ex. 6-11 (Listen)

In Example 6-12 we have a progression similar to the previous one with a mixture of more coloured chords, but the functional fundamentals continue forming htonal and Phrygian relaxions.

Ex. 6-12 (listen)

Or as in Example 6-13 that harmonizes an ascending whole tone scale in the soprano voice using htonal and Phrygian sequences between functional fundamentals.

Ex. 6-13 (Listen)

6.1.4 Using the full palette of chords
In the previous sections we have seen only a small part of the possible sequences of chords with htonal and Phrygian homotonic relaxions. In Table 3 we have a

summary of htonal and Phrygian relaxions between the main chords with a single functional fundamental (still more chords, more exotic and dissonant, could be shown).

This table expands Tables 1 and 2, which incorporated only major and minor triads, although they also contained secondary relaxions.

Table 3

Any chord with C as functional fundamental can link to any chord with fundamentals F or B (functional fundamental or not, uppercase or lowercase) producing homotonic relaxion although in some cases tonal and sonance tensions can be produced (all applied to all 12 notes and their transpositions —C, of course, is an example).

Sonance (consonance/dissonance) of chords is another kind of harmonic tension. A chord can be resolved homotonically towards another more dissonant chord, as in Figure 65. We have homotonic resolution against sonance tension (the harmonic resolution chord is more dissonant). Many times resolution by consonance can "win" in relaxion to an homotonic resolution, but using homotonic resolution against sonance tension can be a way of sweetening dissonant chords.

Fig. 65

In example 6-14 we can see a new sample of the use of htonal and Phrygian homotonic relaxions, using this time somewhat more dissonant chords: a chromatic sequence of chords within a weak tonal field (indeterminate before the final cadence).

Ex. 6-14 (Listen)

In Figure 66 we have represented some of the possible combinations of Table 3 where htonal and Phrygian homotonic relaxions take place. In Figures 65 and 66 we have indicated the inversions by simply placing a line below the symbol (see 3.4) (in the other examples we do not distinguish between inversions in the symbology).

In the next chapter we can see many examples of works by composers using htonal and Phrygian combinations, which are the most significant local homotonic relaxions.

Fig. 66

6.2 Relaxions with one or more functional fundamentals

6.2.1 Two functional fundamentals separated by a tritone (symmetrical dominant chords)
The chord formed by two M3 at a distance of tritone (see 3.1.5) is very interesting because it resolves homotonically towards other fundamentals in a htonal and phrygian way at the same time. If one fundamental resolves htonaly, the other resolves Phrygianly and vice versa. Also two tonal vectors are created (separated by a tritone) since the major third of one fundamental is the minor seventh of the other one and vice versa, that is to say, we have in this chord two 7M3 structures at tritone distance,

therefore is a useful chord for modulating to "distant" keys, as we have seen in Figure 64(e). From a classical point of view would be a chord consisting of two seventh dominant chords with a lowered fifth. The French augmented sixth chord has the structure of this chord, enharmonizing one of the minor sevenths as an augmented sixth.

A 7 over the fundamentals can be placed or not, since the seventh is implicit. To make these examples cleaner we have preferred not to draw it now although, if this chord is found isolated, it is preferable to do so.

In Example 6-15 we have a progression with intermediate resolutions towards major triads. A major chord also resolves homotonically to a symmetrical chord in a htonal and Phrygian manner at the same time. In the latter case we have harmonic relaxion, but tension of sonance.

Ex. 6-15 (Listen)

As this chord is symmetrical the progression can be made by changing the direction of resolution of the fundamentals. What was htonal is now Phrygian and vice versa:

Ex. 6-16 (Listen)

In order for the sonance of chords to be more similar, seventh chords could be used instead of triads, with the same homotonic result:

Ex. 6-17 (Listen)

Resolutions of this type can be seen in the next chapter, among others, Examples 7-16 (bar 137), 7-21 (bar 13) and 7-45 (bar 8) (Brahms, Chopin and Mompou).

We could also resolve in minor chords (example 6-18), that is, towards non-functional fundamentals (in lower case). Let us remember from other chapters that a fundamental in lower case can serve as a «resolutive» fundamental but not as fundamental «to be resolved» since it does not represent tensions of M3 or tritone; the one that has these tensions is the (upper case) functional fundamental of the minor chord (the fundamental that have its upper M3).

