The following essays are now available:

 A Visit to a Translator's Workshop

 A Painting as Revisionist History

 Schenkerian Pre-History

 Instrumental Technique and Compositional Design

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 Unique challenges confront anyone translating poetry including unconventional word order, special meanings of particular words, and specific cultural references. These problems cause great difficulties with translations of poetry from Asian languages to Western ones. One instance found frequently in Chinese poetry will illustrate. Some poems mention a guqin (古琴) a musical instrument of the traditional literati. Although this instrument is a zither, translators nearly always render it as a lute. This distortion serves an important cultural reference. Zithers are marginal in Western music, and do not convey a sense of cultivation and a high degree of competence. The lute does invoke the sense of cultured and accomplished players, so from a cultural viewpoint this seems preferable to a more literal translation.

 Special challenges confront translating very short poems such as Japanese haiku, since a good translation must attempt to evoke elements which the poet has left unsaid. To illustrate some issues and pitfalls I will discuss a well-known haiku by Matsuo Basho, perhaps the most famous of the traditional haiku poets. I do this with considerable hesitancy and trepidation, since my reading takes issues with some very distinguished and admirable scholars.

umi kurete 海くれて
sea becoming dark  
kamo no koe 鴨の声
duck(s) ‘s voice(s)  
honoka ni shiroshi ほのかにしろし
faint to/in/by/with white(ness)  

The sea darkens:

 Voices of the wild ducks

  Are faint and white

The sea darkens:

 The voices of the wild ducks

  Are faintly white.

Seas slowly darken

 And the wild duck’s plaintive cry

  Grows faintly white

 These translations come from the early scholar R.H. Blyth1, the late Donald Keene2, and Sam Hamill3. They differ only slightly from each other and from several other published translations. Despite this consensus, these versions seem to miss certain essential points.

 The word ni occurs often in Japanese. It means ‘in’, ‘to’, ‘into’, ‘by’, or ‘from’, but never ‘and’. Thus one should not join ‘faint’ and ‘white’. All these translations link “white” to voices, but Japan has no tradition of describing sounds with colors. To understand the reference for ‘white’, we must answer two other questions.

 Nearly all haiku have seasonal references, some obvious and others simply long-established conventions. To aid both poets and scholars there are dictionaries which list hundreds of words or phrases with their accepted seasonal references. The book which I have4, lists kamo no koe as a winter reference, and cites this particular haiku as an example.

 The haiku form, with lines of five, seven, and five syllables is well-known, but this poem has five, five, and seven5, surely by intention. Although the line lengths do not balance, the structure does. The first line gives us an image of either late afternoon or gathering clouds, but no other context. The middle line conveys an important descriptive element which establishes the season. It provides a context to interpret the two separate adjectives in the final line. ‘Faint’ does refer to voices, but ‘whiteness’ suggests an expanse of snow, more common in the seventeenth century than now.

 Some other points merit discussion. Japanese does not have articles, so using them in English may create better flow. Japanese rarely distinguishes between singular and plural. In English one or the other may reflect subtle implications, but sometimes the choice becomes arbitrary. For honoka, distant and faint are equally valid. Some translations use ‘cries’ instead of ‘voices’. However the word naki(鳴き)denotes cries, while koe (声)represents voices. Here koe implies communication, supporting ‘ducks’ rather than ‘duck’. Kurete is a participle, which is why I prefer ‘darkening’ to ‘darken’. Both are better than ‘dark’6, which does not convey the passage of time, and also connotes a menacing character not found in the original.

 Haiku does not use punctuation, but to clarify meanings or emphases, it seems helpful to use it in translations, especially given the challenge of transmitting multiple meanings in a handful of words. Thus a comma after “The sea” implies that ‘darkening’ applies to a larger expanse; the darkening of the sea as a consequence of late afternoon or worsening weather7.

 The following two translations have different objectives. The first one tries to express meanings as concisely as possible, with fewer than seventeen syllables. The second version retains the traditional syllable count.

The sea, darkening; The sea, darkening;
  distant voices of ducks―   the distant voices of ducks--
    whiteness.     on the land, whiteness.

  1. Zen in English Literature and Oriental Classics, p.46, Hokuseido, Tokyo, 1941
  2. World Within Walls, p.86, Grove, New York, 1978; and Appreciations of Japanese Culture, p.102, Kodansha, New York and Tokyo, 1971
  3. The Pocket Haiku, p.27, Shambala, Boston and London, 2014
  4. Kigo hayabiki jiten, p.62, Gakken, Tokyo, 2000
  5. Keene comments on the irregular syllabic scheme, but does not see it in structural terms.
  6. Bownas and Thwaite, The Penguin Book of Japanese Verse, p.111, Penguin ,Middlesex, 1964
  7. A possible visual reference for Basho’s image might be the Hiroshige print Fukagawa Susaki Jumantsubo, in which a desolate snow scene with only hints of human habitation is shown from a bird’s-eye view, with a large eagle in the foreground. The location is not very far down river from the place where Basho’s cottage had existed. Hiroshige wrote poetry, and very possibly knew this particular haiku.

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 As one enters the Nagasaki Prefectural Museum’s section devoted to local artists, one of the first works we see is a painting from 1903 by Teitoku Sakaki (1865-1936)1 entitled ‘Concert of Japanese and Western Instruments’. Although some Western music became known in Japan in the late 19th century, the earliest documented instance of a ‘combined’ performance took place around 1930, when a visiting European violinist played the shakuhachi part of Haru no Umi on violin with its composer Michio Miyagi2 playing the koto part. If the rehearsal depicted in this painting led to an actual concert, then we would have to shift the ‘honor’ of being earliest by more than a quarter of a century. However there are reasons to consider this as a fantasy of wishful thinking rather than as historical depiction.

 In the right foreground we see a young man (possibly the artist’s son) playing shakuhachi, an instrument with a long history, starting with solo pieces for Buddhist meditation, and continuing with folk music and a role in jiuta, vocal chamber music composed for voice with sangen and/or koto, to which shakuhachi parts were often added later. Behind him is another young man, inspecting and cleaning another shakuhachi. In the left foreground a young woman plays a violin, but sits in seiza, the traditional kneeling posture used in performing traditional music. Behind her sits a slightly young woman trying to tune another violin.

 Several aspects of this image reveal its implausibility. The shakuhachi player reads from a vertical folded score placed on the floor, standard format for traditional music, published through the Kinko school of shakuhachi playing3. The violinist plays without a score. In the center of the picture we see another woman, without an instrument, perhaps a few years older than the others, listening very intently. A teacup is placed next to her. These circumstances suggest she is a singer, ascertaining how her part will blend with this unfamiliar instrument.

