In the late Sixty Years George Crumb’s music had an enormous diffusion in the world, and its original vision opened to the composers new ways on technical possibilities in the use of the musical instruments. Its abilities to create endless timbral tones as well as ampleness and plurality of its extramusical references and particular perception of a special worldliness, done of silences and echoes, put him among the most important composers of the contemporary time. Vox Balaenae for Three Masked Players, for Electric Flute, Electric Cello and Electric Piano1, composed in 1971 for the New York Camerata2, belongs to a better defined “oceanic kind” based on the whale’s song, an equivalent, so to say, of the Olivier Messiaen’s “call of the birds”. Writing for amplified flute, cello and piano (together to “ancient cymbals” alternatively played by the flutist and the cellist), the piece has been inspired, for explicit admission of the composer, by the song of the whale that Crumb felt in 1969 on a tape recording. Latin title gives an ancient and cultured perspective to the composition, that is an example of organization of the phenomenon of the life “from the beginning to the end of the times”, according to the principal divisions of the geology, the science that occupies him of the physical nature and the history of the earth3. Further philosophical implications are added with two quotations of Richard Strass’s Also Sprach Zarathustra, the first in Vocalise, the second in Cenozoic, to symbolize the man’s appearance on the earth. Crumb affirms on the subject: «I meant to convey by means of music an idea of the development of he human race from its origin, through the various phases of evolution, religious as well as scientific, up to Nietzsche’s idea of the Superman».

Vox Balaenae is a very evocative and full of metaphors piece. The richness of the color makes it unique: from the mysterious unison that results when to the flutist is demanded to play and sing simultaneously, to the glissandos on the piano chords and the sinister sound of the percussive textures that recall the timbre of some prepared John Cage’s piano pieces, to end with a variety of glissandos, pizzicati and cello’s harmonics tuned in “scordatura” (B-F#-D#-A), what perhaps evoke the most vivid images. According to Annie Labussière, what it strikes it’s: «les échos se détachent de leur source sonore […], la progression implacable du temps se volatilise en instants de couleur»4.

Vox Balaenae shows the extraordinary ability of the composer to recall the sounds of the nature in acoustic terms through a kaleidoscopic example of original instrumental techniques and the wish to suggest through extramusical communications the theatricality of the sonorous gestures, using himself of precise indications of production: in fact, it’s expressly imposed to each musician to wear masks on the face to cancel the individual human aspects and symbolically represent the powerful and impersonal strengths of the nature. The composition represents a sentence to the arrogant dominion of the man on the mighty inhabitants of the ocean, mostly underlined by the suggestion of blue stage illumination. Vox Balaenae is hardly describable by a formal or purely diastematical analysis although, as in other pieces of the same composer, besides the poetic and sensorial aspect, exists a very strong constructive logic, based on the use of “motivical cells“, fixed discreet loudness, well definable rhythmic forms that, for juxtaposition, repetition and variation, cross the whole evolution of the three movements. The analytical procedures used for the study of this composition mainly make reference to those proposed by Nicholas Ruwet in 1972, even though with substantial changes. They show their effectiveness in the description of manifold aspects as the search of pitch structures, for which there has used also some indications furnished by the Pitch Class Set Theory,5 the search on the timbre or the evocation of poetic constellations. Therefore such procedures offer the possibility to furnish a picture somehow exhaustive of the complex musical plot.

In Vox Balaenae, the concept of identity cannot directly be connected to the diastematical or rhythmic repetitions, in how much the meaningful events of the musical surface don’t seem to entirely be determined by such aspects. Therefore in the segmentation I have not held so much in mind of the diastematical and rhythmic profile, how much above all of the timbral aspects. Besides I have used there some possibility to operate simultaneously to different levels of segmentation, to the purpose underline least unity that results from repetitions, exchanges and transformations. Finally, I have observed the criterions proposed by Ruwet to individualize a modal hierarchy inside the passages, underlining the presence of notes that have a greater weight in comparison to others, in their initial or final position of the sections, in function of articulation or passage. Vox Balaenae, as shows the Graph I, is constituted by three movements.

The first two movements are based on two modal scales, eptachordal and exachordal scales, while the third movement is in B major. Analysis has been conducted on pieces particularly exemplified of the Crumb’s compositional style: Vocalise, Sea-Theme, Archeozoic e Mesozoic. The first of the three movements, Vocalise, Wildly fantastic; grotesque, is written in a form that juxtaposes a limited number of figures usually varied and prolonged or shortened. The piece (Graph II) is separated in four sections (capital letters) articulated by six figures or meaningful unity (lower case letters). The different meaningful unities are articulated by brief or long breaths and some second of pauses (Table 1).

