Circle Of Fifths Research Articles

A Real-Time Key-Finding Algorithm Based on the Signature of Fifths

The signature of fifths is a special kind of music representation technique enabling acquisition of musical knowledge. The technique is based on geometrical relationships existing between twelve polar vectors inscribed in the circle of fifths, which represent individual pitch-classes detected in a given composition. In this paper we introduce a real-time key-detection algorithm which utilizes the concept of the signature of fifths. We explain how to create the signature of fifths and how to derive its descriptors required by the algorithm, i.e., the main directed axis of the signature of fifths, the major/minor mode axis, the characteristic vector of the signature of fifths, the characteristic angle of the signature of fifths, and the angle of the major/minor mode. We performed a series of experiments to test the algorithm’s effectiveness. The results were compared with those obtained using key-detection approaches based on key-profiles. All experiments were conducted using works composed by J.S. Bach, F. Chopin, and D. Shostakovich. The distinctive features of the presented algorithm, with respect to the considered key-detection approaches, are computational simplicity and stability of the decision-making process.

A Thread of Recitative Ruffs

This article presents a schema-driven analysis of the accompanied recitative “Erbarm es, Gott” from Johann Sebastian Bach’s St. Matthew Passion. Arguing that the scene is best understood in dialogue with the conventions of the Italianate “galant” style of eighteenth-century recitative, we suggest that the music is guided by a simple thread of modulation upward through the circle of fifths. This thread is driven by repeated application of a particular harmonic schema—the “ruff,” as described in Sherrill and Boyle (".fn_cite_year($sherrill_2015).")—which prototypically displaces a consonant harmony with a ".fn_sharp('')."4 figure over a static bass. Tonal arrivals are repeatedly implied and then evaded, such that the recitative’s simple tonal thread is best understood in terms of events that do not actually occur. This interpretation counterpoints Heinrich Schenker’s analysis of the recitative in Der Tonwille, whose details are explored in conjunction with the schema-based perspective. The final portion of the article advances an expressive interpretation of the music, engaging also with scholarship on the Passion such as Chafe (".fn_cite_year($chafe_1991)."). In our reading, the recitative’s tonal shape stages an experience of progressive engrossment in the diegetic world of the Passion, broken off by an abrupt narrative negation at its end. In doing so, the article synthesizes divergent music-theoretical methodologies, hoping to show how the stylistically and culturally informed perspective of schema theory offers a site where the formal insights of Schenkerian theory can be integrated with semiotic criticism.

Phasic Dissonance, Timbral Contrast: Analysing 20th‐Century Guitar Harmonies with Discrete Fourier Transform

ABSTRACTOne of the six‐string classical guitar's most distinctive features is its unbalanced timbre. That is to say, only four pitches of the chromatic aggregate are overtonally supported by the resonances of the guitar's three bass strings. A common pitch‐ and structure‐generating strategy, exploited by twentieth‐century composers who wrote for the guitar, is bound up with this affordance. Local and global harmonic contrasts are often built around an antagonism between open‐string sets and fretted harmonies that largely avoid duplicating notes associated with the open strings. Open‐string/fretted binaries derive their sonic antagonism not just from timbral distinction, however, but from the fact that their pitch materials often lie at diametrically opposed positions on the circle of fifths. In many instances, this colouristic conflict replaces harmonic‐functional tension as a structure‐generating agent; the latter is reduced to a surface‐level topic. Representative works by Frank Martin and Leo Brouwer are taken as exemplifications of this trend.In demonstrating the subtle interactions between instrumental affordance and abstract pitch relationships and hierarchies, an invaluable theoretical approach developed by Ian Quinn and finessed by Jason Yust entails the study of harmonic quality through the discrete Fourier transform. This mathematical formula has been processed into an easy‐to‐use but powerful web application by Jennifer Harding – the Pitch Class DFT Vector Calculator – making its explanatory power available to music analysts without mathematical expertise. Part of the raison d’être of this article is to show how this application might be used and, more important, how the information that it yields might be interpreted.