 This setting suggests typical preparation for jiuta performances. The singer and string player (who might be the same person) learned their parts by rote. If they used a score, it would only serve to reinforce memory. Shakuhachi players would join them shortly before a concert, usually playing a particular piece for the first time, so performing from score became necessary4.

 Why should we suspect that this painting is a fantasy, that such a concert never took place? We can rule out the possibility of music composed for the occasion. No Western composer at that time had the knowledge of either Japanese music or instruments. It seems highly unlikely that a Japanese composer of that time would have known enough about the violin to compose for it. Even if such a piece had existed, it would not have been published. Thus the only possible music would have been an arrangement of a traditional jiuta piece, assuming that some Japanese musician had the requisite knowledge to make such an arrangement. Yet there are reasons why this also seems highly improbable. It seems unlikely that a violinist would have known such an experiment well enough to play from memory. Moreover, even in Japan, no one teaches anyone to play violin in the kneeling posture. The young woman on the left, tuning a spare violin, also arouses suspicions. A violinist, other than a beginner, does not entrust tuning of an instrument to someone else, although koto and sangen players routinely do so.

 I do not wish to accuse Sakaki of deceit or fraud. We have no evidence that he intended this painting as an historical document. If we regard it as a fantasy, it engenders a considerable degree of sympathy. Sakaki grew up during the period when Japan underwent an extremely rapid process of Westernization, deemed necessary to avoid Western dominance under which China suffered at that time. Japan’s self-imposed isolation ended from 1853-1868. From the early seventeenth century onwards Nagasaki had enjoyed a unique status as the only port of entry for Western ships (limited to the Dutch)5 and had become a center for studying Western culture and science. With the opening of the country, the Tokyo-Yokohama and Osaka-Kobe areas became the centers of Western studies, and Nagasaki increasingly became a backwater, a circumstance which must have saddened many of its culturally-minded citizens. In such a context we can easily imagine Sakaki as fiercely loyal to his hometown, trying to imagine an event which might restore the town’s status as a leading force in the introduction of Western culture, even though the event was no more than a fantasy.

  1. Sakaki lived in Nagasaki his whole life except for two or three years each in the U.S. and Europe. Very few of his paintings are known to exist. The Nagasaki museum possesses a moonlit seascape, and a folding screen with several miniature paintings attached has been attributed to him, but the disparate styles of those paintings supports the idea that some of those paintings might have been done by other artists.
  2. Miyagi (1893-1956) was a celebrated composer and koto player, generally credited with being the first person to bring many aspects of Western music into the world of traditional Japanese music. He taught widely. Among his students was Shinichi Yuize (1923-2015) who was one of my teachers.
  3. At that time nearly all shakuhachi players belonged to the Kinko school, the Tozan school not having started.
  4. In Western music an analogous circumstance occurs with a soloist who plays from memory while an accompanist plays from score.
  5. The crews and their agents were almost entirely limited to Dejima, a small enclave, and contact with Japanese persons was strictly limited. Despite these obstacles a ‘school’ of translators existed, and Japanese from other parts of the country made a long trip to Nagasaki to learn whatever they could. The artist Shiba Kokan traveled from Tokyo (the Edo), and apparently gained admission to the enclave. His diary includes seven pages of accurate sketches of the interior of the captain’s house, craftsmen at work, and a detailed view of a Dutch ship.

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 Before demystifying this title let us revisit a distinction between invention and discovery. Recent advances in astronomic knowledge effectively illustrates: inconceivably distant galaxies, binary stars, supernovas, planets orbiting distant stars, and black holes. Since these objects have existed for billions of years, awareness and understanding of them represent discoveries. Astronomers obtained this knowledge using telescopes and photography, which did not exist prior to their initial construction, and thus comprise inventions.

 In the general sense and in specific compositions, music falls into the category of invention, unless if we believe in the music of the spheres. Yet fundamental analytic concepts such as structure and hierarchy belong to the realm of discovery, since these principles (or laws) may be said to have governed how music was constructed long before these principles had been articulated. Yet the language of analysis, both vocabulary and the symbolic means used to communicate and preserve our analytical insights, had no previous existence and therefore comprise invention.

 It is not a large leap of faith to believe that if those concepts and laws have indeed governed the creations of the great masters, then those composers must have had an awareness of such principles, perhaps below a conscious level, and that we can find some instances which might confirm that awareness. I have selected several examples to illustrate this principle.

Marais Suite in D minor Book 1 (1686) Gavotte and Sarabande. These dances appear to be rounded binary forms even though returns to the tonic within the second half do not include thematic returns. In both of these dances (and in many others, including the minuet and gigue from the same suite) the structure occurs within that return as we would expect, and after the second statement of the second half this return is stated for a third time (petite reprise), thus audibly emphasizing the structure. Non-bounded binary dances have structures which span the entire movement, thus Marais does not use petite reprise repeats in those movements.

Bach Partita No.2 for violin and Suite No.2 for cello Sarabandes. Bach composed thirty two sarabandes1 of which only the two just mentioned have a coda. It seems unlikely that the sarabande is unique in this respect; similar proportions probably exist for other common dance types. However given the rarity of a coda in such pieces naturally leads one to ask what aspects of these movements led to this choice. In both dances, the first part strongly establishes the third as the main melodic tone. In both dances the cadence at the end of the binary form proper has an unmistakable descending fifth. A non-Schenkerian does not see a problem here, but a Schenkerian immediately questions how even the clearest descending fifth can have a structural status without significant preparation. In both sarabandes the coda provides a descending third which we can properly designate as structural because of the earlier preparation.2

Mozart Rondo in A minor K.511. In traditional rondo movements the structure must occur within the main theme, and not in one of the episodes. However if one examines the first period of the Mozart rondo, one finds a perplexing melodic issue. Mm.1-4 do precisely what we expect, descending from 5 to 2 above a harmonic motion I -II6-V, but the consequent descent to the tonic in mm.7-8 lacks the fourth degree. Every subsequent statement of this theme until the very last statement has the same dilemma despite a wealth of melodic ornaments. Mozart sensed this lack of the fourth scale degree, so at the end of the movement he entirely recomposes the theme. Now each phrase resolves to the tonic with a clear melodic descent, in the soprano the first time and inner voices the second.

