The following is an abridged preface from an upcoming publication entitled The Pastorale in Western Music (published by Carl Fischer Music).

​Introduction
     In 1703, at the tender age of fifteen, Alexander Pope made his first visit to London, intent on improving his understanding of French and Italian. Precocious, brilliant, supremely confident and arrogant, despite a variety of physical deformities, Pope would compose in the following year his first major work, Pastorals, later published in 1709. The fact that Pope’s Opus 1 would be a set of poems that explore the splendors of nature suggests that his first exposure to the hustles and bustles of urban life was a shock that would give way to nostalgia and longing for the bucolic environs of his childhood. What is especially significant about this work is the opening “Discourse on Pastoral Poetry,” a brief preface that introduces the reader to the poems. In the third and fourth paragraphs of this preface, he offered an extraordinarily insightful definition of the pastorale element in poetry:
 
A Pastoral is an imitation of the action of a shepherd, or one considered under that character. The form of this imitation is dramatic, or narrative, or mix’d of both; the fable simple, the manners not too polite or too rustic: The thoughts are plain, yet admit a little quickness and passion, but that short and flowing: The expression humble, yet as pure as the language will afford; neat, but not florid; easy, and yet lively. In short, the fable, manners, thoughts, and expressions, are full of the greatest simplicity in nature.
The complete character of this poem consists in simplicity, brevity, and delicacy; the first two of which render an eclogue natural, and the last delightful.[1]
 
Like Virgil, Pope began his career as a composer of pastorale poetry; Virgil’s Eclogues, a collection of pastorale poems, remained in Pope’s reading library his entire life. Also, like Virgil, Pope’s work was composed in reaction to dramatic and troubling social events; the turbulence and unrest of Virgil’s Rome finds an uncanny parallel in the clamor of Pope’s London.
     Pope’s reference to the “action of a shepherd” is crucial, for the derivation of the word pastorale is from the Latin “pastoralis,” meaning shepherd. The association, of course, is with nature, for shepherds, by virtue of their vocation, live and work in the country, in expansive and solitary areas where their sheep may safely graze. Regarding the relation between the shepherd and poetry, Pope observed, “And as the keeping of flocks seems to have been the first employment of mankind, the most ancient sort of poetry was probably the pastoral. ‘Tis natural to imagine, that the leisure of those ancient shepherds admitting and inviting some diversion, none was so proper to that solitary and sedentary life as singing.”[2] Further, he remarked, “And since the life of shepherds was attended with more tranquility than any other rural employment, the Poets chose to introduce their Persons, from whom it receiv’d the name of Pastoral.”[3]
     The focus of Pope’s observations in the Discourse is on tranquility and simplicity of the shepherd’s existence, an evocation of an idyllic and idealized “Arcadian” world in contrast to the frenzied reality of urban life. Undoubtedly, Pope and many others of the time regarded this lifestyle with deep nostalgia, a sense of having lost contact with nature with the rise in industry, population and, consequently, urban dwelling. Shepherds thus were viewed with envy and for centuries were held in high regard because of the presumed leisure of their lifestyles. This high regard is reflected in their biblical roles, for the Annunciation of Christ’s birth by the angel Gabriel, a crucial world event by any standards, was made not to nobility or religious leaders, but to shepherds. This, however, was a misinformed view, and as a result few, if any, pastorale poems addressed the hardships endured in the life of a shepherd, such as the wayward temperaments of sheep and extreme weather conditions.
     The pastorale is unique, in that it has encompassed a number of different disciplines of the arts, namely, poetry, drama, painting and music. Its appearance in music was typically late, but no less impactful and no less relevant. It is likewise unique in that it has retained its enchantment in these disciplines over many centuries, from the ancient Greeks and Romans through myriad changes in beliefs and attitudes in modern times. Few other styles, forms or genres can boast such longevity, and clearly its appeal to us over such drastically changing times is deserved of consideration.
 
Elements of the Pastorale Style
     The style of the pastorale in Western music evolved over centuries and then, as in many other instances, became a stylized tradition, whose elements could be recalled for a variety of suggestive purposes. Its origins may be traced to the songs of courtly love of the troubadours and trouvères of the twelfth century, extending to and culminating in the sixteenth-century frottola and polyphonic madrigal. The Italian madrigalists of the late sixteenth century, in particular, exploited pastorale themes in abundance, inspired by the pastorale texts in Torquato Tasso’s Aminta and Giovanni Battista Guarini’s Il pastor fido.[4] Inevitably, the development of opera in seventeenth-century Italy would continue with the pastorale themes that fascinated the Italian madrigalists, particularly the myths of Orpheus and Dafne. These early operas incorporated the pastorale tradition through important roles played by shepherds and nymphs, whose merriment and ease of life recall Pope’s descriptions.
     Musically, the pastorale tradition was only loosely defined in these madrigals and early operas; text and dramatic action were primarily responsible for evoking it. That would change at some point in the seventeenth century, as the pastorale gradually moved into the realm of instrumental music and incorporated specific rhythmic and harmonic figures that, in the absence of words, would become defining trademarks in its style. Scholarship remains indefinite, but elements of both the siciliana, a moderate-paced Italian dance in compound meter, and the musette, a French dance named after a small bagpipe with bellows, were absorbed into the pastorale. Each provided different ingredients. The siciliana shared its compound meter (typically, 6/8 or 12/8) and a characteristic rhythmic figure, often referred to as the “dotted siciliana rhythm.” Trochaic figures, a poetic foot composed of a long and short rhythm (such as quarter note/eighth note), were also prevalent, particularly as accompaniment patterns. The musette, imitating the open drone fifths of its namesake, introduced stability of unchanging harmonies that projected repose and tranquility. Accordingly, the pastorale was founded on two independent dances (one Italian, the other French), during a period in which dance forms became important genres of instrumental music.
     However, the pastorale did not become a dance, but rather, a sentiment of tranquility. Little time would pass before the pastorale became stylized, a result of a uniform musical character and style that were maintained throughout the course of the seventeenth and eighteenth centuries. Uniformity bred familiarity, for which reason the pastorale was commonly used for suggestive purposes. This would become especially popular during the Christmas season, given the role played by the shepherds in the Nativity; hence, the pastorales by Corelli and Heinichen, which were intended for a Christmas Eve service. Both Bach and Handel drew on this familiarity to evoke images of the Annunciation of Christ’s birth to the shepherds in the absence of words. The Sinfonia Pastorale that opens Part 2 of Bach’s Christmas Oratorio paints an unmistakable image of an idyllic countryside inhabited by shepherds and their flocks through repeated use of the dotted siciliana rhythm, trochaic rhythmic figures and an obbligato chorus of oboes. Handel’s “Pifa” from Messiah does the same, though for strings alone, in conjunction with harmonic drones and melodies doubled in thirds.
     This uniformity slowly crumbled at the onset of the nineteenth century as composers sought to expand on the eighteenth century’s understanding of the pastorale tradition. Beethoven was probably one of the first to re-evaluate the elements of the pastorale style by modifying its rhythm, tempo and character to suggest personal recollections of experiences in the country. Both Franz Liszt and Jean Sibelius incorporated the pastorale style for programmatic reasons to conjure impressions of the harvest, while Dvořák’s pastorale evokes the sounds of indigenous folk music. Such reassessments entailed varied musical techniques, and thus the staple features of the style, compound meter, the dotted siciliana rhythm and harmonic stability, were often disregarded.
     Attitudes and approaches would change once again in the period following World War I, as neo-classical pursuits swept through the Western world. The desire to recapture elements of seventeenth- and eighteenth-century styles encouraged a renewed interest in the basic elements of the pastorale style. Ernő Dohnányi, for example, would draw on the pastorale’s heritage by weaving an arabesque of dotted siciliana rhythms around a Hungarian carol celebrating the Annunciation, thus restoring the Christmas association as well. Frank Bridge’s Pastorale projects the tranquility and the association with nature of the baroque pastorale, most probably as a means for the composer to escape from the emotional devastation of the war. The pastorales by Beach and Hindemith both recall two fundamental elements of the style—compound meter and the dotted siciliana rhythm—without, however, any extra-musical associations. Both pieces exist successfully as pure music.
 
Rhythm and Meter
     The dotted siciliana rhythm is perhaps one of the most characteristic and recognizable features of the pastorale, the baroque pastorale, in particular. Indicative of compound meter, its three components are a dotted eighth note, sixteenth note and eighth note:

​In terms of articulation, the three notes are generally slurred together, with a slight lift after the third eighth note to provide some degree of separation:

​This manner of articulation treats the figure as a single, composite unit and lends itself well to the gracefulness demanded of the style. How the figure came to be associated with the siciliana and, thus, the pastorale is difficult to trace, yet its origins may lie with traditions of folk music. The pifferari, mountain shepherds who would perform on bagpipes and oboes (pifferi) in the streets of Rome during the Christmas season, may have contributed to or continued this tradition.
     Being a three-part subdivision of the beat, the dotted siciliana rhythm is generally found in compound meters, and most pastorales are indeed set in compound meter. 6/8 and 12/8 are especially common (9/8 less so), the latter clearly favored over the former in the seventeenth and eighteenth centuries. The pastorales by Corelli, Heinichen, Bach, Handel and Scarlatti in this collection (eight in number) were composed in 12/8 meter, and the preference of 12/8 over 6/8 may be due to a wider degree of separation between metric stresses; that is, 6/8 supposes metric accentuation every two beats, 12/8 every four beats, which would be better suited to the tranquil character of the pastorale. Beethoven’s evocation of a scene by the brook, the second movement of his Symphony No. 6, was also composed in 12/8, perhaps for similar reasons, for the listener should scarcely be aware of any metric scheme in the undulating string figures. Sibelius likewise adopted this approach, although using an atypical 12/4 in his Pastorale from Pelleas et Mélisandé. Alternatively, early examples of 12/8, Corelli’s, for example, may represent a call for a slow tempo, contrary to 6/8 which generally presupposes a quick tempo.
     In time, compound meter and the dotted siciliana rhythm would not be regarded as the exclusive options, and other meters, such as simple duple time and simple triple time, and other rhythmic figures would be explored. Composers in the nineteenth century, such as Fanny Mendelssohn and Franz Liszt, composed pastorales in simple triple time and could evoke the pastorale sentiment through a musical ease and grace, without recourse to the dotted siciliana rhythm. Dvořák branched out into 4/4 time and dactylic rhythmic figures (i.e., eighth and two sixteenths) in a pastorale that relied on the musette’s drone fifths and harmonic stability for realizing its effect. Nineteenth-century practices demonstrate that while rhythm and meter were contributing elements to the pastorale tradition, they were not the sole means for realizing it.
 