Vocalise introduces a most fascinating timbral effect: simultaneously the flutist plays and sings inside the instrument; the combination of this sing-flute produces a mysterious and surreal timbre, not dissimilar by the sounds of the whale. The particular timbre of the harmonics are emphasized by the pedal of resonance of the piano that is always kept lowered. In Vocalise motivical material, also being strongly unitary, produces neither immobility nor monotony; in fact, Crumb operate with ability stylistic choices that, to different levels, make multiform the musical texture. The compositional strategy founds him on the juxtaposition of cell-base whose principal scale of reference is the eptachord [2,4,5,8,9,10,11] formed by the notes D-E-F-A flat-A natural-B flat-B natural (Example 1) what, not being entirely used also never, allows a notable series of imbrications6. All the exploited imbrications make entirely reference to the suitable eptachord.

In the example the eptachord is pointed out in real notes because in a large part of the cases the used subsets are based on such pitches. The principal subsets are the exachord [2,4,5,8,9,10], repeated four times in the section A and the exachord 6-30 [0,1,3,6,7,9] does not point out with the real notes in the example 1 but following the normal order, or rather the most compact representation, because such exachord is introduced in the reiterations of the section D in four different transpositions7.

Graph III related to paradigmatic analysis, show the six meaningful figures found of which, in the columns that point out the parametric unities marked by the lower case letters, the distinctive and peculiar lines have been indicated during the repetitions. The four sections, in which the piece is articulated, have been pointed out to the left. Every meaningful unity has been selected for diastematical pitches, sound production’s way, register, dynamics and for rhythmic figuration. Principal unities in a and b impress themselves together or alternatively the various sections, since the composer proceeds for repetition and variation.

The exachord [2,4,5,8,9,10] synthesizes a large part of diastematical material of the unity a. Different reiterations always introduce the same notes, even if apparently they don’t follow a fixed order. Kenneth Timm, in A stylistic Analysis of George Crumb’s Vox Balaenae8, identifies the scale on which the unity a is based in the “Gipsy scale”, used in a lot of oriental cultures and in the Indian Bhairava-raga. Here how much he says on the subject:

The pitches D, E, F, G sharp (A flat), A, and B flat are repeated many times […] and form a scale common to the music of many cultures. This scale has appeared frequently in Hindu, Hungarian, Turkish, Jewish and Greek church music, and is similar to so the so-called “Gipsy” scale. In India this scale is widespread, very old, and fundamental in both Hindustani and Carnatic music. It is known as the Bhairava raga, often used in invocations expressing reverence, and belongs to dawn9.

For what concerns the sound production’s way, the unity a is featured by sing-flute, or rather by the simultaneous play and song of the flute (figure 1); the register is low, the dynamics completes a rapids excursion from the ff to the pp and the rhythmic figuration is a long note of some unities of time, (three semi-quavers), follows or not by a series of groups of three semi-quavers. Accordingly, all the repetitions of the unity a listed in the Graph III point out the type of variation that the same unity suffers in the piece.

In the section A, for example, melodic direction is alternated in the first four exposures, while the fifth reiteration takes back the descending course of the fourth grade. It will be opportune notice that lines common to the different meaningful unities (a, a1, b, etc.) - inserted in the superior part of the Graph III – are the constant parametric elements that allow to apply the criterion of repetition of the different unities. Methodologically, the lines that constitute the unity are considered equal and not the notes that can be bound to other principles of variation.

In the section A, the reiterations of the unities are based, over that on the whole exachord [2,4,5,8,9,10] also on some subsets of his, the pentachord [2,4,5,8,9] and the tetrachord [2,4,5,8] shown in the example 2.

The unity b (figure 2) is characterized by the followings lines: low register, dynamics that again proceeds from the ff to pp and rhythmic figuration constituted by quatrains of harped semibiscrome or fuse in form of ascending and descending scales, situated on a fixed note and followed by a long note some unities of time (in such way to speculate to the unity a). The line that mostly delineates the unity b and that differentiates it from the preceding unity, is the type of sonorous issue constituted by the articulation of the notes fixed plays and sings by the flutist, with the keys of the instrument.

