7. Examples of homotonic and tonal analyses

In this chapter we will give examples of works by different composers that will show the utility of separating the chords according to the fundamentals that define them, as we have seen in chapter 3, since they provide us with information about their internal harmonic tensions, mainly the "quasi-fifth" tensions of M3 and tritone intervals. In turn, thanks to the structure of the significant harmonics for the auditory system (which, in relation to the fundamental, also form intervals of M3 and tritone), are fundamental tones representing the chord.

The local homotonic relaxions between chords (see summary in 4.4) are represented by an arrow. When they are secondary relaxions (Locrian or Dorian) or when the «quasi-fifth» tensions resolve in complex chords or in secondary fundamentals, we draw the arrow with a dashed line, although we do not always indicate the secondary homotonic relaxions, especially the Doric one, since they are very numerous.

We have looked for examples with great amount of homotonic relaxions in few measures. Searching in the musical literature is not at all difficult to find them, but normally these relaxed progressions between chords are not so concentrated.

To the right or below the title of the musical work we have placed a diagram of the homotonic relaxions of the analyzed fragment and below the last staff (in some examples on the first staff) we can see the detailed homotonic analysis. Sometimes, we have united sequences of chords with fundamentals in the same tonal axis with a "vertical square bracket line".

In most examples we have also added a line (or two) with the analysis of the tonal functions according to the functional symbology explained in 5.3. Sometimes we place a second line or another symbol below to show an alternative functional symbology. We also draw an arrow when there is tonal relaxion, which may or may not coincide with the homotonic one. We feel tonal relaxion basically when we hear the tonic chord (which may be local) or when music rests on the dominant (which can also be local). If the tonal relaxion is weak (because its tonal field is also weak), we also indicate this with a dashed line. Transient modulations are indicated with brackets and inside them we show the function of the chords from the point of view of the new tone, which is specified in the lower part of the first bracket. At the bottom of the second bracket, which closes the transient modulation, we put the new tone function with reference to the main tone/key. Sometimes we put the functional symbology from the point of view of the two tonalities, in two lines, one with brackets and the other without them.

Something similar happens with secondary dominants. The two main homotonic relaxions, honal and Phrygian, since they discharge their tension in the chord in which they resolve, are considered with dominant function (secondary or not) and when they do not resolve into the tonic, we put their function (D) in parentheses with a small arrow: (D)| in case of honal resolution, and (D')| in case of Phrygian resolution. Seen from the tonic, when they resolve into the dominant, they have a subdominant function and alternatively we can also put the symbols S§ (dominant of the dominant) and S¶ (phrygian dominant of the dominant), respectively (without the parenthesis since this is the function from the point of view of the main key).

In most examples, in order not to complicate the symbols, we do not distinguish between inversions of chords, since they do not normally vary their harmonic function (except in specific cases, such as cadential æ). In fact, we have only indicated the inversions, as explained in 3.4, in Examples 7-5, 7-33, 7-37 and 7-40 (apart from Examples 7-3 and 7-6 where all chords are in root position).

Nonchord tones like appoggiaturas, suspensions, passing tones, etc., have been indicated by changing the note-head with a white diamond figure.

If somewhere appears the equal sign (=) between two fundamentals, it indicates that they are on the same tonal axis (and there is no tension or relaxion between them).

As in the previous chapter, examples can be heard in a playlist on youtube (https://www.youtube.com/user/llbape/playlists).

* * *

At the beginning of the Symphony No. 7 of Beethoven (Example 7-1) we find five consecutive htonal relaxions and then a Phrygian one resolving into the dominant.

In bar 5 we have homotonic relaxion but tonal distancing (actually in bar 6, since G} of bar 5 only lasts an eighth note) since a tonal vector appears towards D (major). Something similar occurs in bar 8 towards C. In bar 10 we have homotonic and tonal relaxion, because it returns to the original key of A by the Phrygian sequence (D')|Dµ or, which is the same from the point of view of A: S¶|Dµ. (D') is the Phrygian dominant of the dominant, but has a subdominant function (S¶, Dorian sixth) from the point of view of the tonic or the main key.

Ex. 7-1 (Listen)

In the following examples we will focus on the symphonies of Brahms where there are plenty of homotonic relaxions.

Ex. 7-2 (Listen)

In the second movement of his first symphony (Example 7-2) we find six consecutive htonal relaxions between bars 85-87 —in bars 86-87 between chords with virtual fundamentals (diminished seventh chords).

Then follow Phrygian and htonal relaxions until reaching the tonic of this fragment (which is the dominant of the initial key in E). We have tonal relaxion with functional progressions similar to the previous ones in bars 89 and 90 (apart from bar 86), using the dominant of the dominant: (Dµ)|Dµ|T or, alternatively (taking as a reference the tonic B): S§|Dµ|T.

A fragment with 10 chords communicated by homotonic relaxions, including four consecutive htonals (five if we consider the htonal side that every augmented sixth chord resolution has), is found in the fourth movement of Symphony No. 1 (Example 7- 3a).

In bars 181 and 182 we also find two Locrian sequences (G|B) towards the dominant of this passage, in E. An identical sequence is heard beyond in bar 363 (Example 7-3b), this time in C. In both cases the succession begins with the Phrygian dominant of the dominant in its form of an Italian augmented sixth chord. As we have seen in previous chapters, augmented sixth chords also take advantage of the other tritone's tensing direction (all tritone has two possible htonal and Phrygian resolutions), so the minor seventh is enharmonized as an augmented sixth.

