Music theory


Music theory

Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods. In a grand sense, music theory distills and analyzes the fundamental parameters or elements of music—rhythm, harmony (harmonic function), melody, structure, form, texture, etc. Broadly, music theory may include any statement, belief, or conception of or about music.[1] A person who studies these properties is known as a music theorist. Some have applied acoustics, human physiology, and psychology to the explanation of how and why music is perceived.

Contents

Fundamentals of music

Music has many different fundamentals or elements. These include but are not limited to: pitch, beat or pulse, rhythm, melody, harmony, texture, allocation of voices, timbre or color, expressive qualities (dynamics and articulation), and form or structure. In addition to these "fundamentals" there are other important concepts employed in music both in Western and non-Western cultures including "Scales and/or Modes" and "Consonance vs. Dissonance."

Pitch

Pitch is a subjective sensation, reflecting generally the lowness (slower wave frequency) or highness (faster wave frequency) of a sound. In a musical context, some people have what is called "perfect pitch" and can assign an isolated tone to its place on a musical scale. Human perception of pitch can be comprehensively fooled to create auditory illusions. Despite these perceptual oddities, perceived pitch is nearly always closely connected with the fundamental frequency of a note, with a lesser connection to sound pressure level, harmonic content (complexity) of the sound, and to the immediately preceding history of notes heard.[2] In general, the higher the frequency of vibration, the higher the perceived pitch is, and lower the frequency, the lower the pitch.[3] However, even for tones of equal intensity, perceived pitch and measured frequency do not stand in a simple linear relationship.[4]

At and below about 1,000 Hz, the perceived pitch of a tone gets lower as sound pressure increases, but above approximately 2,000 Hz, the pitch increases as the sound gets louder.[5]

In Western music, there have long been several competing pitch standards defining tuning systems. Most made a particular key sonorous, with increasingly remote ones more and more problematic; the underlying problem is related to the physics of vibrations.

In addition, fixing notes to standard frequencies (required for instrument makers) has varied as well. "Concert A" was set at 435 Hz by France in 1859 while in England, concert A varied between 439 and 452 Hz. A frequency of 440 Hz was recommended as the standard in 1939, and in 1955 the International Organization for Standardization affirmed the choice.[6] A440 is now widely, though not exclusively, used as the A above middle C.

The difference in frequency between two pitches is called an interval. The most basic interval is the unison, which is simply two of the same pitch, followed by the slightly more complex octave, which indicates either a doubling or halving of the fundamental frequency.

Scales and Modes

Notes can be arranged into different scales and modes. Western music theory generally divides the octave into a series of 12 notes that might be included in a piece of music. This series of twelve notes is called a chromatic scale. In the chromatic scale, the interval between adjacent notes is called a half-step or semitone. Patterns of half and whole steps (2 half steps, or a tone) can make up a scale in that octave. The scales most commonly encountered are the seven toned major, the harmonic minor, the melodic minor, and the natural minor. Other examples of scales are the octatonic scale, and the pentatonic or five-toned scale, which is common in but not limited to folk music. There are scales that do not follow the chromatic 12-note pattern, for example in classical Ottoman, Persian, Indian and Arabic music. Arabic and Persian classical traditions often make use of quarter-tones, half the size of a semitone, as the name suggests.

In music written using the system of major-minor tonality, the key of a piece determines the scale used. (One way of showing how various keys relate to one another may be seen in the circle of fifths.) Transposing a piece from C major to D major will make all the notes two semitones (or one full step) higher. Even in modern equal temperament, changing the key can change the feel of a piece of music, because it changes the relationship of the composition's pitches to the pitch range of the instruments that play the piece. This often affects the music's timbre, as well as having technical implications for the performers. However, performing a piece in one key rather than another may go unrecognized by the casual listener, since changing the key does not change the relationship of the individual pitches to each other.

Consonance and Dissonance

Consonance can be roughly defined as harmonies whose tones complement and increase each others' resonance, and dissonance as those that create more complex acoustical interactions (called 'beats'). A simplistic example is that of "pleasant" sounds versus "unpleasant" ones. Another manner of thinking about the relationship regards stability; dissonant harmonies are sometimes considered to be unstable and to "want to move" or "resolve" toward consonance. However, this is not to say that dissonance is undesirable. A composition made entirely of consonant harmonies may be pleasing to the ear and yet boring because there are no instabilities to be resolved.[citation needed]

Melody is often organized so as to interact with changing harmonies (sometimes called a chord progression) that accompany it, setting up consonance and dissonance. The art of melody writing depends heavily upon the choices of tones for their nonharmonic or harmonic character.[vague]

Rhythm

Rhythm is the arrangement of sounds and silences in time. Meter animates time in regular pulse groupings, called measures or bars. The time signature or meter signature specifies how many beats are in a measure, and which value of written note is counted and felt as a single beat. Through increased stress and attack (and subtle variations in duration), particular tones may be accented. There are conventions in most musical traditions for a regular and hierarchical accentuation of beats to reinforce the meter. Syncopated rhythms are rhythms that accent unexpected parts of the beat. Playing simultaneous rhythms in more than one time signature is called polymeter. See also polyrhythm.

