Revised: 17August 2009

When I created the harmonic school of Bimodalism in the late 1950s, I thought it should have occurred as a musical discipline before the advent of atonality. But, as we know, it did not.

Perhaps, Bimodalism should have been a middle road between the followers and abandoners of tonality. But this did not happen either.

In retrospect, we can observe how the development of harmony correlates chronologically to the phenomenon of the harmonic series: As each period in music history came about, harmonic intervals of sounds, corresponding in order and pitch to those of the series, gradually integrated a chordal column of superimposed thirds, a column whose capital was crowned with the thirteenth chord.

Harmonic evolution, based on the phenomenon of the harmonic series, may thus be seen as teleological in nature: Sound carried within itself the harmonic components of the chord, to be revealed as representatives of each period throughout the history of music.

But regardless of whether this teleological conjecture is confirmable, the fact is composers did progressively superimpose the seven different sounds carried in the harmonic series until they attained the culmination of tertian harmony.

This prompted me to ask, Why then did they not also superimpose the major and minor modes in a triad, before the dissolution of tonality? Would this not have been the culmination of tonality?

Perhaps, composers have been too enthusiastic to follow the orientation of other contemporary schools (such as the Second Vienna School), overlooking the possibility of modal unity that Bimodalism proposes.

In this sense, by composing in bimodal harmony, I believe I have not only filled a hitherto unnoticed gap in music history, but have contributed as well in renewing the development of harmony from the point where late Impressionism left it.

In this era, where serialist and polytonal decadence force us to invent extramusical avant-garde procedures, or lead us to excavate the remote past—as the minimalists have done—in search of magical formulas that might aerate the blood of our contemporary music, the chance to return to the realm of harmony presents its most novel and piquant option in Bimodalism.

To return is not always to retreat, but sometimes to resume.

The essence of the harmonic discipline of Bimodalism lies in the simultaneous blending of major and minor modes in triads with the same fundamental root.

This blend—the begetter of a new ethos in harmony—is applied generally throughout an entire musical work, from beginning to end; or in sections where the musical thought lasts a significant duration of time; or in a symbiotic form, sharing roles with other existing chordal entities: symbiotic Bimodalism.

(N.B. Any attachment to a bimodal chord nullifies its harmonic effect [ethos]: The polychord, for instance, although it consists strictly of bimodal chords, actually loses this Bimodal harmonic effect because of the chordal juxtaposition of notes extending beyond the triad. Therefore, any use of polychords is discarded in Bimodalism.)

When we listen to music in bimodal harmony, we hear something reminiscent of the old tonality.

There is a reason for this: Bimodalism forges the two modes that characterize western music since the seventeenth century, in order to form a single chord.

It is precisely the bimodal chord. (Although the bimodal chord does in fact consist of four sounds [such as C-Eb-E-G], I will refer to it as a triad, in keeping with the lexicon of harmony).

By joining major and minor triads (and only these triads) of the same root on each degree of the chromatic scale, we obtain 12 bimodal chords of four notes with their three corresponding inversions—a total of 48 chords.

In turn, these 48 bimodal chords—as each has 44 possibilities of chordal progressions within the chromatic scale—yield a total of 2112 chordal progressions for the entire harmonic system, whose chordal progressions bear a close tonal interrelationship, given the chordal equality that Bimodalism establishes along the chromatic scale.

Consequently, both remote tonalities and modulation concepts are senseless in Bimodalism, as it is intrinsically a chain of 12 equivalent and harmonically linked chords, forming, in fact, a true unitonality—in other words, a Bimodal chord over each degree of the chromatic scale.

N.B. Although the appearance of certain harmonic entities throughout contemporary musical history (such as whole tone chords, fourth or fifth chords, the mystical chord, tone clusters, and others) have made an impression on our ears, composers have only been able to use these harmonic entities as accessories to an eclectic compositional style (Puccini), or as the harmonic emblem of a paradigmatic work (Scriabin), because these harmonic entities cannot generate a rich musical literature on their own within the domain of tonality, such as was the case during the heyday of tertian harmony.

