Comparisons are, by their very nature, delusive. Still, I would like to venture the following — comparative — statement: classical harmony is to modal harmony what a geocentric world concept is to a heliocentric one. Let us arrange the tone set of the Jupiter Symphony in a circle of fifths. In a traditional representation, the tonic – C – will be placed at the top of the fifth-circle, in the center of symmetry of the system.

However, earlier we came to the conclusion and one glance at a piano keyboard (with the white and black keys) will make us understand that the axis of symmetry of the C major scale is not C... but D (or A^b), in relation to which every note of the C major scale has a symmetrical counterpart both upward and down. Of course, C major, with its zero key signature, occupies a special place in our system of musical notation. Moreover, as we can see on the second diagram below, F major with 1 flat and E minor with 1 sharp (or B major with 5 sharps and B^b minor with 5 flats ) are symmetrical opposites in relation to a D center note. 

Wouldn’t it be more appropriate, in a depiction of the circle of fifths, to place the D at the top of the system and the A^b at the bottom? We would thus illustrate the actual relationship between the notes:

Fig. 215

Now let us replace the preceding fifths with their corresponding keys. In the upper half of the circle, the chords F major and D minor, as well as G major and E minor belong to the scale of C major (or A minor). Similarly, the chords D^b major and B^b minor, as well as B major and A^b (G^#) minor all belong to the G^b (F^#) major and E^b minor scales in the lower half of the circle. (See on next page.)

The relative C major and A minor keys rest on the same notes and (as it appears from Fig. 216 on p. 114) have identical weight!

The symmetry thus obtained remains flawless even when the major triads are replaced by their parallel minor triads, and vice versa (for example, D minor and G major by D major and G minor respectively; D major and G minor will still occupy symmetrical positions in relation to the D center). The above figure will henceforth be referred to as the basic formula of our system.

Fig. 216

The subject of our study will be Verdi’s Don Carlos (we must be satisfied with a sketchy summary only, as a detailed analysis would require a separate volume).

According to our basic formula the symmetrical counterpart of 

Is it not striking that the opera begins in B^b major — with 2 flats, and ends in B minor — with 2 sharps? Or, why is it that in the nocturnal garden scene the settling effect of the F major terzett is followed by Eboli’s E minor revenge-aria? And why is the effect so convincing? As has been seen, 

The dramatic turning-point of the dialogue between Posa and Filippo: Filippo’s confession in F minor, is likewise preceded by an E major play with ’colours’. The thought of Carlos’ ’salvation’ in A^b major (Eboli’s aria) and Carlos’ ’fall’ in C^# minor display a similar relationship (see our basic formula). Following Posa’s D^b major farewell, the revolution breaks out — strikingly enough — in A^b minor: in this system the counterpart of 

The clue to this tonal riddle comes from Verdi himself (he places, so to speak, the clue into our hands). The KEY-SENTENCE is sung by the ’Monk’ (whose disguise of anonymity conceals Charles V bound for the monastery). The first half of this sentence is about ’world cares’, the second half about ’heavenly consolation’. The C^b major tonality representing earthly concerns is contrasted with the B^b minor tonality denoting heavenly affairs. The mirror image of C^b major is (as can be seen in our basic formula) B^b minor:

Fig. 217

Incidentally, it was Verdi in his late works who brought this system to the highest perfection. If we were to classify (to ’catalogue’) the scales and themes of Don Carlos, we would discover not only the special meaning of each key, but also the multidimensional relationships between the keys. Let us take as an example a D minor triad, which has the following characteristics:
 
 

Therefore, the meaning of a chord changes according to its relation with another harmony. In the case of counterpoles, a difference of plus 6 = minus 6 key signatures equalize, compensate one another — thus, these necessarily imply the existence of common traits. The upper and lower halves (or left and right halves) of our basic formula also conceal surprising symbols.

Each of the tables below represents a specific tonal relation and, more important still, one of ’content’. Any one of the 24 keys can form 23 direct relations with the rest of the possible keys. Moreover, one is also led to realize that second- and third-degree relations perforce produce  identical results — indeed, they even reinforce each other! In other terms, should we only know the ’meaning’ of 23 keys, we could deduct the meaning of the unknown 24th key from these relations.