Gauge symmetries of musical and visual forces

After reviewing the physicalistic or metaphorical accounts to musical and visual forces by Arnheim and Larson, respectively, which were inspired by the basic tenets of gestalt psychology, I present a novel, naturalistic, mathematical framework, based on symmetry principles and gauge theory. In musicology, this approach has already been applied to the phenomenon of tonal attraction, leading to a deformation of the circle of fifths. The underlying gauge symmetry turns out as the SO(2) Lie group of a musical quantum model. Here, I present an alternative description in terms of Riemannian geometry. Its essential constraint of invariance of the infinitesimal line element leads to a deformation of the circle of fifths into a heart of fifths. In vision, the same approach is applied to Fraser's twisted cord illusion where concentric circles are deformed to squircle objects by means of an optical gauge field induced through a checkerboard background.

Teaching the connections between music and mathematics

Mathematics is often perceived by students as being difficult. However, when they see how math and music are intimately related, their attitude often changes. Some of the most fundamental aspects of music are described by math. For example, the Circle of Fifths closely reflects musical structure and is used to create scales and chord progressions. The Circle is structured around the mathematical relations of frequency ratios of the corresponding notes in a 12-tone scale. The reactions to music of consonance and dissonance are related to the relative ratios of the notes in a chord or note sequence in a piece. The connection between music and math forms the basis for creating instruments that play the music. I will suggest ways to present these concepts to students for their greater appreciation of music and math.

A Key-Finding Algorithm Based on Music Signature

The paper presents the key-finding algorithm based on the music signature concept. The proposed music signature is a set of 2-D vectors which can be treated as a compressed form of representation of a musical content in the 2-D space. Each vector represents different pitch class. Its direction is determined by the position of the corresponding major key in the circle of fifths. The length of each vector reflects the multiplicity (i.e. number of occurrences) of the pitch class in a musical piece or its fragment. The paper presents the theoretical background, examples explaining the essence of the idea and the results of the conducted tests which confirm the effectiveness of the proposed algorithm for finding the key based on the analysis of the music signature. The developed method was compared with the key-finding algorithms using Krumhansl-Kessler, Temperley and Albrecht-Shanahan profiles. The experiments were performed on the set of Bach preludes, Bach fugues and Chopin preludes.

Singing Planets Don't Sing; They Speak

Ancient Greek philosophers conceived a theory called Music of the Spheres. This ancient theory progressed for almost one thousand years before finally proving itself untrustworthy. However, this examination uncovers an overlooked fact: the large amount of natural order in sound and music existing before the creation of man. Scripture reveals that God is a God of order, and an extensive amount of natural order is found in the universe. Evidence points to God being the creator of the universe. Specific examples of such evidence are the inherent order of sound laid out in pitches, interval ratios, the overtone series, the circle of fifths, and the effect of consonance and dissonance. The structure of music from the ground up reveals a pre-existing model that man developed and turned into music. Man harnessed natural components of sound to create music full of emotion and reaction. Though scientifically inaccurate, Music of the Spheres offers a pathway to discover that music, as a concept and system, exhibits God as the intelligent designer.

«24 Children's Pieces for Piano» by Viktor Kosenko in the European Music History for Children