Ex. 7-3 (a) (Listen)

Ex. 7-3 (b) (Listen)

In this same movement we can also see three consecutive htonal sequences (Example 7-4).

Like all minor triad chords, D minor chord has two fundamentals. The non-functional fundamental D resolves the tension of the "quasi-fifth" C{-A of the previous chord. The functional fundamental of this D minor chord (F) creates the "quasi-fifth" tension A-F, which is resolved by the fundamental B¬ (fifth B¬-F) of the following chord, which in turn creates the "quasi-fifth" D-B¬, resolved by the fundamental E¬ (E¬-B¬) of the next chord. This is, as we have seen, how homotonic htonal relaxions work.

In bar 50, tonal relaxion is added to the htonal relaxion, in this typical local modulation towards F (S-...-D-T), with the intermediate suspended chord f~ ^c as appoggiatura of the dominant chord.

Ex. 7-4 (Listen)

Ex. 7-5 (Listen)

In the second movement of his first symphony we find similar successions. We have in bar 109 (Example 7-5) the «dominant of the dominant» in the form of a diminished seventh chord (\S§⁷⁹) reaching the subdominant by three htonal relaxions (F{-B-E-A), with some passing tones and appoggiaturas and the passing chord (E^G{), ending the fragment with the tonal cadence S-D-T. Every S-D sequence implies a

Dorian secondary homotonic relaxion, indicated in this example (although this secondary relaxion is usually not shown in the examples). The sequence of bar 110 (E-G{) could also be considered a Locrian homotonic succession, but here it is nor marked because G does not have its fifth (see 4.1).

Returning to the fourth movement of this symphony, in bar 368 we find four consecutive Phrygian homotonic relaxions, the first chords being in arpeggiated form occupying two bars each (Example 7-6).

Ex. 7-6 (Listen)

Brahms begins this movement (Example 7-7) with a dominant pedal point and above relaxed successions towards the dominant chord appearing at the end of each bar, forming a kind of deceptive cadence at the beginning of next bar (G-A¬). In all four bars the basic sequence of functional fundamentals is E¬|D|G. That is, using two secondary dominants, the first Phrygian and the second honal. E¬ resolves Phrygianly on the dominant of the dominant, which in turn resolves on the dominant. Seen from C minor, we have a functional sequence T¶|S§|D. In the second bar, if we do not consider D as a passing tone, we have a passing chord (B¬_g) that forms two Locrian sequences with the anterior and posterior chords. Something similar occurs in bar 4, in this case forming an augmented passing chord with Locrian relaxion towards the next chord.

Apart from the homotonic relaxions we have tonal and sonance resolutions at 

the end of each measure, when we reach the dominant chord (on the dominant pedal).

Ex. 7-7 (Listen)

Ex. 7-8 (Listen) 

In this same movement, from bar 69 we have a long series of htonal, Phrygian, Locrian and Dorian relaxions (Example 7-8). In bars 71-73, a tonal vector is formed towards A using the typical Phrygian-htonal sequence we have seen before F|E⁷|a (bars 71-72), that is to say, Phrygian dominant of the dominant, dominant, and tonic, one of the more repeated successions used in romantic and post-romantic music. Being a short transient local modulation I put the tonal functional symbols from the point of view of the original tone/key (C), in addition, being relative tones between them (C-a) they have coincidences in tonal function (they are in the same tonal axis). But we could also have put this fragment, as in Example 7-1, between brackets: _a[Dµ|t D|t]_Tp instead of D§ µ|T_p D§|T_p.

In the first movement of the same Symphony No. 1 we find again a dominant pedal and above some interesting progressions (Example 7-9). The sequences of functional fundamentals are A¬|G and B¬|A|D being repeated later a lower fifth from bar 281 (D¬|C and E¬|D|G). That is, the former go to the dominant (G) and the dominant of the dominant (D) and the second one to the (Picardy) tonic (C) and the dominant (G). From the second half of bar 284 the fragment is repeated an octave lower (only the upper voices). Brahms also uses sequences of chords with fundamentals within a same tonal axis, which, according to theory, produce neither tension nor homotonic relaxion.

Ex. 7-9 (Listen)

In bar 15 of the same movement (Example 7-10) we can see a chromatic fragment with a complex homotonic relaxions interlacing, sequences within the same tonal axis and melodic and sonance resolutions in off-beat chords. In bar 18, the functionalities are clarified with a tonal relaxion to the dominant (Gµ A¬|Gµ) followed, in bar 19, by a htonal relaxion towards the Phrygian dominant of the tonic (chord D¬), which lasts two bars, going afterwards to a dominant section in bar 21 (what could be scholastically understood as a dilated succession of Neapolitan sixth) (the example shows only up to measure 19).

Ex. 7-10 (Listen)

Let us now turn to the first movement of the second Brahms symphony (Example 7-11). From bar 248 we have three bars in B major and then a transition bar that leads to C (major and minor). The harmonic structure of bars 248-249 and 252-253 is the same, transposing the fundamentals an upper semitone. In both cases confirming its tonal vector (T D|T D|T).