Beethoven String Quartet Op.130 2nd movement. This scherzo has a typical formal organization with a main section and a trio both in binary form. The first half of the A section of the main section comprises an antecedent-consequent phrase pair in which there is a clear descending fifth to the cadence on the tonic. The B section prolongs a lower neighbor to the tonic. The main section concludes with a single phrase in which the tonic returns, but the melody stays on the tonic. Having the only descent before the double bar seems unsatisfactory from a Schenkerian perspective. Therefore Beethoven adds a coda after the da capo restatement of the main section. In the coda the descent from the A statement returns in a piquant manner, and some tantalizing repetitions of the motion from third to second degree, seem to create false doubt that the resolution to the tonic will actually happen.

Schubert String Quintet 2nd movement, 1st section. This section is divided asymmetrically into units of fourteen, nine, and four measures. The first two units end with a two-measure I-IV-V-I progression, and the last unit simply states this progression twice, without any other material. This repetition provides both emphasis and some semblance of balance. The result is that Schubert has given us an actual reduction, progressively excising everything which does not contribute to the structure. One should note that all significant melodic activity occurs in the unobtrusive second violin part, not in the deceptively prominent first violin.

Brahms Sarabande and String Quintet Op.88 2nd movement. This example involves two closely related pieces: a very early quasi-Baroque dance and a late chamber work which vastly expands the material of that dance. The first part of the Sarabande largely conforms to Baroque formal expectations; it begins with a tonic prolongation (although perhaps a bit ambiguous of mode) and moves to a strong dominant at the double bar underneath a descent from 5 to 2. The second part also seems conformal; a developmental sequence leads to a brief modified restatement of the opening idea. Thus we might consider this to be a very short rounded binary. However the harmony of the second part does not support this reading; the return to the tonic (m.13) is accomplished without a dominant, and the final cadence reaches the tonic through a diminished seventh chord, thus giving us a symmetrical similarity to the end of the first part at the expense of omitting a dominant. Therefore the Schenkerian reading of the dance departs from the formal reading. By default, the descent at the end of the first part is structural, reaching the tonic in both melody and harmony at the beginning of the second part. In this reading the entire second part is a coda, creating proportions far more appropriate to Brahms’ own time than to the Baroque3.

 Brahms never published this piece, but twenty six years later he reused it for the main material in the quintet movement. He used only the first part of the Sarabande for almost the entire quintet movement. The material from the second half of the Sarabande appears only briefly in the quintet. It begins the coda of that movement, strongly implying that Brahms recognized the coda status of the second half of the original Sarabande4.

 Our last example brings us through time and space to ancient China, where composers worked within a system of pentatonic modes. Western treatises referring to pentatonic usage often quote only the most basic form, such as F-G-A-C-D. However in China, and also in works by composers such as Dvorak, Debussy, Mahler, and Bartok we often find this scale starting on D: D-F-G-A-C (These scales can be transposed up or down by any interval). Note that while the first scale contains the pitches necessary for a 3-2-1 descent, the second scale lacks a second degree. Ancient Chinese music in the first of these modes rarely uses notes outside of the mode, but pieces in the second mode often add E, specifically to make it possible to have stepwise descents to the tonic.

 I do not intend to leave us stranded in ancient China, tempting as that might seem. Instead we should examine a Western Medieval example, the Gloria tibi trinitas from the English Sarum chant repertoire. We must remember that traditional music (both ‘composed’ and ‘folk’) from several Western regions often uses pentatonic scales, including Hungary, indigenous areas of the Andes and of North America, and the Celtic region.

 This chant appears continuously in Taverner’s “Gloria Tibi Trinitas’ mass. It subsequently attained special significance as the cantus firmus of more than two hundred “In Nomine” composed for viol consort or keyboard over the following 150 years. The chant is in the second pentatonic mode with only two notes outside the mode. The sixth degree near the midpoint is merely decorative; I prefer to read it as a passing tone as part of a stepwise descent from 8 to 5 rather than as an upper neighbor5. However the second degree near the end is vital, since it makes a stepwise descent to the tonic possible. Thus composers in both ancient China and Medieval Europe found it necessary to introduce this ‘outside’ note to create a stepwise melodic descent of a third or fifth to reach the tonic.

 It seems highly improbable that the composers of all of these pieces took their unusual steps or procedures merely by chance. Unfortunately we have no way to know how they contemplated or verbalized these matters. When they taught composition or conversed about them with performers, some of these aspects must have been discussed in conjunction with planning works, although without our current analytical vocabulary developed in comparatively recent times.

  1. Keyboard 21, cello 6, violin 2, lute 2, and orchestra 1. It is interesting to note that in the violin, lute and orchestra works sarabandes appear only in the minor key works.
  2. The cello sarabande is particularly interesting in the way the third is treated. It appears in an ascent from the tonic in mm.1-2. The idea appears again in mm.5-6, but the initial tonic is now a 6/3 so the third is now introduced in a lower register and brought upwards through a voice exchange. The ascent idea appears in augmentation in mm.21-23 in a highly modified return, but the third is again in the lowest register, simultaneously with the beginning of the descending fifth. The descent from the third in the coda is somewhat submerged, but this allows it to occur in the same register as the beginning.
  3. The Brahms song ‘Maedchenlied’ Op.107 No.5 is an instance of a large two-part form in which the entire second part must be understood as a coda.
  4. The movement as a whole is in an A-B-A-B-A-Coda form. The B sections are also based on a very early pseudo-Baroque Gavotte. If one listens to the second B section, it sounds like a variation of the first B section, but in the actual compositional process the second section is an arrangement of the Gavotte and the first B section is the actual variation.
  5. A similar passing tone reading of a sixth degree applies to Mahler, Das Lied von der Erde, ‘Der Einsame in Herbst’, m.6.

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 Charles Ives once complained about “a fence of sounds, thoraxes, catguts, wires, wood and brass… “…The instrument! There is the perennial difficulty―there is music’s limitations”1. Perhaps a few other composers at different times have felt similarly constricted, but somehow composers have usually managed to surmount these obstacles most effectively.

 Should we regard these impediments solely as limitations obstructing free compositional expression? Instead I would contend that instruments possess finite natures which often channel composers’ ideas, but in doing so giving those ideas shape and a sense of direction, and sometimes even inspiring compositional choices which might not have arisen otherwise (see below, Beethoven Op.31 No.2). Before going further we should differentiate between limitations of instruments and limitations of players. As a simple illustration, there is nothing about the piano which precludes playing a double octave with one hand, but nobody has hands large enough to do that. On the other hand, no violinist, even with limitless virtuosity and perfect physical attributes will ever be able to play two notes as a double stop if both notes can only be played on the G string.