Relevance of research. The long-awaited comprehensive study of the work of the classic of Ukrainian music V. S. Kosenko led to the appeal to "24 children's pieces for the piano" not only as (with the approach still widespread in musicology) to the object of children's, purely oriented on didactic-methodical tasks of musical literature, but as a deeply meaningful work of art. A scientific novelty is the use of comparative intonation analysis of Kosenko's and Chopin's music. The goal is to introduce the piece to the broadest context of European piano music. The research methodology is also based on the application of historical and comparative methods, source analysis, elements of complex musicological analysis, etc.
 The history of how musical literature FOR children of didactic purpose was replenished with the characteristics of musical literature ABOUT children was considered. The library created by I. S. Bach (small preludes, two- and three-part inventions, "Little Books") and sheet music compiled by L. Mozart to teach children (the first stage of their own) profession, were enriched with music about life situations, dreams, and feelings of children of the times of R. Schumann ("Album for Youth") and P. Tchaikovsky ("Children's Album").
 "24 children's pieces for the piano" by V. Kosenko are written for children and about children of a new, modern author's era. The work is dedicated to "young pianists of our flourishing socialist Motherland." The heroes of the plays, like children of all times, have children's fun, toys, fantasies, fairy tales, lullabies in the family circle, and also life in a large team, hiking with pioneer songs, playing their contemporary heroes. Therefore, the dances "Waltz", "Mazurka", "Polka", "Ballet scene" are perceived as imported from another life -small in size, melodious, elegant, elegiac-dreamy pieces, moreover, they are all written in minor keys - dances that did not exist in the everyday life of that era. This music is a memory of one's own childhood, of the lost past.
 Conclusions. The main innovation that distinguishes the work from previous children's literature is the use of 24 tonalities laid out in parallel major and minor on the circle of fifths. The cycle "24 Preludes" by F. Chopin, the favorite composer, the companion of Viktor Kosenko's entire musical life, was written according to the same principle. The comparative analysis of monotone pieces (№№ 4, 6, 8, 11, 15, 19, 20, 22) of the two composers, making impossible the opinion about the external similarity and randomness of the organization of the musical material, proved the depth of the Ukrainian artist's understanding of the "creative spirit" of the Polish genius.
 Kosenko's appeal to Chopin's music is multifaceted. Often this is the appearance in children's plays of Chopin's intonation — the shortest melodic turn or expressive interval that carries the emotional idea of the work and performs this function in Kosenko's plays. Viktor Kosenko re-intonates, transposes, rhythmically changes the intonations of Preludes. They become elements of polyphony, enrich the harmony, the organization of phrases appears to be similar in the works, the metrorhythm, saturation of the texture is the same.

The Musical Language of Rock

The Musical Language of Rock

A Hardware-Oriented Algorithm for Real-Time Music Key Signature Recognition

The algorithm presented in this paper provides the means for the real-time recognition of the key signature associated with a given piece of music, based on the analysis of a very small number of initial notes. The algorithm can easily be implemented in electronic musical instruments, enabling real-time generation of musical notation. The essence of the solution proposed herein boils down to the analysis of a music signature, defined as a set of twelve vectors representing the particular pitch classes. These vectors are anchored in the center of the circle of fifths, pointing radially towards each of the twelve tones of the chromatic scale. Besides a thorough description of the algorithm, the authors also present a theoretical introduction to the subject matter. The results of the experiments performed on preludes and fugues by J.S. Bach, as well as the preludes, nocturnes, and etudes of F. Chopin, validating the usability of the method, are also presented and thoroughly discussed. Additionally, the paper includes a comparison of the efficacies obtained using the developed solution with the efficacies observed in the case of music notation generated by a musical instrument of a reputable brand, which clearly indicates the superiority of the proposed algorithm.