Rudiments and Table of Bimodal Chords

FIG. 1. The example at right shows two notations of the bimodal chord in its closest position: (a) as it should be written currently; and (b) as it would appear if we were to use this accidental sign of my own creation. I propose such a sign for altered and unaltered notes of equal name and placement that are played simultaneously—in the case of the bimodal chord, only its third. Similarly, I propose that the third degree of the diatonic scale be termed as modal, despite the mediant (a merely ordinal term). If the first and fifth degrees of the scale are denominated by their tonal function, its third degree, likewise, should then be denominated by its modal function, since it—and only it—determines the mode in music. Thus, the first, third and fifth degrees of the scale could be denominated, tonic, modal, and dominant, respectively.

FIG. 2. The example at left illustrates Bimodal chords on the 12 degrees of the chromatic scale in enharmonic notation.  Notice how each chord displays the double-letter symbol representing the fusion of both major and minor modes: Cc, Cc#, Dd, and so forth. They are, however, to be referred to solely by their root and (mixed) modality: Bimodal C, Bimodal C#, and Bimodal D, respectively. And in the Guidonian nomenclature: Bimodal Do, Bimodal Do#, Bimodal Re, and so forth. (Where you see the asterisk in the illustration at left, note the following: To avoid any confusion, I employ the letters B and H as symbols for the B flat and B natural notes, respectively, following the usage in German nomenclature. Consequently, the bimodal chord over B flat is written as Bb, and the bimodal chord over B as Hh.)

FIG. 3. The example below illustrates first, second, and third inversions of the bimodal chord—close and open positions, respectively. All three inversions, including the defective ones, are beautiful and well balanced. The first two inversions (chords of the sixth, open or close), however, are more characteristic of Bimodalism in their harmonic soundscape (the Bimodal ethos).  I have therefore used them abundantly in my own bimodal scores, practicing, in fact, a kind of bimodalized sixth-chord style ("fauxbourdon").

 Click here to play Figure 3.

N.B. Where you see the asterisk in FIG. 3 above (*), please note that in Bimodalism, the ever-present double third degree alone defines unmistakably the identity of a chord. Therefore, you can omit the fifth—and even the first!—degree of a chord without losing or confusing the harmonic identity of the same chord. This means you can easily tell if the omitted degree is the first or the fifth, once you find the the double third of the chord. I refer to this type of chord as a "minus-one chord" (my own coinage).

FIG. 4. The illustration below shows that in Bimodalism the 12 fundamental chords—each with three corresponding inversions—yield a total of 48 bimodal chords. These 48 chords, when multiplied by the 44 possibilities of chordal progressions that each may have, yield a total of 2112 chordal progressions for the entire system.

Bimodalism does not ignore the dynamic law of tension and relaxation that rules tonality, as do both serial music and polytonality.

Bimodalism, however, does not apply this law through exclusive scalar degrees, depending on a gravitational center or tonic, as does tonality.

Bimodalism, barring the influence of tonal rules and its metric restrictions, applies this law to a very free melodic wave that reveals its proper points of tension and relaxation on any bimodalized chordal degree of the chromatic scale.

In other words, bimodal tonality allows all sorts of prerogatives in the syntax of musical discourse.

Nonetheless, although Bimodalism has expanded the harmonic radius of tonality to its limits, this system can freely resort to the most common chordal progressions of traditional harmony— by way of Bimodalizing its triads. (Of course, you could also include nonharmonic tones in bimodal harmonizations).

In doing so, we experience these chordal progressions harmonically attired in the bimodal style.

Although Bimodalism is generally applied to work on the degrees of the chromatic scale, this system can also be applied to any known or invented scale. In effect, you can then form bimodal chords over the degrees of any such scale to accompany a given melody by following the guidelines of a homophonic style.