In recent years, rhythm and meter have become an important area of research among music scholars. Recent work in these areas includes books by Bengt-Olov Palmqvist, Fred Lerdahl and Ray Jackendoff, and Jonathan Kramer.

Melody

A melody is a series of tones sounding in succession. The tones of a melody are typically created with respect to pitch systems such as scales or modes. The rhythm of a melody is often based on the inflections of language, the physical rhythms of dance, or simply periodic pulsation. Melody is typically divided into phrases within a larger overarching structure. The elements of a melody are pitch, duration, dynamics, and timbre.

Harmony

Harmony is the study of vertical sonorities in music. Vertical sonority refers to considering the relationships between pitches that occur together; usually this means at the same time, although harmony can also be implied by a melody that outlines a harmonic structure.

The relationship between two pitches is referred to as an interval. A larger structure involving more than two pitches is called a chord. In common practice and popular music, harmonies are generally tertian. This means that the interval of which the chords are composed is a third. Therefore, a root-position triad (with the root note in the lowest voice) consists of the root note, a note a third above, and a note a third above that (a fifth above the root). Seventh chords add a third above the top note of a triad (a seventh above the root). There are some notable exceptions. In 20th century classical music, many alternative types of harmonic structure were explored. One way to analyze harmony in common practice music is through a Roman numeral system; in popular music and jazz a system of chord symbols is used; and in post-tonal music, a variety of approaches are used, most frequently set theory.

The perception of pitch within harmony depends on a number of factors including the interaction of frequencies within the harmony and the roughness produced by the fast beating of nearby partials. Pitch perception is also affected by familiarity of the listener with the music, and cultural associations.

"Harmony" as used by music theorists can refer to any kind of simultaneity without a value judgement, in contrast with a more common usage of "in harmony" or "harmonious", which in technical language might be described as consonance.

Monophony is the texture of a melody heard only by itself. If a melody is accompanied by chords, the texture is homophony. In homophony, the melody is usually but not always voiced in the highest notes. A third texture, called polyphony, consists of several simultaneous melodies of equal importance.

Texture

Musical texture is the overall sound of a piece of music commonly described according to the number of and relationship between parts or lines of music: monophony, heterophony, polyphony, homophony, or monody. The perceived texture of a piece may also be affected by the timbre of the instruments, the number of instruments used, and the distance between each musical line, among other things.

Timbre

Timbre, sometimes called "Color", or "Tone Color" is the quality or sound of a voice or instrument.[7] The quality of timbre varies widely from instrument to instrument, or from voice to voice. The timbre of some instruments can be changed by applying certain techniques while playing. For example, the timbre of a trumpet changes when a mute is inserted into the bell, or a voice can change its timbre by the way a performer manipulates the vocal apparatus, (e.g. the vocal cords, mouth and diaphragm). Generally, there is no common musical notation that speaks specifically to a change in timbre, (as "pianissimo" would indicate "very soft" for a change in dynamics).

Expressive Qualities

Expressive Qualities are those elements in music that create change in music that are not related to pitch, rhythm or timbre. They include Dynamics and Articulation.

Dynamics

In music, the term "dynamics" normally refers to the softness or loudness of a sound or note: e.g. pianissimo or fortissimo. Until recently, most dynamics in written form were done so in Italian, but recently are sometimes written or translated into English. Another sense of the word refers to any aspect of the execution of events in a given piece; either stylistic (staccato, legato etc.) or functional (velocity) are also known as dynamics. The term is also applied to the written or printed musical notation used to indicate dynamics.

