When I was to give a lecture on the computer analysis of music, my colleagues, having been asked to give me a ’subject for examination’, chose The Magic Flute’s Sprecher scene. The question was whether the ’artificial brain’ could add anything new or astonishing to the known interpretations of the work. This analysis is no more than the result of a comparison between the computer-made ’map’ and the libretto.

What can a computer do? Its inventor, the Hungarian-born John Neumann said: ’the computer is the most stupid animal — but it works terribly fast.’ To put it more precisely, the computer is capable of what the human eye is useless for: it can find a needle in a hay-stack. For some time it has been recognised that the sound dramaturgy of music can only be partly understood through the rules of classical harmony. Therefore we know that instead of conforming to the accustomed rules of music, Verdi’s or Wagner’s operas are rather like organic cells where every component is organically related to every other element. Earlier, we sought the logic of how one element is born from another one: what logic governs the succession of chords. At the historic moment of abandoning the heptatonic system, the tonal system immediately became multi-dimensional: so complex and indecipherable for the human mind that without computer programs it would be hard to give an answer to such seemingly simple questions as to what the true meaning of the dialogues in The Magic Flute is.

Let us start our investigation with a few ’test-bars’. Mozart constructed each section of the memorable dialogue around a dramatic ’breaking point’ — which also determines the tonal structure of the scene.

The major and harmonic minor scales differ from each other in two tones: the minor scale contains the minor third and minor sixth (instead of the major third and major sixth). For example, modifying the D major by ordering a minor third and minor sixth to the root D (i.e. the notes F and B^b), we obtain a B^b major triad which (compared to D major) gives the impression of the most violent ’major-minor’ contrast. In the scene, the sharpest contrast comes with the command "Zurück!" sounding from behind the temple gates: it all but gives the feeling of a physical blow: the D major harmony is knocked down, shattered by a B^b major chord.

It is worth noting: a similarly sharp major-minor contrast is generated by the A^b major chord within the C major tonality.

A scenic and a tonal surprise appear in each section of the dialogue. The first genuine astonishment accompanies the sighting of the temple columns. Earlier, the ’effect mechanism’ of modal thinking was explained in the following way: with each coming chord a comparison is made between the chord which would be expected, according to the natural logic of music, and the chord which actually occurs. The tension difference between the two determines the message, the meaning (modal quality) of the music.

What happened now was the A major dominant being followed by a D minor tonic: "Wo bin ich? was wird mit mir?". Now the A major dominant is unexpectedly followed by F^# major (instead of the D minor):

Fig. 1

That implies at least three consequences:

     (a) F^# major — in the context of the C major scene — performs the role of the ’counterpole’.

     (b) More important still, after the dominant A major the F^# major a minor third lower registers a 3-sharp rise; to put it in another way, an ’axis-rise’ takes place in the positive direction (F^# major also appearing as a dominant). As was said, the inner elevation captures the moment when Tamino catches sight of the columns of Sarastro’s temple.¹⁾

     (c) It must not be forgotten, however, that the F^# major appeared instead of D minor, and significantly, D minor and F^# major are complementary keys: they ’destroy’ each other. As is indicated by the Allegro theme, our hero gets into a state of excitement.

That is how we arrive (via the resolution of the F^# dominant) into the B minor key which depicts Tamino’s character in vivid colours (Allegro). What do this B minor and the following E minor key mean (forte: "mutig zur Pforte hinein?") As we know, in Romantic music the greatest contrast is created by complementary keys that neutralize each other. 

