Circle Of Fifths Research Articles

The influence of height and key on the perceptual similarity of transposed melodies.

In two experiments, the perceptual similarity between a strong tonal melody and various transpositions was investigated using a paradigm in which listeners compared the perceptual similarity of a melody and its transposition with that of the same melody and another transposition. The paradigm has the advantage that it provides a direct judgment regarding the similarity of transposed melodies. The experimental results indicate that the perceptual similarity of a strong tonal melody and its transposition is mainly determined by two factors: (1) the distance on the height dimension between the original melody and its transposition (pitch distance), and (2) the distance between keys as inferred from the circle of fifths (key distance). The major part of the variance is explained by the factor pitch distance, whereas key distance explains only a small part.

The Beethoven quartet companion

The Quartet Companion Edited by Robert Winter and Robert Martin. Berkeley: University of California Press, 1994. 300 pp. $30 THE BEETHOVEN QUARTET COMPANION ADMIRABLY FULFILLS THE TWOFOLD PURPOSE LAID OUT IN THE BOOK'S INTRODUCTION. The first of these is enhancement of the listening experience; the second is enriching our understanding of the That term, cultural context is clearly meant the broadest possible context. It includes not only economic, societal, and historical factors, but also such technical/musicological information as the nature of earlier string instruments and the interplay among a quartet's four performers. Although the entire volume will enhance the listening experience, the book's second part focuses directly on the works themselves. Part II is entirely the contribution of that doyen of program annotators, Michael Steinberg. Readers fortunate enough to have read his notes the program booklets of the San Francisco Symphony, for example, know that he is a most learned, yet most readable (not to say, amusing) guide. In chapters, Steinberg brilliantly analyzes the early, middle, and late quartets. Steinberg also contributes a detailed glossary which by itself is all but worth the price of admission. The entry under Chord, for example, is page long and includes among other subjects inversions and figured bass. An illustration from the Fortepiano Sonata, Opus 27, No. 1, first movement, illustrates the former. Likewise the Circle of fifths is illustrated as well as explained. The same is true for the tunings of the quartet instruments. Sonata form is covered two pages of fascinating analysis. The glossary, sum, is a most useful guide, not only to the terminology utilized by Steinberg's fellow essayists, but also for the uninitiated to an appreciation of music. Perspectives is the title given to the first part of the book which contains five superb essays. The first of these, Joseph Kerman's Beethoven quartet audiences: actual, potential, ideal, is a riveting, enlightening history of the quartet medium during the period spanned by Beethoven's life. The principal audience for the String Quartets, Opus 18, according to Kerman, was the players themselves. They were in the best sense, amateurs, who played same passion which we until recently played bridge. The appearance of the Razumovsky Quartets, Opus 59, marked the emergence of the professionalization of the string Virtuosi string players led by such performers as Beethoven's friend Ignaz Schuppanzigh played not for themselves but for the new and growing audience for chamber music. And finally, the last quartets represent the composer fulfilling his own needs, meeting new musical challenges, expressing his interior vision. At the risk of oversimplification, Kerman writes, one can say that whereas Beethoven's first period the essential audience for his quartets had been the quartet players, and his second period the concert public, the late period the audience was primarily the composer. at the end of his life achieved the privatization of the string quartet. If, as Kerman puts it, we can see three phases the history of the quartet during Beethoven's lifetime the amateur, the public, and the private, the second essay, Performing the quartets their first century by Robert Winter, logically carries the argument on through the nineteenth century. It is a most absorbing story. Winter begins by detailing the contributions of Ignaz Schuppanzigh, who is credited with pioneering of the permanent string quartet and the presentation of regular series of chamber concerts, thereby changing landscape of quartet playing forever. An astounding fact is the youth of the period's leading players. Schuppanzigh was seventeen years old when he presided at Prince Lichnowsky's over colleagues ranging downward to fourteen years of age! …

Chromatic Notation of Music: Transforming Bach and Webern into Color and Light

The author describes the development of her color tone system, which began with her coloring of the harmonic structures in the Prelude in C Major from The Well Tempered Clavier , Part I, by Johann S. Bach. She describes her Extended Circle of Fifths, which condenses the harmonic system into one gestalt. Besides converting Bach’s Prelude to color, she has converted three other preludes by Chopin, Busoni and Scriabin and an early 12-tone study by Webern. Burgmer describes and evaluates her experiment in composing music with colors: She used her color:tone system to create a new piece of music as an addition to Webern’s “Piano Piece.” She also discusses her transformation of music graphics into holograms. She contends that these music holograms have their own aesthetic value and exist independently as artworks, parallel to the musical pieces on which they are based.