Ex. 7-11 (Listen)

From bar 254 comes a long chain of homotonic relaxions to the dominant (G) of this transient tone.

In all chords with fundamental A¬, the minor seventh appears as augmented sixth (F{), as leading-tone of the dominant, manifesting the two faces of the tritone contained in the chord. The main fundamental resolves Phrygianly (A¬|G) and the other (hidden virtual one) htonally (\Dµ|G) (remember that \Dµ = F{C). Exceptionally, at the end of bar 251, the augmented sixth chord resolves Locrianly in the chord whose fundamental (C) will be the next local tonic, but with the dominant in the bass.

In bar 355 of this movement (Example 7-12) we find a chord of the cluster chords family resolving both htonally and Locrianly towards the D minor chord, which resolves again Locrianly to chord F{µ, preparing the entrance of the arpeggiated B minor chord in next bar, producing a slight local modulation to the relative tone of D.

Ex. 7-12 (Listen)

In Example 7-13 we have a long fragment of the first movement of Symphony No. 3. Here Brahms, htonal relaxions aside, uses chord sequences whose fundamentals are on the same tonal axes (i.e., they are separated by m3 or tritone intervals), which creates a great chromatism without this affecting too much harmonic tension, since in the theory of Lendvai (and partially in ours), fundamentals of the same tonal axis have the same tonal function and there is no homotonic tension or relaxion between them (but in this example there are differences of sonance, since we have chords with different intervalic structures). For example, at the end of the fragment, in bars 132-133, we have three different chords on the dominant tonal axis, resolving in the tonic in bar 134, the latter dominant chord being the most used and powerful one (Dµ) since it alone contains the 7M3 structure, which determines the key. The next one most used before the tonic is D'⁽⁷⁾ accompanied by D⁽⁷⁾ as we have seen and will see in more examples. In addition, in this example, during this sequence of chords in the same axis, the diminished seventh chord appears

many times, which is, by antonomasia, the chord with its four virtual fundamentals in the same tonal axis.

Ex. 7-13 (Listen)

Another long fragment analyzed is the beginning of the third movement of this Symphony No. 3 (Example 7-14). Again almost all chords are linked with homotonic relaxions. In this example we also mark the Dorian ones. In bar 3, note G in the melody, can be considered an appoggiatura of F minor or not (or, to put it better, both can be perceived together, that is why I have put the htonal relaxion E¬|A¬ with a dashed line). If it is not considered an appoggiatura, we have the added

relaxion gµ|c (which with the chord of the second beat of this measure forms the sequence: dominant|tonic). Will this double interpretation make it so pleasant to the ear?

Ex. 7-14 (Listen)

We can show more examples of almost consecutive homotonic sequences in the same movement, in bar 70 (example 7-15). Here Brahms uses a curious and remote succession of chords to finally reach the dominant of A¬.

Basically it consist of using (in bar 76) two htonal relaxions to reach B (C{|F{|B), which is the Phrygian dominant of the `dominant of the dominant' (B|B¬), but then, once they have arrived to the `dominant of the dominant' (subdominant S¶ seen from A¬), instead of linking directly with the dominant (B¬|E¬), it connects

with a chord of the same tonal axis at tritone distance (F¬), which is the Phrygian dominant of the dominant (subdominant S¶ seen from A¬), a function we have already seen used in so many previous examples (i.e. the augmented sixth chord). But this time, to resolve in the dominant, it uses an intermediate chord (appoggiatura), which is the tonic minor chord in second inversion or, scholastically, a cadential æ seen from A¬ (minor), although after hearing the dominant, music does not continue with the tonic A¬. This intermediate chord could also be seen as two Locrian relaxions to reach the dominant (F¬|a¬ and C¬|E¬).

Ex. 7-15 (Listen)

Let's go to the fourth movement of this symphony (Example 7-16). We depart from the dominant to return to the dominant using a long homotonic chain.

The harmonic sequences of the first system (bars 134-137) and the second one (bars 138-141) are similar. In both cases there is a small transient modulation to C¬ (F¬-G¬|C¬), returning to F¬ as a Phrygian dominant of the dominant (helped by B¬, the dominant of the dominant) (bars 137 and 140 ). That is, the combination (D+D')|D again. In the first system with a dominant pedal point.

If this fragment were considered a modulation to E¬ instead of being in A¬ (as always is a mixture of the two things), then the functional language would change the term `dominant of dominant' to dominant. The Phrygian dominant and the htonal dominant would resolve in the tonic. Note that using a scholastic language we would

be talking about converting the resolution of the French augmented sixth in a Neapolitan cadence (placing the correct notes in the bass, of course, we are thinking in fundamentals and chord-classes). That is to say, equaling inversions, the French augmented sixth is to the dominant what the Neapolitan sixth + dominant is to the tonic. In C would be: (A¬+D)|G, transposing a lower fifth we have: (D¬+G)|C. In both cases, we obtain the Phrygian and honal resolutions of the respective dominants (A¬ and D —Phrygian and htonal— dominants of G // D¬ and G —Phrygian and htonal— dominants of C).

Ex. 7-16 (Listen)

Example 7-17 of the first movement of Brahms Symphony No. 4 is very interesting. Without a clear key at the beginning of the first measures of this fragment it is a sample of how the homotonic relaxions can be useful to give fluidity and sense to chromatic chord sequences.