While we are on the subject of invented scales, I would like to propose a scale that I have dubbed the overtone scale (C-D-E-F#-G-A-Bb-B) because it encompasses the eight distinct sounds of the harmonic series in an octave.

With this scale, we could stack all eight sounds thereby creating the overtone chord. This chord contains a harmonic spectrum consisting of waves that interfere with one another, resulting in a sound that is reminiscent of the dominant seventh chord, but with a dense and quasi-electronic aural quality.

I liken the overtone scale and overtone chord to gems whose facets consist of the components of the harmonic series and that now shine with equal relief and radiance.

We can think of the overtone scale and chord as the legacy of a long history that began with Pythagoras and that culminated with the birth of the founding father of harmony, what we call the natural chord.

Although Bimodalism can resort to other chordal entities with which it can share harmonic roles, I advise against alternating bimodal chords with those of tertian harmony beyond the perfect triad.

Neutral fourth chords can also play a compatible role owing to their tonal neutrality.

There is, however, a specific chord that can share the best harmonic role with the bimodal chord in a symbiotic relationship: the major chord with an appoggiatura without resolution.

This non-invertible chord always has its appoggiatura of second minor or major in the bass, and it can be stored in any of its close or open melodic positions of root, third, or fifth. (The third position, open or close, is the best balanced of the three.)

Although this chord is not bimodal, it has two possibilities to become a bimodal chord: (1) Its appoggiatura can either be resolved in a bimodal chord, or (2) its three upper voices (the major triad) can serve as a triple appoggiatura resolved in another bimodal chord in a descending movement.

The major chord with an appoggiatura without resolution in the bass may owe its spatial sound identity to a phenomenon of mere acoustic illusion: We perceive a sensation of dimensional growth in this chord because the root of the chord is elevated only in the bass while the upper major triad is unchanged. (The open triad position intensifies this effect.)

This illusory acoustic phenomenon acts as a sort of harmonic lever, lifting the major triad pillar onto a higher plane without muddling its sound identity as a major triad. Because its root is transposed to the supertonic degree, this chord could easily be dubbed the Paratonal Major Chord (or Paratonal Chord, for short); and so, by virtue of symmetric paratonality, the first degree of the triad (in the bass) climbs onto the second degree and assumes its permanent role as the new root of the chord from that position.

Moreover, contemporary audiences can easily withstand a passage consisting solely of Paratonal Major Chords, owing to the close relationship of this chord with its originator, the major chord. (This is yet another harmonic déjà vu brought about by Bimodalism.)

FIG. 5. The example below illustrates: (1) The major chord with an appoggiatura without resolution; (2) the major chord with an appoggiatura resolved in a bimodal chord; and (3) the major chord with an appoggiatura, whose three upper voices are used as a triple appoggiatura resolved in another bimodal chord. (N.B. It is a theoretical inconsistency to denominate the major chord with an appoggiatura without resolution in the bass as an "eleventh" chord [a common misconception in pop music]: A tertian chord, which is what an eleventh chord really is, must be integrated by its generic third degree [the modal] so that its modality may be determined. There are eleventh chords, both major and minor, but if they lack their corresponding third degree, it is impossible to determine which mode they belong to.)

For instance, a minor eleventh chord could be: C-Eb-G-Bb-D-F [11]; whereas, the major, raising its third, seventh, and eleventh by one halftone could be : C-E-G-B-D-F# [+11]. And the augmented eleventh chord, whose prominent effect makes it the ace of the group, could be: C-E-G-Bb-D-F# [7+11].)

 Click here to play Figure 5.

As we can conclude from this discussion, an aesthetic aim of Bimodalism is to evoke tonality in its most remarkable of fundaments.

Bimodalism dresses the chordal progressions of traditional harmony in the attire of the bimodal harmonic style, allowing us to use any known or invented scale as a fertile ground where bimodal harmony can also flourish.