THE “24 PRELUDES” FOR THE PIANO BY IVAN KARABYTS AS AN ENCYCLOPAEDIA OF THE UKRAINIAN RENAISSANCE OF THE 1970S

Background. The research is devoted to revealing the semantic analysis of the dramaturgy of one of the large-scale compositions in the creative work of IvanKarabyts – the cycle “24 Preludes” for the piano. The composition was written by Ivan Karabyts in 1976 and today it is of great interest to concert performers and fans of modern piano music. The attention of pianists to the cycle “24 Preludes” by I. Karabyts is attracted, firstly, by the distinctive, original musical language, secondly – by a wide range of performing capabilities and means of expression, and thirdly – by vivid images that inspire pianists to reproduce artistic ideas, hidden philosophical implications. The object of research is the cycle “24 preludes” for the piano as a musical encyclopaedia, reflecting the artistic era in the context of the Ukrainian renaissance of tthe 1970s, and the aim is to identify stylistic patterns by means of the semantic analysis of the dramaturgy of the cycle, finding the intersection in a kind of dialogue with a diverse, significant fund of the music of the 20th century. The methodology of research is focused on the relationship of special methods of analysis: functional-structural, intonation, genre, style, semantic and interpretative one. Results. Ivan Karabyts chooses for his cycle a model of tonal dramaturgy of the cycle “24 Preludes”, introduced by F. Chopin and later by D. Shostakovich, namely – the movement along the circle of fifths in the ratio of major-minor. From the point of view of musical semantics of the preludes of the cycle they can be divided into 5 thematic groups (contemplative and introspective lyrics; grotesque and dance; sound imitation and spatial-visual; stylistic allusions; and tragedy ones), varied in genre-stylistic sense (according to the criteria of modelling the awareness of the lyrical hero (I – the world around me.) The dramaturgy of the cycle is built through their correlation, while forming a certain plot, which begins with the image of the lyrical hero, and ends with a demonstration of the society which is ambiguous and problematic for a human. The composer chooses the prelude as a genre with a historical memory of culture, which allows performers and listeners to experience the range of psychological moments of the human spirit in the turbulent world of events of the last third of the 20th century. The composer is fascinated by this genre not by chance, because the prelude allows reflecting in miniature numerous states of “fixed” moments of existence, the inner balance of the artist and the world. Each prelude in the cycle is a kind of creative laboratory, a field of creative experiments. It reflected both already developed and new methods and principles of the composer’s thinking. While performing one prelude after another as a whole composition, one realizes that this genre expresses the freedom of creativity, the element of existence: it is a fantasy, and a story of the heart, and the revelation of the spirit, and at the same time – bright genre sketches. Conclusions. The analysis of the musical semantics of I. Karabyts’s piano cycle “24 Preludes” testified to the presence of 5 genre-stylistic groups in the cycle (according to the criterion of the dual world notion “psychology I – the world around”). Thus, the genre-semantic analysis of the piano cycle “24 Preludes” has shown that I. Karabyts does not lose touch with history and time, by paying tribute to the masters of the 18th–20th centuries, continuing to develop the type of tonal dramaturgy, laid down by J. S. Bach. In the cycle there is a special “counterpoint” of the “blues” stylistic. The dramaturgy of the cycle has a detailed plot, which begins with the image of the lyrical hero, and ends with a demonstration of the society ambiguous and problematic for a human (“I – World”). The dramaturgy of the romantic dual world turns into a harmony of the modern world with multiple images, echoes of time and inner drama. The genre semantics and its analysis allow the performer to comprehend the large-scale cycle as an artistic picture of the world, and its stylistic unity – as a spiritual universe which belongs to the Ukrainian art of the 21st century.

Dialectical spiral of the circle of fifths

Dialectical spiral of the circle of fifths

Chord-Interval, Direct-Familiarization, Musical Instrument Digital Interface, Circle of Fifths, and Functions as Basic Piano Accompaniment Transposition Techniques

This research article aimed to examine and describe different research-based practical and transmedia transposition techniques as basic transposition methods for piano accompaniment. The need for transposition in music, particularly in piano accompaniment, whether in a single line melody, a scale, or an entire composition, is to provide the constituents in the music industry like musicians, pianists, singers, composers, and arrangers with ease and comfortable playing or singing performance using both traditional and/or technological (transmedia) transposition techniques. The most practical way of reading a music sheet is the combination of a lead sheet which composed of the melody and its respective chord/s for each measure or within the measures. Sometimes, song compositions such as folk songs and other traditional songs are just written in a single staff with its defined lyrics. The concern in this matter is the chord that would be used in accompanying the song. In making an accompaniment for this type of music, we could use the three basic types of harmonic progressions in a downward or upward manner by seconds, thirds, and fourths. After learning the basics of chord progressions, the five identified research-based techniques might be used for the process of transposition by church clavinovist and piano enthusiast namely Chord and Interval Transposition Technique, Familiarization and Direct Transposition Technique, Musical Instrument Digital Interface Transposition Technique, Circle of Fifths Transposition Technique, and Functions Transposition Technique.