 Composers’ awareness of correlations between instrumental technique and compositional technique can be readily shown. The familiar “horn fifths” (otherwise ‘incorrect’ similar motion to a perfect fifth between two natural horns due to their being restricted to the notes of a single overtone series) appear not only in horn writing, but also in keyboard and string writing to simulate the sounds of horns.2 Notably this practice largely ended when valve horns supplanted natural horns, when this association would have seemed less current and less recognizable.3

 Awareness of these correlations has also been a part of scholarly approaches at least since the late nineteenth century. Consider the Bach keyboard concerto BWV.1052 in D minor, one of the earliest works of his to be ‘rediscovered.’ Based on the nature of the solo part4, it was quickly and universally regarded as a reworking of a lost violin concerto, an instance of musical “fingerprints” so revealing that this judgment has never been questioned.

 Correlations can be found as early as the beginning of the seventeenth century, when the extraordinary chromaticism and harmonic freedom found in some highly adventurous choral music could not be replicated in the keyboard music of the time, due to the keyboard tuning systems then in use. In the later Baroque period many string techniques influenced compositions, including patterns based on fifths and the prevalence of transpositions of melodic ideas up or down a perfect fifth. Underlying all of these were the continuous advances in the playing techniques of almost all instruments5, so that the resemblance between instrumental and vocal writing decreased markedly. However these additional resources brought a new problem: in rapid configurations individual notes can not communicate independent meanings since the ear can not absorb separate bits of information at high speeds. This required composers to find ways to organize successions of rapid notes into patterns which moved slowly enough for the ear to follow. These patterns included arpeggiation and scalar passages (sometimes in combination). The pace of these patterns (often the harmonic rhythm), and the diverse kinds of connections between them, allowed these florid instrumental pieces to relate to traditional patterns of vocal writing. Closely related to arpeggiation (and sometimes including it) is the technique of polyphonic melody, another way to create audible events which move noticeably slower than the surface flow of notes. Polyphonic melody can be found as early as 1612, in the Gibbons Galiardo [The Earl of Salisbury]6. The use of a polyphonic melody as a varied repeat of an idea originally stated in two voices suggests very strongly that Gibbons understood exactly what he had done. A much later example of polyphonic melody occurs in the Brahms Clarinet Quintet IV. The entire texture of the beginning of the theme is replicated by the cello alone in the first variation7, again suggesting awareness.

 A word of caution seems appropriate here: the necessity to refrain from drawing conclusions without considering the complete context. One such ‘devious’ instance is another work by Gibbons from the same Parthenia collection, the Fantazia in Foure Parts8. The cadence in mm.8-9 recurs a fifth higher in mm.12-13, with an added voice the second time. One might conclude that the added voice did not occur in mm.8-9 because it would have created an unduly thick texture, but the same cadence occurs still higher in mm.17-18 without an added voice which rules out the texture/register argument. The influence of transpositions will be found in many of the other examples below.

 For purposes of clarity and comparison, the examples presented here have been organized into several categories of examples. These comprise range, open strings and tunings, playability of chords, natural horns and trumpets, and special aspects of large ensembles. A final section includes a few other examples which do not fit conveniently into any of these categories.


 Historically range has been a frequent constraint, as demonstrated by the gradual expansion of range through improved instrument construction: enlarged keyboards, additional keywork for woodwinds and improved performance techniques (especially violins). One unintended consequence of these evolutions is that certain works originally composed as virtuoso pieces are now often given excellent performances by advanced students.

 The simplest solution to a range problem involving a single note is to transfer that note an octave inwards. Yet even such an small-scale procedure can raise interesting issues. The last four notes in the bass line of the Purcell Fantazia No.2: Ab1- D1- Db - F, would not seem out of place in a 20th century serial piece, but this was composed in 1680. D1 is the lowest note on the six-string viol, which accounts for the peculiar displacement.

 The bass line in the Bach Suite for Cello No.4, ‘Prelude’ in Eb, mm.70-78 is more complex. The V chord in m. 70 moves to a minor IV6 in m.74. The motion to a minor cadential 6/4 in m.78 is decorated by a VII/7 of V. The bass line pitches are Bb-cb-C-Bb. The C occurs in a lower register, as an open string to facilitate fingering. The other bass tones can not be played in the lower register9.

 Still another cello bass line issue occurs in the Brahms Sonata for Cello and Piano Op.99 II, m.24. Again descent below C1 is impossible, but here taking advantage of the ease of playing the octave double stop smoothes over the transfer of register.

 In the Bartok String Quartet No.4 III, mm. 55-63, there is an unusually free canonic treatment. Interval size, interval direction, and duration of notes are all altered in imitation, and imitation changes from inverted to normal in m.60. A sequence in mm.60-63 descends too low for the first violin to imitate the final descending fourth in the cello, so the first violin line ends with an ascending fourth. Descending fourths occur at phrase endings in mm.13, 21, 57 and 59, in each instance evoking the traditional recitative ending. This association is more important than the specific pitch produced, which is why the first violin does not ascend to F. Contrast this choice with a similar range issue in m.62 where Bartok answers a fourth with a major third. Since this point is not a phrase ending, the recitative association does not apply.

 In Bach Cantata 21, the tenor aria ‘Baeche von gesalznen Zaehren’ m.34, a cross-relation between the second violins and the bass line appears to result from a simple range limitation in the second violins. However in the next measure the same cross-relation occurs between the same two parts an octave higher. Did Bach merely develop a previous idea, or is the second cross-relation evidence that the first cross relationship might have occurred even if no range problem existed.

 Range problems involving more than one note will often have more audible results, but should not be any more difficult to understand. Compare Beethoven Sonata Op.31 No.2 I, mm. 55-62 and 185-192. In m.55 the first prolongation of the A minor I/6 is prolonged in the same register as the preceding cadence, and eventually ascends to the piano’s highest F in m.192. But at the corresponding point in m.185 the cadence on the I/6 of D minor is too low to allow clear-textured prolongation in that register, so the first prolongation is an octave higher, which creates a range problem in mm.191-192. Beethoven could have simply omitted the octave doubling, but he offers a far more attractive solution. The soprano pedal D creates an attractive succession of apparent sevenths. Note that as soon as the range problem disappears Beethoven resumes exact transposition of the exposition. Would this change have occurred if Beethoven decided to make use of the expanded range of the new pianos which began to be manufactured in Vienna a few years earlier.