Britten and the Circle of Fifths

Britten‘s music is usually thought of as tonal, even if the term is sometimes qualified.’ Much of the drama of his operas (at least, up to and including A Midsummer Night's Dream, op. 64 (1960)) has been discussed in terms of ‘tonal action’, as have the structures of individual songs, song cycles and purely instrumental works. Yet his earliest published works are not tonal in the same way as later, more widely known ones. Not until the works of his American years (1939–42) do we begin to see the kind of tonal structure that enabled Peter Grimes, op. 33 (1945), to be conceived – structures like the highly focused tritonal opposition of Les illuminations, op. 18 (1939), or the E/C ambiguity (eventually resolved in favour of E) of the Hymn to St Cecilia, op. 27 (1942). Nevertheless Britten's first three works with opus numbers, written while he was a student at the Royal College of Music from 1930 to 1933 (the Sinfonietta, op. 1, completed in July 1932; the Phantasy Quartet, op. 2, completed in October 1932; and A Boy was Born, op. 3, completed in May 1933), are quite clearly diatonic in basis. This essay explores the ways in which this diatonicism is structured, and sketches the subsequent history of the main procedure involved.

A Theoretical Work of Late Seventeenth-Century Muscovy: Nikolai Diletskii's Grammatika and the Earliest Circle of Fifths

The earliest circles of fifths appear in a Russian-language treatise on composition, Nikolai Diletskii's Grammatika (Grammar), written in the late 1670s. This article discusses the origins of Diletskii's treatise and its long-standing influence in Russia, where it circulated for one hundred years, the theoretical content of the work, and the complex relationships among the three surviving variants (from Smolensk 1677, from Moscow 1679, and from Moscow in versions from 1679 and 1681). Diletskii's circles of fifths, although certainly unknown to contemporary theorists, fit into a Western theoretical tradition centering on practical instruction using keyboard instruments.

A Theoretical Work of Late Seventeenth-Century Muscovy: Nikolai Diletskii's "Grammatika" and the Earliest Circle of Fifths

A Theoretical Work of Late Seventeenth-Century Muscovy: Nikolai Diletskii's "Grammatika" and the Earliest Circle of Fifths