Tempo and “Tempo Giusto”
     “Tempo,” wrote Alexander Malcolm in his Treatise of Musick of 1721, “is a various and undetermined thing and indeed the Determination of them must be learnt by Experience from the Practice of Musicians.”[1] The establishment of a natural or suitable tempo, or tempo giusto as seventeenth-century musicians called it, is generally derived from the character of the music. Leopold Mozart offered such a definition of the term in his Violinschule of 1756: “Tempo commodo and tempo giusto…tell us that we must play the piece neither too slow nor too fast, but in a suitable, convenient and natural tempo. We must therefore look for the true pace of such a piece in the piece itself.”[2] Practicing musicians understand that music of brilliance demands quick tempos, music of expressiveness and grace demands slower, broader tempos. Yet numerous other factors, for example the performance medium or the make-up of the ensemble, bear influence as well, demonstrating that a true tempo giusto can only be relative in relation to other such variables.
     Understandably, the pastorale style would depend on slow, broad tempos to realize its sentiment properly. The uniform character of the pastorale during the baroque period entailed a uniformity in tempo, as can be seen in the tempos of the nine baroque pieces in this collection; all are in compound meter, and all can be performed comfortably at dotted quarter = 50 to 60. Time-words may be found in these pieces as well. The Largo in Corelli’s Pastorale and Larghetto in Handel’s Pifa probably represent requests both for slowness and tranquility. Couperin’s unique marking “Näivement” is an instruction to perform the music with simplicity, an important element of the baroque pastorale. Scarlatti’s direction Allegrissimo in the Sonata in F Major is exceptional, though most likely referring to mood and not speed.[3] By contrast, Frescobaldi, Heinichen and Bach provided no tempo indications, suggesting that these composers assumed, as Leopold Mozart stipulated, that the performer would be able to establish a tempo from the musical character.
     Divergences begin to appear in the nineteenth century as the style itself diverged, and establishing a true tempo giusto for the pastorale becomes fraught with difficulties. Beethoven would offer two different tempos for his pastorales, one Allegro, the other Andante molto mosso. Further into the nineteenth century, Allegretto becomes a frequently encountered indication, as in the examples by Liszt, Gounod, Elgar, Chaminade, Bridge and Dohnányi. The marking requests a tempo slower than Allegro, yet requires that the lively and joyful character of Allegro should be maintained. Quantz in his Versuch einer Anweisung die Flöte traversiere zu spielen of 1752 defined Allegretto as quarter note equals 80, a marking he also recommended, appropriately enough, for the musette.[4] Charles Rosen has convincingly argued that Allegretto in the time of Mozart and Beethoven works well in many instances with a metronome marking of quarter note equals 76, a marking that corresponds closely to that given by Quantz.[5]
In essence, the pastorale in the nineteenth century became (generally) somewhat faster than its baroque ancestor and carried with it a sense of liveliness that was generally not a part of the seventeenth- and eighteenth-century traditions.
 
The Subdominant
     Johann David Heinichen, whose Pastorale may be found on p. xxx of this collection, is credited as the first to describe the circle of fifths in his seminal treatise Der General-Bass in der Composition of 1728.  This scheme, familiar now to most musicians, is a beneficial means for explaining the relative harmonic remoteness of one key from another; specifically, the number of fifths separating two notes on the circle. It is also beneficial in explaining the tension in tonal music created from the dominant-subdominant relationship. To illustrate this, the following diagram, a linear rendition of Heinichen’s circle, shows fifths ascending from the note C (right side) and fifths descending from C (left side):
 
                                   <------Subdominant                              Dominant----->
Dbb-Abb-Ebb-Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C     C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B#
 
Clearly, this scheme assumes an equal-tempered tuning system. The pattern to the right of C establishes the dominant, or sharp keys; those to the left establish the subdominant, or flat keys. Every other note in the dominant direction is the dominant of the previous note, and every other note in the subdominant direction is the tonic of the previous note. Thus, the dominant direction strengthens the tonic, and the subdominant direction weakens it, since the tonic becomes the dominant of the following note. In short, the dominant implies tension, the subdominant implies relaxation.[6]
     Mozart, with ears of extraordinary sensitivity and precision, was perhaps one of the first to grasp the full implications of this relationship, for many of the slow movements of his sonatas, symphonies and concertos are appropriately in the work’s subdominant key, better to enhance the relaxing character of the slow movement and to release the tension of the work’s overall tonality. This might explain why flat keys, F major in particular, have been favored for pastorales. Charles Rosen offered the provocative observation that, “Donald Francis Tovey ascribed the idea that keys have defined characteristics in their relation to C major, unconsciously treated as basic, since that is the first one everyone learns as a child. F major is, therefore, by ‘nature’ a tonality with a subdominant quality or a release of tension relative to C major, and most pastorals are, indeed, written in F.”[7]
     Certainly, a remarkable observation, but one that remains speculative, for this phenomenon is a subjective one that results from conditioning. However, there must be a reason, subconscious or otherwise, why Beethoven chose the key of F major for his Pastoral Symphony and the subdominant of F (Bb major) for the work’s second movement (although he did the same in his “non pastoral” Symphony No. 8). Would the work promote a different effect if composed in the key of D major (incidentally, the key of Beethoven’s Sonata for Piano, Op. 28, nicknamed “Pastoral”)? Similarly, Heinrich Schütz’s exclusive use of F major for his striking telling of the Nativity in Historia der Geburt Jesu Christi of 1660 must have been a subtle, yet effective means on the composer’s part to capture the tranquility and serenity of the birth as told in the Gospels of Luke and Matthew by means of the subdominant key. Closer to modern times, Cécile Chaminade’s choice of F major for her Pastorale could have been made for similar reasons, as were Dohnányi’s and Sibelius’ choice of Ab major for theirs. Perhaps this is simply how these composers “heard” music.
     What frustrates the elegance of Rosen’s proposal is that pastorales have been composed in a variety of other keys, without doing detriment to its character. G major (or simply “G”) has been a popular key, as in the pastorales in this collection by Frescobaldi, Couperin, Corelli, Bach and Wesley, as well as later ones by Beach and Hindemith. The examples by Corelli and Bach may have been determined in terms of suitability for the string instruments of their time, although Bach’s G major is the subdominant in the work’s overall key of D major.[8] Amy Beach’s choice of G could have been based on her personal associations between colors and keys, and Hindemith, deep in a neoclassical nostalgia, probably drew on examples from the repertoire. Both Heinichen and Haydn composed their pastorales in A major, as Fanny Mendelssohn and Liszt later did, although by this point in history the individuality of keys had become a forgotten relic. D major may be found in the pastorales of Dvořák and Bridge, leaving Stravinsky with the unique and equally effective choice of F# major. How these pieces would differ in terms of their effects and our responses to them if they had been composed in flat keys remains one of speculation.
     What is indisputable, however, as all composers in this collection realized, is that a major key is better suited than a minor key for the pastorale, which has to do with our general response to and reception of the qualities of the minor key. The tranquility and serenity of the baroque pastorale or the liveliness of nineteenth-century examples would be poorly realized in a music composed in a minor key with its general acoustic insecurity and associated pathos. We will encounter sicilianas composed in minor keys, such as the well-known example from Bach’s Sonata for Flute in Eb Major, BWV 1031 (G minor), the second movement of Mozart’s Concerto for Piano, K. 488 (F# minor) or Fauré’s famous Sicilienne in G minor. However, the siciliana is a dance, not a tradition that evokes a specific, familiar sentiment. And a dance depends more on its metric and rhythmic organization than its tonality for realizing its intended effect.
 
About This Collection           
     The twenty-six pieces in this collection feature the music of composers from differing time periods and differing nationalities to illustrate the varied heritage of the pastorale tradition. Those pieces not specifically written for the piano, including works for organ, harpsichord, chamber ensemble and orchestra, have been transcribed for it in such a way as to be idiomatic for the keyboard and to preserve the sentiment and expression of the original version. Only those pieces that have been specifically entitled “pastorale” (or a derivative of the term) by their composers have been included in this collection, rather than pieces that may be suggestive of the pastorale style (which would certainly be a subjective decision). There is no question that these pieces were intended to evoke a pastorale sentiment; more relevant questions might be exactly what sentiment and how it was done.
     Editorial markings have been provided in instances in which decisive indications were not given. Many of the early pastorales lack tempo markings, for which reason metronome markings in brackets have been notated. These have also been indicated in instances in which composers notated only time-words as tempo markings. Dynamic markings are also often missing in original versions, again, particularly those from seventeenth and eighteenth centuries. Markings in brackets are suggestions for performance, and the general tendency is to assume a moderate or soft dynamic in most instances. However, caution should be exercised in the interpretation of forte and piano markings in baroque music, for they do not convey the same meanings as they do in later music. Corelli’s forte indicates the “normal” volume for performance; piano is a special effect. Beethoven’s forte and piano are different requests.
Suggestions are also provided for fingerings and pedalings in instances in which these may not be obvious. Some composers, such as Liszt and Dohnányi, notated their own recommendations for fingerings and pedaling, and these should be observed. Pianists should bear in mind that all such editorial markings are suggestions for performance; other markings or substitutions can be or may need to be introduced as the situation requires.
     A collection of this nature, featuring pieces from different time periods and composers of different nationalities, shows that the pastorale has maintained its appeal through dramatically changing times. Its longevity is undoubtedly a result of its flexibility and adaptability to varying tastes and styles. It might also say something about the human condition. Perhaps, despite our hectic pace of existence and eagerness for technological advancement, we still feel a sense of nostalgia and a longing—surely, an impossible one—to escape from it all; perhaps a deep sense of regret at having lost contact with nature.

[1] The Poems of Alexander Pope, ed. by John Everett Butt. (New Haven: Yale University Press, 1963), 119–120.

[2] Ibid., p. 119.

[3] Ibid., p. 119.

[4] The New Grove Dictionary of Music and Musicians, ed. by Stanley Sadie, vol. 14 (London: MacMillan Publisher, Ltd., 1980), 290.
​
[1] The New Grove Dictionary of Music and Musicians, vol. 18, 678.