Bimodalism also allows us to resort to other chordal entities to share harmonic roles that are symbiotic with the bimodal chord.

Taking into account that Bimodalism can be freely applied with no previous compositional scheme, we see how the bimodal chord can also generate textures of an extratonal nature, applying a procedure of musical writing of my own design, whose aural results I coined "symmetrical paratonalism." (This is not to be confused with the single-term "paratonality" as employed by John Vincent in his book, The Diatonic Modes in Modern Music, published in 1974.)

My term defines the simultaneity of a voice with its doubling systematically transposed to another tonality by only a major or minor third, or by only a major or minor sixth.

But symmetrical paratonalism should not be judged as a procedure of bitonality, when this sort of twin paratonalism works within a homophonic texture of bimodal harmony, without any adherence to extratonal elements that may attenuate or nullify its pure effect, such as may sometimes occur in bitonality. (Cf.: Milhaud’s "Le boeuf sur le toit" [Part H, from measure 188 through 202]).

In applying symmetrical paratonalism, music originally composed for two voices (e.g. melody and its counterpart, or melodies in counterpoint) will be magnified and tonally enriched: Transpositions and doublings of every voice departing from every note of a bimodal chord obtain this effect.

Thus, the melody and its transposed doubling might depart from two notes of the bimodal chord, a third or sixth apart, whereas the transposed counterpart with its retransposed doubling depart from the two remaining notes of this chord.

(Restriction: A symmetrical paratonal combination that originates harmonic intervals of diminished or augmented octaves, in the doubling of a same voice, must be avoided: It destroys the aural effect and ethos of Bimodalism).

FIG. 6A. This illustration shows how symmetrical paratonal writing magnifies and bimodalizes a tonal passage originally written for two voices.

 Click here to play Figure 6A.

FIG. 6B. This is a strict eight-measure canon derived from the former theme appearing in Fig. 6A. If certain notes, notwithstanding, from symmetric paratonal writing were incompatible with the Bimodal ethos, you could change them in favor of Bimodalism.

Click here to play Figure 6B.

In addition, a simpler variant of symmetrical paratonalism can generate a full bimodal harmony when scales, arpeggios, or broken chords of identical patterns charmingly transfigure their old look, as they depart from each voice of a bimodal chord in parallel motion.

FIG. 7. This illustration below shows how a scalar symmetrical paratonalism can be made from each note of a bimodal chord. In this example, the inverted Cc chord originates and juxtaposes the Eb, G, C, and E scales. The same procedure can be used on arpeggios, broken chords, and so forth.

Such a magnification of the musical writing for two voices, forming symmetrical paratonal duos with harmonic intervals of thirds and sixths only, places these traditional consonancies in the new aural dimension of Bimodalism.

From this dimension, they evoke a singular flavor of Mediterranean idiom that is as copious as it is core to our western music.

The terms Bimodalism and bitonality are by no means interchangeable. Bitonality (or polytonality) is a contrapuntal style of writing music, where each of two or more diatonic melodies is written in a different tonality, so that all may sound simultaneously.

But because each melody shares equal prominence, creating a dissonant field of voices in collision, the ear cannot differentiate the tonality of each melody nor can it find the resulting tonality from a summation of combined tonalities. This is evidence that polytonality is, in fact, atonal.

The reality is that the concurrent function of various tonalities that polytonality prescribes betrays the aesthetic idea behind its conception, and loses the defining ethos that would otherwise differentiate it from atonality.

Only when two tonalities are superimposed a tritone apart (such as C to F#), and the counterpoint between them is carefully treated, can our ears sense a certain feeling of tonality, derived from that bipolar interaction. (Bear in mind that between the C and F# tonalities there are only two notes in common: B and E# [F].)

Hence, polytonal composers have so often used this bitonal formula, above all, in the orchestra, where different plans of timbres and nuances favor this stratiform style of writing.