Ex. 6-18 (Listen)

But this chord can also resolve in another chord with the same structure. In this case four homotonic relaxions are produced at the same time (!). It is the link that occurs at the famous beginning of Tristán e Isolda by Wagner between the chords formed in the last eighth note of the third measure and the first eighth note of the fourth (see Example 7-26). Either of the two fundamentals is (htonal or Phrygian) relaxion of any of the two previous fundamentals. In Example 6-19 we have only put two arrows, but in reality they should be four.

Ex. 6-19 (Listen)

We could also resolve this family of chords using virtual fundamentals in the resolution chords and in general utilizing all the types of chords that appear in Table 3. In Figure 67 we have some examples. In this figure we only put an arrow although the resolution is double, and we also mark the inversions.

Fig. 67

6.2.2 Two functional fundamentals separated by a M2 (dominant chords family)

The unitonal dominant chords contain a single 7M3 structure and their natural resolution is towards chords that have the fundamental pointed out by the tonal vector they create (the two fundamentals are the local dominant and subdominant of the fundamental to be resolved). They can also relax Phrygianly. Thus, a chord type C^B¬ can resolve or discharge the tension in chords having the fundamental F/f or B/b. Secondarily, taking into account B¬, they would also provide a certain relaxation the fundamentals E¬ and A, as htonal and Phrygian resolutions of the second fundamental B¬.

As happens with all chords that have two functional fundamentals, they can resolve in another chord of the same family producing a double htonal or Phrygian relaxion, as in Example 6-20, where we have double htonal relaxions and we take advantage of a double Phrygian relaxion in the third bar to return to the chord of the beginning and resolve at the end in the "tonic" functional fundamental E¬ while A¬ resolves secondarily in C (lower case) according to a Locrian relaxion.

Ex. 6-20 (Listen)

The Tristan chord also belongs to this family of unitonal dominant chords (F} enharmonized to E{), with the characteristic that its main fundamental is virtual.¹ In Example 6-21, in the first bars, we have a sequence of chords with the same "Tristan structure" (with double htonal homotonic relaxion). In the link between mesasures 3 and 4 it is the secondary fundamental that resolves Phrygianly (C|B) in another dominant chord, which resolves to "its tonic" as in the previous example (but with another inversion of the chord taking advantage of the bass melodic Phrygian relaxion: A-G{). The process of tonicization of E is also facilitated by the melody of the soprano voice.

Ex. 6-21 (Listen)

These dominant chords with two fundamentals also link well with the dissonant augmented chords (type C^E) since it is the resulting chord if we make a htonal leap from one fundamental and a Phrygian leap from the other fundamental, as can be seen in Example 6-22. Here we have also placed the fifth of the main fundamental (dotted on the symbol) and in the case of the last chord we have placed both fifths of the two fundamentals. This last chord is quite dissonant, but at the same time, harmonically, it is a very resolution chord (in spite of its tonal instability) because of the htonal and Phrygian homotonic relaxions between fundamentals that are formed with the previous chord. We do not need to repeat here the difference between sonance resolution and harmonic or tonal resolution.

¹ In my book La convergència harmònica, 1994, I devote a whole section to show examples of the use of the Tristan chord in 14 musical works, with their different resolutive combinations.

Ex. 6-22 (Listen)

In Example 6-23 we have a sample of relaxed Phrygian progressions between chords of the same family, setting in the middle (measures 2-3) a htonal relaxion to break up the monotony of the descending Phrygian movements; ending the fragment with a chord of two fundamentals of the family of major chords, which we will see in the next section.

Ex. 6-23 (Listen)

The relaxed combinations of this chord with other chords following the htonal and Phrygian resolutions between fundamentals can also be very varied. In Figure 68 we can see some examples.

Fig. 68

In the following chapter we can see resolutions of this type, among others, in the Symphony No. 4 of Brahms (Example 7-17, bars 229, 233, 241, 243-246), Wagner's

Wesendonk lieder (Examples 7-30 [bar 78] and 7-31 [bars 5-7]) or Rachmaninov's Prelude (Example 7-37).

6.2.3 Two functional fundamentals separated by a P5 (major chords family)

These chords have no tonal tension because they do not contain the 7M3 structure and in addition the second fundamental is the main harmonic (the fifth) of the principal functional fundamental. They have the relative tension of the two M3 they contain, especially that of the main fundamental and the sonance tension of the m2/M7 interval being formed.
As usual, tensions can be solved htonally or Phrygianly (secondary resolutions apart). Using htonal relaxions between chords of this family we obtain a curious and sweet progression of fourths and fifths, as shown in example 6-24, broken at the end by a Phrygian relaxion (D¬^A¬|C^G) that gives it a cadential air, surely reinforced by the Locrian secondary relaxion that is formed (A¬|C).