[2] Quoted in Robert Donington. The Interpretation of Early Music. Revised Edition. (New York: W. W. Norton & Co., 1992), 389–90.

[3] This marking is discussed on p. xxx.

[4] Quoted in Donington, The Interpretation of Early Music, 400–403.

[5] Charles Rosen. Beethoven’s Piano Sonatas: A Short Companion. (New Haven and London: Yale University Press, 2002), 48ff.

[6] This feature is discussed in great detail in Charles Rosen. The Classical Style. (New York: W. W. Norton & Co., 1972), 23ff.

[7] Ibid., 28.

[8] The subdominant aspect of this movement in Bach’s Weihnachts-Oratorium is discussed on p. xxx.

​2. “Tombeau”: Harmonic Form
 
    As the preceding blog entry has shown, the means that Boulez created for organizing pitch in Pli selon pli is an immensely intricate and original technique for harmonic organization in a non-tonal context. Despite its theoretical complexity, this technique is unarguably less complex from an audible standpoint, for the ear can come to appreciate the harmonic logic in the music, due to the nature of the frequency-multiplication technique of interval associations. What is also appreciable in “Tombeau” is the incredible sensation of forward motion in the first 500 measures and the way in which this motion drives inexorably to the structural climax in m. 517, both of which are audibly clear events in the music. This part of the analysis of “Tombeau” illustrates how pitch, instrumentation and tempo promote these sensations and how pitch, in particular, defines musical form: “Serial harmony,” in essence, becomes “harmonic form.”
    The roles of instrumentation and tempo in the musical form are relatively straightforward. “Tombeau” is composed of three distinct sections: Section I extends from mm. 1 to 257, Section II from mm. 258 to 517 and Section II from mm. 518 to 548. In Section I, Boulez divides the orchestra into five principal instrumental groupings (labeled as groupe principal, the vertical ordering of instruments in the score corresponding to these groupings), and each group is introduced successively. Section I is also organized around five different tempos. In this way, instrumentation and tempo contribute to the design of Section I as five distinct sub-sections:

Each sub-section concludes with a rallentando that “modulates” seamlessly to the following tempo. The principal instrumental groups, introduced by the piano in the opening thirty measures, are maintained throughout the entire work. Example 16 also shows that the brass section of the orchestra is not given a principal role in this section. Rather, the brass instruments generally double the pitches of other instrumental groups, thus establishing bonds between the different groups. In Section II, however, the brass is treated more as an independent instrumental group.
    Despite an exact correspondence between changes in instrumentation and tempo in Section I, these changes occur inversely; that is, the number of principal instruments increases, while the tempos decelerate. However, this is reversed in the fifth sub-section: here, the number of principal instruments is reduced, and the tempo accelerates. Also of importance is that the length in the number of measures of each sub-section increases (39, 40, 46, 46, 82 measures, respectively) and thus the duration of each sub-section.
    Pitch (specifically, the array of sets reproduced in Example 10) assumes a more substantive role in this formal scheme. The principal instruments in each of the five sub-sections perform harmonic units from the five basic sets (sets x-00 from Example 10), beginning with sets 1-00, 2-00 and 3-00 in the piano in sub-section 1. Sub-section 2, featuring the xylophone, timpani and tubular bells as the principal instruments, continues with sets 4-00, 5-00, 1-00 and an incomplete 2-00. The seven winds of sub-section 3 resume with the third unit of set 2-00, the first two units of which are performed at the end of sub-section 2. Each of the five sub-sections continues with this methodology, resulting in five statements of each of the five sets in Section I. This is reproduced in Example 17. Note the significance of the number five in organizing the differing parameters of the piece.

Incomplete sets between sub-sections 2, 3, 4 and 5 are means of establishing links between these sub-sections. Thus, these harmonic links are elisions between formal units.
    By contrast, frequency-multiplication sets are distributed to the instrumental group that no longer functions as a principal group. The first of these sets is given to the piano in m. 43 (sub-section 2) with the appearance of the second instrumental group (xylophone, timpani and tubular bells) in m. 41. Here, the piano becomes background, at which point frequency-multiplication sets (sets x-22) appear for the first time. At m. 81 (sub-section 3), with the appearance of the third principal group, frequency-multiplication sets are distributed to the piano (x-33) and to the xylophone, timpani and tubular bells (x-22, those sets previously performed by the piano). In essence, each instrumental group moves deeper into the array of sets as Section I progresses:

Inevitably, as Section I progresses and the number of instrumental groups increases, the number of distinct pitch layers increases as well. Whereas the first sub-section of Section I contains just units of the twelve-note sets performed by one principal group, the fifth sub-section comprises all five levels of the array, performed by one principal group and four non-principal groups. To co-ordinate these various layers, Boulez establishes a unique registral fixation for each pitch, defined by units of the twelve-note sets that are performed by the principal groups, as reproduced in Example 17.
    Boulez’s strategy for selecting sets in “Tombeau,” one that is similarly used in cummings ist der dichter from 1970, is determined by a diagonal progression through the array of harmonic units, matching the diagonal pattern of Example 18. The following example charts the sets performed by the piano in Section I:

When the piano is the principal instrument, it begins, expectedly, with the parent set and moves successively to the following sets on the same level. When it is no longer principal (beginning in sub-section 2), it moves to the second level of the array, yet rather than beginning with the first set of this level, it moves to the set that is diagonally to the right of the parent set (i.e., set 2-22). It then continues reading the second level of sets successively, until a new principal group appears. At this point (sub-section 3), it moves to the third level of the array, again diagonally to the right of the first set performed on the second level (i.e., set 3-33) and continues on the third level.
    When an instrumental group succeeds another group on a level of the array, the former begins with the same set, occasionally the following set, performed by the latter. Example 20 lists the sets performed by each instrumental group in Section I. Sets in parentheses represent incomplete sets.

The incomplete sets in Example 20 are means for establishing links between instrumental groups and formal units, as with the incomplete sets in Example 17. For example, the piano performs only the first four units of set 4-22 in sub-section 2. The final unit of this set is then given to the xylophone, timpani and tubular bells, as this group moves to the second level of the array in sub-section 3. This is done in nearly every instance when an instrumental group changes levels in the array and is also done between Sections I and II and between Sections II and III. Additionally, sets on each level of the array appear a unique number of times: There are twenty-five x-00 sets, eighteen x-22 sets, twelve x-33 sets, fourteen x-44 sets and eight x-55 sets.
    The justification for classifying mm. 258–517 as Section II is that these measures, with a length approximating that of Section I—Section I is 257 measures; Section II is 259 measures—represent another formal unit, one that continues and expands on ideas introduced in Section I. However, while Section I comprises five sub-sections, determined by unique tempo markings, principal instrumental groups and levels from the array of sets, Section II comprises a single irreducible section, defined by one tempo indication, no principal instrumental group and one set-array level for each group for the duration of this section. At the beginning of Section II, each instrumental group advances one level deeper into the array and retains these positions for all of this section. The brass group, now an independent instrumental group, performs the five basic sets (sets x-00), rather than doubling other instrumental groups. The piano, having reached the fifth and final level in the array (sets x-55) at the conclusion of Section I, returns to the first level, with the brass, although in a different manner. Example 21 shows the initial set performed by each instrumental group in Section II:

    Sections I and II feature a variety of similarities, in addition to an approximate correspondence in length. Both sections open with a piano solo, the only instances in these sections in which the piano, or any instrument, assumes such prominence. In both cases, the piano introduces the other instrumental groups by pitch doublings. Further, both sections show a dramatic unfolding of textural density and motion, that of Section II, continuing where Section I left off, being far greater. Lastly, each section concludes with an extended rallentando, by far the two largest modifications in tempo in the entire piece (largest in terms of range of metronome values). In fact, the metronome marking of Section II takes the reduction introduced in Section I one additional step and completes it:

Additionally, Section II could be regarded as a formal expansion, equivalent to the length of Section I, of the third sub-section (mm. 81–127) of Section I. For example, both share the same metronome marking, and both feature six statements of the twelve-note sets by the respective principal instrumental groups.
    In contrast to Section I, however, Boulez does not categorize a principal instrumental group in the score in Section II. This may be interpreted in one of two ways. One, the brass section, performing the five basic twelve-note sets, is now the principal group, inasmuch as the principal group invariably performs these sets in Section I (although the piano likewise performs these sets). Two, none of the groups is principal, whereby each would be considered equivalent. Also of distinction in this section is the structuring of pitch. Registral fixation continues to be used in this section, but whereas each of the twenty-five twelve-note sets in Section I works with a different fixation, that of Section II is defined by the parent set (set 1-00) performed by the piano at the beginning of this section (mm. 258–274). Example 23 illustrates the registers of the notes of this set:

Most of the pitches in Section II conform to this scheme, although certain pitches stray somewhat and do not appear to adhere to any system. Section II, thus, becomes a massive instance of harmonic stasis.
    As noted previously, each instrumental group moves one level deeper into the array and maintains this level for the entire section. Example 24 lists all of the sets performed by each of the instrumental groups in Section II:

As in Section I, the number of sets performed by each instrumental group in Section II differs.
    As Example 20 shows, certain sets are left incomplete at the conclusion of Section I. Boulez transfers the units of the incomplete sets to the instrumental group that occupies the new level in the array in Section II (compare Examples 20 and 24). In m. 280, the harps, vibraphone, celesta and guitar are given the fifth unit of set 4-22, left unperformed by the strings at the end of Section I. In m. 283, the strings, somewhat more meticulously, take the one note, C-sharp, that is not included in the second unit of set 4-33 of the winds (mm. 254–55) and remain on this level. In m. 276 the winds open with the third unit of set 2-44, left unperformed by the xylophone, timpani and tubular bells in m. 257. This group, in turn, begins with set 3-55 in m. 287, following the piano’s conclusion on this level with the fourth unit of set 2-55 (the fifth unit of set 2-55 is seemingly disregarded). Again, harmonic units act as elisions between formal components.
    Section II and Section III, mm. 518–548, the final section of “Tombeau,” are separated by a general pause, being, in fact, the first complete silence until this point in the piece. Unquestionably, the greatest distinction between Section III and the previous two sections is the absence of metrical rhythmic notation. Section III is grouped into time fields, designated as “senza tempo,” the durations of which are left to the discretion of the conductor on the basis of prescribed cues. Curiously, this aspect of the piece was not subsequently rewritten by the composer, as similar passages in the opening movement, “Don,” were subjected to drastic revisions in the 1980s.
    This dual temporal construction, strictness versus freedom, is a pivotal feature in the music of Boulez’s mature period. He has noted, “I regard the two categories—smooth and striated time—as capable of reciprocal interaction…I can say that my whole formal time system is based on them and them alone.” (Pierre Boulez, Orientations, p. 87) His terminology is based on a division of musical time into two categories: pulsed time (i.e., metrical time) and amorphous time (i.e., proportional time). He has observed that in pulsed time, durations are related to chronometric time; this constitutes a pulsation, of either the smallest unit of the system of a simple integral multiple of the system (such as 2 or 3). Amorphous time, however, is related to chronometric time only in a global sense: Durations, either with or without designated proportions (i.e., unfixed time values), occur within a prescribed time field. However, only pulsed time is susceptible to temporal modifications, i.e., acceleration and deceleration; amorphous time can do this only by deviating in density according to a number of events that occur within a chronometric time field (i.e., events occurring more slowly or more quickly). Amorphous time, for Boulez, is comparable to a smooth surface, pulsed time to a striated surface; hence, the terms smooth and striated time. (Pierre Boulez, Boulez on Music Today, pp. 88–89)
    The introduction of smooth time in Section III can be understood in terms of the objective of Sections I and II. Since temporal deceleration and the drive to the structural climax in m. 517 are important factors in the first two sections, an application of an amorphous time system would be impractical. However, once the process of deceleration has been completed, as after the climax in m. 517, an amorphous temporal system may be incorporated, as Section III appropriately does. The five movements of Pli selon pli are pivotal in Boulez’s output, in the sense that they are virtually the first works to exploit the reciprocal interactions of these two temporal systems, although this was prefigured in the second book of Structures. In fact, most works after Pli selon pli, in their original, unrevised forms, combine both temporal systems in various ways.
    Section III is composed of three sub-sections, each of which is distinguished by differences in texture. The first sub-section, mm. 518–529, is composed entirely around a homophonic (or chordal) texture; the second sub-section, mm. 530–535, is predominantly polyphonic with two brief insertions of the homophonic texture of the first sub-section; the third sub-section, mm. 536–548, recapitulates and concludes with the homophonic texture of the first sub-section. Section III, as this organization suggests, complements the design of Section II. Whereas Section II expands on the micro-structure of a sub-section of Section I, Section III duplicates the three-part macro-structure of “Tombeau” in general, by presenting in miniature (thirty-one measures) the three-part form of the work. Thus, in a structural sense, Section II elaborates on a formal unit of Section I, while Section III summarizes what takes place in the entire work.
    Section III maintains the six-part instrumental grouping of the previous section, yet the six groups are aggregated to form two larger groups (piano, xylophone, gongs, bells, harps, vibraphone, celesta, guitar and winds, brass, strings). In the first sub-section these groups are differentiated by the way harmonic units are performed. The first group performs these units with differing attacks and simultaneous decays; conversely, the second group performs units with simultaneous attacks and differing decays. The array of sets likewise contributes to defining this two-part orchestral structuring. The piano, xylophone, gongs, bells, harps, vibraphone, celesta and guitar perform harmonic units from the basic twelve-note sets (sets x-00), while the strings, winds and brass perform units from the frequency-multiplication sets:

The sole exception to this is the harmonic unit that opens Section III in m. 519, performed by the harps, vibraphone, celesta and guitar (notated at sounding pitch):

Remarkably, this eight-note harmonic unit is a collection of the units left unperformed by each instrumental group (except for the strings) at the conclusion of Section II. F-sharp of the piano (unit 5 of unit 5-00), G, E, F-sharp of the brass (unit 5 of set 1-00), and B, A, B-flat of the xylophone, vibraphone 2 and bells (units 4 and 5 of set 5-55), A of the harps, vibraphone 1, celesta and guitar (from unit 4 of set 2-22), and C and E of the winds (from unit 5 of set 4-44) are combined to form this harmonic unit. As in Sections I and II, pitch is a significant ingredient for linking Sections II and III.
    In the second sub-section of Section III, mm. 530–535, the soprano appears for the first time in the work with the final line of Mallarmé’s sonnet Le tombeau de Paul Verlaine, accompanied by horn and guitar (i.e., one instrument from each of the two instrumental groups). However, as a consequence of the virtuosic writing for the voice, featuring wide leaps and a demanding tessitura, the five words, “Un peu profond ruisseau calomnié…” (“A shallow stream and slandered…”) are virtually unintelligible, much like the vocal writing in the preceding “Improvisation III sur Mallarmé.” The conclusion of this sub-section features a different approach. In mm. 546–548, the soprano performs the two closing words of the sonnet, “la mort” (“death”), set to the pitches F-sharp and D; the performance of the latter is to be “…spoken without timbre, solely on the breath.” Contrary to the unintelligibility of the first five words of the text, these two words are entirely comprehensible, perhaps as a means of highlighting the significance of death in this portrait of Mallarmé.
    In mm. 531–32, the soprano is interrupted by two harmonic units, the first performed by xylophone, gongs, bells and strings, the second by harps, vibraphone, guitar and winds. The two-part orchestral structure is still evident here on the basis of performance; the strings and winds perform their units as trills, while the harps, vibraphone, guitar and winds do not. What is especially noteworthy is that these are units 4 and 5 from set 3-33, the two units left unperformed by the strings in m. 517 (see Example 24):

    The third and final sub-section of Section III, mm. 536–548, serves as a preparation for the final twelve-note unit that concludes the work, the same unit that opens the first movement, “Don.” Moreover, this sub-section preserves the two-part instrumental grouping presented at the beginning of Section III. All harmonic units in these measures are taken from the fifth column of sets in Example 10, as illustrated in the following example:

Note from Example 28 that the order of entry of instruments adheres to the two-part orchestral grouping: The strings and winds perform first, followed by the bells, harps, celesta, piano, vibraphone and guitar.
    The twelve-note unit that concludes “Tombeau” in m. 548 likewise concludes the motion generated in the first two sections and, in a sense, in Pli selon pli as a whole. The pitches of this unit and their registral positions are identical to the unit that opens “Don,” yet the instrumentation of each is different. Contrary to the explicit structuring of instrumentation in the preceding measures, all instruments of the orchestra, as well as all pitches, act as one in this unit:

The voicing of this twelve-note unit is the key to its configuration. Observe, for example, the notes given to the piano:

This six-note unit is the second hexachord of the parent set (units 3, 4 and 5 from set 1-00) from Example 1. The first hexachord of this set (units 1 and 2) is positioned between the innermost notes of this unit:

The complementary hexachordal make-up of this unit is also represented in each hexachord. The four inner pitches of the first unit in Example 31, B-flat, B, D and C-sharp, are unit 2 of set 1-00, and the two outer notes, E-flat and F, are unit 1 of this set. In the second hexachord of Example 31, units three and four of set 1-00, plus the G of unit 5, are placed in the treble range, while the E and F-sharp of unit 5 are placed in the bass.
    Section III of “Tombeau” is a remarkable and an audibly explicit conclusion to the two large-scale evolutions that have taken place throughout the entire piece. The first is that of the orchestra. In Section I, each of the five instrumental groups is successively designated as principal. In Section II, the five orchestral groups are maintained, joined by the brass section as a sixth group. Section III then reduces this six-part division to two parts, reduced further to one part in the final measure, where all groups merge on the final twelve-note unit. Nowhere in the piece prior to this is there an event in which all instruments of the orchestra perform the same gesture. Consequently, this evolution is organized around a gradual aggregation of the independent instrumental groups, from six groups in Sections I and II, to two groups in Section III, and finally to a single group in the last measure. In fact, it is an evolution from an ensemble of ensembles to a single ensemble.
    The second evolution, independent of the first, yet following a similar course, is that of pitch. In Section I each instrumental group is assigned different levels from the array of sets that alternate over the course of this section. Section II continues this procedure, yet each group is fixed on a specific level for the duration of this section. In Section III, corresponding to the orchestral structure, individual set levels are disregarded—Section III, in a sense, recycles and completes those sets left incomplete in Section II—and lastly it is the final unit in m. 548 in which all pitches are joined. As there is no simultaneity of performance until this unit, there is similarly no single event in the music prior to this that is built on all twelve pitches.
    Elaborate objectives require elaborate means to realize them. More importantly, practically all conclusions made in this analysis can be explained in terms of the musical and poetic correspondence, as the third and final part of this blog entry will show.

1. “Tombeau”: Serial Harmony
           
(Note: Due to copyright restrictions, examples from the score of Pierre Boulez’s “Tombeau” have not been included. The full orchestral score is available from Universal Edition.)
 