What it’s perceived to the listening is a trembling sound with microtonal oscillations, of which Crumb often suffers the charm and whose interest is declared in the interview of 1980, in reference to the evolution of the musical language in the contemporary epoch. In this interview he affirms:

In the last years the experimentation in the field of the microtonis has been considerable but, at least for a western ear, the structural use of the microtonis is so much difficult to be frustrating. It seems that the most frequent use of the microtonis both rather of character coloristico what, for example, in the oscillation of the heights10.

The unities b, considered in their complex, refer to a subset of the eptachord [2,4,5,8,9,10,11], or rather the tetrachord [4,8,10,11] shown in the example 3. The structural connection among the unities a and b is strong, as every unity b makes part of the whole that has preceded it, to exception of the last reiteration of the unity b (B natural) that appears in this context for the first time but that, united to the tetrachord previously quoted, form the pentachord [2,4,5,8,11] related once more to the eptachord [2,4,5,8,9,10,11].

The section B is entirely constituted by nine reiterations of the unity a of which the first six timbrically connotates by the flatterzung and the remaining three again by the sing-flute. The first six unities are based on pitches that articulate the structure of the exachord [2,4,5,8,9,10], or rather the trichord [2,4,5] constituted by a second and a minor third and by the trichord 3-1[0,1,2] formed by two minor seconds (example 4).

Dynamics is the same one of the unities met in precedence, with small variations to the fourth and fifth reiteration. The last three repetitions are almost identical to the last three of the section A, to exception of the ascending direction of the next to last succession and a light difference of heights. The used sets (example 5) are the exachord [2,4,5,8,9,10] and the tetrachord [2,4,5,8] noticed in precedence and the pentachord [2,4,8,9,10], all related to the eptachord [2,4,5,8,9,10,11].

For a sort of opposition in the structure of the compositional plain, after having founded the section B on the unity a, Crumb uses in the section C the unity b, strengthened by a cultured quotation and by strongly conflicting elements gave to the piano, the unities c, d and e. Such unities have the purpose to interrupt the melodic flow and produce an atmosphere of suspension and mystery. The unity c is constituted by a smaller triad of B minor (major in the following reiteration), preceded by a major triad in the form of acciaccatura, or by two sets [11,2,6] and [5,9,0] both related to the primary form 3-1111. The triad is not casually introduced but it is the point of arrival of an ampler gesture (figure 3) what realizes, through the quotation of Richard Strauss’s Also Sprach Zarathustra, the climax of the piece, more underlined still by the dynamics fff dramatic.

The quotation, besides repeated, suggests and anticipates the premonition of the man’s appearance that will happen in the Cenozoic era. The same quotation reappears, in form less prolonged, in the fifth variation of the second movement, not by chance entitled Cenozoic. If Vocalise has the principal purpose to specifically evoke the poetic content of the piece through some flutistical sonorities as the sing-flute, the writing of the piano realized as if it entirely dealt with a proper instrument to produce resonances of every kind, intervenes with elements of breakup what the pizzicato on the mute strings, marked in the Graph III as unity d and the gliss over strings, the glissando on the strings pointed out as unity e, become peculiar of the Crumb’s compositional style since their apparition12.

The section D, further to be closed by reiterations of the unities a and e, introduces a new melodic configuration gave to the flute played in ordinary way, f (figure 4), based on two trichords 3-7 [0,2,5] situated to mirror, that form the exachord 6-30 quoteds in precedence (example 1), transposed on various degrees. The unity f follows the double quotation of Also Sprach Zarathustra and its reiterations proceed toward a loosening of the dynamic tension, producing a sort of effect of echo13.

Structural use of motivical elements showed in the analysis conducted in Vocalise, is a peculiar aspect of Crumb’s compositional style. As we have been able to observe, in this piece is concretized in a multiplicity of combinations and variations, inside repetitions and annulments of the same meaningful unities, creating a complex musical plot. Respect this Timm adds:

The resultant complexity of pitch organization within phrases and phrase groups tends to obscure rather than clarify the formal design. Because the overall form of pitch organization in the “Vocalise” agrees with the more lucid timbral design14.