Temporal order of musical keys and subjective estimates of time

The aim of this study was to observe the influence of the temporal order of musical keys involved in sudden modulations, which implies compositional developments in clockwise and counterclockwise directions of the circle of fifths, on subjective time estimations. Seventy-five undergraduate students from Universidade de Ribeirão Preto participated in this experiment, which consisted of listening to a modulating musical stimulus and retrospectively reproducing the duration with the aid of a stopwatch. The results showed that reverse sudden tonal modulation in the counterclockwise direction or, for instance, the temporal order from the original key of A-major to the arrival at the destination key of C-major, elicited shorter time estimations than the clockwise direction or, for instance, the temporal order from original key of C-major to the arrival at the destination key of A-major. These data were interpreted using the Expected Development Fraction model that describes the development of expectations when an inter-key distance is traversed during a certain time interval. This expected development is longer than the perceived duration, leading to the underestimation of time.

Quantum approaches to music cognition

Quantum cognition emerged as an important discipline of mathematical psychology during the last two decades. Using abstract analogies between mental phenomena and the formal framework of physical quantum theory, quantum cognition demonstrated its ability to resolve several puzzles from cognitive psychology. Until now, quantum cognition essentially exploited ideas from projective (Hilbert space) geometry, such as quantum probability or quantum similarity. However, many powerful tools provided by physical quantum theory, e.g., symmetry groups have not been utilized in the field of quantum cognition research so far. Inspired by seminal work by Guerino Mazzola on the symmetries of tonal music, our study aims at elucidating and reconciling static and dynamic tonal attraction phenomena in music psychology within the quantum cognition framework. Based on the fundamental principles of octave equivalence, fifth similarity and transposition symmetry of tonal music that are reflected by the structure of the circle of fifths, we develop different wave function descriptions over this underlying tonal space. We present quantum models for static and dynamic tonal attraction and compare them with traditional computational models in musicology. Our approach replicates and also improves predictions based on symbolic models of music perception.

Decoding MEG responses to musical pitch reveals the dynamic emergence of tonal structure in human cortex

Tonal music is characterized by a hierarchical structuring of pitch, whereby certain tones appear stable and others unstable within their musical context. Despite its prevalence, the cortical mechanisms supporting such a percept remain poorly understood. We examined the neural processing dynamics underlying pitch-structure in Western Tonal Music. Listeners were presented with tones embedded within a musical context whilst their magnetoencephalographic (MEG) activity was recorded. Using multivariate pattern analysis, decoders attempted to classify the identity of tones from their corresponding MEG activity at each peristimulus time-sample, providing a dynamic measure of their cortical dissimilarity. Time-evolving neural distinctions were then compared with the predictions of several acoustic and perceptual models. Following the onset, a temporal evolution was witnessed in the representational structure in cortex. While MEG dissimilarities between tones initially corresponded to their fundamental frequency separation, distinctions beyond 200 ms reflected their status within the hierarchy of perceived stability. Transposing dissimilarities corresponding to this latter period into different keys, neural relations between keys correlated with the well-known circle of fifths. Convergent with fundamental principles of music-theory and perception, results detail the dynamics with which the complex perceptual structure of Western tonal music emerges in human cortex within the timescale of an individual tone.Tonal music is characterized by a hierarchical structuring of pitch, whereby certain tones appear stable and others unstable within their musical context. Despite its prevalence, the cortical mechanisms supporting such a percept remain poorly understood. We examined the neural processing dynamics underlying pitch-structure in Western Tonal Music. Listeners were presented with tones embedded within a musical context whilst their magnetoencephalographic (MEG) activity was recorded. Using multivariate pattern analysis, decoders attempted to classify the identity of tones from their corresponding MEG activity at each peristimulus time-sample, providing a dynamic measure of their cortical dissimilarity. Time-evolving neural distinctions were then compared with the predictions of several acoustic and perceptual models. Following the onset, a temporal evolution was witnessed in the representational structure in cortex. While MEG dissimilarities between tones initially corresponded to their fundamental frequency ...