Modeling the Perception of Tonal Structure with Neural Nets

ly summarized by the circle of fifths. Analogously, a net exposed to Indian ragas fills in expected tones when presented with subsets of the raga tones. An auto-associative net trained on the scales of one culture can be tested with the scales of another, making predictions about tonal implications 46 Computer Music Journal This content downloaded from 157.55.39.231 on Thu, 06 Oct 2016 04:43:25 UTC All use subject to http://about.jstor.org/terms generated in the minds of listeners hearing an unfamiliar form of music. A net trained on the Western major and minor scales seems to assimilate some Indian ragas to the Western scales, sometimes shifting the tonic (Bharucha and Olney 1989). Learning Hierarchical Representations Through Self-Organization Hierarchical relationships, such as between tones, chords, and keys, can be learned passively by algorithms for self-organization (Kohonen 1984; Linsker 1986; Rumelhart and McClelland 1986; Carpenter and Grossberg 1987). Most self-organization mechanisms assume the prior existence of abstract units into which the input units feed. These abstract units initially have no specialization, since the links from the input units are initially random. However, repeated exposure to commonly occurring patterns causes some of these abstract units to tune their responses to these patterns. One of the more straightforward, self-organization algorithms, called competitive learning (Rumelhart and McClelland 1986) accomplishes this as follows. For any given pattern some arbitrary abstract unit will respond more strongly than any other, simply because the weights are initially random. Of the links that feed into this unit, those that contributed to its activation are strengthened and the others are weakened. This unit's response will subsequently be even stronger in the presence of this pattern and weaker in the presence of other, dissimilar patterns. In similar fashion, other abstract units learn to specialize to other patterns. This process can be continued to even more abstract layers, at which units become tuned to patterns that commonly occur in the lower layer. The overwhelming preponderance of major and minor chords in the popular Western musical environment would drive such a net to form units that respond accordingly. Furthermore, the typical combinations in which these chords are used would drive units at a more abstract layer to register larger organizational units such as keys. The notion that individual neurons specialize to respond to complex auditory patterns has some preliminary empirical support from single-cell recording studies on animals (Weinberger and McKenna 1988). Once these chord and key units have organized themselves, the net models the implication of tones, chords, and keys given a set of tones. A hierarchical constraint satisfaction net built on this organization has been reported in earlier work (Bharucha 1987a; 1987b). In this net, called MUSACT, activation spreads from tone units to chord and key units and reverberates phasically through the net until a state of equilibrium is achieved. At equilibrium, all constraints inherent in the net have been satisfied. Given a key-instantiating context, the unit representing the tonic becomes the most highly activated. The other chord units are activated to lesser degrees the further they are from the tonic along the circle of fifths. Two behaviors of the net illustrate its emergent properties. First, the above activation pattern does not require the tonic chord to be played at all. An F major chord followed by a G major chord will cause the C major chord unit to be the most highly activated. Second, the circle of fifths implicit in the activation pattern cannot be accounted for on the basis of shared tones alone. If a C major context chord is played, the D major chord unit is more highly activated than the A major chord unit, even though the latter shares one tone with the sounded chord (C major) and the former shares none at all. A careful tracking of the net's behavior as activation reverberates and before it converges to an equilibrium state reveals a lower initial activiation of D major over A major, reflecting an initial bottom-up influence of shared tones. As activation has a chance to reverberate back from the key units (a top-down influence), this advantage is lost, and D major overtakes A major. So the circle of fifths is truly an emergent property of the simultaneous satisfaction of elementary associations between tones and clusters of tones. See Bharucha (1987a; 1987b) for details. Learning With Sequential Nets Some of the schematic expectancies that are essentially sequential, as in chord progressions, can be modeled with sequential nets. The architecture

Musical Scales and the Generalized Circle of Fifths

(1986). Musical Scales and the Generalized Circle of Fifths. The American Mathematical Monthly: Vol. 93, No. 9, pp. 695-701.

Musical Scales and the Generalized Circle of Fifths

Musical Scales and the Generalized Circle of Fifths

Pitch Relations and the Formation of Scalar Structure

The experimental study of the perception of pitch structure in music requires the use of a framework that is not necessarily tied to a particular musical culture or period. This article describes three sets of experiments in which rules governing the production of musical sequences are formally defined in terms of a closed system, the "circle of fifths." Experimental results indicate that important aspects of musical judgment are well accounted for by logical consequences of such formal definition, without the necessity of invoking either simplicity of frequency ratios or musical "well-formedness."

Relations between multiple divisions of the octave and the traditional tonal system

Abstract Multiple divisions of the octave are described as extensions of the traditional tonal system, by building them from chromatic and diatonic semitones with certain numbers of steps assigned to them. Except for some multiple division systems with relatively few tones per octave, this procedure always works. The number of circles of fifths in the system is equal to the largest common divisor of the numbers of steps contained in the two types of semitones. The description provides a consistent notation and a method for evaluating the qualities of the fifths and the major thirds in the systems. A list of systems with 5 up to 129 tones per octave is given, with summary references to instruments, compositions, and literature.

The detection of notes incompatible with scalar structure

A fundamental question in the perception of melody is how listeners use pitch information to arrive at a sense of musical scale. We investigated possible ways in which this might be achieved. Listeners were required to detect notes incompatible with scalar structure in the course of melodic sequences. Within the parameters of our experimental sequences, the results showed that detection of nonscalar notes could be based on identifying nonscalar interval combinations (e.g., F♯, A, Bb), as well as by matching successive notes to a developing scalar schema. The strength of the scalar schema was influenced by relations between notes in the circle of fifths, with notes closer together in the circle of fifths (e.g., C, G, D, A) leading to a stronger schema than notes further apart (e.g., C, G, E, B).

Perceived Structure of Triads

This study explored the perception of major, minor, diminished, and augmented triads in two experiments. In the first, subjects judged whether pairs of these triads were of the same or different chord type. In the second, subject rated the similarity of these pairs of triads. Subjects were grouped by degree of musical training. We looked for effects of several variables on the subjects' judgments: the "musical key" suggested by major and minor triads, whether a triad was presented in root position or in first inversion, and whether the notes of the triad were presented successively (i. e., arpeggiated) or simultaneously. Among the three groups of subjects, only the judgments of those who were highly trained appeared to be influenced by the above factors. For these subjects, judgments were influenced by relative position in the circle of fifths. In addition, judgments appeared to incorporate the perceived degree of consonance or dissonance, rather than relative interval sizes.