In contrast, the superimposition of major and minor triads of a same tonality (such as C and Cm) is not proper of bitonality; if this were to occur, it would be done following the technique of bitonality, where each of two autonomous melodies would be written in a different mode of the same tonality, forming a contrapuntal plot exempt from harmonic ties; in other words, a counterpoint where every occurrence of Bimodal harmony would always be welcome, and not preceptual, as it is in Bimodalism.

Consequently, if two melodies were written in separate modes of a tonality with no harmonic coupling they would sound discordant and turbid, as in any other bitonal combination. (A rich harmonization, notwithstanding, of Bimodal chords over a basso ostinato of a specific tonality could well emulate the optimal effect of the well-known Boléro.)

Indeed, when bitonality does superimpose major and minor tonalities, it always does so by superimposing remote tonalities of different roots (such as C and Ebm), whose tonal disparity in the scalar degrees may produce the most contrasting display of aural plans, which is the primary objective of bitonality.

As we can see, case by case, bitonality differs from Bimodalism as much in theory as in practice.

Bimodalism, in contrast, is the new harmonic style of homophony. In it, there is but one melody, harmonized with Bimodal triads.

Notwithstanding, Bimodalism could lend itself to counterpoint, provided one were to keep the polyphonic texture within a bimodal harmonic framework, where a certain sensation of tonality is always present.

This sensation of tonality, owing to the fusion of two modes into one, may incline one to think of Bimodalism as the maker of a sort of neotonality, but never as an offspring of bitonality—as some might mistakenly consider it.

Bimodalism, moreover, bears no resemblance to the twelve-tone technique in its use of the chromatic scale. The differences between Bimodalism and the twelve-tone technique are fundamental:

• The twelve-tone technique—envisioned by Schoenberg as well as by Hauer—organizes the 12 degrees of the chromatic scale to nullify tonality. Bimodalism, in contrast, builds a new chord system on the same degrees, serving to restore and harmonically enrich tonality to its limits.
  
• The twelve-tone technique substitutes the melodic strand of a work by a formulated succession of sounds (the series or the trope), whose interval order engenders an atonal language. Bimodalism, in contrast, not only rejects all forms of serial composition, but it also indiscriminately uses any scalar degree to create melodies of a traditional style. 

To direct the musical flow to all scalar degrees (while ensuring that no degree assumes greater prominence than another), one does not need to nullify tonality, as the twelve-tone technique has done.

Bimodalism can direct the musical flow to all scalar degrees without nullifying tonality, by reducing tonality to its harmonic quintessence: the bimodal chord.

By harmonizing each degree of the chromatic scale with the bimodal chord, each degree assumes equal prominence. From this ground, homogeneity (a quality that is proper of tonal harmony) becomes a common attribute of all chordal relationships within the scale.

In other words, all chordal relationships become closer; but these relationships sound even closer, when they keep notes in common or when they are formed by chords whose roots are separated by consonant intervals (such as perfect fourths or fifths). In both cases, such relationships are consequently easier for the ear to identify.

In fact, Bimodalism creates a sort of harmonically equalized unitonality, synthesizing modal variety in the bimodal chord, where chordal relationships built only with bimodal chords show just as much homogeneity as those of the classical tonality, and where the trace of all elementary chordal progressions from the classical tonality are easy to follow by ear, although they are now made strictly of bimodal chords.

Intending to put an end to the chaos of atonality that late post-Romanticism brought about, the Second Vienna School systematized atonality in an aesthetic dogma, whose restrictive rules are comparable to those of early polyphony in history.

Thus, by inventing a system of musical composition, the twelve-tone technique created the serial form in music. A form that shines like a lone star, once tonality, melody and harmony were extinguished in the musical sky.

Soon thereafter, by applying the serial form to all compositional elements in music, the Second Vienna School brought this to its apotheosis: Form, so proclaimed by the School, is the most important matter in music!

As for serial form, however, one may argue that its perception is more visual than aural: When one reads a serial music score, the perception of its form is clear to the eye; but when one then listens to the same score, the ear no longer perceives its form as did the eye.