The Sprecher’s A^b major utterance followed by the E^b major reply (it is still the Sprecher who strikes the tone of "Heiligtum") constitutes such a contrast compared to B minor and E minor that it has no match in expressiveness even in Romanticism. E minor and A^b major, as well as B minor and E^b major are complementary keys: they extinguish each other. Has it ever struck anyone why Bartók gave E minor to the right hand and A^b major to the left in Burlesque No.2 (or in the piano piece See-saw)? Or why he put the movement II. of the String Quartet No.4 in ’E minor’ and its variation — movement IV — in ’A^b major’? (Later it will also come to light why the former assumes a ’Phrygian’ character, and the latter, necessarily, a ’Lydian’ character.)²⁾

Let us play Tristan’s love duet in C major and we shall immediately get a hint of the electric charge in the contrast between B minor and E^b major. The major tension of G major can be hightened by B minor (=positive substitute chord), the minor tension of G minor by E^b major (=negative substitute chord).³⁾

Fig. 2

On the one side there is Tamino’s youthful passion, spontaneous, almost militant bravado: the challenging activity — a combination of physical strength and a keen mental interest, fresh receptivity. We must feel from our hero's enraptured and incalculable outbursts, his enthusiasm, his brave resolution, that he would not balk at the obstacles in his path. — On the other side there is the world of the Sprecher’s mature age, the attitude of thoughtful and profound wisdom. The always predictable spiritual power draws on a deep experience of life — while bridled passion hides ascethic perspicacity.

At a superficial glance, B minor and E minor might suggest a ’minor’ world, and A^b major and E^b major a ’major’ world — but actually, it is the other way round. Just like in the above example, B minor appears in the C major scene as the substitute chord of G major of the 5th degree (cf. "das Laster nicht leicht") — positive substitute chord, we must add. Similarly, E minor occurs as the (positive) substitute key of the tonic C major (the resolution of the former G major chord is not C major; instead, the monologue shifts toward E minor).

On the other hand, the A^b major chord indicating the Sprecher’s entry upon the stage is introduced as a ’deceptive-cadence’, taking the place of C minor. As a result, A^b major appears as the negative substitute key of C minor. What, however, calls for some more words is the perception of E^b major (first of all) as the substitute chord of the minor dominant — G minor — and not as the relative key of C minor.

An example from Romanticism comes to mind. In the Prelude to Parsifal (at the second entry of the theme, see Fig. 42 on p. 30) A^b major appears as the suspension of C minor while the E minor answer motif is enveloped by C major. It can clearly be seen that 

Fig. 3

To close the circle: what gives the pith and marrow to A^b major and E minor — the negative and positive substitute chords — is that the two neutralize each other because they enter into a complementary relationship.

As for the relationship between B minor and E^b major, B minor replaces the dominant G major (positive substitute chord), while E^b major substitutes for the minor dominant G minor (negative substitute chord; earlier we said that E^b major = the deceptive cadence of G minor).

Returning to the Tristan love duet it should not pass unmentioned that although the second step of the sequence is identical with the first, it is heard like this:

Fig. 4

Here, E minor and A^b major are none other than the substitute chords of the tonic C major and C minor, respectively. I often used to ask why Bartók insisted on the E minor and A^b major keys? The central turn of Cantata Profana — "A fáklyák már égnek..." [The candles are all lit...] — was clearly conceived in C major, yet the melody is accompanied by E minor and A^b major chords alternating from bar to bar. One colours the motif as the positive substitute chord of C major, the other as the negative substitute chord of C minor. And what weighs most is that the two are united in a 1:3 model, that is, their interrelation is complementary.

Fig. 5

In the second movement of Bartók’s Dance Suite, the Ritornell melody in G is similarly buttressed by B minor and E^b major chords as the substitute chords for G major and G minor. (See Fig. 201 on p. 104)

What marked the most significant turn in the development of classical heptatonic diatony was the birth of the SI degree as the leading note of the minor scale (upon the model of the major scale). The ’fermentation’ of diatony, the Romantic slackening of the tonal system was indeed caused by the difference between SO and SI. A palpable example of this process is the slow movement of Beethoven’s 7th Symphony (in A minor) in wich the C major triad appears as the substitute chord of the minor (E minor) dominant, and not as the relative key of the tonic A minor. This is aptly shown by the refrain of the theme:

Fig. 6

In the first four bars the G^# note means SI, in the next four bars G means SO. The SO note gets a separate leading note thanks to the FI-SO (F^#-G) melodic step. The difference between SI and SO gives the impression that the character of SO is far more ’clouded’, blunter (darker), more sensitive. The same takes place when the refrain melody is repeated (in bs. 15-18):

Fig. 7

in bs. 15-16 we hear SI, but in b. 17 the more ’cloudy’SO appears again.