Musical key structure and its influence on perceived harmonic relations

A structural description of musical key distance was derived from the tone‐profile technique in which listeners rate how well each chromatic‐scale tone follows a key‐defining element (scale, tonic triad chord, or chord cadence). The rating profiles, reflecting the music‐theoretic hierarchy of tonal stability, were shifted to all possible tonics, intercorrelated, and analyzed using multidimensional scaling. The four‐dimensional solution placed the 24 major and minor keys on a torus simultaneously accommodating the circle of fifths and relative and parallel relations between major and minor keys. Two subsequent experiments investigated the effect of interkey distance on perceived harmonic relations between chords. The magnitude of three independent context effects (measured in both direct relatedness judgments and memory confusions) was found to depend systematically on the distance between the keys in which the chords function and the key of the broader context in which they are embedded. The results provide strong evidence that listeners interpret chords in their tonal functions and have internalized structure at the level of abstract tonal centers. [Work supported by NSF.]

Recognition of musical key: Exploratory study

The initial 5 s of each of the 12 major-key preludes from J. S. Bach’s Well-Tempered Clavichord was recorded by an expert pianist, either as written or shifted by plus/minus 1, 4, 6, or 7 semitones, and a randomized tape consisting of two repetitions of each of the 108 items was prepared. Twenty musicians, mostly pianists, and two nonmusicians judged whether each item was played in the correct key, transposed up, or transposed down; a simplified version of original (correct) score was printed on the answer sheet. Only four subjects failed to discriminate, at the 5% confidence level, the correct rendition from the transposed ones, even when only the ±1-semitone transpositions are considered. Errors are dependent on the ’’linear distance’’ of transposition (i.e., pitch height offset) rather than ’’harmonic distance’’ (i.e., offset on the circle of fifths). The subjects claiming to possess absolute pitch performed slightly better than the best of the nonpossessors, but the results suggest that absolute identification of tonality is an ability that is more widespread than commonly supposed.

Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys.

The cognitive representation of harmonic and tonal structure in Western music is investigated using a tone-profile technique. In this method listeners rate how well single tones (any one of the 12 tones of the chromatic scale) follow a musical element such as a scale, chord, or cadence. Very stable rating profiles reflecting the tonal hierarchies in major and minor keys are obtained, which, when intercorrelated and analyzed using multidimensional scaling, produce a four-dimensional spatial map of the distances between keys. The keys are located on the surface of a torus, in which the circle of fifths and the parallel and relative relations between major and minor keys are represented. In addition, single chords (major, minor, diminished, and dominant seventh) are found to be closely associated with the major and minor keys in which they play harmonic functions. The developing and changing sense of key during sequences of chords is traced by obtaining probe tone ratings following each chord in 10 different sequences, 8 of which contain modulations (changes) between keys. Modulations between closely related keys are found to be effected more immediately than are modulations between relatively distant keys. In all cases beyond the initial chord, the sense of the prevailing key is stronger than that produced by the last heard chord in isolation. Thus, listeners integrate harmonic functions over multiple chords, developing a sense of key that may need to be reevaluated as additional chords are sounded. It is suggested that the perceived relations between chords and keys and between different keys are mediated through an internal representation of the hierarchy of tonal functions of single tones in music. Music consists of tones varying in pitch, serve to highlight rhythmic patterns, further duration, loudness, and timbre, but the per- emphasize phrase structure, and distinguish ception of music extends well beyond the between tones constituting the primary meregistration of these physical attributes of lodic line and tones serving more ornamental the musical stimulus. Indeed, music contains or harmonic functions. Timbral characterconsiderable structure even in the relations istics may additionally provide important that obtain among the individual tones. For cues for the overall structure of the musical example, the durations are such that metri- composition.