On the contrary, the ear actually loses all trace of form, perceiving only a disordered mass of sounds in perpetual collision.

Therefore, is serial music for the eye or for the ear? More realistically, can one say that what is appealing on paper is equally appealing when played? I think not.

The horizontal path of sounds deliberately chained to align them perpendicularly with other parallel sounds either produces a concordant or discordant effect. If the result is concordant, as you would have in tonal counterpoint, the form from the placement of these sounds stands out easily to the ear; but, if the result is discordant, as you would have in serial atonalism, the form (serial or tropal) is overshadowed and the ear cannot perceive it; the trace of form is lost, in other words. (Atonality rendered unintelligible the only language we all understood.)

Hence, the elements useful in an aesthetic judgment of mere graphic forms play no role in music by themselves. The ear can only perceive musical form acoustically in the more or less near presence of tonality: The greater the tonal presence, the greater the perception of form.

Herein lies the paradoxical dilemma of atonalism, where the musical form—the prime compositional element to be judged in this kind of music—is actually not catchy to the ear. This paradox may have been signaling over the decades that form, in music, is inherent to tonality.

Therefore, it is unwise to separate form from tonality as serial and polytonal music have done, as there is no way to discern form, if one does not present it to the ear within a certain atmosphere of tonality.

Perhaps, in lacking tonal sensation, no kind of atonality has ever stimulated worldwide audiences over time, although this style of music has been represented by some of the most conspicuous personalities of twentieth-century music.

Bimodalism, contrary to all mathematical formulation or aleatoric practice in music, adopts and develops thematic form, as only this form can bring out the musical wealth that tonality bequeathed us.

Therefore, Bimodalism has no place in serial or aleatoric formulation of form. (Art is emotion: A work of art is merely a dexterous formulation, if it does not pour out from the fonts of the subconscious, where emotion is transmuted into metaphorical language.)

A musical style such as minimalism, however, which marks an abrupt return to thematic form within tonality, may employ bimodal harmony in its monothematic expositions ad infinitum.

In fact, Bimodalism and minimalism have two aesthetic principles in common: Both represent a reaction against atonality, and both return to the tonal thematic form.

But in returning to tonality and thematic form, Bimodalism and minimalism differ greatly in their procedures: Minimalism returns to tonality without reforming it, whereas Bimodalism revives tonality by radically reforming its harmonic organization.

Moreover, as minimalism returns to thematic form without developing it, Bimodalism returns to all the forms of thematic development, rejuvenating them with a hitherto unheard harmonic touch. (Perhaps, Bimodalism is a dimension and ethos that tonality reserved for its very own survival).

In short, while the twelve-tone technique and polytonality are schools of modern counterpoint (aimed at developing polyphonic textures that nullify tonality), Bimodalism, in contrast, is a harmonic system whose achievement is to bring tertian harmony to its ultimate consequence: the modal unity of the perfect chord (the unique harmonic element that Bimodalism uses throughout the chromatic scale).

There have been sparse instances of bimodal chords adrift on the musical ocean of the twentieth century, either by modern contrapuntal incidence or for a sought-after color effect.

Notwithstanding, Pietro Raimondi’s Esempio No.4, from Due Fughe In Una (1849) for two mixed choruses and organ, cannot be considered a case of early Bimodalism in history, as the composer consciously eludes the simultaneity of both thirds on the modal degrees of the work in question.

As Raimondi admitted, "Molto difficile mi è riuscito questo esempio, per causa delle terze ora maggiori, ed ora minori, secondo le circostanze, onde non guastare il canto." [This example has been quite challenging for me, first in handling the major third, then the minor, such that the singing would not be impaired in any case.]

His procedure was ingenious: As one chorus sings in D major, the other sings in D minor, neither chorus coinciding harmonically in the modal thirds throughout the piece.