The second movement of Vox Balaenae, Variations on Sea-Time is constituted by the Sea-Theme and by five variations inspired to the geological eras. Sea-Theme, Adagio; solemn, with calm majesty, is gave to the cello (table 2), realized completely in harmonics and accompanied by the glissandos on the chords of the piano, by dark and fatal timbre, that articulate the different entrances of the thematic unities. The slow gait of such glissandos give to Sea-Theme a processional connotation. The theme, as the Graph IV shows, is constituted by three melodic unities that without solution of continuity are followed for juxtaposition, resulting in the following form:

The unity a is constituted by trichords 3-7 [0,2,5] and 3-9 [0,2,7], both related to the eptachord [2,4,5,8,9,10,11], that are used for realizing the Sea-Theme gave to the cello (figure 5). The second trichord differs by the first one for the presence of the interval of ascending just fifthy instead of fourth grade. In the Graph V related to the paradigmatic analysis of Sea-Theme, the column formation in the same unity explains it with how much I affirmed before: in the segmentation I have not only considered the diastemazia, but also the rhythmic profile and gesture.

The unity b, gave to the piano, it is also founded on a series of 3-7 trichords. The chord succession realizes the particularly evocative timbral effect defined “Aeolian Harp Effect” (example 6), what is gotten pressing the corresponding keys to some accords, without making to play them, and contemporarily glissando with the fingers on the corresponding chords.

The unity c represents a brief elaboration of the Sea-Theme and it is mainly composed of trichords [0,2,5], even if uses altogether heights referred to the exachord [0,1,3,4,6,8] not contained in the eptachord of reference (figures 6). In total or partial way, the unity c will be more times taken back during the variations and in the final movement.

If it has been possible to underline some structural aspect through the analysis of Vocalise and Sea-Theme, the analysis of Archeozoic (table 3), the first of the five Variations on Sea-Time inspired to the geological eras, instead offers the possibility to gather, besides the elements that compete to determine the structure, the elements related to the poetic program of Vox Balaenae, that is motivical unities that makes reference to the evocation of the sounds of the nature.

Preceded by an introduction, Archeozoic, Timeless, inchoate [Timeless, inchoate], begins with the “Seagull Effect”, and introduces it in tripartite and to blocks form, gotten for accumulation and growth of five unities marked with the lower case letters in the Graph VI related to the formal structure.

The successions what are shaped as meaningful unities for identity of sonorous issue, register, dynamics and rhythmic figuration, have shown in the relative superior part of the paradigmatic analysis of the Graph VII.

The compositive procedure adopted in Archeozoic, unlike that of Vocalise, is based on the interaction among trichords related to different scales , procedure found in Makrokosmos I and II. The structural elements are represented by the unities b, based on some cells contained in the Vocalise’s scales of reference Vocalise, that are the basic-trichord 3-7 [0,2,5] and 3-8 [0,2,6] and by the unities e, what instead are related to the esatonal scales being constituted by the tetrachord 4-24 [0,2,4,8].

It is necessary to add that the first succession of the unities d and e is based on the pentachord 5-33 [0,2,4,6,8], subset of the esatonale scale, and that the trichord 3-8 are also it a subset of such scale.

Instead the onomatopoeic effects are constituted by the unities a “Seagull Effect” realized with a particular glissando of the cello and by the unities d and e what musically reproduce the song of the whale that Crumb had listened in 1969. A peculiarity of the unities b and c is the timbral effect “Chisel Piano”, a “beat-glissato” that is got making gliss on the chords a chisel of crystal. Then it’s very interesting the trill of quarters of tone realized in the unity d, that shows the attention of the composer for the microtonal oscillations.

The fourth variation, Mesozoic, Exultantly! (table 4), shows the interest of Crumb for a certain type of writing that could be defined “contrappuntistica” for the overlap of different meaningful unity. In fact Mesozoic is constituted by two identical sections built in the shape of joint between antecedent (unity a) and consequent (unity b) that runs after them each other to canon, as shown in the Graph VIII.

The unity a is given to the piano with an very sonorous timbral effect, gotten leaning a fillet of glass on the piano chords correspondents to the keys to crush with strength. The unity a is based on the trichord 3-7; the different rhythmic profile of its iterations makes to separate the unity a in two different elements (x and y with micro-variations) what are treated in double counterpoint15.

The unity b, consequent in comparison with the unity a, is constituted by ample melodic phrases gave to the cello and flute that doubles to the superior octave. Such figurations finish with some fast final articulations on two notes, with a measured trill. The initial part of the unity b is based on the trichord 3-7. The unity c is characterized by a figures of five trichords on fixed heights, with triple acciaccatura, situated alternatively on the inferior and superior lines of the piano. Its function is of opening and seal of every excursion of the unity a, to exception of the first one. The acciaccatura and the trichords are also them sets 3-7. The Graph IX related to the paradigmatic analysis shows the succession of the meaningful unities.