It can be regarded as a rule that moving from a minor to a relative major means progressing in the ’negative’, unnatural direction, because the leading note (SI) does not resolve upwards, but unnaturally downwards (towards SO). In Tristan’s love motif, too: the B leading note, instead of guiding us to the expected C, goes downwards towards the B^b note.

Fig. 8

Let us return to The Magic Flute. In the dialogue the first rhetoric contradiction — also in the libretto — is the emphasized word "allein", which might be translated as ’only,...’. The Sprecher ’raises’ his voice: but no matter whether the key is interpreted as E^b major or C minor, the B^b note has the meaning of SO and the B note means SI: 

Fig. 9

But let us go back to an earlier point. (Formerly, I devoted a detailed study to this problem.) In a major tonality the tonal centre is DO. That causes the predominance of the TI-DO leading-note steps. In minor keys, however, it is not necessarily LA that constitutes the tonal centre, but much rather the note MI: the core of a melody mostly comprises FA-MI steps (think of the main idea in Mozart’s great G minor Symphony, or the main theme in Beethoven’s Appassionata sonata: D^b-C turns in bs. 1-24). In our tonal system, the symmetry-pair of the TI-DO leading note step is the FA-MI step; Mi is the mirror image of DO. 4)

The note MI (note E in A minor) can most effectively be approached from two sides: from the directions of FA and RI (F and D^#). What is the ’augmented six-five‘ chord? It is a special chord typical of the minor tonality obtained when the V. (MI) degree is prepared not simply by the IV. degree,

Fig. 14

but the MI centre is approached from both sides with chromatic steps: with FA-MI and at the same time RI-MI ’direction notes’, (that is, with F-E + D^#-E turns). In such a case this tight harmonic relation will almost produce a ’physiological’ effect, as it were!

Fig. 15

Between each statement of the three-times repeated motto-theme ("Sobald dich führt..."), which in terms of form expresses the attainment of the goal, the augmented six-five chord occurs, lending special weight to the MI centre. The chord first accents the word "Licht", then underscores the sentence "saget mir: lebt denn Pamina noch?"

Fig. 16

The nucleus of this harmony — F major — is the VI. degree of the tonic A minor. B major preparing the development is, on the other hand, the counterpole — the tritone — of F major. The model of the polar turn must be traced back to the Neapolitan chord. The Neapolitan chord and the V-I step following it serve to circumscribe a tonal centre note, with semi-tone steps, to boot; in A minor key, the notes B^b and G^# chromatically head towards the note A: ⁵⁾

Fig. 17

By the way, we know from earlier experience that the B^b major Neapolitan chord is nothing else but the substitute chord of the subdominant D minor.

The above quoted polar turn ("Man opferte... sie schon?") is based on a similar attraction — proved unquestionably by the renderings of Toscanini, Bruno Walter or Karajan — : the melody falls over the central E note.

Fig. 18

This time, too, FA and RI play the role of the ’leading notes’ from two sides (I prefer to use the term ’direction notes’: the meaning of both FA and RI is expressed by the sphere of attraction of the MI centre).