    Pierre Boulez’s Pli selon pli “(Fold by fold”), composed in 1957–62 for soprano and orchestra, is a masterpiece in associations between text and music. Planned as a large-scale portrait of the French symbolist poet Stéphane Mallarmé (1842–1898), the piece’s five movements approach and solve the problems of text setting in different ways. Composed immediately after the Sonata No. 3 for piano, Boulez’s first essay in formal mobility, the piece is also a masterpiece in the establishment of a system of harmonic order that contributes both to a correspondence of melody and harmony, and to formal coherence. “Serial harmony”—the term is not Boulez’s and has no reference to principles of tonal harmony—is a convenient means for describing his approach at this time: A twelve-note set serves as a “parent set,” from which other sets are derived and from which an array of harmonic units is established. The approach focuses less on the linear control offered by serial operations and more on the coherence that can be achieved with intervals and harmonic units, which are subjected to serial techniques.
    “Tombeau,” the fifth and final movement of Pli selon pli, is an audibly explicit demonstration of the efficacy of serial harmony and the ways in which it can contribute to form. Boulez’s concerns with serial harmony may be traced to his critiques of the inadequacy of serialism to relate vertical and horizontal structures, as in the way in which “accidental harmonies” could result from the emphasis on horizontal aspects. He has been harshly critical of the absence of control over vertical structure in much of Schönberg’s twelve-tone music and his own early works, yet has credited Webern’s achievement in the fifth movement of the Cantata No. 2, Op. 31 with composing “…the four melodic lines such that they meet at the same point to form a specific harmony. The counterpoint becomes convincing because the vertical, horizontal and diagonal aspects are controlled by the same laws.” (Célestin Deliège. Pierre Boulez: Conversations with Célestin Deliège, p. 94) Regarding his own music, he noted, “I have often been absorbed by a concern to discover clear harmonic relationships. If we can write harmony with melodic line under laws common to both, then we can begin to find a solution that will considerably enrich the musical vocabulary.” (Célestin Deliège. Pierre Boulez: Conversations with Célestin Deliège, p. 91)
    His solution in the early 1950s, coincident with the composition of Le marteau sans maître, was a technique for generating an array of harmonies by means of operations on a twelve-note set. In an interview in the 1970s, he observed, somewhat obliquely, that:

In my most recent works, practically all the pitches are deduced from each other by means of harmonic systems such as those which can be multiplied by each other. I believe it is impossible to write in two different dimensions following two different sets of rules, and that one must in fact follow rules that apply reciprocally to the horizontal and the vertical.
​(Célestin Deliège. Pierre Boulez: Conversations with Célestin Deliège, p. 90)
 
Introduced and discussed in his article Èventuellement… from 1952, the technique to realize serial harmony has come to be known, somewhat erroneously and perhaps from a misguided translation from the French, as “frequency multiplication.” (Pierre Boulez, Boulez on Music Today, pp. 39–40) One advantage of this technique as a means for systematizing pitch is that the total pitch content of a work, both the horizontal and vertical dimensions, is based on a single twelve-note set; another is that it provides a cogent system for non-tonal harmonies. Of particular note is that the same array of harmonies in “Tombeau” was used in a number of subsequent compositions, up to cummings ist der dichter of 1970; succeeding works would explore different resources for pitch organization. Although Boulez has never explained why he has worked with the same array over a period of approximately twenty years—and then disregarded it—it is unquestionably this recycling of material that contributes to what one could designate as the “Boulezian” sound.
    From a theoretical standpoint—arguably, from an audible one as well—“Tombeau” unfolds its process of serial harmony in an exceptionally clear manner (one might even say “fold by fold”) over the course of its fifteen minutes. The first eight measures of the piece introduce a twelve-note set (the “parent set”) in the piano, which is identified in the score as a principal instrument (piano principal). In these opening measures the twelve pitches of the parent set are grouped into five harmonic units of two, four, two, one and three notes. Measures 9–19 present a second statement of this set, though varied by harmonic units of four, two, one, three and two notes. Measures 20–39 continue with a third statement of this set, whose harmonic units comprise two, one, three, two and four notes. This is followed by a fourth statement in mm. 40–48 with harmonic units of one, three, two, four and two notes, and a fifth statement in mm. 53–61 of three, two, four, two and one notes. The five statements of the set in these measures are:

    ​Example 1.
 
The linear rendition of the parent set, though not given in this work in any instance, can be deduced from Example 1 as:

    Example 2.
 
    The four sets in mm. 9–61 are created by varying the harmonic groupings of the parent set, though not the pitches themselves; for this reason they could be designated as “derivative sets.” The number of notes in each harmonic unit, but not the notes themselves, is modified by rotation, whereby the number of notes in the first harmonic unit of a set is moved (or rotated) to the end of the following derivative set. The “magic square” in Example 3 illustrates this, in which each number represents the number of notes in each of the five harmonic units:
 
                        Parent Set:                   24213
                        Derivative Set 1:          42132 [“2” of the Parent Set is moved to the end of this set]
                        Derivative Set 2:          21324 [“4” of Derivative Set 1 is moved to the end of this set]
                        Derivative Set 3:          13242 [“2” of Derivative Set 2 is moved to the end of this set]
                        Derivative Set 4:          32421 [“1” of Derivative Set 3 is moved to the end of this set]
 
    Example 3.
 
Continuing past the fourth derivative set would reproduce the parent set. The emphasis is placed on the ordering of the harmonic units in these sets, rather than on the ordering of individual pitches; hence, the serial aspect of the system.
    Each of these five sets serves as source material to produce frequency-multiplication sets. In this process, the intervals of one harmonic unit are transposed to each of the pitches of a second harmonic unit; thus, units of greater harmonic density are created from units of lesser harmonic density. Clearly, the technique works more successfully with harmonic units, rather than with single pitches, which explains why the pitches of the five sets in Example 1 are grouped into five units. Presumably, its nomenclature can be attributed to the nature of the technique. If a unit of three pitches is multiplied with another unit of three pitches, then the result is nine pitches (3x3), duplicate pitches being excluded. This is illustrated in Example 4, where the intervals of one three-note unit are multiplied with (or transposed to) the pitches of another three-note unit, producing nine pitches:

    Example 4.
 
D is the only pitch that is duplicated in the three steps of multiplication; a harmonic unit comprising eighth unique pitches is the result.
    Because of the importance of interval associations in this technique, set-theory analysis can be instrumental in determining the presence of the harmonic units in the music and in revealing the interval structure of these units. To illustrate the latter, the two harmonic units from Example 4 will be used. Firstly, determine the prime-form designation of each unit (see Alan Forte, The Structure of Atonal Music, pp. 3–7):

    Example 5.
 
Then, add, from left to right, each number of the first unit to each number of the second unit (any resultant numbers greater than twelve are reduced by twelve):
 
                           0 1 3
                        + 0 2 6
                         _____
                         0 2 6 (0 + 0 2 6)
                         1 3 7   (1 + 0 2 6)
                         3 5 9 (3 + 0 2 6)
 
    Example 6.
 
Lastly, arrange the numbers in arithmetical order, with duplications eliminated. The result is the pitch-class (pc) set [0,1,2,3,5,6,7,9] (or 8-Z29, according to Forte’s classification), and this, as Example 7 shows, is the prime-form designation of the definitive harmonic unit in Example 4:

    Example 7.
 
Note that the process is commutative in terms of interval content; which is to say, switching the order of the units in Example 4 will produce the same pc set:
 
                           0 2 6
                        + 0 1 3
                         _____
                         0 1 3 (0 + 0 1 3)
                         2 3 5 (2 + 0 1 3) = [0,1,2,3,5,6,7,9]
                         6 7 9 (6 + 0 1 3)
 
    Example 8.
 
    Unquestionably, the most practical aspect of frequency multiplication from a compositional standpoint is the relationship that is established amongst various harmonic units, as Boulez has emphasized. The process, in fact, centers primarily on intervals, rather than frequencies; in essence, the intervals of one harmonic unit are “mapped” onto the pitches of another unit. In Example 4, the definitive eight-note harmonic unit is created by mapping the intervals of the first unit on the pitches of the second unit. As a result, the definitive unit will have a superstructural association with both units; that is, each of the initial units will be embedded in multiples instances in the definitive unit. For example, the definitive unit [0,1,2,3,5,6,7,9] comprises five [0,1,3] pc sets (the first unit) and five [0,2,6] pc sets (the second unit). Interval associations between harmonic units are thus the essence of this technique and of Boulez’s serial harmony, in general; for this reason, the term “interval mapping” might be a more appropriate designation, though one that is not in current usage.
    Boulez realized an array of frequency-multiplication sets by applying the process in Example 4 to each of the five sets (the parent set and the four derivative sets of Example 1). Each of the five harmonic units of a set was multiplied with every other unit of the set including itself, in which each unit was used as a “fixed multiplicative factor”. For example, unit 2, as the fixed multiplicative factor, was multiplied with units 1, 2, 3, 4 and 5; unit 3, as the fixed multiplicative factor, was multiplied with units 1, 2, 3, 4 and 5, and so forth. Boulez provided a schematic diagram to illustrate this process, as well as to illustrate the resultant associations in the array, what he called “isomorphic relations.” In the following diagram, the letters “a,” “b,” “c,” “d” and “e” represent the five harmonic units of the parent set, and the letters “m,” “n,” “p,” “q” and “r” are the number of notes per unit.
 
a{m                   b{n      c{p       d{q      e{r
 
m x [aa           ab        ac         ad        ae]
n x  [ba           bb        bc        bd        be]
p x  [ca             cb        cc         cd        ce]
q x  [da             db        dc        dd        de]
r x  [ea            eb        ec         ed        ee]
 
    Example 9.
    (Pierre Boulez, Boulez on Music Today, p. 79)
 
The diagram shows that every harmonic unit appears twice in the array (i.e., “ab” is the same as “ba,” as part of the commutative process), except for units multiplied by themselves (i.e., “aa,” “bb,” “cc,” “dd” and “ee,” read diagonally). This example also shows the relationships between multiplied units of a single frequency-multiplied set. For example, all units on line “m” share the common unit “a” (each of which is duplicated in column “a”), yet each is distinguished by unique groupings with other units of the set. The significance of the letters “m” through “r” will be discussed subsequently.
    Following the diagram in Example 9, the parent set, the four derivative sets and the array of frequency-multiplication sets in “Tombeau” are reproduced in Example 10.

    Example 10.
 
Every pitch in “Tombeau” can be explained in reference to this example, an astonishing instance of adherence to a pre-compositional plan and the significance of the plan.
    The numbers in Example 10 are means to designate specific sets and harmonic units, and to designate the ways in which they are formed. Sets in the same column (read vertically) are derived from the twelve-note set on the uppermost staff. For example, set 1-22 (column 1, line 2) is derived from set 1-00 by using the second unit of set 1-00 as the fixed multiplicative factor, where this unit is multiplied with every unit of the same set, including itself: 2x1, 2x2, 2x3, 2x4 and 2x5, or “ba,” “bb,” “bc,” “bd” and “be,” according to Example 9. Similarly, sets on the third staff (-33) use the third unit of each twelve-note set as the fixed multiplicative factor and so forth. Note from Example 10 that Boulez disregarded using the initial unit of each set as a fixed multiplicative factor, which would result in sets x-11 from line “m” of Example 9, for reasons that are not apparent.
    A crucial aspect of this technique is that the registral distribution of notes in the harmonic units of the twelve-note sets is decisive in determining the pitch transpositions, though not the interval content, of the definitive units. Boulez has noted, “Each of the superimpositions of frequencies is evidently susceptible to modification each time it is reproduced, all the notes being novel as to tessitura and susceptible to permutations along a vertical line in relation to the intervals of definitions.” (Pierre Boulez, Notes of an Apprenticeship, pp. 167–68) This is illustrated in Example 9, where the letters “m,” “n,” “p,” “q” and “r” indicate the number of different transpositions available for each multiplied unit based on the number of notes in each of the original harmonic units. For example, the first unit of set 1-00, if multiplied with the second unit, will produce two different transpositions of the definitive harmonic unit based on the registral distribution of notes in the first unit (F/E-flat vs. E-flat/F):

    Example 11.
 