This fragment begins with the tonic chord (in major) and take advantage of the Phrygian relaxion to go to Eµ, in the following measures use chords of the same tonal axis to go to other regions using htonal and Phrygian relaxions.

In bars 229, 233, 241 and 243 we see the Phrygian resolution of the second functional fundamental of the chord (Tristan chord subfamily from the dominant chords family); and from bar 243 to resolve on the dominant in bars 246-247. It is

a variant of the typical Phrygian resolution to the dominant that we have seen in the other examples. In addition, bars 232-235 are a progression (an upper tone) of bars 228-231. The same thing happens between bars 236-237 and 238-239 (to htonally reach the dominant in bar 240). When, at bars 240-241, the same progression seems would be repeated, Brahms changes the chord in bar 241 in order to initiate again the cadence to the dominant.

Ex. 7-17 (Listen)

In the third movement of this same symphony we reach the dominant after five consecutive Phrygian relaxions (bars 278-281), after the tonic chord (in second inversion, in bar 278) has directly linked with another chord of its tonal axis (G¬). In bars 275 and 277 we have two clear Locrian relaxions (A¬µ|C). If we look at the bass line, from bar 276 we have eight melodic Phrygian movements until we reach the dominant of the dominant (D).

Ex. 7-18 (Listen)

Let's go to Chopin's music now. In his Mazurka Op.17 No.4 we can also see a Phrygian sequence of fundamentals until we reach the dominant (Example 7-19).

Ex. 7-19 (Listen)

Most of Prelude No. 20 is intertwined with homotonic relaxions (Example 7-20). The third system is a repetition of the second. In this example we place the symbology of the chords on the first staff and below the tonal analysis.

All bars have tonal distension in the last beat of the bar, either as a tonic or as a local half cadence (except perhaps in bar 5 [and 9] when the last beat rests in the minor dominant after a htonal homotonic relaxion). To these relaxions in last beats the very important sonance relaxion (in almost all bars) has to be added due to the resolution of the appoggiatura in the previous beat.

In bar 6 (and 10) we find the formula (D+D')|D that we have already seen (in the form of French augmented sixth) and at the end the formula (D'+D)|T, which does not become Neapolitan cadence because D' is not in first inversion (it does not have the subdominant in the bass).

Ex. 7-20 (Listen)

In Example 7-21 we have a homotonic analysis of its Nocturne Op. 27 No. 1. See again (in bar 13) the use of D+D' to reach the tonic by means of its characteristic double homotonic resolution, sum of a Phrygian resolution and a htonal one.

Ex. 7-21 (Listen)

In the last bars of Etude Op. 25 No. 4 by Chopin (Example 7-22) we have an ending with a Phrygian relaxion directly on the tonic (with Picardy third). A tonic pedal could be considered in this passage. If we do not consider A a pedal note, a

Locrian relaxion would be added to the Phrygian one, from the second fundamental of the previous chord (F|A).

Ex. 7-22 (Listen)

Example 7-23 shows the beginning of Mozart's Quartet No. 19, nicknamed «Dissonance». To the chord A¬ of the beginning follows (with the entrance of the first violin) a «Tristan» chord-class (\F^E¬, being virtual fundamentals F and A¬ in the same tonal axis). The second fundamental Phrygianly resolves to the dominant of the dominant (S§ €) and then htonally towards the dominant (Dµ|G —A^G as an appoggiatura chord—). In bar 5 music apparently move away from key C by repeating the beginning a lower M2, forming a chord G¬ with the entrance of the viola (the link is Phrygian), but only apparently because, repeating exactly the same notes as the three first bars (a lower second), in bar 7 we come back to the key of C when we hear the fundamentals corresponding to the dominant and the subdominant, and therefore we hear again the 7M3 structure of C.

Ex. 7-23 (Listen)

Ex. 7-24 (Listen)

In the work The Swan by Camil Saint-Saëns (Example 7-24) notes in bars 1-2 and 5-6 are the same (apart from the tail of the cello theme in bar 5), basically within the tonic chord. In first instance the tonic chord resolves htonally towards the (S_p type) subdominant (G|C_a, in bars 2-3), but the second time resolves Phrygianly (G|\F{⁷⁹-F{µ, in bars 6-7) to the chord that later will be dominant of B minor (F{, Phrygian subdominant S' seen from G), making fit the second part of the cello theme within this new harmony (F{|b) retaining the melodic drawing, although the intervals are not exactly the same.

In Example 7-25 we have the beginning of Catalonia by Albéniz. The first bars show (under a tonic pedal point) different types of htonal resolutions from the dominant to the tonic. From bar 9 there is a local modulation to the relative tone/key B¬ (the relative tones are always on the same tonal axis), and in measure 12, it takes advantage of usual htonal sequences in the new key (F|B¬|E¬) to make a Phrygian relaxion towards the dominant D of the initial key (E¬|D). Typical cadence of the minor mode.

Ex. 7-25 (Listen)

Now let's inquire into Wagner's music. As it could not be otherwise, we will begin with the beginning of the Prelude of Tristan and Isolde (Example 7-26), perhaps the musical fragment of the whole history of music that has been more often analyzed by theorists, with numerous divergences between them.¹ Many times they are not divergences but the consequence of the fact that the ear listens to several harmonic structures at the same time.