Ex. 6-24 (Listen)

A similar but more chromatic progression can be seen in Example 6-25, this time resolving in a symmetrical dominant chord by means of the htonal and Phrygian relaxion (at the same time) of the second fundamental without changing the main fundamental (C^G|F{_C) which resolves Phrygianly (C|B) (and the other htonally [F{|B]) by returning to a chord of the same family we are studying (F{_C |B^F{).

We take advantage of the tonal tension of the symmetrical dominant chord adding at the end one of its two possible resolution tones in the soprano voice (obtaining the local leading tone - tonic sequence), which gives this semi-conclusive air despite the dissonance of m2 sounding in the upper voices.

Ex. 6-25 (Listen)

In Example 6-26 we resolve this chord several times into minor chords by means of a htonal relaxion between functional fundamentals. In measure 3 we use a symmetrical intermediate chord (D¬_G) and a major-minor type chord (C_E¬) (blue chord). In order to make a somewhat more conclusive ending —in this short fragment without a clear key— we end with a dominant chord resolving again in a chord of the family we are studying in this section, that is, major with two fundamentals separated a fifth, in this case —to give color— in second inversion.

Ex. 6-26 (Listen)

Harmonically relaxed htonal and phrygian combinations of this family of chords, like the previous ones, are very varied, as shown by some examples in figure 69.

Fig. 69

6.2.4 Two functional fundamentals separated by a M3 (augmented chords)

These chords are symbolized with two capital letters, but all chords with two fundamental at a distance of M3 actually have another hidden functional fundamental, also separated by a M3. Therefore, in fact, these chords would be symbolized with three capital letters representing the three fundamentals. For practical reasons, we place the two that best fit the tonal or harmonic context in which they are immersed, but this third fundamental must always be taken into account.

This chord, in its simplest version, that is, with three fundamentals without fifths, is a chord of three notes that divides the octave into three equal parts. Of the three fundamentals there is no principal one, unless one has his fifth. From a scholastic point of view it is a triadic chord with an augmented fifth, but, without a tonal context of reference, any of the three fundamentals could be the augmented fifth/minor sixth.

We could consider this chord as a chord that shares simultaneously six appoggiaturas (or suspensions if they come from another chord) to six major or minor triads and therefore one of its possible relaxions would be the resolution of the appoggiatura to one of these six chords, as shown in Figure 70. Actually, using these links there are only two different types of relaxions, raising or lowering by semitone some of the three fundamentals, which result in a htonal relaxion or in a Locrian one (resolved melodically Phrygianly).

Fig. 70

If we resolve htonally the two (three) fundamentals of this chord, we obtain another chord of the same family (as it happens in all other types of chords since the intervalic relation between fundamentals remains the same); if the two resulting fundamentals are then Phrygianly resolved, we get the chord of the beginning, as shown in Example 6-27 (the beginning of each measure have the same chord). In this example we have, therefore, the harmonization of a melody using only two chords, except for the final chord where we have added a lower minor third; this ending chord remains in the same family since no new functional fundamentals are created, but acquiring genes of F minor.

Ex. 6-27 (Listen)

If we resolve htonally one of the fundamentals of the augmented chord, and the other resolves Phrygianly, we obtain a chord of the dominant family. If we do the same with this resulting dominant chord, that is, one fundamental resolves Phrygianly

and the other htonally, we again get a chord of the augmented chord family, as we have already seen in Example 6-22. Depending on the fundamental resolving Phrygianly and the one resolving htonally we can also obtain again the original chord, as shown in Example 6-28.

In this example we have added to the augmented chords the fifth of one of its fundamentals, which gives us a more dissonant chord since there is a minor 2nd collision between two notes of the chord. The example is finished, in last bar, with one of the relaxions (appoggiaturas) that appear in Figure 70, in this case resolving in a minor triad with (minor) 7th or, if it is to say in other words, resolving in a major triad (~B¬) with an added lower minor 3rd (of the fundamental) (in this case in the bass).

Ex. 6-28 (Listen)

In Debussy's Proses lyriques Example 7-39 we can see, between bars 14 and 17, a Phrygian succession of augmented chords (the two —three— fundamentals resolve Phrygianly).

As with the other chords, the possible relaxed combinations of this chord, by means of htonal and Phrygian resolutions between fundamentals, are very varied. In Figure 71 we have some examples.

Fig. 71

Also we can see, among others, resolutions of a chord belonging to this family in Examples 7-46 (bar 19) and 7-47 (final) (Mompou and Schoenberg).