However, the definitive units in Example 11 are transpositionally equivalent, inasmuch as each preserves the same interval content. Consequently, from a theoretical standpoint, the pitches of sets x-22 through x-55 in Example 10 are relative; however, these pitches can be justified as absolute in “Tombeau,” for these are the transpositions that Boulez used. Ultimately, this would signify that Boulez was more concerned with the interval associations produced with this technique, rather than with the actual pitch transpositions, suggesting, again, that the term “frequency multiplication” is a misnomer.
     One final step remains in the production of the harmonic units in Example 10. Boulez transposed units that are multiplied with a fixed multiplicative factor in relation to the interval between the lowest pitch of the fixed multiplicative factor and the lowest pitch of the first unit of the set. This can best be illustrated when one pitch is used as a fixed multiplicative factor, resulting in a transposition of the twelve-note set (e.g., set 1-44 of Example 10, an interval-class (ic) 3 transposition of set 1-00). Prior to multiplying the fourth and first units, Boulez transposed the first unit ic3 to A-flat/G-flat, ic3 being the interval between the fourth unit (A-flat) and the lowest note of the first unit (F):

    ​Example 12.
 
The remaining units were treated similarly, thereby resulting in an ic3 transposition of set 1-00 (cf. Example 2):

    Example 13.
 
This transposition is applied to each process of multiplication in Example 10, presumably to establish pitch commutativity (in addition to interval commutativity) between harmonic units in the array. For example, if the order of units in Example 11 is reversed, the same harmonic unit will be produced, yet at a different transposition (i.e., at ic5):

    ​Example 14.
 
Transposing the second unit ic5 in relation to the interval between the lowest notes of units one and two will yield the identical unit:

    Example 15.
 
Seemingly, Boulez applied this transposition specifically for this reason and applied it in every instance for purposes of consistency.
    Such is the methodology of serial harmony in Boulez’s “Tombeau.” It is unquestionably a cogent strategy for establishing interval associations between simple and complex harmonic units, in which “progressions” between these units are governed by a serial order. The next entry will address how this array of serial harmony is instrumental in defining the form of “Tombeau.”

(The following is an excerpt from a forthcoming publication on Glenn Gould's 1981 recording of the Goldberg Variations by Johann Sebastian Bach.)



  Johann Sebastian Bach’s Aria mit verschiedenen Veränderungen (Aria with Diverse Variations), BWV 988 was published in 1742 as the final part of his Clavierübung series, a four-part collection of keyboard works intended, as the title proposes, as pieces for “keyboard practice.” It is only the second piece by Bach in the variation form, by name, at least, the other being the earlier composed Aria variata in A minor, BWV 989 (an “unpretentious set,” as Glenn Gould unfavorably described it[1]). Certainly, Bach was no stranger to variations, for his output is rich with variation-type pieces, though with alternate names, such as chaconne and passacaglia. Bach would compose in the variation form one more time in the momentous Five Canonic Variations on “Vom Himmel hoch, da komm ich her”, BWV 769 of 1747, which likewise bore the word Veränderungen in the title. But none of these pieces is the type of theme and variations that would become wildly popular in the period after Bach’s death.
   Johann Nikolaus Forkel, who published the first biography of Bach in 1802, seems to have been responsible for propagating the account of the work’s commission by an insomniac Dresden nobleman, who counted Johann Gottlieb Goldberg, a former student of Bach, as one of his employees:

We are indebted to Count Kaiserling, formerly Russian Ambassador at the Court of the Elector of Saxony, who frequently resided in Leipzig, and brought with him Goldberg, who has been mentioned above, to have him instructed by Bach in music. The Count was often sickly, and then had sleepless nights. At these times Goldberg, who lived in the house with him, had to pass the night in an adjoining room to play something to him when he could not sleep. The Count once said to Bach that he should like to have some clavier pieces for his Goldberg, which should be of such a soft and somewhat lively character that he might be a little cheered up by them in his sleepless nights. Bach thought he could best fulfill this wish by variations, which, on account of the constant sameness of the fundamental harmony, he had hitherto considered, as an ungrateful task.[2]

Hence, the moniker “Goldberg” Variations, which, for better or worse, has remained to the present day. However, recent scholarship has dismissed this anecdote as apocryphal. Christoph Wolff, for example, has noted:

However, all internal and external clues (lack of any formal dedication to Keyserlingk as required by eighteenth-century protocol, and Goldberg’s tender age of fourteen) indicate that the so-called Goldberg Variations did not originate as an independently commissioned work, but were from the outset integrated into the overall concept of the Clavierübung series, to which they constitute a grandioso finale.[3]

The title page of the score bears no dedication to the Count (or anyone else), and there is no evidence that Bach was financially compensated for the work. Other sources opine that Bach composed the work, in fact the Clavierübung collection as a whole, as keyboard practice for his eldest son, Wilhelm Friedemann Bach, whose prowess as a harpsichordist was widely celebrated.[4]
   The bass line of the opening Aria, originally composed in 1725 as part of the Notebook for Anna Magdalena Bach, assumes the role of the “theme” in each of the variations; otherwise, all of the Aria’s thematic material is disregarded.[5] This bass line, reduced to its basic form,
is presented in various guises in the succeeding variations and is one important ingredient that unifies the heterogeneous pieces of the collection. Or as Gould observed, somewhat belittlingly, “Indeed, this noble bass binds each variation with its own inexorable assurance of its own inevitability.”[6] However, the harmonies that accompany the bass line in the Aria are not invariably maintained. For this reason the piece as a whole could be described as a passacaglia, though lacking the formal continuity common to baroque passacaglias.
   In addition to the unity provided by the Aria’s bass line, the Goldberg Variations proliferates in various forms of structure and coherence, factors which undoubtedly were of enormous appeal to Gould. The Aria, a sarabande, is in simple binary form, each of the two parts composed of sixteen measures, thus thirty-two measures in total. Each of the thirty variations is similarly in simple binary form, and surely it is no accident that the number of pieces in the collection totals thirty-two (Aria, thirty variations and a concluding Aria da capo). Thus, Bach composed a large-scale form derived from the small-scale form of the Aria.
   Structural cycles are additional types of coherence that are pivotal in unifying the variations. One such cycle is based on a three-part grouping of pieces. Each of these cycles opens with a character piece (generally a stylized dance), is followed by an etude-like piece that exploits various keyboard figurations and concludes with a canon. There are ten such cycles, starting with Variations 1, 2 and 3 and concluding with Variations 28, 29 and 30. Another cycle, one that corresponds to the binary form of the Aria, is based on a grouping of sixteen pieces. The first part of this cycle begins with the Aria and extends through Variation 15. The second part opens with Variation 16 (an Overture in the French Style) and concludes with the Aria da capo. Thus, each part of this cycle comprises sixteen units, like each part of the Aria, another example of large-scale form determined by small-scale form. It is noteworthy the role played by the French overture in each of the Clavierübung sets, as centerpieces that mark the beginning of a new cycle or division. This is the case with the opening of Partita No. 4 from the first part of the Clavierübung (the only one of the six Partitas with an overture), the Overture in the French Style from the second part of the Clavierübung and the fughetta (without the fast fugal section) on “Wir glauben all an einen Gott” at the approximate center of the third part.
   The nine canons in the collection deserve special attention. Each is distinguished by a unique harmonic interval (i.e. the interval of transposition between the first and second voices), beginning with a canon at the unison (no. 3) and concluding with a canon at the ninth (no. 27). Thus, this interval is determined by the variation number divided by three, a concept that is not musically relevant, but is appreciable and pleasurable in its own right. All canons comprise two voices with an accompanying bass line, with the exception of no. 27, which is in two parts without a bass line. Moreover, two of the canons are in contrary motion (nos. 12 and 15) and two are in the parallel minor (G minor, nos. 15 and 21).
   One additional point of interest in the Goldberg Variations is Bach’s specifications of the number of manuals of the double-manual harpsichord for each variation. Donald Francis Tovey has observed something telling about the uniqueness of these markings in Bach’s music:

The directions “a 1 Clav” or “a 2 Clav”, which we find at the head of each variation, show not only that Bach was writing for a harpsichord with two manuals, but that he was unusually anxious that the player should use both manuals together at the appropriate passages—for there are, at least, three variations headed “a 2 Clav” which would be quite as easy to play on one manual; and in no other clavier-work does Bach trouble to make such indications, though he frequently wrote for a double-manual harpsichord. Bach is, then, writing here with even more than his usual attention to the circumstances of actual performance.[7]

Tovey’s comment regarding “the circumstances of actual performance” could be supplemented to read “the circumstances of keyboard practice,” for it is evident that all of the pieces in the Clavierübung sets were composed with this intent in mind. Those pieces with the directions “for two keyboards” generally feature voices that cross or overlap. The differing keyboards facilitate execution of parts and clarification of voice leading, and both matters can be compromised when these pieces are performed on a single piano keyboard.
   The Goldberg Variations is certainly not the traditional type of theme and variations; in fact, there are few other examples of its kind. Bach seems to have composed a collection of “diverse variations” (without specifying the number of variations in the title) and then devised a number of elaborate organizational schemes to unify them. This might suggest that these schemes were afterthoughts, once Bach realized how truly diverse the pieces were becoming. There is no evidence that Bach ever envisioned a complete, integral performance of all of the variations at one time—that seems to be a modern practice—yet these thorough structural systems and cycles might suggest otherwise. The piece is an extraordinary example of diversity bridled by unity, factors which undoubtedly contribute to the power and enchantment of the piece as a whole.