Therefore the fourth variation represents the quintessence of the application of the mosaic technique founded on the cell-base use. In the three unities found in Mesozoic, the inside cohesion is not only given by the common affiliation to a basic trichordal model, but also by a sort of harmonic tie created by the composer (figure 7), what consists in a commune note among the unities a and b (the A) and among the unities b and c (the F).

On the ground of the suggestions of Ruwet as regards the attribution of modal hierarchies in the passages took in examination, I’ve produced the Graph X, weighing the notes from beginning to end section.

Through this analysis has been deduced that the B natural has a greater structural weight in comparison with the other notes, and this involves the consideration that a thin thread ties Vocalise to the second movement, Variations on Sea-Time, what emphasizes the D sharp, and to the last movement, Sea-Nocturne, built in the greater bright tonality of B major16. As regards the temporal organization, Vox Balaenae is primarily ametrica, except that in Sea-Theme. Every meaningful motivical unity is suspended from a breath or some second of stop in relationship to the extramusical programmatic elements. The silence has an importance equal to the other structural elements. Timm affirms:

When phrases or smaller units are repeated, formal significance is inherent, but the reasons for such events stem more from melodic and extra-musical considerations than from rhytmic organization. The overall effect is arhythmic. Motion swells and recedes slowly to give a feeling of vastness. Rhythmic activity tends to gradually increase to a climax and recede17.

The analytical methodologies used in Vox Balaenae have allowed to face an uncommon repertoire, thanks to the possibility to effect a flexible segmentation in the identification of the meaningful unities. The analysis has shown Crumb’s peculiar attitude to draw food by a small trichordal cell in order to create forms of great ampleness articulated according to tensiv and relaxing arcs. Furthermore, he has allowed to activate a procedure able to distinguish the structural aspects, deeply tied up to the music in itself, from the extramusical aspects. It has offered the possibility to understand how much the inside coherence is strong that subtends such composition, without besides leaving from the poetic wealth, the symbolic and evocative aspects, also them determinant for the understanding of the poetic meaning of the Crumb’s music.

Books and Thesis:

Antokoletz, Elliott, Twentieth-Century Music, Englewood Cliffs, Prentice-Hall, 1992.

Azzaroni, Loris, Canone infinito. Lineamenti di teoria della musica, Bologna, CLUEB, 1997.

Bass, Richard W. Jr., Pitch structure in George Crumb's Makrokosmos, Volumes I and II, PhD diss., Music, University of Texas, Austin, 1987.

Baur, John, Music Theory through Literature, New York, Englewood Cliffs, Prentice-Hall, 1984.

Boretz B. - Cone Edward T., Perspectives on Contemporary Music Theory, New York, W. W. Norton & Company, 1972.

Childers, Joseph - Hentzi, Gary, Columbia Dictionary of Modern Literary and Cultural Criticism, New York, Columbia University Press, 1995.

Chlopecki, Andrzej, George Crumb. Musica humana - Musica mundana, Ruch Muzyczny, Poland, X, Maggio 1976.

Cone, Edward T., Musical Form and Musical Performance, New York, W. W. Norton & Company, 1968.

Cook, Nicholas, Guida all’analisi musicale, ed. it. a cura di Guido Salvetti, Milano, Guerini e Associati, 1991.

Drabkin, William –Pasticci, Susanna – Pozzi, Egidio, Analisi schenkeriana. Per un’interpretazione organica della struttura musicale, Lucca, LIM, Libreria Musicale Italiana, 1995.

Fouse, Kathryn Lea, Surrealism in the piano music of representative twentieth-century American composers with three recitals of selected works of Ives, Cowell, Crumb, Cage, Antheil, and others, DMA doc., Music, University of North Texas, 1992.

Forte, Allen, The Structure of Atonal Music, New Haven and London, Yale University Press, 1973.

Gann, Kyle, American Music in the Twentieth Century, New York, Schirmer Books, 1997.

Gartner, Kenneth Ralph, The expansion of pianism since 1945, PhD diss., New York University, 1979.

Gillespie, Don, George Crumb: Profile of a composer, New York, Peters, 1986.

Griffiths, Paul, Modern music and After: Directions since 1945, Oxford, Oxford University Press, 1995.