‘Well-tempering’ the clavier: five methods

Nearly a third of a century before J. S. Bach assembled his first collection of 24 preludes and fugues ( 1722), the organist, organ maker and theorist Andreas Werckmeister had written of ways wol temperirt stimmen ('to tune well-tempered') and had even used the expression ein wohl temperirtes Clavier ('a well-tempered clavier').' For writers like Werckmeister (1645-1706), Johann George Neidhardt (1685-1739), and Francesco Antonio Vallotti (1697-1780), a good temperament was one whose circle of fifths was closed (wolf-free) and thus playable through all the keys, and whose major and minor intervals [thirds and sixths] varied in size systematically: in order to favour the more common scales and triads, the most heavily tempered of these intervals were placed in keys with the greatest number of sharps or flats, and the purest in those with the least. The effect of this irregular construction is quite striking; it enlivens the harmony and emphasizes modulation by contrasting key qualities. Such effects are well suited to the late baroque repertoire, most of which is not only more authentically but also far more expressively performed in these temperaments than in any others.

The Recognition of Major and Minor Keys in German Theory: 1680-1730

German theory on modes and keys during the 17th and early 18th centuries is rife with paradox. It was a German theorist who first presented a comprehensive theory of majorminor polarity and who differentiated two types of mode almost exclusively according to the quality of tonic triad (Johann Lippius, in works from 1610 to 12). Yet his German contemporaries and several generations of succeeding German theorists persisted in maintaining the traditional modal theories as the basis of contemporary music during their gradual but total demise in several other countries. Second, several theorists writing in German who did begin to recognize keys as opposed to modes around the turn of the 18th century related those keys to foreign practices (Antoni Berthali to French practice,1 and Johann Mattheson to Italian2). Yet the earliest presentation of 24 keys as equally original (not with some keys considered transpositions of other more original ones) and the earliest recognition of the circle of fifths appeared in German theory. And finally, despite these firsts in German theory, it is only in Germany that a major

Paintings Based on Relative Pitch in Music

The author’s approach to painting changed direction in 1961 when she began formal studies of music. She adopted relative pitch as a measure for quantitatively differentiating musical tones represented in her paintings. Specifically, the pitch of a tone was designated in terms of the ratio of its frequency to that of middle C on a piano. She chose a scale with the aid of the Lambdoma diagram or Pythagorean table. The values of the resulting relative pitches and their reciprocals were appropriate for the representation of tones by radii or by areas in her paintings. In addition, for each musical tone she has assigned a specific color, a designation that she generally keeps constant. All the paintings of this series are of a geometrical character. She calls attention to two interesting results that have a relationship to musical tones in a circle of fifths. For example, her assigned colors in a circle of fifths appear as a sequence of complementary color pairs. The planets assigned by G. Arnoux to musical tones, when they are arranged in the order of ascending fifths, occur in a sequence corresponding to the positions of the planets from the Sun.

Intervallic Procedures in Debussy: Serenade from the Sonata for Cello and Piano, 1915

THE FIRST two measures of the Serenade make clear that, of the various musical aspects, it is the intervallic that is most significant. The interval has acquired a new self-sufficiency and independence as an entity in its own right. Isolation of the interval from a motivic or contrapuntal context permits its exploitation in more systematic fashion through symmetrical organization, planned alteration, and patterning. In particular, this isolation allows the type of symmetrical molecular arrangement that appears here. A nucleus of two consecutive half-steps, constituting together the interval of the whole step, is defined by the first three pitches, played pizzicato by the cello. This nucleus is then repeated, but transposed to leave a gap of a minor third between the two occurrences. After another gap of a minor third, the nucleus is heard for the third time, now in the piano, but without its middle term, so that its composite interval, the whole step, is presented explicitly. The total distance covered in this systematic fashion is an octave; two successive presentations of the nucleus cover the distance of a fifth. The standard unit of calibration, the octave, seems to be divided into three equal parts by the successive placements of the intervallic nucleus, since the points of division fall in each instance within the gap of a minor third that separates them. The fact that these points of division occur at two thirds of the first gap, at the pitch level of C, and at one third of the second gap, at the level of E, shows that the placements of the nucleus are not symmetrical in relation to the octave, however; rather do these placements begin a series that would not be completed until the entire great circle of fifths has been traversed, since each nucleus begins at the equivalent of a fifth below the preceding one. To close off within an octave three successive statements, as Debussy does here, is a less obvious and more interesting procedure than would be a true tripartite division, as their relationship to the octave is unexpected. Since the fifth itself is two thirds of an octave, it encompasses two of the placements. It therefore is divided in turn into two halves that