Esempio No.4 therefore does not consist of the simultaneity of the two modes, chord by chord, which is precisely what Bimodalism is about.

Moreover, we may not allude to the first part of Stravinsky’s cantata, Le roi et les étoiles (1911), as a case of contemporary Bimodalism, as the principles of harmonic analysis classify the chord of the major and minor thirds that characterize its harmony, as a simple dominant seventh chord with an augmented ninth added: C-E-G-Bb-D# or Eb (9+).

To assume that this latter augmented ninth interval is the minor third of the chord in question is theoretically incorrect, because this assumption ignores the order and variable qualities of the chordal degrees in the tertian harmony.

Therefore, chords such as those in Stravinsky’s cantata, which are integrated by other voices beyond the perfect triad, are not purely bimodal: In this case, the presence of a minor seventh changes the denomination of the bimodal triad and its proper identity of sound (the bimodal ethos). (It is worth noting that, as long ago as young Gershwin's day, this augmented ninth chord has found a place in the harmonic palette of Jazz.)

Stravinsky did, notwithstanding, use purely bimodal chords on occasion, but this was mostly in its embryonic phase. One can therefore find specimens of these chords scattershot throughout the body of his work.

We must not assume, however, that such a deliberate use of these chords within isolated segments from a few of Stravinsky's many works implied an underlying purpose in developing a new school of harmony based on Bimodalism.

Far from this, the ongoing change in style that Stravinsky experimented with throughout his career—be it polytonalism,  pandiatonicism, neoclassicism, or even Webernian serialism (to which he was lately inducted)—clearly demonstrates that these meant nothing more than a mere compositional resource at the service of his aesthetic thinking.

As far as I know, a prototypical work, deliberately written to represent Bimodalism, has never been published or mentioned in the accredited literature of music.

But even if such a paradigm were to exist as an isolated laboratory experiment, it would in no way cast a shadow over the value and originality of this invention: to create a harmonic system that gives music a new dimension and ethos embodied in Bimodalism.

If some bimodal work did exist previously, rather we ought to agree with Goethe that, once again, "future events cast their shadows in advance."

In light of this sense of forecasting, we must recall that

• four centuries ago, Luca Marenzio (1553-1599) and Carlo Gesualdo (1560-1613) independently wrote madrigals that could have been written by a Richard Strauss;
  
• as for Hans Neusiedler's (1508-1563) work, Der Juden Tanz (The Jew’s Dance), there are numerous records in the seventeenth century where the police of Prague leveled charges against Jewish musicians for improperly playing "scales, modes, and functional harmony" in this dance, which prompts us to file this work among the earliest cases of modern bitonality, and not as a case of scordatura;
  
• both Chopin and Liszt, each intending to expand tonality, wrote ostensible passages in a frank atonality;
  
• Giovanni Battista Vitali (1644-1692) fully constructed the cyclical form of the sonata, two centuries before César Franck accomplished it in his Sonata in A for piano and violin;
  
• it is Ernest Fanelli (1860-1919), not Debussy, who we should acknowledge as the father of musical Impressionism, upon hearing his Tableaux Symphoniques (1882) recently recorded on compact disc;
  
• the unfairly forgotten Pietro Raimondi (1786-1853) deliberately and systematically wrote an ingenious collection of fugues in four and six simultaneous tonalities, apart from his Three Oratorios In One, juxtaposing contrary tempos and meters, long before Milhaud, Prokofiev, Bartók (or even Ives) began to experiment with polytonality and Stravinsky, Carter, and Blacher, with polymeter, metric modulation and variable meters, respectively;
  
• the virtuoso pianist Charles Kunkel (1840-1923) used a tone cluster effect in his work, Alpine Storm (1888), before Charles Ives used this effect in 1911 in the second movement of his Concord Sonata, and before Henry Cowell — its principal user — coined the term in 1915;
  