Jiorle-Nagy, Linda A., A study of phrase structures and unifying devices in George Crumb's Makrokosmos I and II, DMA doc., Boston University, 1993.

Kessner, Dolly Eugenio, Structural coherence in late twentieth-century music: The linear-extrapolation paradigm applied to four American piano compositions of diverse musical styles (Martino, Rzewski, Crumb, and Adams), PhD diss., Music, University of Southern California, 1992.

Kingan, Michael Gregory, The influence of Bela Bartók on symmetry and instrumentation in George Crumb's Music for a Summer Evening, DMA doc., University of North Texas, 1993.

Kinney, Michael, Perceptions of developmental influences as contributing factors to the motivation for musical creativity of eminent twentieth-century living American composers, EdD diss., Syracuse University, 1990.

Kramer, Jonathan D., The Time of Music, London, Schirmer Books, 1988.

Krarup, Bertel, American music the from avant-garde of the twenties to the traditionalism of the seventies, [Amerikansk musik fra 20’er -avantgarde til 70’er-traditionalisme], Dansk musiktidsskrift, Denmark, LIX, 3, 1984-85.

Lanza, Andrea, Il secondo Novecento, in Storia della musica a cura della Società Italiana di Musicologia, Torino, E.D.T., XII, 1991.

Ledbetter, Robert Bryan, An examination of the percussion writing in the chamber works of George Crumb, 1960-1980, DMA doc., University of North Texas, 1993.

Lester, Joel, The Rhythms of Tonal Music, Southern Illinois University Press, 1986.

Mac Lean Harkins, Suzanne, A study of constructional principles in George Crumb's Makrokosmos, Volume I, MA diss., Music, The American University, Washington, D.C., 1974.

Machlis, Joseph, Introduction to Contemporary Music, second edition, New York, W. W. Norton, 1979.

Malgorzata, Rosinska, Symbolism in George Crumb’s Madrigals, [Symbolika Madrygal¢w George’a Crumba], Muzyka, Kwartalnik Instytutu Sztuki Polskiej Akademii Nauk, XXXV, 1, 1990.

McGee, William James, An expanded concept of timbre and its structural significance, with a timbral analysis of George Crumb's Night of the four moons, PhD diss., University of Arizona, 1982.

McKay, John Robert, Notational practices in selected piano works of the twentieth century, DMA diss., Piano, Eastman School of Music, 1977.

Morgan, Robert P., Twentieth-Century Music: A History of Musical Style in Modern Europe and America, New York, W.W. Norton, 1991.

Ott, David Lee, The role of texture and timbre in the music of George Crumb, DMA diss., University of Kentucky, 1982.

Perle, George, Serial Compositions and Atonality, An Introduction to the musica of Schoenberg, Berg and Webern, second edition, University of California Press, 1968.

Podhajski, Marek, Celestial Mechanics (Makrokosmos IV) of George Crumb, [Celestial Mechanics (Makrokosmos IV) George’a Crumba] Muzyka fortepianowa, Gdansk, Akademia Muzyczna im. Stanislawa Moniuszki, VIII, 1989.

Rostkowski, David, Makrokosmos by George Crumb: three significant American piano works, [Makrokosmos George'a Crumba - trzy wyr¢zniajace sie zbiory kompozycji fortepianowych w muzyce amerykanskiej], Muzyka fortepianowa, (see RILM 1981-05929-bc).

Ruwet, Nicholas, Langage, Musique, Poésie, Paris, Seuil, 1972, trad. it., Linguaggio, musica, poesia, Torino, Einaudi, 1983.

Sarup, Madan, An Introductory Guide to Post-Structuralism and Postmodernism, second edition, Athens, University of Georgia Press, 1993.

Schultz, Donna Gartman, Set theory and its application to compositions by five twentieth-century composers, PhD diss., Michigan State University, 1979.

Schwartz, Elliott – Godfrey, Daniel, Music since 1945: Issues, Materials, and Literature, New York, Schirmer, 1993.

Siedlaczek, Grzegorz, The piano in George Crumb's oeuvre (based on an analysis of selected works), [Fortepian w tw¢rczosci George'a Crumba (w swietle wybranych utwor¢w)], Zeszyty naukowe, Akademia Muzyczna im. Fryderyka Chopina w Warszawie, Z prac II katedry fortepianu, 29, 1995, (MA thesis, History of Music: Akademia Muzyczna im. F. Chopina, Warszawa, 1994, Iss. 29, 1995, (Warszawa, Akademia Muzyczna im. Fryderyka Chopina, 1995).