Perception of Pitch and Time in the Structurally Isomorphic Standard Rhythmic and Diatonic Scale Patterns

The three experiments reported here investigate how pitch and time interact in perception using the standard rhythmic pattern and the diatonic scale pattern, which share the intervallic structure of 2 2 1 2 2 2 1. They share a number of theoretical properties, including being cyclic with seven unique rotations. Experiment 1 measured rhythmic stability by dynamically accenting each of the events in each rhythm, called the probe accent; listeners rated how well the probe accent fit the rhythm. Listeners heard the rhythms in subgroups and with reference to a syncopation-shifted metrical hierarchy. Experiment 2 used the probe tone technique to measure the tonal stability of each tone in each mode beginning and ending on C. Higher ratings were given to tones earlier in the contexts and tones closer to C on both the chroma circle and the circle of fifths; influences were also found of tonal hierarchies of diatonic scales with corresponding accidentals. A measure of similarity derived from the probe ratings found the same order for the rhythms and modes which matched theoretical proposals of inter-rhythm and inter-mode distances. Experiment 3 presented all combinations of rhythms and modes; listeners judged how well the rhythm fit the mode. These judgments did not depend on the intervallic isomorphism between tone duration and interval size. Instead, the judgments depended on whether tonally stable events occurred where accents were judged as fitting well with the rhythm. Overall, the standard and diatonic patterns follow different perceptual hierarchies while sharing similar cognitive principles between rhythms, between modes, and across dimensions.

Polarization, Modal Orientation, and Voice Leading

This article seeks to define tonality as an interaction between different levels of structure in terms of a conjunction of intervallic distance, polarization, modal direction, and voice leading of the harmonic progressions. Intervallic distances depend on the modal direction of transformations as well as on polarization on the circle of fifths in any hierarchy level. Voice leading creates pitch spaces that both embody and outweigh the abstract concept of tonal event according to transformational theories. This model of tonality is particularly relevant to music of the later nineteenth and earlier twentieth centuries, where works are based on complex, developed structures composed of non-symmetrical recurring units. The potential for analysis is demonstrated with several passages from Dohnányi’s Piano Quintet, op. 26, first movement.

Variety and multiplicity for partitioned factors in Christoffel and Sturmian words

The fundamental results in mathematical music theory due to Clough and Myerson, Cardinality equals Variety (CV) and Structure yields Multiplicity (SM), correspond to statements about Christoffel words. These results may be stated and proved in a purely word-theoretic environment by means of the concept of a partitioned factor. In this setting, the generalized circle of fifths of Clough and Myerson is replaced by the lexicographic order of word theory, and their notion of the generic description of a musical line is replaced by that of a partitioned factor of a circular Christoffel word. Sturmian words are the infinite counterparts to Christoffel words, characterized as aperiodic but of minimal complexity, i.e. words such that for all there are n+1 factors of length n. Berthé showed that the factors of a given length have at most three frequencies (probabilities). Drawing on previous work, we extend to results on factors under a fixed partitioning (decompositions of factors of length n into concatenations of words whose lengths are given by a composition of n into k parts). Any factor of a Sturmian word thus partitioned into k components belongs to one of k+1 types (varieties). We show how to compute the frequencies of the varieties. These results recapture CV and SM in Sturmian words, and we suggest musical interpretations, parallel to those in the finitary case.

Psychoacoustic Approaches for Harmonic Music Mixing

The practice of harmonic mixing is a technique used by DJs for the beat-synchronous and harmonic alignment of two or more pieces of music. In this paper, we present a new harmonic mixing method based on psychoacoustic principles. Unlike existing commercial DJ-mixing software, which determines compatible matches between songs via key estimation and harmonic relationships in the circle of fifths, our approach is built around the measurement of musical consonance. Given two tracks, we first extract a set of partials using a sinusoidal model and average this information over sixteenth note temporal frames. By scaling the partials of one track over ±6 semitones (in 1/8th semitone steps), we determine the pitch-shift that maximizes the consonance of the resulting mix. For this, we measure the consonance between all combinations of dyads within each frame according to psychoacoustic models of roughness and pitch commonality. To evaluate our method, we conducted a listening test where short musical excerpts were mixed together under different pitch shifts and rated according to consonance and pleasantness. Results demonstrate that sensory roughness computed from a small number of partials in each of the musical audio signals constitutes a reliable indicator to yield maximum perceptual consonance and pleasantness ratings by musically-trained listeners.