[1] “The ‘Goldberg Variations’,” in The Glenn Gould Reader, 22.
[2] “Forkel on Bach’s life and Works,” in The Bach Reader, ed. by Hans T. David and Arthur Mendel. (New York: W. W. Norton & Company, 1972), 338.
[3] Christoph Wolff. Johann Sebastian Bach: The Learned Musician. (New York: W. W. Norton and Company, 2001), 377.
[4] See Peter Williams. Bach: The Goldberg Variations. (Cambridge: Cambridge University Press, 2001), 28ff.
[5] The attribution of this Aria to Bach has been a matter of dispute, and he may have disregarded the Aria’s thematic material, simply because he did not believe it worth preserving. Some scholars believe that Bach could not conceivably have been the composer of the Aria, and his decision not to preserve its material, other than the bass line, lends support to this argument. See David Schulenberg, The Keyboard Music of J.S. Bach. (Routledge, 2013), 377. In this sense, his apparent low regard of the theme of his variations is similar to Beethoven’s low regard of the theme Diabelli that provided for his variations.
[6] “The ‘Goldberg Variations’,” in The Glenn Gould Reader, 24.
[7] Donald Francis Tovey. “Aria with Thirty Variations” from Essays in Musical Analysis: Chamber Music. (New York: Dover Publications, 2015), 31–32.

   Poetic verse is commonly organized according to patterns based on alternations of stressed and unstressed words and syllables. Meter, a term familiar to any musician, is an effective means for coordinating these patterns in poetic verse and for providing cohesion to the pace and flow of the words. It is generally organized in terms of groupings, whereby stressed and unstressed words or syllables together form a single metric unit (or “foot”). Most English verse is based on five types of poetic feet: iamb, trochee, dactyl, anapest and amphibrach.
   A dactylic foot consists of one stressed word or syllable followed by two unstressed words or syllables. The term is taken from the Greek word for “finger,” since fingers are composed of one long and two short joints. This three-part poetic foot is naturally energetic, inasmuch as the two unaccented units provide a powerful forward motion to the single accented unit, and is often incorporated in verses to suggest joy or vitality. It can also be advantageous for poems with vigorous subject matters. This might explain why Alfred Lord Tennyson introduced the dactylic foot in the fifth stanza of his poem The Charge of the Light Brigade, even purposely omitting words so that the dactyl might be realized:

Can-non to right of them,
Can-non to left of them,

And later:

Came thro’ the jaws of Death
Back from the mouth of Hell,
All that was left of them,

Trisyllabic words can be dactylic as well, such as ra-di-o, sub-ma-rine and bas-ket-ball, as are such proper names as Bee-tho-ven, Men-dels-sohn and Ten-ny-son.           
   Poetic feet may also be encountered in music, each foot serving a very specific purpose. Dactyls are often incorporated in musical contexts, as in poetic contexts, to suggest joy or to project vitality and may be realized in one of two ways. They may appear in the guise of a rhythmic pattern, for example, an eighth note followed by two sixteenth notes. This could be described as a “rhythmic dactyl,” in that the pattern is realized by means of duration. Dactyls may also appear in the guise of a metric pattern, for example, a stressed unit followed by two unstressed units. This could be described as a “metric dactyl,” in that the pattern is realized by means of accentuation.
   Regardless of the form, the two unstressed or shorter units are instrumental in providing a forward thrust to the stressed or longer unit. Numerous instances may be found in musical literature in which the motion that dactyls create establishes sensations of joy or contributes to a vitality that no other means could create as effectively. Examples are evident in modal music (for example, the third rhythmic mode is dactylic) and non-tonal music, although with far less frequency than in tonal music, suggesting that somehow the dactyl is most effective in tonal contexts.
   Rhythmic dactyls may be found in countless examples in isolated instances, but examples in which they are the primary rhythmic units are far more telling. Johann Sebastian Bach seemingly delighted in the power of the rhythmic dactyl, and he was likewise fond of inverted dactyls (i.e., short-short-long, or an anapest). For example, the tenor aria “Erwäge,” no. 20 from the St. John Passion, is composed predominantly with dactylic rhythmic figures, both in the tenor voice and the accompanying violas d’amore and continuo. Focusing on the scourging of Jesus by the Roman soldiers, the aria urges the listener to “Consider how his blood-stained back in every aspect is like heaven, upon which, after the watery deluge was released on our flood of sins, was placed the most beautiful rainbow as God’s sign of grace.”
   Although dealing with an intensely horrific episode in the telling of the passion, the aria is a bittersweet meditation on Jesus’ self-sacrifice. Ironically, Bach encourages the listener to regard the whole event as something beautiful, creating a metaphor with a rainbow appearing in the sky as a symbol of divine grace. Albert Schweitzer regarded this aria as “indescribable felicity,” and undoubtedly much of this felicity is realized through the ubiquity of dactylic rhythms (in this instance, sixteenth note, two thirty-second notes) in an elegiac C-minor tonality:

   The first movement of Beethoven’s Sonata No. 14 for Piano, Op. 27, No. 2 (“Moonlight”) is quite certainly one of his most famous pieces, if not the most famous piece of “serious music,” rivaled perhaps only by the first movement of the composer’s Symphony No. 5. The fame of the “Moonlight” Sonata was established immediately and to such an extent that the composer expressed anxiety that the piece’s popularity would eclipse his other works. Nineteenth-century composers stood in awe of the Sonata’s opening movement. Hector Berlioz, for example, aptly wrote that the first movement “…is one of those poems that human language does not know how to qualify.” Carl Czerny purportedly likened the opening melody to the voice of a ghost sounding from the distance. And Frédéric Chopin must have had the Sonata’s tumultuous third movement in mind when composing the Allegro agitato of his well-known Fantaisie-Impromptu, Op. 66 of 1834. The fame and popularity of the piece have never waned.
   Today, more than 200 years after the piece’s composition, the massive popularity of the “Moonlight” Sonata has not eclipsed Beethoven’s other works, as the composer feared, but certainly has detracted from the astounding novelties of the piece. Composed in 1801 on the threshold of Beethoven’s highly individualist second stylistic period, the piece comprises a variety of startling innovations that are scarcely to be found before and arguably would never be found again. Regrettably, the warhorse that the piece has become detracts us from these innovations, yet it is worth re-assessing its originality and the way that it has appealed to successive generations of listeners and performers. What follows is a discussion of four of the piece’s most remarkable innovations.

1. “Sonata quasi una fantasia”
   Both of the Sonatas of Op. 27 are designated “Sonata quasi una fantasia,” a designation that Beethoven would use only in this instance. The term is generally translated as “sonata in the manner of a fantasy” but is perhaps more correctly translated as “sonata that is almost a fantasy.” The latter carries slightly different connotations, for it indicates that the elements of sonata form are the starting points and are modified to such an extent that they approach the free form of a fantasy. In Op. 27, No. 2, this can be understood in two ways. The first way applies to the schematic layout of the Sonata’s three movements. The piece opens with a slow movement (Adagio sostenuto), which is followed by a dance-like Allegretto and a concluding Presto agitato. The Sonata thus accelerates over the course of its three movements, with the dramatic weight thereby placed on the final movement. This is an unusual format for this time (fast-slow-fast is the norm), although a precedent may be found in Mozart’s Sonata for Piano, K. 282, whose three movements are similarly designed.
   The second way applies more specifically and more importantly to the unorthodox layout of the first movement, and it is here that few, if any, precedents may be found. What makes this movement “almost a fantasy” is the complete absence of thematic contrast: The musical idea and character introduced in the opening measure are maintained for the remaining 68 measures of the piece. The movement is thus monothematic, and as a result the traditional divisions of the classical sonata (exposition, development, recapitulation) are obscured. The piece is more in the style of a baroque improvisatory prelude with touches of classical components, and it is significant that at the movement’s conclusion Beethoven provided the indication “attacca subito il seguente,” meaning no pause between the first and second movements.
   Traces of sonata form may be found on the basis of tonal areas. For example, mm. 1–22 are an exposition, the second-theme group beginning at m. 10 with a brief modulation to the relative major (E major). Measure 23 marks the beginning of the development section, in which the opening theme is stated in the subdominant (F-sharp minor). Measures 28–41, the conclusion of the development, comprise a dramatically extensive dominant preparation for the recapitulation at m. 42. Twenty percent of this movement is devoted to this preparation, exemplifying, perhaps, Beethoven’s fondness for intensifying the drama of the recapitulation. The recapitulation at m. 42 is audibly the clearest division of the Sonata, thanks to this preparation and to the literal repetition of mm. 5–9 of the exposition. Measures 60–69, lastly, provide a coda to the movement, whose finality is expressed by three dominant-tonic progressions with the opening melody transferred to the left hand.
   Beethoven clearly adopted eighteenth-century formal concepts in this movement, as he did in all of his works, but varied them to such a remarkably individual extent that no one could possibly duplicate them.

2. C-sharp minor
   The key of the “Moonlight” Sonata, C-sharp minor, is an unusual key for the eighteenth century. Its key signature of four sharps demands an equal-tempered tuning system, for which reason few examples of pieces in this key may be found before 1770 (the pieces in the two books of Bach’s Well-tempered Keyboard being the obvious exceptions). Only two of Domenico Scarlatti’s 550 keyboard sonatas are in C-sharp minor (K. 246 and 247). Haydn wrote in C-sharp minor on one occasion in his Sonata for Piano No. 36 (according to the Hoboken numbering) composed at some point in the 1770s. Mozart resisted the challenge, and Beethoven used the key in only one other exceptional instance, the String Quartet Op. 131, perhaps the first non-keyboard piece composed in this key. As a result of a scarcity of examples, the key was unable to establish its own specific character at this time.
   Why did Beethoven choose this key for this Sonata? Why did he not choose a more orthographically friendly key, such as C minor or D minor, both of which are often encountered with modern, “easy” piano arrangements of this movement? I have to believe that his choice of key was made primarily from considerations of keyboard technique; specifically, ease of performance for the fingers in the articulation of the melody in the opening movement. What C-sharp minor does for the pianist’s hands that few other keys (such as C and D minor) do is force them into a naturally elevated position above the keys, due to the layout of the black keys on the keyboard. An elevated position is immensely beneficial for the right-hand fingers to articulate clearly the repeated G-sharps of the melody in m. 5:

The hand, now withdrawn from the black keys, is no longer naturally elevated, and the fourth and fifth fingers of the right hand may well experience some awkwardness (and tension) on the repeated Gs in the melody. Note also the drastic difference in sound of the left-hand chords in the third measure of these examples, separated merely by a semitone.
   Beethoven’s choice of key in this Sonata is a remarkably shrewd consideration for piano technique, perhaps no surprise given his virtuoso abilities as a pianist (and, of course, the key, once established for the opening movement, would have to be maintained for the entire Sonata). The key may have lacked a character prior to 1800, but surely this Sonata, intentional or not, did much to establish one that proved to be deeply appealing to nineteenth-century composers.