Shuffett, Robert Vernon, The Music 1971-1975 of George Crumb: A Style Analysis, D.M.A. thesis, Peabody Conservatory of Music, 1979.

Skowron, Zbigniew, New American music, [Nowa muzyka amerykanska] Studia et dissertationes, Uniwersytet Warszawski, Instituti Musicologiae Universitatis Varsoviensis, 2, Krak¢w, Musica Iagellonica, 1995.

Simoni, Mary Hope, The computer analysis of atonal music: an application program using set theory, PhD diss., Michigan State University, 1983.

Strickland, Edward, American composers: Dialogues on contemporary music, Bloomington, Indiana University, 1991.

Takenouchi, Aleksei, Numbers and proportions in George Crumb’s solo piano compositions, DM doc., Music, Northwestern University, 1987.

Terse, Paul, The symbolism of “Myth” from George Crumb’s Music for a Summer Evening (Makrokosmos III), [Zur Symbolik in “Myth” von George Crumbs Music for a summer evening (Makrokosmos III)], Die Sprache der Musik, Festschrift Klaus Wolfgang Niem”ller zum 60, Geburtstag, Kassel, Bosse, 1989.

Timm, Kenneth N., A stylistic Analysis of George Crumb’s Vox Balaenae, Ph.D. diss., Indiana University, 1977.

Tomaro, Robert John, Contemporary compositional techniques for the electric guitar in United States concert music, PhD diss., New York University, 1993.

Weber, Horst, George Crumb: Amplified piano-amplified tradition. A critique of “postmodern” composing, [George Crumb: Amplified Piano-amplified Tradition, Zur Kritik “postmodernen” Komponierens], Das Projekt Moderne und die Postmoderne, Kassel, Bosse, 1989.

Articles and Essays:

Bass, Richard, Sets, Scales, and Symmetries: The Pitch-Structura1 Basis of George Crumb’s Makrokosmos I and II, «Spectrum of Theory Music», XIII, 1991, pp. 1-20.

Bass, Richard, Model of Octatonic and Wholetone Interaction: George Crumb and his Predecessors, «Journal of Music Theory», XXXVIII, 2, 1994, pp. 155-186.

Brooks, Richard, Orchestral Music: By George Crumb, Luciano Berio, Arvo Pärt, «Notes», Ser. 2, XXXXV, 3, Marzo 1989, p. 619.

Bruns, Steven Michael, George Crumb in Prague and Boulder 1992: A tale of two festivals, «American Music Research Center Journal», III, 1993, pp. 3-8.

Bruns, Steven Michael, “In stilo Mahleriano”: Quotation and allusion in the music of George Crumb, «American Music Research Center Journal», III, 1993, pp. 9-39.

Burge, David, Instrumental Solo and Ensemble Music: By George Crumb, «Notes», Ser. 2, XXXVI, 3, Marzo 1980, p. 748.

Burge, David, George Crumb, in George Crumb: Profile of a Composer, a cura di Don Gillespie, New York, Peters, 1986, pp. 6-7.

Burge, David, Keyboard Music: By George Crumb, «Notes», Ser. 2, XXXXV, 4, Giugno 1989, p. 859.

Carbon, John, Astrological symbolic order in George Crumb's Makrokosmos, «Sonus, A journal of investigations into global musical possibilities», X, Iss. 2, primavera 1990, pp. 65-80.

Carraro, Mario, George Crumb, poeta del suono, «Musica/Realtà», XXXXVI, Marzo 1995, pp. 71-96.

Castanet, Pierre-Albert, George Crumb: Poetics and musical analyses, [Georges Crumb: Poetique et analyse musicales], «L’idée musicale», Saint-Denis, Université de Vincennes, 1993, pp. 211-224.

Cohn, Richard, Inversional Symmetry and Transpositional Combination in Bartók, «Music Theory Spectrum», X, 1988, pp. 19-42.

Cook, Nicholas, Analizzare la musica seriale, in Anton Webern, Spunti analitici: interpretazioni e metodologie, «Quaderni di Nuova Consonanza», n° 1, 1991, pp. 137-155.