However, the circumscription of the note LA cannot be missing, either. Before the outburst of the ’motto-theme’ repeated ritually three times ("Die Zunge bindet Eid und Pflicht... schwinden") the notes B^b and G^# perform a similar task: ⁶⁾

Fig. 19

In terms of taxonomy, the ’symmetry-pair’ of the G major dominant seventh is the B-D-F-A subminor chord. The character of the latter is far more lyrical and melodious (’singing’): thanks to the LA note. Instinctively, that is how it became the backbone of the repeated motto-theme:

Fig. 20

The dialogue has a recurring ’leitmotif’ which owes its existence to the direction-notes — precisely to an element that is the opposite of the leading note step. Let us do ’violence’ upon the leading note of the V. degree (e.g. the note B of G major) by modifying the note B to B^b (instead of the tonic resolution): that is, let us have the major third (B) replaced by a minor third (B^b). This charges the note B^b with tension, turning it into a dissonant element, a ’direction note’ requiring further leading — towards D major. From that point onwards the function of the dominant is not performed by the G major but by D major (= the secondary dominant: the dominant of the dominant). The task of the tonic belonging to D major (i.e. G minor) is simply to restore the equilibrium — as it was expressed by a great conductor in connection with "Ja, ja, Sarastro herrschet hier".

Fig. 21

Following the onerous admonition "Tod und Rache dich entzünden" (concealing an augmented six-five again), Tamino surfaces with the G major dominant:

Fig. 22

The old priest suddenly tones down his voice: instead of G major we hear G minor and the continuation conforms to what was described above.

The G minor and D major chords shown in Fig. 21 constitute a symmetry-pair, mirroring each other taxonomically. In this sense, the mirror image of the note B^b (TA) is F^# (FI). It would be redundant to mention this, had Mozart not made use of its opposite upon the return of the motif. This time the emphasis is laid not on TA but on FI, which immediately gives the motif a ’challenging’, provocative character: it becomes a threat to murder, striking the key of passionate protest, of indignation. The key is F major (or F minor): "das ist mir schon genug". Were this to occur in a classical oratorio, the homophonic melody would be answered in the following way:

Fig. 23

What accounts for the sharp ’challenge’ is the replacement of the C major domi-nant by C minor (the passionate minor dominant), and just like earlier, the natural continuation is in G major — where the note B means FI, striking the tone of danger and threat.

Fig. 24^*)

The scale-degrees FA and FI have a distinguished place even among the ’direction-notes’. FA pulls downwards (to MI), while FI pushes upwards (to SO). This is a very modern idea in dramaturgy. The introductory part of the dialogue is tied to C major, its middle part to C minor and its related keys, while the closing section returns to the keys without accidentals: first to A minor and finally to C major.

The question arises what scenic moment elicits the reprise. In Tamino, the image of the "unglückliches Weib" (disconsolate mother) evokes the tragic G minor. For the old priest, however, the "Weib" means something quite different. At first one might think he is only envious (that is why the sarcastic intonation) but it shows through the music more and more apparently that she is the opponent, or even the ’enemy’. Isn’t it peculiar that "Ein Weib" as well as the image of the enemy is associated with the C major chord? Let us take a closer look at what is happening here. In G minor the E^b–D steps give the impression of emotional FA-MI steps; the guiding thread of the harmonies is made exclusively of E^b–D steps (earlier I pointed out that FA–MI is the ’emotional’, introverted element in music):

Fig. 25

As against that, when the Sprecher begins to speak, — what a turn! — FA is replaced (ousted, to be precise) by FI, which ushers the dialogue towards SO:

Fig. 26 (Play before it the former example!)

In connection with the mentioned G minor, B^b7and E^b major turns let me refer to another idiomatic turn also behaving like a leitmotif. In my analyses of Verdi and Wagner I termed this element the turn-motif. In Mozart’s music the role assigned to it is to give emphasis (emotionally charged emphasis) to a word. When, for instance, the root of the A minor chord is raised a semitone higher, that is, modified to B^b (NB: the modification giving accent to the chord), a major seventh (C⁷) is gained which automatically leads into the substitute key (F major).

Fig. 27

That’s how we arrive from the first "Zurück!" to the second gate, from G minor to E^b major:

Fig. 28

Later we move from the G minor of "Sarastro herrschet hier" to E^b major in the same way: raising the basic note of G minor gives edge to Tamino’s violent "nicht".