6.2.5 Other chords

Of the chord families with functional fundamentals, only the family with two functional fundamentals separated by a m3 (major-minor chords) and the family with two functional fundamentals separated by a semitone (cluster chords) would remain to be studied. They are dissonant chords, but we can "soften" their sonance if we make the auditory system clear what is the "identity" of each of their notes.
The major-minor chords have two functional fundamentals at a distance of minor third or, what is the same, are formed (at least) by a major triad and a minor triad, both starting from the same scholastic root (e.g. CE¬E}G, which is symbolized with E¬_C), which gives two functional fundamental separated by a m3 (CE/E¬G). One is the fundamental of the two major-minor triads (C) and the other is the functional fundamental of the minor triad (E¬_c), that is, a minor third above the first fundamental C.

The most common use of this chord is incorporating the fifth of the upper fundamental as this gives support to it and sweetens the sonance, although the chord has more notes (e.g. CE¬E}GB¬). However, in this layout of five different notes (three complete triads with fifth) a 7M3 structure is created (the main fundamental obtains the minor seventh), which causes the chord to acquire "genes" of dominant chord. In this way this chord is usually used in jazz, particularly in blues; although always with the main fundamental in the bass.

In Example 6-29 we have this chord at the beginning of each bar and we do htonally resolve its second fundamental towards the lower fundamental of a chord of the major chords family (as we saw in 6.2.3). The last chord is the sum of a major-minor chord and an augmented chord, with three functional fundamentals (four if we consider the third hidden fundamental of the augmented chords). In isolation it is a very dissonant chord (we have the cluster G-A¬-A}), but here, as it proceeds from a double htonal homotonic relaxion, the ear somehow "justifies" its dissonance because it perceives internal harmonic resolutions from the previous chord. This last chord of 6 different notes has genes from three families of chords: that of the major (F^C, which gives stability and sustains the chord), that of the augmented (A¬^C) and that of the major-minor (A¬_F).

Ex. 6-29 (Listen)

The members of the cluster chords family are very dissonant because they have two functional fundamentals separated by a m2 and, having only one semitone of difference between them, the ear does not perceive any harmonic relationship, except a very remote relation between a fundamental and its fifteenth harmonic or, rather, the fifth harmonic of his third harmonic (the major third of his fifth). In addition we have, at least, two semitone shocks: between its fundamentals and its thirds (as long as its fundamentals are not virtual). If any of its fundamentals are virtual, the dissonance is significantly reduced (see, for example, the resolution of the cluster chord in bar 355 of the first movement of Brahms Symphony No. 2, Example 7-12).

Of the two htonal homotonic resolutions of the two fundamentals, the relaxion of the second fundamental (the fundamental corresponding to the lower semitone) produces more relaxion than the other, because the fundamental to which it resolves is closer harmonically. For example, if we take C^B, these two fundamental resolves htonally in F and in E. C accepts the resolution B|E because E is one of its main harmonics, whereas B «does not like» the resolution C|F because F is at tritone distance with respect to B and forms with it a «false-fifth». Therefore, the resolution C^B|E is more relaxed than C^B|F (apart from having the Locrian relaxion C|E, see Brahms' previous example). In the symbology we put the B above because it is a harmonic of C, but functionally it would also be good to put B^C. The symbols of the cluster chords can be put either way, that is, by placing the upper seventh major or the upper second minor above.

In Example 6-30 we have a succession of cluster chords with Phrygian and htonal relaxions. We take advantage of the strongest htonal relaxion of the second fundamental to resolve the penultimate chord to a fundamental based on F{ (C{|F{) resolving the other fundamental Phrygianly (D|C{), which, in addition, gives us a chord of the family of major chords (F{^C{), suitable for an ending. But this would give us a too consonant and resolutive ending that would contrast with the dissonances of the previous chords, so we have added to the final chord a third fundamental (another upper fifth) (G{: —F{/C{/G{—), which gives us a chord with genes from the family of the major chords (if we think from F{) and of the dominant

family (if we think from G{), in this example, the first configuration predominates, due to the fifth F{C{ in the bass.

Ex. 6-30 (Listen)

We have shown only two examples with major-minor chords and cluster chords, but, as in the other sections, the harmoniously relaxed combinations between these chords and other chords of other families remain very large. In addition, due to the dissonance in these two families, we can easily obtain an added relaxion by sonance resolution.

We would only have one family of chords to study, that of suspended chords. These chords are defined precisely because they are the only ones that do not have functional fundamentals, therefore they do not have homotonic relaxions between them, since the homotonic relaxions only resolve the tensions of the functional fundamentals (representatives of the "quasi-fifths" of the chord, M3 and tritone intervals). But they can be chords of resolution if they come after a chord belonging to other families.