Crumb, George, Music: Does It Have a Future?, «The Kenyon Review», Estate 1980, trad. it. La musica ha un futuro?, a cura di Lucia Cristina Bartolucci, in Da New York a New York. La musica moderna tra vecchio e nuovo continente. Rapporti, influenze e migrazioni di artisti tra i due continenti, 36° Festival di Nuova Consonanza, a cura di Egidio Pozzi, Roma, Nuova Consonanza, 1999, pp. 13-18.

Culver, Anna M., «Denver Rocky Mountain News», 25 Ottobre 1973, in George Crumb: Profile of a Composer, a cura di Don Gillespie, New York, Peters, 1986, p. 31.

DeDobay, Thomas R., The Evolution of Harmonic Style in the Lorca Works of Crumb, «Journal of Music Theory», XXVIII, 1984, pp. 89-111.

Forte, Allen, Debussy and the Octatonic, « Music Analysis», X, 1/2, 1991, pp. 125-169.

Harris, Simon, Chord Forms Based on the Whole-Tone Scale in Early Twentieth-Century Music, «Music Review», XXXXI, 1, 1980, pp. 36-51.

Labussière, Annie, George Crumb: Ancient Voices of Children, «Analyse musicale», Issue, 24 Giugno 1991, pp. 55-80.

Mac Lean, Suzanne, George Crumb, American Composer and Visionary (“The Phantom Gondolier”), in George Crumb: Profile of a Composer, a cura di Don Gillespie, New York, Peters, 1986, pp. 20-25.

Narmour, Eugene, De Oraculo Crumbi, in Don Gillespie, George Crumb: Profile of a Composer, New York, Peters, 1986, p. 74-75.

Neubert, David, The extended family: Electronic bowed string instruments, «American String Teacher», USA, XXXIV, 1, Inverno 1984, pp. 46-49.

Odegard, Peter S., Avant-Garde Editions: by George Crumb, Vinko Globokar, and Tera de Marez Oyens, «Notes», Ser. 2, XXIX, 3, Marzo 1973, p. 539.

Pasticci, Susanna, Teoria degli insiemi e analisi della musica post-tonale, «Bollettino del GATM», II, 1, 1995, pp. 111.

Perle, George, Scriabin's Self-Analyses, «Music Analysis», CCCXII, 1984, pp. 104-109.

Price, Harry E., Yarbrough, Cornelia; Kinney, Michael. Eminences of American composers: University faculty attitudes and symphony orchestra programming, «Bulletin of the Council for Research in Music Education», Iss. 106, 1990, pp. 37-48.

Riis, Thomas Lawrence, A conversation with George Crumb, «American Music Research Center Journal», III, 1993, pp. 40-49.

Sanguinetti, Giorgio, Il primo studio teorico sulle scale octatoniche: le “scale alternate” di Vito Frazzi, «Studi Musicali», XXII, 2, 1993, pp. 411-446.

Schuldt, Agnes Crawford, Molecules and music, «Virginia Quarterly Review», USA, LVII, Summer 1981, pp. 486-97.

Spitz, Ellen Handler, Ancient voices of children: A psychoanalytic interpretation, «Current musicology», XL, 1985, pp. 7-21.

Terse, Paul, »Makrokosmos I« 12 Fantasiestücke über den Tierkreis für elektronisch verstärktes Klavier von George Crumb, in Americanische Musik selt Charles Ives: interpretationen, quellentexte, komponisten monographien, a cura di Hrsg Von H. Danuser, D. Kämper und Paul Terse - Laaber, Laaberverlag, 1993 (2a ed.), pp. 191-199.

Thomas, Jennifer, The Use of Color in Three Chamber Works of the 20th Century, «Indiana Theory Review», IV, 3, 1981, pp. 22-40.

Tomaro, Robert, Contemporary Compositional Techniques for the Electric Guitar in United States Concert Music, «Journal of New Music Research», IV, 1994, pp. 349-367.

Warburton, Thomas Andrew Jr., New piano techniques for Crumb's piano music, «Piano Quarterly», USA, LXXXVII, 1974, pp. 15-16.

Wile, Kip, Communication and Interaction in Stravinsky's Scherzo fantastique (1907-1908), «Indiana Theory Review», XIII, 1, 1992, pp. 87-112.

White, John, Systematic analysis for musical sound, «Journal of musicological research», USA, V, 1-3, 1984, pp. 165-90.

Whittall, Arnold, Tonality and the Whole-Tone Scale in the Music of Debussy, «Music Review», XXXVI, 4, 1975, pp. 261-271.

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