Hence it follows that the system holds true only if every dimension is verified and confirmed by the work’s dramaturgical content or poetical meaning. Our ’serial’ study — our musical ’Rubik cube’ — as it appears in the tabulation below, is based on the five-act version of Don Carlos, 1886 (Ricordi Edition, 1982). The tonic is C major.
 
 

Actually, our traditional method of notation also reflects the above principles (see the basic formula). Those keys in which the number of sharps and flats is identical occupy a symmetrical position in the system.
 
 

In the case of parallel major and minor keys one of the two keys always belongs to a family of ’natural’ keys, and the other to one of ’modified’ keys. C major, for example, differs from C minor in that the natural DO-MI-SO chord is altered to DO-MA-SO.

The major and minor keys displaying a difference of 6 accidentals reflect the same contrast, in terms of content, as the pole–counterpole relations. These correspondences may be found in our first tabulation — if the connections below are considered:

TONIC chords reflect static motionlessness. This applies not only to the tonal pillars of C major (and A minor), or F^# major (and E^b minor), but to the variations of these as well:
 

are also characterized by the fact that they cannot be further developed — there is no way out of them.

DOMINANT keys differ from tonic ones by the ’active’ force they exert (complying with the principles of the axis system). This is how
 

The dominant minor chords can engender a rise, like the E minor and B^b minor keys, but a violent and tragic turn as well, like C^# minor and G minor.

SUBDOMINANT chords function in a similar manner:
 

The minor subdominant chords are distinguished by their passivity:
 

The upper half of our basic formula (p. 114) makes up the diatonic scale (7-degree scale). The diatonic scale contains six perfect triads:
 

This is to say:

are in fact

All of the above applies to the lower half of our basic formula as well (in which case F^#=DO).

If we change the ’gender’ of the keys, that is, if we substitute a minor key for its parallel major and a major key for its parallel minor, we may note that all ’natural’ thoughts correspond to unmodified keys and ’stimulated’ emotional states to modified ones.

Even in Eboli’s revenge melody, E minor is associated with ’natural’ (i.e. instinctive) emotions — while E major radiates an ’artificial’, intensified light (Auto-da-fé Scene). In the same way, A^b minor denotes self-destruction, but A^b major suggest a heroic sacrifice at the cost of one’s own life (as in the finale of Eboli’s aria or in the Peace Song of the Auto-da-fé Scene).

The system is ’closed’. The upper and lower halves of our basic formula are mirror reflections of one another. This means that, for instance, F^# minor is dialectically related not only to F^# major — but to the C major tonic as well. In the first case we perceive it as a DO-MA-SO chord, expressing ’despair’ — as in Elisabeth’s heart-rending lament ("Ben lo sapete...") after the chest has been forced open. But in relation to the C major tonic, ’F^# minor’ represents the most immaterial harmony to be found in Verdi’s music: that is why we sense the FI-LA-DI formula — Elisabeth’s F^# minor melody — as being so "immacolata" (as indicated in the libretto).

Symbolically speaking, the upper and the lower halves of our basic formula relate to each other as the empirical Aristotelian world to the Platonic one — to the world of Ideas.

A similar ’dictionary’ may be compiled from the relations between relative major and minor keys — or what is even more fascinating, those between axis-related major or minor keys. Let us cite only one example here. 

The cheerful B^b major of the Royal Hunt in Act I is transformed into D^b major: into a love-scene. (If B^b major is a DO-MI-SO tonic, then the D^b major must be interpreted as MA major!) — In Act II, too, the finest moments of the Carlos—Elisabeth duet are marked by a B^b major—D^b major turn. In the nocturnal garden scene it is just the opposite what happens: Carlos and Eboli – hidden behind masks – declare their ’love’ to each other in D^b major; but at the moment of unmasking, we immediately return to B^b major!

According to the classical definition, the major triad consists of the 3 closest overtones of a basic note — while the minor triad is made up of the 3 closest basic notes of a common overtone:

Fig. 218

This alone helps us understand why C major represents the external (visible) world and F minor the internal (invisible) one. Using the above principle as a new basis, we may discover a new dimension of our tonal system (which also corresponds to the ’Western’ and ’Eastern’ ways of thinking):