If we take a look at the fundamentals of the example, this beginning is an entanglement of htonal and Phrygian homotonic relaxions towards the local dominant of A minor (B|E and F|E).

We do a parenthesis to say that, harmonically, enharmonic notations could be used in several notes; Wagner uses the ones that best fit melodically, but both coexist at the same time. For example, in bar 3, there is no doubt that it is better to put F on bass, but the ear also listens this note with its function as E{, forming part (as local leading tone) of the chord (C{)E{G{B(D{) and which would melodically link with note A{ of the soprano in the downbeat of the next bar (E{|A{). But, at the same time, this A{ could also have been written as part of the chord B¬D(F¬A¬) as the htonal resolution of the previous chord (the last eighth note of bar 3) (Fµ|B¬).

Closing the parenthesis; in the last quaver of bar 3 and the first one of bar 4, two chords of the symmetrical dominant family (two 7M3 structures at tritone distance) are formed. This succession has the characteristic that four homotonic relaxions are produced at the same time (!), two Phrygian and two htonal (F|B¬, F|E, B|B¬, B|E), thus greatly softening the entry of bar 4 appoggiatura to the dominant chord. One might even consider the Tristan chord of bar 3 as a long appoggiatura to FBD{A chord and summarize this beginning of the prelude as the only two-chord link (F^B|Eµ), which is no more than the resolution of the French augmented sixth towards the dominant, i.e. the simultaneous htonal and Phrygian resolutions (F|E and B|E). In addition, secondarily, almost as an added anecdote, we have the Locrian relaxion of the virtual fundamental of the Tristan chord to one of the fundamentals of this augmented sixth chord (C{/D¬|F).

Ex. 7-26 (Listen)

¹ Der Tristan-Akkord und die Krise der Moderner Harmonielehre. M. Vogel (1962).

In the introduction of the third act of the same opera we have a long homotonic chain to reach the dominant of the relative tone/key of A¬, passing through intermediate chords with htonal and Phrygian relaxions. In addition, the bass walks descending steps of m2, Phrygian melodic movement.

Ex. 7-27 (Listen)

In the Pilgrim's chorus from Tannhäuser, from the end of bar 17, we have three progressions (of two bars, beginning in the last beat) that repeat exactly at a distance of m3, that is, they go through the tonal axes. The first two resolve htonally (F=A¬_f|D¬ and A¬=C¬_a¬(B)|E), but the third progression does so Phrygianly (B|B¬), to go to the dominant. Within each progression we can observe two Locrian relaxions. The return to the dominant (in bar 25) and to the initial key in order to repeat the progressions again consists of two new htonal and Phrygian relaxions from the same mentioned dominant (B¬µ|E¬¼C¬_a¬ |B¬).
Ex. 28 (Listen)

Note that in all the examples we have been showing in this chapter, Phrygian resolution to the dominant is more common than the htonal one (as dominant of the dominant). We have placed tonal functions, in this and in some other examples, in two lines, the first one is the symbology from the point of view of the key of the piece (E¬), and the second one an alternative notation that indicates, in this case, the functions within each transient modulation (in brackets, to D¬, F¬/E and G).

In the Wesendonk lieder (of Wagner), we also find numerous homotonic sequences.

Ex. 7-29 (Listen)

In Der Engel, broadly, we find the homotonic chain G_e|C|F|E|Aµ (3 htonal, 1 Phrygian), to go to Aµ, that is, to go to the dominant of the dominant (S§ from the point of view of G), though then does not immediately resolve to the dominant (we have to wait three minims, up to bar 12).

Note B in the melody in bar 10 could be considered appoggiatura of A; If we consider B to be part of the chord its function does not vary, because both the chord

A^G and the chord Aµ contain the 7M3 structure (in this case towards D); as we have seen in 3.1.3, both belong to the unitonal dominant chords family. The resolution of A^G in Aµ is basically by sonance (the second is more consonant), but one could also speak of a weak Dorian relaxion between fundamentals (G|A).

In Stehe still!, from the same lieder collection, we find a homotonic chain consisting only of htonal relaxions (Aµ|Dµ|G^F|Cµ|F_d ...Gµ|C).

Using scholastic language we may say that it begins with the secondary dominant of the `dominant of the dominant'. Using our symbology we write (Dµ)|S§ µ. Continues, in bar 78, resolving towards the dominant (with the subdominant as second fundamental) (S§ €|D^S), to resolve in the tonic chord with the minor seventh, which in turn resolves htonally in (scale 2nd degree) S_p (D^S|Tµ|S_p), considering notes E of bar 82 appoggiaturas of notes D of S_p. Finishing the fragment with the authentic cadence (Dµ|T).

Ex. 7-30 (Listen)

We find more homotonic sequences in Matilde Wesendonk lieder; for example in Im Treibhaus (Example 7-31), also to reach the dominant (harmonically the part

of the piano is almost identical to the introduction of the third act of Tristan, although at 6x8 metre instead of 4x4).

The piece starts with three htonal resolutions to the tonic chord from a chord of the dominant chords family (\\C^B¬, «Tristan» chord-class) (resolution to the functional fundamental of D minor chord: \C^B¬|F_d).