 Problems involving range hardly seem fertile ground for humor, but Stravinsky succeeds in doing just that. In his Symphony of Psalms II, a fugue begins in a fairly straightforward manner with a modulating subject. The answer is real rather than the expected tonal, but the third entrance (mm.13-17) begins to introduce more serious problems. Assigned to the flute in the low register Stravinsky soon runs out of space so to speak, and by applying invertible counterpoint within the subject, momentarily gets out of trouble. Of course one should not do this in an exposition, so the fourth entrance provides a ‘cover up’. The same invertible counterpoint is applied to an oboe, which could have played the subject in its original shape, thus ‘proving’ that the changes in the third entrance were deliberate, not the result of drastic orchestral oversight. Many subsequent elements in this movement involve similar highly sophisticated and subtle touches of humor, even within the final pedal point.

 The influence of range is not limited to instances in which changes occur to avoid exceeding instrumental limits. The eighteenth century piano had a range of five octaves, with an F at each end, thus possessing only that pitch in six registers from F1 to f3. Beethoven in Sonata Op.31 No.3 I mm.44-45 prepares the second theme with a cascade of all six Fs, covering the entire range of the piano in an instant. In the recapitulation (mm.168-169) this is no longer possible, so we hear only five Bbs. Beethoven calls our attention to this difference with a very interesting detail. After the Fs in the exposition we have an eighth rest-- very short, but just enough to catch our breath after this maximum possible descent. The corresponding passage in the recapitulation does not have as dramatic an aspect, so Beethoven replaces the rest with a passing seventh in the bass, perhaps his way of telling us that Bb is not the bottom of the world.

 Mozart in the Rondo K.494 uses these six Fs over a more extended context. In mm.152-158 a stretto passage based on the main theme ascends from a very low register up to the highest F, using the four middle Fs in different ways. A few measures later the original theme returns, stated for the first time in the lowest register, using the lowest F four times.

 In Mozart Piano Concerto K.467 II mm. 77-78 are a transposition of mm.28-29 up a minor third. In the later passage the resolution from Db to C is transferred down an octave instead of using the higher decoration of the earlier passage to resolve in the same register. This might seem like one of the preceding range examples, but if Mozart had simply transposed the original passage it would have gone up to the high F, just within the range but not beyond. In this piece it seems that Mozart wished to avoid the tense and somewhat penetrating character of the highest notes on the fortepiano10.


 Imaginative resources of open strings seem inexhaustible, despite a long history. Even something as elementary as drones accompanying simple melodic figures can have an electrifying effect when played by most of a string orchestra11. The compositional implications can occur at any level from smallest details to determinative aspects of entire movements.

 In Bach, Sonata for Violin Unaccompanied No.3 iii, the sense of resolution of the 2-1 suspension in m.19 is stronger than the 2-3 suspension in m.21. The resolution F#-E moves to an open string, so it is not particularly difficult, but a 2-1 resolution E-D against D in the same register is very difficult, especially as regards intonation. Making it 9-8 with the D an octave lower is easy, but that would destroy the symmetry of texture.

 In Brahms, Quintet, Op.88 I, an ascending stepwise sequence in mm.150-155 uses open strings C -D -E as roots. Since these must ascend by ninths, the open strings migrate from bass to middle voice to top voice as the sequence unfolds. The changes in texture derive from the registers of the open strings.

 In Haydn, Quartet Op.76 No.4 I, m.58, the cello combines the bass tone of a cadential 6/4 with arpeggiation of that chord. At the corresponding point in the recapitulation m.160 the bass tone is no longer an open string. Consequently the cello plays only the pedal tone and the arpeggiation is assigned to the viola. One should note that in the keys of F, C, or G this change would not have been necessary.

 In Haydn Quartets Op.76 No.1 II, mm.87-88 and Op.77 No.2 I, mm.143 augmented sixth chords resolve with dynamic emphasis on the chord of resolution. However he reverses this seemingly natural dynamic usage in Op.74 No.3 II, mm.7-9. Here the augmented sixth chord contains the pitches of a G major chord, atypically facilitating the use of open strings. This piece may have been the first instance of Haydn placed a slow movement in a remote key, and the emphasis on the augmented sixth chord may have been intended as a reference to G major, the main key of the work. Whether this interpretation is correct or not, the emphasis on the augmented sixth is facilitated by the availability of open strings.

 Possibly as an influence from this quartet, a similar circumstance occurs in Beethoven Quartet Op.131 I, mm.112-116 where a cello open C string supports a diminished third chord which functions twice as an altered VII7 with a diminuendo to the resolution. The C is of course notated as B#, as happens frequently in the main motive of the last movement, even with octave doublings as in m.3012.

 In Beethoven Trio Op.9 No.3, I, m.103 a forte and a decrescendo help to elide a fortissimo resolution to an applied dominant. The two lower open strings of the downbeat chord must be allowed to echo long enough to clarify that function, but the player must reach the top G soon enough to make a convincing melodic connection from the previous F#, and do all this while going from f to pp in a single short measure. In m.120 the viola and cello open C strings under the continued outlining of the V7 in the violin serve to clarify the function of the reappearance of the initial motive as the beginning of the recapitulation although the resolution to I does not come until four measures later.

 From the Baroque period onwards many examples of repeated tones alternating between open and stopped strings (or in unison) can add color. When the tones in question have functions other than root or fifth, some interesting harmonic implications can result. In Ravel, String Quartet I, mm. 50-54 such unisons serve as a typical dominant preparation, but in mm.180-183 the sane figure untransposed now functions as a sixth degree within a dominant chord, moving directly to I without the customary 6-5 motion.

 In Brahms Quartet Op.51 No.1, III (trio) alternations on the same A occur as a third degree in mm.85-99, 119-126, and 135-144. As a consequence this imposes a unique palette of chords: I, III, and VI. Chromatic variants permit the VI to function also as V/VI and the III as VII/IV. Factors such as the bass line suggest that we should read the apparent III6 in m.142 as V, with the typical 6-5 resolution within the V bypassed, anticipating the similar procedure discussed in the Ravel.

 In Brahms, Sextet Op.36 I, the open string G in the G-F# figuration creates a sense of tonic even before a full I chord appears in m.3. This figure later serves to elide the end of the development and the beginning of the recapitulation. G-F# in mm.341-342 occur at the end of a long dominant prolongation. Since this represents a 4-3 motion over the V, the G is dissonant. For this reason, the open string indication does not appear over the G until the I chord appears in m.343. Note how effectively Brahms reverses this relationship in the development. The A-G# figure from m.251 has the A as an open string in the context of an A major chord functioning as a backwards-relating dominant. In m.253 the A major chord moves unexpectedly to C# minor. The A-G# figure continues, but since the A is no longer a chord tone the open string indication disappears. This section in C# minor (and the transitions to and from there) occupies much of the development, on a broad scale it functions as a lower neighbor to D, so the harmonic plan of the development constitutes a broad expansion of the movement’s opening figure.