Suspended chords are basically chords by fourths or fifths (up to three if there are four different notes)² also known as quartal and quintal chords. In the case of three consecutive fifths or fourths a very weak tonal vector is formed since in the chord we have included two fundamentals (two fifths) at a distance of M2. For example, let's take the chord FCGD. FC could be considered a subdominant part and GD a dominant part (without the leading tone) towards the key C. This chord could «resolve», then, with some relaxion, towards a tonicized fundamental C. As we have said, it is not properly a htonal homotonic distension, but it could be taken into account, as shown by the sequences of suspended chords of Example 6-31, where the chords «resolve» locally in this weak way that we have commented towards the first fundamental of the next chord. All chords are chords with three consecutive fifths (or fourths) (in different inversions), except the one in the end formed only by two fifths and therefore is a more stable and consonant chord. It would have been possible to put a simpler example —probably with more relaxion between chords— by putting the sequence of fifths in the bass (as in Examples 6-1 to 6-4 of the beginning). We leave it as an exercise for the active reader.

² If we put a fourth fifth, we would get a functional fundamental.

Ex. 6-31 (Listen)

Samples of this small dominant character that quartal/quintal chords have can be seen in Examples 7-33 (bar 29) and 7-38 (bar 18-19) (Bruckner and Debussy).

6.3 Locrian homotonic relaxion

We have already seen in Chapter 4 an introduction to secondary homotonic relaxions. In our opinion, the one that implies a greater relaxion is what I denominate Locrian relaxion, that is the fundamental jump of an (upper) M3. The first fundamental must be functional, the fundamental of resolution can be functional (upper case) or not (lower case), but the resolution fundamental should have its fifth since, as we have seen in 4.2, it was an important note when we did the demonstration by means of the resolution of the "quasi-fifths".

It is not a sequence too used in tonal music because it does not fit with the chords that are formed in the diatonic scales except in an important succession: the passage from tonic to dominant in the minor mode, taking into account the functional fundamental of the minor triad. In A minor we would be talking about the succession C_a|E. This sequence has a left-to-right (Locrian C|E) relaxion but a right-to-left (htonal E|a) resolution. The htonal relaxion is more powerful, but if in the tonicized chord we add a little tension of sonance, for example, putting it in second inversion, then the resolutive effect of the two relaxions may be similar, as shown in Example 6-32, where the piece rests cadentially in any of the two chords. In the last five bars of Vexilla regis of Bruckner (Example 7-35) we can see a variant of this sequence (E without the third), which is also introduced by the Locrian relaxion A¬|C in bars 30- 31.
Secondary relaxions are indicated by a dashed arrow.

Ex. 6-32 (Listen)

Therefore, in the minor mode, the authentic cadence using cadential æ, from the point of view of the fundamentals is a Locrian + htonal sequence and in the same minor mode the half cadence from the tonic is a Locrian relaxion.

In more chromatic passages we can link Locrian sequences with other chords in many ways. As the fragment of Example 6-33. All voices are moved by semitone and the descending progression may continue indefinitely, but in bar 8, instead of resolving Locrianly, we resolve the chord F_d htonally (F|b¬).

Ex. 6-33 (Listen)

Three successive Locrian homotonic sequences take us to the same starting chord (or to the same functional fundamental), since the interval M3 divides the octave into three equal parts. We have two samples in Example 6-34.

Ex. 6-34 (Listen)

In Example 6-35 we have Locrian relaxion between the two seventh chords of each bar combined with other Phrygian and htonal relaxions.

Ex. 6-35 (Listen)

Like the other homotonic relaxions, the possible Locrian sequences between chords are very varied. In Figure 72 we give some examples between chords with a single functional fundamental.

Fig. 72

Some of these sequences would also produce relaxion in the opposite direction, that is, if the two chords were played in the reverse order, since «false-fifths» or «quasi-fifths» are resolved in both directions.

In chromatic music, Locrian successions occur constantly, but often it can simply be something inevitable, the result of chance. Samples where it seems to take advantage of its local relaxing property (including the half cadence `i-V' in the minor mode) can be found, inter alia, in Examples 7-11 (bars 251-252), 7-12 (bars 355-357), 7-18 (bars 276 and 278), 7-28 (bars 18, 20 and 22), 7-35 (bars 25-27) and 7-48 (bars 41 y 45) (Brahms, Wagner, Bruckner and Schoenberg).

Chapter 7