Fig. 29

The sentence "Er ist ein Unmensch" is stressed by a similar motif (G minor, B^b7, E^b major).

Fig. 30

The same takes place after "Erklär’ dies Räthsel" (E minor, G⁷, C major; the raising of the root note falling on the sentence "täusch mich nicht!"). ⁷⁾

Before sketching the tonal structure of the scene, let us remember an analogy: Tristan’s ’dream chords’ (Brangäne’s first monologue in Act 2: "Einsam wachend..."). Each group of chords springs from C^# major — and the nadir is reached when C^# major is followed by A minor and E minor triads:

Fig. 31

In the opera C^# major signifies the ’mother’s lap’ (the womb), the ’dream’, the ’night’ — the Nirvana. The A minor and E minor nadir presupposes the renunciation of this, too: after C^# major A minor establishes an annihilating relation (not to speak of the fact that the leading note within the C^# major chord [E^#] is resolved unnaturally downwards), while E minor represents a polarly distant relationship with C^# major.

If in the cited dialogue of the Magic Flute D major symbolizes the ’gate’ of Sarastro’s temple, the concept of the ’non-gate’ — the moment when Tamino gets ready to leave disappointed and ’renounces’ the gates — will be represented by B^b minor and F minor, according to the above logic. D major and B^b minor are complementary keys: renouncing each other, while F minor is removed to the other pole from D major: in our tonal system (e.g. the circle of fifths) the largest possible distance is expressed by a difference of 6 accidentals. As if it were the model for the Wagnerian technique: the very point where Mozart noted in the score "er will gehen" (and the Sprecher asks Tamino: "Willst du schon wieder geh’n?"), we find ourselves in the B^b minor key, followed four bars later by F minor.

The contrast becomes even sharper when it is considered that D major is prepared by a salient, conspicuously emphatic A major dominant (almost 9 bars in length); attraction and repulsion are made even more apparent:

Fig. 32

A similar contrast was noted earlier: Tamino’s youthful, spontaneous utterances were expressed in B minor and E minor, while the old priest opened up the tonal scope of A^b major and E^b major:

Fig. 33

As a result of the above-described connections, a dual, intertwining ’spiral system’ is created in which each element is balanced off by its counterpole (its tritone-pair):

Fig. 34

Another two arguments are elicited by the transformation in the wake of "Zurück!". In C major tonality, the symmetry centre of our tonal system is marked by the D or the A^b pole. The gate is evoked by D major, the Sprecher appears on the counterpole: in A^b major. For Bartók, the inversion of the major pentatony — DO pentatony — became incarnated in MI pentatony, (projecting the DO pentatonic scale built on the note D downwards of D, a MI pentatonic scale is produced). The basic idea of Cantata Profana cannot be separated from the fact that our tonal system (notation, the stringing of the keyboard instruments, and in many respects the string instruments) is based on the ’d’ symmetry axis. Bartók contrasts the DO pentatony based on D with MI pentatony also based on D. The starting scale of the work rests on the D=MI pentatonic frame, while the acoustic scale closing the work unfolds from the D=DO pentatonic scale: (see Fig. 85 on p. 49) ⁸⁾

Thus the above two scales are the exact mirror images of each other. The D=MI pentatonic scale incorporates the G minor and B^b major triads. The D=DO pentatonic skeleton of the finale contains first of all the D major triad, but also the B minor triad. In The Magic Flute, D major symbolizes the ’gate’ (cf. also Tamino’s rapture in B minor upon sighting Sarastro’s temple). The D=MI pentatonic scale, on the other hand, contains the B^b major and G minor triads: at the turning point, as a consequence of the word "Zurück!", these very ’reversed’ triads get legitimation: first B^b major, followed by G minor (the former after "Zurück", the latter after "Glück").