3. Triplets versus Dotted-eighth/sixteenth Notes
   Various performances and recordings of this piece show that there is no universal agreement on the execution of the melody that is introduced in m. 5. The misinterpretation stems from the correct interpretation of dotted-eighth/sixteenth notes in the context of triplet notes; specifically, where does the placement of the sixteenth note occur? According to the baroque tradition of note inequality, the sixteenth note would, in effect, be “absorbed” into the triplet pattern: “Duple notes set against triplets in baroque music are not meant for cross rhythms; they are meant to accommodate themselves to or to be accommodated by the rhythm of the triplet.” (Robert Donington, The Interpretation of Early Music, p. 464). Thus, the following example, the opening of the Courante from Bach’s Partita No. 1 in B-flat Major:

In the first movement of the “Moonlight” Sonata, however, the half note is the unit, and it is most likely the case that Beethoven intended for a cross rhythm (4:3) in this instance, a striking subtlety that produces a momentary point of tension. It likewise sets the melody in relief over the ubiquitous triplet-note accompaniment (provided that no interruption or hesitation is made in the execution of the triplets) in a way that could not be realized if the passage had been notated in 12/8.
   Carl Czerny, who had the great honor of studying with Beethoven, stipulated on p. 89 of his Piano-Schule, Op. 500 of 1839 that such a passage should be executed exactly as notated; that is, as a cross rhythm with the sixteenth note after the third triplet note. And in 1802, Daniel Türk observed that, “Two against three is a beauty to which we have to grow accustomed.” (Donington, p. 478).

4. The Damper Pedal
   The damper pedal on the piano followed a slow evolutionary progress in the eighteenth century. The pedal as we know it today came into being only in the late 1790s, decades after the invention of the piano and just prior to the composition of this Sonata. Because of the relatively late arrival of the pedal, its use in the eighteenth century is more an exception than the rule that it would become in the nineteenth century. Thus, the infrequent specifications for pedal before 1800 seem to be requests for a momentary and dramatic contrast in musical timbre and texture, rather than a means for enriching the piano sonority; meaning, pedal should otherwise be used sparingly. Such is the case in two instances in the first movement of Haydn’s Sonata in C Major No. 50 of 1794, one of the earliest instances of a specification of the damper pedal, in which the composer indicates the pedal to blur changes in harmony. Undoubtedly, eighteenth-century listeners would have been quite startled by this effect.
   Beethoven must have been familiar with Haydn’s sonata, for he incorporated the same technique, though somewhat differently, in the first movement of the “Moonlight” Sonata. Above the opening measure, he specified “Si deve suonare tutto questo pezzo delicatissimamente e senza sordini” (The entire piece should be played with the greatest delicacy and without mutes), followed by the indication between the staves “semper [sic] pp e senza sordini.” “Senza sordini” is an instruction to the performer to play the instrument “without mutes,” that is with the dampers raised off of the strings; that Beethoven stated it twice is testimony to its importance (Debussy made a similar specification in his Clair de lune, although this probably refers to the una corda pedal). Any pianist familiar with this piece would regard this statement as self-evident, since a generous amount of pedal would have to be used to realize the correct musical effect, but its indication was quite necessary at the time.
   However, musicians disagree on Beethoven’s exact intent. Charles Rosen, for example, interprets this instruction to mean that the pedal should be depressed one time and one time only for the duration of the entire movement:

In short, the pedaled sound is still a special effect for Beethoven as it was for Haydn, and he used it above all for contrast. The first movement of the “Moonlight” Sonata is perhaps the only exception in his work, a unique essay in tone color: here he wanted the entire piece to be played with pedal, to be played, in fact, delicately and pianissimo without ever changing the pedal, that is, without lowering the dampers to the strings. (Charles Rosen, Beethoven’s Piano Sonatas: A Short Companion, p. 108)

Performers today rarely follow this directive in this way; imagine the massive blur created by a damper pedal depressed constantly for approximately four minutes on a modern piano. The problem is that such an immense wash of sound will obscure many of the fine subtleties in the piece (such as the magnificent harmonic changes and the extended dominant preparation in mm. 28–41), even on an eighteenth-century instrument. Moreover, while the damper pedal is clearly requested, there is nothing in the composer’s wording that suggests that the pedal should be used without change. The qualifier “sempre” (or “semper”) in the second instruction certainly applies to pp, but not necessarily to “senza sordini.” If perhaps it does, then “sempre senza sordini” could represent an instruction to use pedal throughout the movement, rather than occasionally, though not necessarily without change. Also, does “tutto questo pezzo” mean that this instruction applies to all three movements or just the first (presumably the first)? Unfortunately, the first page of the composer’s manuscript is lost, thereby denying any opportunity to verify his exact wording.
   However, it should be agreed that this movement is “a unique essay in tone color,” one that was never duplicated, even when Beethoven employed the same effect, though on a smaller scale, in the third movement of the Sonata Op. 53 (“Waldstein”). One can envision Beethoven seated at his piano, experimenting with the newly designed damper pedal and utterly entranced by the wondrous sonic world that opened up to him.
   I can only imagine how marvelously original this piece must have sounded at its first performance. But rather than baffle or bewilder, as is often the case when listeners are presented with musical novelties, these four remarkable innovations, as well as a few others, only contributed to the piece’s astounding fame and popularity that continue unabated even in our drastically different modern times.

This is an aurally shocking and powerful moment, one that is bound to grab any listener’s attention. Surely, the singularity of this choral and orchestral unison in this massive work is no accident. I can scarcely think of a stronger affirmation that can be found anywhere in the musical literature, a brief and relatively straightforward passage made miraculous by its singularity.

2. Rondo-Finale (Movement 5) from Symphony No. 7 by Gustav Mahler
   Mahler’s Symphony No. 7 is perhaps the most complicated and the most original of the nine completed symphonies in terms of structure, tonality and orchestration. The concluding Rondo, set in a sunny, “Meistersinger” tonality of C major, may have been the composer’s attempt to contrast the dark characters of the preceding four movements with one of celebration and fanfare. Some critics see this movement as a disappointment in the context of the symphony as a whole; specifically, the symphony lacks a concluding apotheosis. However, I have always regarded it as one of Mahler’s greatest compositional triumphs.
   Something wondrous happens in the final two measures of this movement. Racing to the conclusion in a blaze of orchestral fanfare, the music abruptly stops on what Mahler must have intended as a joke, a startling Haydnesque surprise. We expect that this musical rush will conclude on a C-major cadence, but Mahler thwarts our expectations by giving us instead an augmented triad on C that brings the orchestral blaze to a sudden and perplexing halt (all parts in this example are notated at concert pitch):

This is a marvelously original effect. The augmented triad in m. 589, by nature of its instability, offers no cadential close, no sense of finality. We wonder and wait for a few seconds as this intrusive triad fades away; surely this is not the ending. Indeed, it is not, and Mahler provides the true cadence and the close to the symphony as a whole on a thunderous C-major triad in m. 590. Measure 589 is a joke. It is a miraculous instance of musical humor, and I cannot help but envision Mahler in a paroxysm of laughter as he completed this score.

3. Region IV from Hymnen, No. 22, by Karlheinz Stockhausen
   Scored for magnetic tape in its original version, Hymnen (“Anthems”) of 1967 is Stockhausen’s supreme compositional achievement, a climactic point in his output to which all prior works point. Regrettably, for reasons that are not altogether clear, much of what followed has represented a gradual decline in quality and originality, perhaps an inevitable result once such a pinnacle has been reached.
   Hymnen is a virtuosic masterpiece in the handling of its materials, namely national anthems and electronic sounds. Featuring, like the St. Matthew Passion, a variety of miraculous moments, the fourth and final movement (or “Region,” as the composer called it), in particular, offers an example of Stockhausen’s heightened compositional prowess at this time in his career. In the opening nine minutes of this Region, a recording of the Swiss national anthem, sung by a four-part chorus, is subjected to a variety of electronic manipulations until it is gradually transformed beyond recognition. Through the sheer patience of these manipulations, the listener experiences not only the compositional process, but also the piecemeal disintegration of the Swiss anthem into the closing part of this Region, the utopian “Hymunion in Harmondy” at 9’17.5”: (This example is taken from p. 45 of the study score prepared by the composer.)

The final chord of the Swiss anthem (notated on the lower two staves just prior to 9’17.5”) is modified, largely through aggressive filtering, reverberation and tape deceleration, into an undulating mass of low-bass sound that ushers in Stockhausen’s visionary paradigm of a utopian (and thus fictitious) realm. Accompanied for another thirteen minutes by electronically synthesized spirals of sounds (commonly referred to as “Shepard Tones”), the sound mass modulates into the sound of breathing at 20’45”, provided by the composer himself. This, and many others in the piece, is a stunning example of the innate relationships between sounds, a crucial pursuit for the composer throughout his career. It is also a miraculous and exhilarating instance of an artistic originality and a creative joy that is unparalleled in Stockhausen’s output, perhaps in anything composed after World War II.

4. “L’ultima prova dell’amor mio” from Don Giovanni, K. 527 by Wolfgang Amadeus Mozart
   The concluding scene of Mozart’s exquisite opera is a watershed moment. Here, we finally witness Don Giovanni’s true character, which in virtually all prior instances has been disguised by the personalities of those in his company, a shrewd means for the nobleman to manipulate others. In previous scenes, Don Giovanni assumes the nervous hesitancy of Zerlina as he seduces her, the steadfastness of Masetto as he unsuccessfully challenges him, and even the devoted subservience of Leporello as he faithfully serves him. In this scene Don Giovanni shows his true self.
   This scene is a moment of truth and is composed around Donna Elvira’s implorations to Don Giovanni to redeem himself for his transgressions. Expectedly, he refuses, in fact taunting her by repeatedly singing “Vivan le femmine, viva il buon vino! sostegno e gloria d’umanità” (Long live women and good wine, the substance and glory of humanity). Set in B-flat major, the first unequivocal instance of Don’s own personal tonality (though anticipated in the famous aria “Fin ch’han dal vino”), we realize that the sole ambition of the nobleman is a life devoted to hedonistic pursuits: