Axis Harmonic Principles

A) THE PENTATONICCHROMATIC SYSTEM

The harmonic types of the chromatic system, that is

(1) alpha chords

(2) 1:2, 1:3, 1:5 models

(3) chords of equal intervals

are characterized by the fact that they unite the tension of the Fibonacci-models with theclosedness of the twelve-tone system.

The most characteristic chromatic melodies and harmonies obey the proportions of theFibonacci sequence. Calculated in semi-tones:

2 means a major second,

3 means a minor third,

5 means a perfect fourth,

8 means a minor sixth,

13 means an augmented octave, etc.

In reality, these numbers express proportion and not semi-tone steps.

(1) ALPHA CHORDS

As Mozart or Haydn had employed primarily major and minor triads or as we had grown

HARMONIC PRINCIPLES http://www.mi.sanu.ac.rs/vismath/lends/ch3.htm

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accustomed to seventh-chords in Romantic music, the use of harmonies built ofFibonacci intervals became just as common and predominant in the works of Bartók andKodály:

Fig. 24

We call this type of chord with the collective designation: the alpha harmony the varioussections of which can be distinguished by letters beta, gamma, delta and epsilon.

Chord alpha consists of two layers. In order to establish tonality, at least two notes arenecessary: the key-note (C) and one of its overtones: i.e. the fifth (G) or the major third(E). In this simple case the G or E reinforces the C although G and E has in itself adominant significance.

Type alpha has a strong tonal, even functional character. When, for example, in the maincadence of the Recruiting Music Kodály looks for a dominant alpha chord (before the

tonic E major), he moulds its upper layer from the melody itself (B-G#-F-D), and the

lower layer from the diminished seventh chord C-D#-F#-A:

Fig. 25

Thus type alpha is nothing less than the axis adaptation of the simple C-E-G or C-Grelation (this is why type epsilon rarely occurs owing to the absence of the fifth andmajor third G and E without which the tonal character of the chord is unsteady); thesole requirement is that the chord should be constructed of two layers (two axes).

Type alpha complies with two requirements. On the one hand it derives from pentatony.The intervals 2, 3, 5, 8 (or their octaves) sound together with every note of the chord.

Fig. 26


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This is how in one of Bartóks choruses the pentatonic DO-SO-LA-MI motif istransmuted into gamma and delta harmonies:

Fig. 27

On the other hand, alpha harmonies are axis models and, as such, express the polymodaltensions of the axis system. (See: Fig. 32 on p. 27)

Let us add: the tension of alpha harmonies may most simply and effectively beexpressed by the symbols of polymodality.

Fig. 28


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Examples

Fig. 29.

(2) MODELS 1:2, 1:3, 1:5

By the chromatic juxtaposition of intervals 2, or 3, or 5, closed scales are produced (i.e.by the periodic repetition of the intervals we are taken back to the starting point).

MODEL 1:2 is an infinite chain of minor and major seconds,

e.g. C-C#-Eb-E-F#-G-A-Bb-C,

MODEL 1:3 is an infinite chain of minor seconds and minor thirds,

e.g. C-Eb-E-G-Ab-B-C,

MODEL 1:5 is an infinite chain of minor seconds and fourths,

e.g. C-C#-F#-G-C.

Fig. 30


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Model 1:2

Model 1:2 should be considered the fundamental scale of the axis system.

In the 12-note system three different 1:2 models may be established, in accordance withthe three functions: a tonic, a dominant and a subdominant. Every further form agreeswith one of these models, e.g. in C tonality:

Fig. 31.

All chords and models appertaining to the same axis constitute a functional unit.

Fig. 32


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Examples

Fig. 33

1:2 model harmonic turns **)

If we combine a major seventh and a subminor chord lying a minor third higher we getthe 1:2 model. This combination became current among the Romantics.

Fig. 34


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In Act I of Tristan, the inner world is visualized by the Tristan chord (F subminor), whilethe outer world by the chorus of the Sailors at all times appearing in the form of a Dmajor seventh chord.

Fig. 35

Model 1:5


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Model 1:5 represents another typical axis sequence: it rests upon polar relations.

Fig. 36

Fig. 37

Model 1:2 may be split up into two 1:5 models.

Fig. 38

In the Mikrokosmos piece From the Island Bali both left and right hands play 1:5models, which together create a complete 1:2 model.

Fig. 39

The closing chords unite these four elements.


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Fig. 40

In the final chord two triads are merged: the difference of six accidentals between thetwo triads create a polar tension and the two triads fuse in a perfect alpha harmony.

Fig. 41

1:3 model

While models 1:2 and 1:5 have a powerful tonal character, the 1:3 model annihilates

tonality due to its augmented triad structure. For instance, the C-Eb-E-G-Ab-B 1:3 modelcomprehends the following triads:

The floating quality of model 1:3 was already recognized by Liszt and Wagner.

Fig. 42


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Examples

Fig. 43

Complementary (annihilating) keys

A 1:3 model can be created by uniting a major and a minor triad (the latter lying a majorthird lower).

In this case one triad neutralizes the other since their notes combine in an atonal 1:3model. This is why such triads express a contrast in their content as well.

Fig. 44.


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The idea of annihilation goes back to Romantic models. When Wotan in the greatMonologue of the Valkyrie prophesies the Twilight of Gods, his words "Das Ende! DasEnde!" evoke E major and C minor which tonally destroy each other.

Fig. 45

A peculiar manifestation of the annihilation idea is Weberns famous Reihe in which two1:3 models are merged:

Fig. 46

A change from major to the complementary minor (E majorC minor) results in anegative effect, it is associated with gloomy, or even, oppressive and irrationalexperience. (See Fig. 45 above)

And conversely, a change from minor key to the complementary major (e.g. C minor Emajor) creates a positive impression, it is inspired by enthusiasm and serenity like the E


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major episode at the end of Beethovens Piano Concerto in C minor.

In the chorus The Aged, Kodály translates the life-and-death symbolism of the piece into

the language of music by means of complementary keys: G major and Eb minor then Bb

minor and D major:

Fig. 47

The entire tonal plan of Bluebeards Castle is built up of such complementary relations.

F# minor is the key of night and C major that of light. C major can be destroyed by

means of the Ab minor key thus the latter is associated with death symbolism. On the

other hand, the nights F# minor can be defeated by Bb major thus it became the symbolof love. The four triads together include every degree of the chromatic scale:

light: C-E-G death: Ab-Cb-Eb

night: F#-A-C# love: Bb-D-F

The basic tonality in Kodálys Psalmus Hungaricus A minor is equivalent to weeping,imploration, despair, curse. The dénouement of the action, on the other hand, takes place

in the complementary key: Db = C# major. It is the task of Db major to absolve from the

weight of the curse: From you he removes your every burden. And that of C# major tobecome the key of elevation and apotheosis: In honour Thou wilt raise him on high!

Complementary keys may appear in a hidden form, too. The "Sündenqual-motiv" fromParsifal exerts a tormenting effect because the authentic sequence (moving authentically

on the circle of fifths) is coloured by complementary chords: F major and Db minor in b.

1; Db major and A minor in b. 2, etc.

Fig. 48


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(3) EQUIDISTANT SCALES

Closed sequences may, however, be created by simpler means by composing equidistantscales from the Fibonacci intervals 2, 3, 5, 8:

from major second intervals (2) whole-tone scale,

from minor third intervals (3) diminished seventh,

from perfect fourths (5) fourth chords,

from minor sixth intervals (8) augmented triad

can be established.

Fourth chords**)

Owing to the folksong inspiration, strikingly frequent is the theme formation andharmonisation with fourth chords. The characteristic fourth-accumulation in our ancientmelodies spurred us on to the forming of fourth chords: we have projected here thehorizontal succession into vertical simultaneity (Bartók: The Influence of Peasant Musicon Modern Music, 1920).

This is how Bartók transforms the fourth melody into a fourth harmony in the ViolinConcerto:

Fig. 49


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Tonicantitonic relations in the pentatonic scale

Thus, fourth chords in Bartók and Kodálys compositions derive from folk-music, and inkeeping with the folksong inspiration, also in the connection of fourth chords the innerlaws of pentatony assert themselves.

To mention the most important one: of two fourth chords which are placed at a distanceof a minor third (3) or major second (2) from each other, the tonic model is always theone which lies a minor third lower or (which means the same) a major second higherthan the other. We call one of them the tonic and the other antitonic model:

Fig. 50

In Fig. below, the tonic model is represented by fourth-degrees C-F-Bb, and the antitonic

by fourth-degrees Bb-Eb-Ab.