 In Bach Partita for Unacompanied Violin No.3 “Praeludium,’ a passage incorporating an open string pedal plays a significant role in determining the harmonic plan of the entire movement. Mm.17-29 comprise a chain of 7-6 suspensions against a pedal open string E. Since arpeggiation of this figure covers the top three strings, the only possible transposition is down a fifth to the subdominant, and this occurs in mm.68-7913. Each statement of this figure occurs in the context of larger prolongations of I and IV which provide much of the I-IV-V-I framework of the entire movement. In principle Bach could have composed this piece in A major which would have placed the E major section afterwards as a large section in V, but this would have deprived the movement’s opening and closing flourishes of much of their brilliant character.

 The preceding examples within this category all pertain to uses of the violin, viola, and cello with standard tunings. The following group of examples concern issues relating to unusual instruments or unusual tunings of conventional instruments.

 The Bach Suite for Cello Unaccompanied No.6 in D was written for an unusual five-string cello, with the added string put a fifth higher than the A string. On a four-string instrument a unison or alternation between open and stopped strings can occur with any of the three upper strings; a five-string instrument offers four options. Bach exploits this resource extensively in the first movement. The main thematic idea begins with such an alternation mm.1-2. This figure recurs in V (A) in mm.12-13, in II (E minor) in mm.21-24, in IV (G) in mm.54-55, and as a means to return to the tonic with statements on V and I in mm.90-91 and mm.92-93. We should note that each statement of this figure lasts two measures except for the E minor, which lasts for four measures as part of a very long prolongation of II. This seems a subtle way to emphasize the presence of that very unusual fifth string.

 The viola d’amore is primarily a Baroque instrument, although there are a few later pieces which also use it. It has seven strings (plus seven more which run under the fingerboard and through the bridge and which vibrate sympathetically) and is tuned to arpeggios in A major, A minor, D major, or D minor (with variants). Nearly all music for that instrument is in one of those four keys, such as all but one of the Vivaldi concerti14. The concerted aria from Vivaldi Nisi Dominus is also in D minor. The Biber Partia (sic) No.7 for two viole d’amore and continuo is in C minor, but he specifies a C minor tuning, so the connection between tuning and key applies here as well.

 Irregular tunings of string instruments were more common in the Baroque period than later15. Many of these involved changing the tuning of one string by a step, but others required a radical retuning. Obviously these alterations create new resources, but also preclude others normally possible.

 One of the most famous instances is Bach Suite for Cello Unaccompanied No.5 in C minor. It begins with a tonic prolongation, and on the downbeat of m.2 there is a VII7 without a third over C in the bass. This piece is often performed with the conventional tuning, which precludes using the top string, so both third and fifth are missing16.

 Fourteen of the sixteen Biber Rosary Sonatas use all manner of bizarre tunings. No.11 is one of the most radical: g -g1-d1-d2. Thus the way that string players usually play fifths will now produce octaves, so stopping two strings with one finger and trilling with the next produces octave trills rather easily. A much less radical example is the Kodaly Sonata for Cello Unaccompanied in which the two lower strings are tuned down a half step. Not surprisingly the piece is in B minor, with many tonic chords, both in three- and four voices.

 A less common option involves retuning all strings by the same interval. This does not create any new technical resources (although it does result in different coloration), but simply transpose those which already exist. Instances include the solo viola part in the Mozart Sinfonia Concertante K.364 and the solo violin part in Mahler Symphony No.4 II. One should take note of the very effective use of the violin piccolo in the Brandenburg Concerto No.1 II and III. This instrument is somewhat smaller than the usual violin and is tuned a minor third higher.


 In keyboard music one often encounters stepwise sequences in which chords move regularly up or down without changes in construction other than those required to preserve tonality. In string music this is often not possible, for reasons of range and open strings and occasionally because changes required by tonality might create fingering problems.

 In Bach Partita for Violin Unaccompanied No.1, Sarabande, m.5 a stepwise descending succession of 6/3 chords in three voices becomes two voices when the register becomes too low to have three voices. The neighbor note figuration at that point compensates for the disappearance of the middle voice. An interesting reversal of this procedure occurs in m.27 where the first two beats have neighbor motion in two-voice sonorities, and the third beat has four voices without the neighbor figuration, as a consequence of the greater separation of the outer voices.

 Mozart Symphony No.36 I, mm.71-72 have a brilliant character, due to the triple stops for all the violins with the open E on top. In the recapitulation, mm.217-218 Mozart had to choose between transposing down a fifth with considerable loss of brilliance or giving up the triple stops by transposing a fourth upwards. The latter choice preserves more of the character of the exposition. The triple stops in mm.83-84 are retained in mm.229-230 by using close rather than open position.

 These passages might have influenced the youthful Mendelssohn when he composed the Midsummer Night’s Dream Overture. In mm.218-221 the violin fifths and the descents from them humorously simulate the braying of a donkey. In the return in mm.538-541 a fourth higher, the fifths are no longer practical, but this is partially compensated for by the open A strings. Transposing down a fifth would have preserved the fifths, but at an unacceptable cost of brilliance.

 In the Bach Partita for Violin Unaccompanied No.2 ‘Chaconne’ mm.89-91 the parallel tenths are played on the A and G strings with the open D in between. In m.90 the open D becomes a bass suspension. Resolving down to C would create parallel fifths; resolving down to C# would resolve into a diminished triad in root position, both theoretically unacceptable, not to mention that neither chord is playable. Bach deftly solves this dilemma by resolving the D upwards to C# on the same string, yielding a diminished 6/3 which is quite playable.

 The Bach Suite for Cello Unaccompanied No.2 in D minor, ‘Sarabande’ is more complex. In mm.1-2 an ascent from the tonic establishes F in the melody. This happens again in mm.5-6 with ascent starting from a I6 so we now also hear the F in a lower register, and it appears in both registers in the cadence at m.12. Mm.19-20 prepare a return to the tonic, but the arrival in m.21 is a deceptive cadence. From this point the ascent to F starts again but more slowly. The delay of the tonic and the slower pace of the melodic ascent create a sense of tension which finally reaches a climactic release with the tonic quadruple stop in m.23, the only quadruple stop in the entire movement. However the release is not complete. The completion of the ascent would require an F in the melody, but that note is too low for the top voice of a quadruple stop. Also, a root position chord with the more practical A on top would result in similar motion to an outer-voice fifth. Therefore Bach uses a I6 which is a practical quadruple stop that avoids voice-leading problems. The desired F appears, although not in the top register, and the tonic sense is strong enough to provide a resolution delayed from m.20, and finally, because of the inversion and the bass interval of a diminished fourth, leaves enough momentum to continue through the cadence in m.24.