• a Clavecin Oculaire, invented in 1734 by Louis-Bertrand Castell (1688-1757), preceded by almost two centuries the synaesthetic inventions of sound and color that Scriabin exhibited in his Prometheus (1910);
  
• microtonality, since its extinction in Greek antiquity, reemerged systematized by the astronomer Christiaan Huygens (1629-1695) in his Tricesimorprimal Temperament, many times before Jacques Halévy practiced microtonality in his cantata, Prométhee Enchaîné (1847), and before Richard Stein (1882-1942), Alois Hába (1893-1973), and Julián Carrillo (1875-1965) with his Sonido 13, made their first experiments fractionalizing the octave into several microtones of different values;
  
• as history now sheds light, it was Charles Ives (1874-1954), who first used a twelve-tone row, before J. M. Hauer, Schoenberg and Jefim Golyscheff were to divulge their own serial techniques amid a paternity contest claiming credit for the idea itself;
  
• in 1914, Erik Satie instructed that sheets of paper be inserted between piano strings in Le piège de Méduse, and Rued Langgaard (1893-1952) instructed that a glissando be played on the piano strings in Sfaerernes musik (1918), before John Cage (following Cowell’s experiments, respectively) could have embodied all these ideas in his Prepared Piano or Klaviergamelan (1940), whose industrialization and universality were owing to the inventions by its performer and promoter, Richard Bunger (q.v.: Bungerack, Pianotation, and so forth);
  
• in his work, Das Eisige Lied (The Icy Song), and in his opera, Cyrano de Bergerac, the Dadaist Jefim Golyscheff (1897-1970) employed a noisy collection of domestic objects, three decades before Pierre Schaeffer (1910-1995) would formulate his theory of musique concrète in 1948;
  
• Ravel's Boléro (1928) should be considered the true forerunner of minimalism, a work that features all the components of this style;
  
• 12 years before George Crumb premiered his work, Ancient Voices of Children (1970), I created musical vocal effects by sympathetic resonance within a piano, in my incidental music composed for Jean Anouilh's drama, Roméo et Jeanette, performed at the Fine Arts Palace of Havana;
  
• and that Mozart — two centuries after Der Juden Tanz had materialized — not only added to the dossier on bitonality as initiated in that dance with his own Ein Musikalischer Spaß, but once again, amusing himself, threw the dice to vary the numerical order of a group of written measures, long before John Cage, Karlheinz Stockhausen, Pierre Boulez, and many other composers were to write aleatoric music in our day.

The challenging character of these historic events, however, should not alter our wise evaluation of the matter at hand: In art, the paternity of an aesthetic idea is not consequently attributed to who, by hazard or by genius, originates the essence of the idea, but to who — regardless of claims to origination — develops the idea to its consecration.

Indeed, ever since I wrote several works employing this harmonic element as an aesthetic principle of musical composition, Bimodalism has finally developed into and become a recognizable system with its own identity, as a new world-class school of  contemporary harmony.

Let history put everything in its place.

 Click here to listen to a complete, albeit brief, composition in bimodal harmony.

Click above for my digital guitar recording of The Sleepless Street. This is a work for solo classical guitar written entirely in bimodal harmony. I composed it to pay homage in dance form to the many jazz clubs that existed along New York's West 52nd Street from the 1930s to the 1950s.

One of my master class composition students, Peter Corey, was also an accomplished classical guitarist, and in 1984 he premiered this work at Merkin Concert Hall. Allen Hughes of the New York Times, who reviewed it back then, felt it was “a pretty little piece with mildly piquant dissonance deriving from bimodal treatment.”

I wish to point out that The Sleepless Street is only one of several compositions I've written in bimodal harmony. I chose it as an example for this web page because its homophonic simplicity unmistakably highlights the harmonic novelty of Bimodalism. As time permits, I intend to add other examples consisting of chamber and orchestral works to showcase Bimodalism with a greater complement of compositions.

| Click here for a separate, new essay on The Augmented 15th Chord. |

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