One more thing should be touched on in connection with the outcry "Zurück!". This is what triggers off the radical change that occurs through the entry from the ’major’ world into the ’minor’ world. On the one hand, C major and its V. degree (G major) ensure the tonal aura that can be schematized in the following way (every second step in the figure rhyme in perfect fifths). The formula also implies the possibility of B minor and E minor (as was mentioned earlier, Tamino’s temper is governed by the positive substitute chords of G major and C major: B minor and E minor).

Fig. 35

To produce the above net of fifths (Fig. 35a), a single C major (or A minor) triad would suffice. Should we replace C major by C minor, the fifth steps would take the shape above (Fig. 35b—the relative key of C minor is E^b major, etc.).

Looking at the two schemata side by side, one realizes that the relationship between the alternating major and minor third is reversed. The second row of thirds goes along the same path as is covered by the music after the shock of "Zurück!":

Fig. 36

In the scheme above C major and C minor, G major and G minor indicate a ’major-minor’ (modal) contrast. A more complicated case of the major-minor relationship is the contrast of the chords with a common third: A^b major and A minor almost present an allegoric contrast (appearance and dissappearance of the old priest). A similar contrast is created between E minor and E^b major, B minor and B^b major. By the same token, the chord with the common third as the enthusiastic B minor emanating from Tamino’s personality is to become the B^b major, which expresses the stout resistance of the gate, the shock ("Zurück!"). Tamino’s flaring up in E minor will also have a chord with a common third: E^b major, which projects to us the state of mind of the old priest together with the spiritual world behind him. And if you add to that, that E minor and A^b major, as well as B minor and E^b major are spheres that negate one another (complementary keys), while B minor and A^b major are polar spheres, then the computer may help you orientate yourself in this multidimensional network of relations.

The broad cadences illuminate the D major episode like spotlights: Tamino is standing in front of the temple gate. The ’map’ of the scene gives similar salience to the importance and frequence of the G minor episodes: "Ja, ja, Sarastro herrschet hier": a G minor arrival, right in the foreground of an explosion. And later: "So gieb mir deine Gründe an!": G minor, before another explosion. And yet another G minor: "das Gram und Jammer niederdrückt."

In C major, the simplest subdominant and dominant are represented by D minor and G major. In taxonomic terms: the symmetry-mirror of D minor is G major. The symmetry remains intact even if we replace D minor and G major by D major and G minor, respectively. In this case, the notes FA and TI are replaced by FI and TA: in our system of notation, the nearest (first) sharp and the nearest flat appears:

Fig. 37

(The frequency of D major and G minor not only illustrates how the mentioned principle became one of the most obvious instruments of ’expanding’ the diatonic system, but also suggests how the ’acoustic’ scale: — DO-scale with FI and TA — was born in Bartók’s style.)⁹⁾

Speaking of the 19th century, let it be mentioned that the dominant of the A minor key: the E major chord also has a symmetry pair, the F minor. In the C major key the symmetry axis of the system is constituted — besides D — by the G^# = A^b note (in the notation, it is marked by the ’middle’, the third sharp or flat, resp.). One points upwards, the other downwards.¹⁰⁾ The nadir of disillusionment is scored by Mozart in the F minor key ("... nie eu’ren Tempel sehn!"), while the development (motto-theme) is born out of E major. F minor and E major not only satisfy the requirements of the ’mirror relationship’, but are also chords with a common third.¹¹⁾

The Mozartian chromaticism is omnipresent — ensuring the organic, unbroken connection between the chords. After the statement "Die Absicht ist nur allzu Klar!" the note B leads to C, then (extended into a major third) gets emphasis from the C^# note so that the latter could proceed to D as the leading note. The role of the neckbreaking polar change (D minor—B major) seems to imply that D should turn into D^# and resolve in E. Via the ’turn-motif’, however, this E also rises to F, which eventually becomes reconciled in the central E note.

Fig. 38

The basic motif of relief, of ’smoothing out’ is the DO-RE-MI motif (also on the stage of Wagner and Verdi). Significantly, the three-times repeated motto-theme is prepared by this phrase (see Fig. 19 on p. 143).