 Bach St. Matthew Passion ‘Komm susser Kreuz’ is a concerted aria for tenor, viola da gamba (bass), and continuo. All sizes of the viola da gamba have six strings (basses occasionally seven) and are tuned in fourths with a major third between the two middle strings. Therefore when a melodic idea is transposed from tonic to dominant the transposition must go down a fourth rather than up a fifth. However the third in the tuning can change the playability of chords. This is illustrated by mm.1-2 and transpositions down a fourth in mm.20-21 and up a fourth in mm.40-41. On the downbeat of m.2 there is a complete II6/5, but in m.21 the chord has only two voices, and only three voices in m.41. It should be mentioned that chords of five or even six notes are often found in the repertoire (especially Marais), but never with more than four stopped notes.


 As mentioned before, the issues concerning uses of these instruments in the Classical period are well-known. Equally well-known but less often discussed is the practice of sometimes omitting the brass from slow movements not only in symphonies, but even in works with brass soloists such as the Purcell Sonata for Trumpet and Strings, the Vivaldi Concerto for Two Trumpets and Strings in C, and the Bach Brandenburg Concerto No.2. The examples discussed below will stress some less obvious considerations.

 Players of natural horns developed the use of stopped tones to fill in ‘gaps’ in the overtone series and to adjust intonation. The nasal quality of those tones is not in itself unpleasant, but they often sound out of place in melodic passages. However in some circumstances they can prove very effective. In Beethoven Symphony No.3, III m.252 the motion to a sustained IV6 just before a V which prepares the return of the main part of the movement requires stopped tones in the outer voices which seem very appropriate for the way that chord seems to delay the return to the tonic.

 Mendelssohn Symphony No.3 I m.6, has a double suspension played by winds, horns and violas. The resolutions in the horn parts are exchanged since not doing so would require stopped tones in both parts. One part seems to create questionable octaves, and the other seems to have an unresolved ninth. Since the winds and violas resolve correctly, Mendelssohn has deftly concealed the horn exchange.

 It is well-known that a particular chord can have different harmonic meanings in different contexts. In Mozart Symphony No.29 I, the A in m.3 is the dissonant tone of the II6/5, resolving to G# in m.4. In the second statement of the theme the A in m.15 is now consonant, part of an apparent seventh. This happens since the A is played by oboes and horns, but since the horns can not play the G# which would be necessary to repeat the II6/5 from the first statement, the horns have changed the harmonic meaning of the chord.


 Large ensembles certainly offer the composer some very obvious benefits. Counterpoint in many voices is readily obtainable, variety of texture is easily attained, and if a range or playability problem arises, there is usually some other instrument which can take over and sidestep the difficulties. Perhaps it is less clear that large ensembles can provide smoother realizations of somewhat laborious patterns, and even make certain improper procedures work properly.

 In Mendelssohn Octet II, mm.41-44, a series of double suspensions (9-8 and 7-6) descend against an ascending 5-6 sequence. Normally this would require at least some of the resolutions to leap upwards above a stationary bass to create new preparations (9-8-10 or 7-6-8). However with the resources of the octet the entrances of each of the soprano tones which prepare the 9-8 suspensions is stated by the four violins in succession. This emphasizes the basic parallel tenth relationship between outer voices and removes the necessity for the melodic leaps. The similarity of color among the violins enhances the impression of a single ascending scale.

 In Mendelssohn Hebrides Overture mm.1-5 a somewhat related procedure occurs. The bass line ascends B-D-F#, and the melodic line ascends F#-A-C#. Normally we would consider this an egregious case of parallel fifths. However the upper voice tones are played in succession by violins, clarinets, and oboes, with the violins and clarinets sustaining their tones against subsequent entrances. Unlike the previous case, the instruments here are noticeably different in color which enhances the sense of separate voices. A very early sketch17, despite being written on only two staves nevertheless shows clearly the eventual orchestral format as a means to evade the voice-leading problem.


 Several other examples do not readily fit into any of the above categories. Precisely for that reason they are of interest and deserve inclusion. Two of them involve kinds of limitations on players rather than on instruments, but different from the ones which were mentioned in the introduction.

 In Beethoven Sonata for Piano and Violin Op.23 I, mm.144-147 the violin holds a low A and simultaneously plays an ascending scale from E to A. The fifth (E-A) is played with the first finger, and the F, G, and A with the second, third, and fourth fingers. It all works smoothly so he immediately repeats the figure. This idea appears again starting from m.228, now in invertible counterpoint. This means starting on a fourth with E over B with a scale ascending from B. However playing the fourth requires two fingers so the ascent to C and D requires the third and fourth fingers. Now there are no fingers left to reach the unison E which would complete the ascent, so he returns to C and does not repeat the figure this time. It is important to understand that the unison E in other contexts is definitely playable; the problem here is simply that the violinist has run out of fingers.

 In Brahms Quartet Op.51 No.2, I, a double canon using elements from the two main themes commences in m.177 and leads imperceptibly into the recapitulation starting in m.183. The cello has single pizzicato notes in mm.179 and 181; these notes are not part of the canon but are interpolated for harmonic reasons. The use of pizzicato as a contrasting color is a way to isolate these tones from the canon. A more sophisticated application of this concept occurs later in the movement. We must look first at mm.18-19 in which an approach to the tonic is slowed down, but after the resolution in m.20 the motion to the VII7/V is suddenly very quick with the I chord lasting only for half a beat. In the recapitulation mm.200-202 the passage is repeated without change, but in the movement’s coda the slowed down version of the approach to the tonic happens again in mm.287-288, but this time all instruments play pizzicato. In m.289 instead of a I chord followed immediately by the diminished seventh, the two chords are now simultaneous. The first violin and cello play a tonic octave18 pizzicato to complete the cadence, while the second violin and viola simultaneously play the diminished seventh chord arco. The use of arco and pizzicato clarify the overlap of events which previously had been played in succession. The same elision occurs in m.297 with all instruments arco, presumably assuming that we will now understand the elision even without the special coloristic treatment.