Fig. 51


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The tonic-antitonic attraction originates in the SO-REMI (DO-SOLA) cadence sofrequent in folk melodies: the SO-RE holds the tension, while the MI corresponds to thetonic resolution:

Fig. 52

It deserves a special attention that this time we are faced with a two-function system (andnot a three-function one, as in classical harmony).

Incidentally, chords based on the SO-LA-DO-RE structure have a floating, soaring effectsince the tonic and antitonic (RE-LA and DO-SO) relation supports not the lower, but theupper note.

Omega chords

In my analyses, the letter omega indicates the whole-tone scale. I have deliberatelychosen the letter farthest from alpha because Bartók himself used them oppositely. Alphais tense in character, omega is dissolved and this quality becomes apparent in that thewhole-tone scale, as opposed to alpha and pentatonic structures, contains not one singleperfect fourth (nor a perfect fifth) without which the tonal character of the chordbecomes unstable.


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In the 12-tone system two omega scales can be distinguished (6+6 notes), they are

mutually geometrical dominants of each other (C-D-E-F#-G#-A# and Db-Eb-F-G-A-B):w1 and w2.

Fig. 53

Kodálys chorus Fancy concludes with a complete omega harmony depicting the peal ofbells:

Fig. 54

The omega harmony due to its fluid character lends itself particularly well fortone-painting (landscape painting). We give a Kodály example:

Fig. 55

Because of the contrast, the direct confrontation of the alpha and omega tonalities is veryeffective. In the third movement of Music for Strings, Percussion and Celesta, the centrallight theme accompanied by the high-pitched cymbal also includes this duality. Themotif is centralized around the C note. Depending on whether it occurs in root position


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or inversion,

Fig. 56

it can be accompanied by omega chords (Fig. 53 on p. 35) or by alpha chord: through theperiodic repetition of diminished triads, a closed alpha structure is created:

Fig. 57

*

In practice, the Fibonacci models i.e. alpha and axis harmonies, 1:2, 1:3, 1:5 models andequidistant scales merge into each other. In Fig. below, the hunting ostinato is quotedfrom the chromatic first movement of Bartóks Sonata for Two Pianos and Percussion:

Fig. 58


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The entire chromatic scale is included in the twelve notes of the ostinato. Apolecounterpole relationship exists between the opening and the closing notes (upper

part A and Eb, lower part F# and C), forming together an axis arrangement: F#-A-C-Eb.

The upper part is composed of the A-B-Db-Eb-F-G omega scale (its formula being

2+2+2+2+2+2), while the lower part of the complementary omega scale F#-G#-Bb-C-D-E. The two parts progress in parallel minor thirds (3). Motivically, each part is built

of minor sixth (8) elements: the upper part of augmented triads A-F-Db and B-G-Eb, and

the lower part of augmented triads F#-D-Bb and G#-E-C (8+8+8). The harmoniccharacter of the ostinato is defined by gamma chords (3+5+3) and 1:3 models:

Fig. 59

The two 1:3 models can be fitted chromatically. Thus all of the structural elements areFibonacci formulas.

B) THE DIATONIC-OVERTONE SYSTEM

ACOUSTIC (OVERTONE) CHORD

The basic form of the diatonic system is the so-called acoustic scale (DO-scale with FIand TA), e.g.

C D E F# G A Bb C

and the acoustic harmony, for example, C major triad with natural seventh Bb, acoustic

fourth F#, and major sixth A (pastoral sixth) which is called acoustic since its notesoriginate in the natural overtone series:

Fig. 60


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Examples

Fig. 61

Other examples see: Fig. 87 b on p. 50 and Fig. 107 on p. 65.

The acoustic character becomes even more effective if it is the result of an expansion.The hunting-fugue in Bartóks Cantata Profana becomes so explosive since the minorthird changes into major and bursts with an acoustic fourth (MA-MI-FI):

Fig. 62

The perfect fifth frame of the theme and its two points the acoustic fourth at thehalf-close (FI) and the acoustic MI-TA step at the full close only enhance the naturalatmosphere of the scene.

The acoustic chord, with the exception of one tone, contains a whole-tone scale (omega)

as well: Bb-C-D-E-F#. Therefore, the acoustic chord can easily be coloured by theomega scale:

Fig. 63

The acoustic harmony is familiar in Kodálys music too although he had an affection for


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the brighter Lydian modality:

Fig. 64

In Kodálys music there is often very little difference between the acoustic harmony and

the polar harmony. For example, in Fig. above the first chord is the combination of Bb

seventh and E seventh. The acoustic chord of Fig. 84 b consists of the C major and F#

major counterpoles.

Now we try to find a new path in deducing the acoustic harmony. Let us start from ourobservations See: p. 77 that the relative of the C major is A minor and that of the latter

is F# subminor:

Fig. 65

The tones have a symmetrical distribution around the virtual RE (=D) symmetry center!If we combine the tones of the three chords we obtain an acoustic harmony. In both casesFI and TA are determinants of character. And what is evident again: FI and TA are exactreflections of each other in relation to the RE symmetry center.

The acoustic scale became a static colour chord because it lacks the two sensitive notes

that characterize the major scale: instead of FA and TI (F and B) FI and TA (F# and Bb)notes occur.

What the spectrum of rainbow-colours is in optics, is the natural overtone scale in music.(The term acoustic scale comes from me 1947).

Hypermajor and hyperminor

Whenever Bartók or Kodály intends to endow the acoustic harmony with more light, the


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minor seventh of the acoustic chord is raised to major seventh.

Fig. 66

We shall call this chord type (major triad with major seventh) the hypermajor harmony:

Fig. 67

The hypermajor embodies the most solemn sound-type in Bartók and Kodálys music (theopening chords of Psalmus Hungaricus and Budavári Te Deum are hypermajorharmonies) it has become the characteristic concomitant of apotheoses.

The origin of the well-known Bartók signature also goes back to the hypermajor:

Fig. 68

The hypermajor owes its light not only to its major character and major seventh, butprimarily to its consisting of two perfect fifths (see Fig. 67 above).

In the most pictorial effects the hypermajor harmony merges with the acoustic fourth(FI):

Fig. 69


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The hypermajor has a counterpart: the hyperminor. Its construction is a minor triad with

major seventh (the Eb-G-B augmented triad adds considerably to its individualcolouration):

Fig. 70

Your leitmotif wrote Bartók in a letter to Stefi Geyer. It is to be found all over where thetext speaks of pain and passion (see: Fig. 70 b above).

By inverting Bartóks leitmotif of love, a peculiar kind of chord arises which in Bartóksworks is associated with the symbolism of death: Eros turns downwards his torch!

Fig. 71.

It appears whenever desire is fulfilled and, as a consequence of the fulfilment, passion

ceases. Through the inversion the augmented triad of the hyperminor (Eb-G-B in the

foregoing example) moves to the bottom of the chord: A-F-C#; this is the source of theneutralizing effect. The final chord of Bluebeards Castle is also a hyperminor-inversion,the whole opera terminates in this symbol of death:

Fig. 72


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Alpha inversion

If it is true that the diatonic system is merely a mirror-image of the chromatic

system,***) then diatonic sound-types can be produced by inverting the layers of thealpha harmonies:

Fig. 73

The diatonic impression is the direct result of the alpha-inversion being ruled by fifths,major thirds and minor sevenths (i.e. the closest overtones) that are precluded by thealpha harmonies.

Oddly enough, the harmony with a major third above the root and with a minor thirdbelow it, evokes the most opened impression:

Fig. 74

And to bring to an end the interconnections: the alpha-inversion carries in itself the seedof the acoustic harmony as well:

Fig. 75

By exchanging the C and F# notes, a polar relationship can be effected (C and F# seventh

chords, or C and F# ninth chords). The recapitulation theme in Kodálys Háry Prelude

(Fig. 84 b on p. 48) may be interpreted equally as a C acoustic or an F# acoustic tonality.


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This 4-note polar formula started gaining ground already in Romantic music. The tonalnucleus of Station 10 in Liszts Via Crucis, or the fortissimo explosion introducing thegreat ensemble of Act III in Verdis Otello, allows the following interpretations:

Fig. 76

*Let us summarize the basic types of the two harmonic systems:

CHROMATIC-PENTATONIC SYSTEM DIATONIC-ACOUSTIC SYSTEM

Pentatony, Fibonacci-models Overtone chord, acoustic scale

Alpha chords Alpha-inversion

Models 1:2, 1:3, 1:5 Hypermajor, third-tower

Equal-degree harmonies Equal-degree harmonies

from intervals from fifths, major thirds

2, 3, 5, 8 and minor sevenths

*) It includes subminor chords, too (see. p. 77) (ed.)

**) Title given by the editors.

***) (see pp. 48-50)

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Axis Harmonic Principles