The cathartic moment follows the introduction of the motto-theme: "O ew’ge Nacht" — Mozart has the tonic (bass) and the dominant sound at the same time —hence the time-paralyzing, static effect:

Fig. 39

The true unfolding — the absolution — is brought along by the flute solo of Andante. The dual — E^b major and C major — stage of The Magic Flute has often been discussed by analysts (E^b major as the tonality of being initiated, being an ’insider’ in the sacred secret); this dual tonal stage prevails in the examined dialogue as well.

Compared to C major, the E^b major chord assumes its expressive character from the notes MA and TA (E^b and B^b). When the flute solo taming the beasts is intoned and Tamino’s aria "Wie stark ist nicht dein Zauberton" is begun, the stage lighting also changes: in our tonal system, the mirror image of TA and MA are the notes FI and DI – the aria is dotted with FI and DI colours, which produce the ’fabulous’ natural aura of the scene:

Fig. 40

In connection with the ’sanctified’ key of E^b major, let me remind you of the Sprecher’s theme "... Heiligtum. Der Lieb’ und Tugend Eigentum" following his introduction. The more so, because it ‘rhymes’ with the closing act of the above Quintet, quoting almost ’note for note’ its B^b major melody "Drei Knäbchen jung, schön, hold und weise, umschweben..." and its smoothingly blissful parallel thirds. Deepening the chords in thirds (a typical feature of the ’motto’-theme as well) is well known from the classical literature: it promotes the constant expansion of the ’inner stage’, it keeps intensifying the radiance of the theme.
 

1. In a major key, the turn of the V-III. degree elicits a similar effect. (see. e.g. Beethoven’s Missa Solemnis from bar 345). Here we are already in a minor key, that is, the dominant effect is enhanced by the dominant chord built upon ’DI’.

2. Bartók: Beginning of String Quartet No.4.

3. Cf. the Command motif in the Tristan.

4. See the beginning of Beethoven’s 5th Symphony.

5. Cf. Verdi’s Don Carlos (Act II), the Monk’s second half-phrase in A: F-B.

6. End of page 60, too (The Magic Flute, piano score, Universal Edition):

*) ‘ ta‘ added by the editors

7. This motif has a counterpart, taxonomically speaking a symmetry-pair, which is created by lowering the SO note of the major chord by a semi-tone, e.g. modifying the G of C major to F^#. The resulting F^# subminor (F^#-C-E, or F^#-A-C-E) is a ’blood-kin’ of the secondary dominant (that is, it is related in meaning to the dominant of the dominant: the D⁷ chord), so much so, that in Mozart’s score the two usually appear simultaneously. Such is the Sprecher’s first sentence where the fifth (E^b) of the A^b major is modified to D, and the simultaneously appearing D subminor + B^b7 chords unite in a B^b9chord: 

Tamino’s very first sentence at the beginning of the finale is also governed by a similar rule: the fifth of the C major is changed into F#, resulting in an F# subminor + D7 (= D9 chord). The ’turn’-motif and the turn described above are mirror images of each other: both belong to the basic stock of Mozart’s idiom. One gravitates towards the subdominant, the other towards the dominant.

8. Bartók himself made mention of this MI scale in his Harvard lectures.

9. Lurking behind the gripping - moved and at the same time exalting - B major breaking point: "Man opferte ... sie schon" and the Bb minor expressing disappointment (see the "er will gehen" instruction) one discerns a similar symmetry relation: compared to the D centre, B major is the mirror reflection of Bb minor (the contrast inhering in the reflection prompted Verdi to create one of his favourite dramatic motifs.)

10. That is how the F minor-C major cadence, which is none other than the counterpart, the mirror image, of the E major-A minor cadence (customary in a minor key), struck root in Romanticism. The D subminor-C major cadence is used with a similar meaning (as the mirror image of the E⁷-A minor cadence).

11. Wagner even composed a theme on the relationship between F minor and E major: the Marke theme is one of the most enigmatic thoughts of the Tristan.