 In Mozart Piano Concerto K.466, I, both the orchestra and solo tuttis begin with the same orchestral writing for the first thirteen measures, characterized by the repeated note syncopations in the strings which often occur in Classical period string and wind writing. When the piano enters in the fifth measure of the solo exposition, its continuous sixteenth note texture covers all of the main notes of both melody and harmony found in the orchestra material, but the piano doesn’t play the syncopations. The rest of the movement continues this circumstance: the soloist never has the main theme in its basic syncopated format, a rare if not unique treatment of the concerto. Mozart recognized that the syncopations so essential to the theme would not sound very effective on an eighteenth century pianoforte19.

 It doesn’t seem unfair to say that for eighteenth-century composers the timpani were essentially noisemakers, often exciting but never truly organic. Beethoven gave considerable attention to timpani. While the solo octave in the 9th Symphony is unquestionably the most well-known example, the slow movement of the 4th Symphony presents a more interesting usage. At the very beginning of the second movement the simple accompaniment in fourths given to the second violins seems innocent enough. Who would expect that at after hearing this figure in the bassoon and then two measures later very quietly in the lower strings, that the beginning of the recapitulation would be heralded by this figure played by the timpani, or that in m.102, just before the end the timpani would play this figure alone, as a pianissimo solo?

 Using timpani in a motivic way became much easier for composers when the standard complement was enlarged to include four easily returnable drums. Hindemith does this in Symphonic Metamorphoses II in various dynamics and contexts. At [V] in a passage solely for percussion, the timpani references to theme provide the only melodic material. This passage returns at the end of the movement, and in a delightful role reversal the timpani provide the melodic material while winds, brass, and strings provide the repeated note punctuations which timpani usually provide.

 We return to the Stravinsky Symphony of Psalms II. Although this is a choral-orchestral work, he has constructed a fugue subject conspicuously unsuitable for voice (another instance of his subtle self-directed humor). As happens so often in that composition he finds an unexpected way out of the dilemma he created for himself. The movement becomes a double fugue; he reserves the new subject entirely for the chorus.


 Many more examples of correlations exist, including works by composers not represented here. However none of them seem to offer any insights which would differ significantly from the compositions already discussed. I would add that instances of these correlations exist in at least some non-Western music, certainly in examples from Japan20 and China21, although any comprehensive discussion of these must await another occasion.

  1. See Essays Before a Sonata (New York: Dover, 1962), pp.168.
  2. Scarlatti Sonata K.159/L.104, m.1; Mozart Sonata K.332 I, mm.13-14, Mozart Quintet K.593 I, m.23; Schubert String Quintet III, m.1. The Brahms Quartet Op.67 I, mm.5-6 might be considered an exception, but it seems consistent with his notorious preference for natural horns long after valve instruments had become standard.
  3. An interesting exception is provided by the poignant opening of the Vaughn Williams Symphony No.5. Composed in 1943 during World War II, these horn fifths combine a nostalgic yearning for a more peaceful era with a dissonant bass tone reminding us of the agonizing present.
  4. First movement: mm.62-75 and 124-137, and third movement: mm.86-89 and 96-99.
  5. There are occasional reverse examples, such as the disappearance of virtuoso high-register trumpet playing in the middle of the eighteenth century.
  6. Cf. mm. 15-16 and mm.24-25. Originally published in Parthenia in 1612, modern edition (London: Stainer and Bell, 1960) pp.34-35. Hardy souls willing to brave six or seven line staves in diverse clefs might consult the facsimile edition of the original publication (New York: Performers’ Facsimiles, [Broude],1985).
  7. Cf. mm.1-4 and mm.33-36.
  8. Op. cit., p.30.
  9. For an instance of upwards transfer of register of a single note in a bass line, see Schumann, String Quartet Op.41 No.3, II, m. 197 or 221.
  10. This is consistent with the mood of both this passage and the movement overall. However in other circumstances the quality of that high register can be exploited most effectively, as in several passages in the Beethoven Piano Trio Op.1 No.3 III.
  11. Bartok Music for Strings, Percussion, and Celesta I, mm.52-54.
  12. However in Op.132 V mm.395-396 Beethoven writes C-C# in the cello and viola in the context of Amajor. One possible explanation for this is that the C was intended as a last reference to A minor, the overall key of the work.
  13. Issues of how changes made in transcriptions lie outside the scope of this study, but this particular instance merits attention since the transcription is Bach’s own. The first movement of Bach Cantata No.29 is a transcription of the violin Praeludium for organ and orchestra. In the violin original, the suspension chains involve a voice crossing between the highest voice and the open string pedal, which create no special difficulties for the violinist. However this is extremely awkward for a keyboard instrument, so Bach drops the pedal tone an octave, putting it out of the way of the suspensions.
  14. One concerto is in F, but it has no chordal writing, so the lack of accommodation to the tuning does not create any difficulties.
  15. Having gut strings at much lower levels of tension facilitated this. Also, at that time string players were accustomed to reading a variety of clefs. In the five-part Sinfonia to Bach Cantata No.4 the second viola part is written in tenor clef, and in his Suite for Orchestra No.1, ‘Passepied II’ the melody from the first Passepied becomes an inner voice for all violins and violas, with all parts written in alto clef.
  16. In Bach’s arrangement of this suite for lute, the larger number of strings allowed him to have this chord in five voices comprising a complete diminished seventh chord above the tonic bass.
  17. This sketch was contained in a letter written to his father from Scotland. It is reproduced in the Baerenreiter edition of the score (p.XXV) and in Mendelssohn, A Life in Letters (New York: International, 1986), p.87. This score edition also includes a reproduction of a pencil sketch which Mendelssohn made of the location.
  18. In m.20 and m.202 the tonic chord is stated by an octave without inner voices, so their absence in m.289 in no way weakens the representation of the tonic chord. One has to wonder whether Brahms already foresaw the elision when he was first composing the exposition and therefore chose to use only an octave for the tonic in the first two passages.
  19. Although Mozart composed cadenzas for many of his piano concerti, he did not compose one for K.466. It would be most interesting to know how he would have reshaped the main theme in a cadenza.
  20. I have addressed several instances in Analytic Aspects of Japanese Koto Music in “The Music Forum” vol. IV (New York: Columbia Univ. Press, 1976) pp.335-393.
  21. One interesting case in Chinese music is Section 11 from “Mist and Clouds over the Xiao and Xiang”, an ancient piece for guqin. Because the tuning of the seven strings includes all the notes of the pentatonic mode it is possible to play a high register melody on the two top strings more than two octaves above a melody played entirely on the other five open strings in the low register. This brief passage is a rare example of true counterpoint in traditional Chinese music.

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