As we have seen in 5.3 we have two alternatives to functionally symbolize the minor tonic chord. One is writing t and the other writing T¶_p (since its functional fundamental is the upper relative major tonic, in this example F=T¶ of D minor). In this case perhaps it would be more convenient to put T¶_p, since we come from its subdominant (S¶ ) and from its relative dominant (\D¶ ) and in fact, in these initial moments, the tonal vectors point more to F than to D.

In bar 9 we reach Phrygianly the dominant with the sequence C^B¬|F_d|B¬|A. The homotonic string until the dominant (Aµ) of bar 13 is even longer (and more complex and chromatic), it is not necessary to write it here, it is the one that appears in the example from chord F in the 2^nd beat of bar 9 (in first inversion). Note also, from bar 10, that there are melodic Phrygian descents in the bass.

Ex. 7-31 (Listen)

The first chord of bar 10 may have several interpretations: the main one is as G minor seventh chord (~B¬_g7) with C{ as appoggiatura to D. But in the second quaver we could also consider C{ as leading tone within the virtual dominant chord (A)C{(E)GB¬, which gives it a dominant touch. But, by enharmonizing C{ as D¬, we could also see the Tristan-class chord (E¬)GB¬D¬F with the M3 tension D¬F which resolves in the fundamental G¬ of the next chord. As always, it is not necessary to choose one of the three interpretations, the coexistence of the three can be considered. Something similar happens to the chord of the first beat of bar 12.

The following example goes to Bruckner, to his four-part motet Christus factus est (Example 7-32). Again we see a long htonal and Phrygian string between fundamentals (with a Dorian one in bar 10) to reach again the dominant in bars 12 and 14. The resolution to the dominant from the previous chord is also Phrygian in both cases, but using different chords.

In the last beat of bar 11 we have a Tristan-class chord and is its second fundamental D¬ which resolves Phrygianly to the fundamental C. In bar 13 it is the functional fundamental (D¬) of B¬ menor minor chord, which resolves Phrygianly to C. Seen from tonic F, this B¬ minor chord is the subominant S¶_p (seen from the dominant C this chord has the function of Phrygian dominant type D'_p).

Ex. 7-32 (Listen)

In this same work of Bruckner we can see in bar 25 (Example 7-33) a modulation to E (major) using, from the dominant of F, a Phrygian and htonal successions to go directly to the tonic E chord (C|B_g{|E). To reaffirm the modulation we have a homotonic string to reach the subdominant A (Bµ=G{µ¼E_c{|A) and then a step to the dominant taking advantage of the small dominant character of the quintal chord (suspended chord f{^e|B), aided by the soprano melodic leap of descending fifth to

the dominant. This suspended chord (being the dominant on the bass) could also be viewed as an appoggiatura of the dominant chord (E|D{ in the tenor voice).

Ex. 7-33 (Listen)

In Example 7-34 we can see the homotonic sequences of bars 21-30 in Bruckner's Ave María (the symbology of chords above the system and below the tonal functions). The scheme to reach the tonic chord in bars 21-25 is A¼F_d|B¬¼~C_aµ |F. Here the Phrygian succession to the `dominant function' of C_aµ is not from D¬ but from the subdominant B¬ which resolves in A minor seventh, chord which has the dominant triad inserted (has functional fundamental C), in addition, in this case, with the dominant in the bass, therefore with a clear dominant function (functional sequence D^S-D_p-T).

In bars 26-28 there is a transient modulation to the subdominant B¬ colouring the chords of the typical sequence T-S-D-T (of B¬), and in bar 30 we return to the dominant C by means of the tonal tension of chord G^F (dominant + subdominant `of C' as fundamentals). Note that in bar 26 we have a double htonal homotonic relaxion.

In Vexilla regis, also by Bruckner (Example 7-35), we see, in last bars, a modal ending with Locrian relaxions (Bµ|D{/E¬, A¬¼E¬_c, A¬_f|C, C_a|~e…), combined with the htonal relaxions E¬|A¬ and ~e¼C_a (the latter very weak since E does not have its M3). The final Locrian relaxion is supported by sonance relaxion.

Ex. 7-34 (Listen)

Ex. 7-35 (Listen)

The beginning of the prelude of Morceaux de fantasie by Rachmaninov (Example 7-36) seems to augur the transient cadences that will appear throughout the work. In fact we are talking about the cadential progression we have seen more often in the previous examples: Phrygian relaxion followed by a htonal relaxion towards a dominant or a tonic (transient or not). The prelude starts with this sequence (ff ) but only with one (fundamental) note, repeated in three octaves (a|g{|c{). In bars 5-6 we found the same progression in the relative key/tone E (C|Bµ|E) and in bars 6-7 towards the minor dominant (E|D{µ¼B_g{). In bars 3-4 there is a variation of this formula. It begins with a Phrygian relaxion (D{µ|D}µ) but instead of going to G}, it links to the most similarly functional chord on its tonal axis (Dµ=G{µ, which shares the same tritone F{-C/B{) to, finally now, release the tension htonally to the tonic chord (D{µ|Dµ=G{µ¼E_c{) (the same thing occurs in bar 4).

Ex. 7-36 (Listen)

As always, the second row of functional symbology (the third, counting all rows) is an alternative notation; both are correct. D in parentheses (D) refers to a secondary 

dominant (htonal (D) or Phrygian (D')), but from the point of view of the tonic is a subdominant (S§ or S¶ ). The fragments between brackets show a transient modulation and the functions within it are seen from the point of view of the new key/tone (indicated at the beginning of the bracket).

Ex. 7-37 (Listen)

At the end of this prelude (Example 7-37) we find a rare cadence to finalize a tonal composition (bearing in mind that the tonic is C{) (\D¶ ^S¶|t), even though we have htonal homotonic relaxion between chords (F{|\B^A|E_c{) (would be a more typical cadence of E major instead than C{ minor). However, since we always have the pedal C{ in the bass, we could also consider an ending with sonance resolution, since all the previous chords are more dissonant and none is in root position giving us a sensation of ending simply by de fact of hearing the tonic chord in its most consonant position, aided by the htonal relaxion towards the functional fundamental of the minor chord (\B|E) and the Locrian one towards the root (A|c{).

In Debussy's La fille aux Cheveuz (Example 7-38) we can see a long homotonic chain. In bars 12, 15, and 18 the same formula is used to create three tonal vectors: this passage take advantage of the slight property of quartal chords to tonicize next chord, which is a major triad in root position.

Quartal chords with four fifths (five notes, for example: FCGDA) functionally have the form d^S, where d is the dominant with its fifth (GD) and S is the major triad of the subdominant (FAC) and form a small tonal vector towards C although the chord does not have the leading-tone. This is what happens in the above-mentioned bars, transposed to G¬, C¬ and E¬, in bar 18 incorporating the semiquaver C} at the end of the bar; if by its small duration we do not incorporate it, we have a chord with three fifths (four notes) that still conserves this small tonal vector (d^s); in this case we have dominant + fifth and subdominant + fifth. With this long chain

Debussy reaches the relative of G¬ mayor (E¬) but with the Picardy third.

Ex. 7-38 (Listen)

The following examples are from Proses lyriques, also by Debussy.

In the first song (De rêve…, example 7-39) we also have a long homotonic progression, in this case, from the Phrygian dominant (B¬) of the dominant (bar 9) to the dominant (Aµ) (bar 18).

In bar 9 we have an example of the double function of the augmented sixth in the chord (G{), as leading-tone of the dominant and as minor seventh of the fundamental. Debussy seems to be aware of this duality and in the melody places a G{ and a A¬ on the piano.

In bars 10-12 we see combinations of fundamentals corresponding to the tonic (D), the subdominant (G) and the dominant (A), the latter being the main fundamental of these bars until in compass 13 it resolves Phrygianly to G{µ, chord that prepares an atonal fragment in bars 14-17 (there are the 12 notes of the chromatic scale every two bars) but, as we shall see, consistently homotonically speaking. In fact, bars 14-15 and 16-17 are harmonically the same and basically are formed by chords of

the family of augmented chords (augmented fifth) and their fundamentals are joined by Phrygian homotonic relaxions. As we have seen in 3.1.4 and 6.2.4, the chords of this family have three fundamentals at M3 distance, but for simplicity we only show two (those that correspond most to the written notes). In this example in some cases we put the three to see more clearly this Phrygian homotonic chain. If we place the three fundamentals, we would see that each is Phrygian relaxion of some fundamental of the previous chord.

In bar 18 we return home with the htonal relaxion towards the dominant and also by the sonance resolution.

Ex. 7-39 (Listen)

At the beginning of the third song (De fleurs..., Example 7-40) we see again what we have already analyzed in other examples: the combined use of the Phrygian (D') and honal (D) dominants to go towards the tonic or the dominant (remember that the Phrygian dominant also has something of subdominant); in our example they go towards the tonic using a variant of the Phrygian dominant (D'_p). Recall also that D, D', D_p and D'_p are in the same tonal axis and in the theory their succession is harmonically neutral (there is no tension or homotonic distension between the chords), although there is a difference of sonance between major and minor chords.

Ex. 7-40 (Listen)

Ex. 7-41 (Listen)

At the end of this song (Example 7-41), Debussy seems to take advantage of tonal axes functional equivalence to go to the tonic. He does something similar to the previous example, but instead of using the dominant (D'_p = D), we see the direct use of the tonic (T'_p = T). We also have, in last chord, a relaxation of sonance (the chord follows the overtones trail). In order to reach G¬_e¬ a homotonic chain is used (Phrygian + htonal + Locrian relaxions) (G^B|G¬_e¬|C¬¼G¬ _e¬).

Ex. 7-42 (Listen)

In Example 7-42 we have a very long homotonic link in the beginning of Pavane pour une infante défunte by Ravel. All the functional fundamentals (real or virtual) correspond to the three functional fundamentals of G major (G, C, D = T, S, D), in different combinations. We do not have any tonal deviation in these initial measures, although the Phrygian resolutions at the beginning of bars 2, 6 and 7 (C¼ D_b) give a touch of B Phrygian mode.

From mid-bar 5 (or even from mid-bar 4) to bar 7 we have the dominant function present in different chord structures (\D^C, G^D and D_b), but we never hear the dominant chord in its usual triadic form. We can say the same thing with the subdominant triad and the tonic triad (except at the beginning).

In the following example (Bartók's Bagatelle No. 4, Example 7-43) we also have (almost) all chords (except in three) using the functional fundamentals of a tonality (in this case of F —F,B¬,C—, although this work is not properly in F major).

The work rests every four bars in the D minor seventh chord, i.e. the triad of the tonic F (FAC) plus the D, which gives a modal eolian air. The fact that the final chord (and the fermata of bar 2) has F as functional fundamental allows this rest.

Most of these chords are also homotonically linked. The ending bars does not have these diatonic fundamentals (bars 8 or 12, because the third system is a copy

of the second). In the second beat of this ending measure Bartók puts a dissonant chord (from the symmetrical dominant chord family: G{_D) which contrasts sharply with the diatonic chords we have been hearing so far, but it is a chord that follows the tonal axes theory of Lendvai, that is, before the final chord of D minor seventh, which functionally (seen from F) is type T_p, we find other tonic tones (fundamentals) of the same tonal axis (D, F, G{/A¬ , respectively T§, T, T¶ ).

Ex. 7-43 (Listen)

Almost all chords are linked with homotonic relaxions in Example 7-44 (To Thee We Sing by Chesnokov).

Ex. 7-44 (Listen)

In the first two bars we have resolution towards the functional fundamental (D) of the B minor chord (\A^G|D_b) and a Locrian one from the second fundamental (G|b), which gives (as in the previous example) a modal air (B Eolian). In fact, in this musical fragment, two tonal vectors alternate towards B and D.

Since we have put all the functional symbology with regards to tone D, the symbols (functional symbology) that indicate redirection of the tonal vector towards B are T_p (tonic of the relative key) and D§ (dominant of the relative key). The fundamental corresponding to the dominant of the relative tone/key B (F{) is usually reached by a Phrygian relaxion of the fundamental G, corresponding to the subdominant of main key D (G|F{).

We can see a long homotonic chain in the Cançó i Dansa n.º 6 (Example 7-45) by Mompou, in this case coloured with numerous appoggiaturas and passing tones.

In bar 8 we have, again, the combined use of (D) and (D') to resolve htonally and Phrygianly towards the dominant.

Ex. 7-45 (Listen)

Ex. 7-46 (Listen)

In La Font i la Campana, also by Mompou (Example 7-46), apart from the Dorian and htonal sequences we have a link that sweetens the arrival of the dissonant chord of bar 19 (B¬^~Dµ, a mixture of the families of augmented and dominant chords). We find a double relaxion between fundamentals: a Phrygian one to the incipient future dominant (E¬|Dµ) and the other htonal towards the local tonic (Fµ|B¬). Indeed, this

chord marks the end of the tonal vector towards B¬ to redirect it to G (minor). It is a mixture of authentic cadence and Phrygian cadence to the dominant, for that reason its dissonance is sweetened.

A very similar chord is found in Schoenberg's Friede auf Erden (F^Aµ, in other inversion) (Example 7-47), but here the harmonic situation is different. We see a chromatic sequence of chords consisting of three consecutive Phrygian relaxions and a final htonal distension at the end of the work.

Ex. 7-47 (Listen)

Ex. 7-48 (Listen)

In Schoenberg's Transfigured Night (Example 7-48) we can also find many homotonic relaxions. In bars 41-45 we have an example. It is a fragment with a mixture of htonal and Phrygian relaxions between fundamentals (with some Locrian ones) to finish in the dominant.

The diminished seventh chord of the second minim of bar 44 (be aware of the C-clef from bar 42) have been symbolized in the form \Gµ_\Eµ (one could also write \E⁷⁹, see 3.1.5) to show that the virtual fundamental G is a Phrygian relaxion from the previous chord (A¬|\Gµ) and that the virtual fundamental E creates the «dominant of the dominant» tension that resolves in the dominant (\Eµ|Aµ). Between these two chords we have a cadential æ, which can be considered as an appoggiatura chord to the dominant chord, although in the minor mode it implies a Locrian relaxion between functional fundamentals (F_d| Aµ).

A few bars later (Example 7-49) we find several chords with two functional fundamentals that are also linked using homotonic relaxions.

Ex. 7-49 (Listen)

We could continue to give many more examples of the use of homotonic relaxions (basically the htonal and Phrygian ones), as a way to achieve relaxed successions between chords or harmonically fluid progressions, with or without the participation of tonal distensions, which we have shown in a lower line in the examples, although, when the tonality is well established (strong tonal field), they are more powerful than the homotonic forces. When the tonal field is weaker, homotonic relaxions acquire more significance, as we have seen in some previous examples.

We have also seen the utility of separating the chords in their fundamentals that define them tensionally, the capital letters indicating functional fundamentals that manifest the «quasi-fifth» tension of the M3 they contain or, when µ also appears, the tension of its tritone, being the fundamental crossed (virtual fundamental) or not (fundamental that exists in the chord). These fundamentals give us a lot of information about the harmonic move and the tonal vectors that are created. For example, to find a tonal vector we simply have to look a capital letter with a µ or a

M2 interval between fundamentals, the first may be lower case or upper case, but the second should be functional (represented with a capital letter). For example, the fundamentals a and B, or A and B, create a tonal vector towards E, which will define a major or minor mode depending on whether its major or minor third appears in that fragment. Also Bµ or B^A directly defines a tonal vector towards E, and in a weaker way also \~Bµ o \~B^A.²

² If one uses a computer, this work of finding the fundamentals could be done by the same score editor software, such as the plugin I made for the Sibelius program. I can provide it free to any reader interested in it.

Annex 1