Articulation

Articulation is the manner in which the performer applies their technique to execute the sounds or notes—for example, staccato or legato. Articulation is often described rather than quantified, therefore there is room to interpret how to execute precisely each articulation. For example, Staccato is often referred to as "separated" or "detached" rather than having a defined, or numbered amount by which the separation or detachment is to take place. Often the manner in which a performer decides to execute a given articulation is done so by the context of the piece or phrase. Also, the type or style of articulation will depend on the instrument and musical period, e.g. the classical period, but there is a generally recognized set of articulations that most all instruments (and voices) have in common. They are, in order of long to short: legato ("smooth, connected"); tenuto ("pressed", "lengthened but detached"); marcato (heavily accented and detached); staccato ("separated", "detached"); "martelé" (or "rooftop accent" or "teepee accent") for its written shape (short and hard). Any of these may be combined to create certain "in-between" articulations. For example, portato is the combination of tenuto and staccato. Some instruments have unique methods by which to produce sounds, such as spicatto for strings, where the bow bounces off the string.

Form or Structure

Form is a facet of music theory that explores the concept of musical syntax, on a local and global level. The syntax is often explained in terms of phrases and periods (for the local level) or sections or genre (for the global scale). Examples of common forms of Western music include the fugue, the invention, sonata-allegro, canon, strophic, theme and variations, and rondo. Popular Music often makes use of strophic form many times in conjunction with Twelve bar blues.

Theories of harmonization

Four-part writing

Four-part voice leading for dominant thirteenth chords in the common practice period.[8] About this sound Play

Four-part chorale writing is used to teach and analyze the basic conventions of "Common-Practice Period music", the time period lasting from approximately 1650 to 1900.[9] In the German musicology tradition referred to as functional harmony. Johann Sebastian Bach's four voice chorales written for liturgical purposes serve as a model for students. These chorales exhibit a fusion of linear and vertical thinking. In analysis, the harmonic function and rhythm are analyzed as well as the shape and implications of each of the four lines. Students are then instructed to compose chorales, often using given melodies (as Bach would have done), over a given bass line, or to compose within a chord progression, following rules of voice leading. Though traditionally conceived as a vocal exercise for Soprano, Alto, Tenor, and Bass, other common four-part writings could consist of a brass quartet (two Trumpets, French Horn, and Trombone) or a string quartet (including violin I, violin II, viola and cello).

There are seven chords used in four-part writing that are based upon each note of the scale. The chords are usually given Roman Numerals I, II, III, IV, V, VI and VII to refer to triadic (three-note) chords based on each successive note of the major or minor scale the piece is in. Chords may be analyzed in two ways. Case-sensitive harmonic analysis would state that major-mode chords (I, IV, V7, etc.), including augmented (for example, VII+), would be notated with upper-case Roman numerals, and minor-mode chords, including diminished (ii, iii, vi, and the diminished vii chord, viio), would be notated with lower-case Roman numerals. When a scale degree other than the root of the chord is in the bass, the chord is said to be in inversion, and this is indicated by numbers written above the roman numeral. With triads a 6 indicates first inversion, and 6 4 indicates second inversion. With seventh chords, 6 5 indicates first inversion, 4 3 indicates second inversion, and 4 2 indicates third inversion. ( I6, IV4/3,V 4/2 , etc.) Schenkerian harmonic analysis, patterned after the theories of Heinrich Schenker, would state that the mode does not matter in the final analysis, and thus all harmonies are notated in upper-case.

The skill in harmonizing a Bach chorale lies in being able to begin a phrase in one key and to modulate to another key either at the end of the first phrase, the beginning of the next one, or perhaps by the end of the second phrase. Each chorale often has the ability to modulate to various tonally related areas: the relative major (III) or minor (vi), the Dominant (V) or its relative minor (iii), the Sub-Dominant (IV) or its relative minor (ii). Other chromatic chords may be used, like the diminished seventh (made up of minor thirds piled on top of each other) or the Secondary dominant (the Dominant's Dominant – a kind of major version of chord II). Certain standard cadences are observed, most notably IIb7 – V7 – I. The standard collection of J. S. Bach's chorales was edited by Albert Riemenschneider and this collection is readily available, e.g. here.

Music perception and cognition

Jackendoff and Lerdahl attempt to develop a "musical grammar". Using Jackendoff's background as a linguist and Lerdahl's compositional and theoretical background, a series of generative rules are defined to explain the hierarchical structure of tonal music. The rules focus on musical grouping, or methods in which rhythmic groups of notes, as well as formal hierarchies, are perceived by listeners. Three sets of rules are given: "Grouping Well-Formedness Rules", "Grouping Preference Rules" and "Transformational Rules". These rules are designed to interpret how listeners group structures in tonal music. These groupings then play into the segmentation of events by listeners, which in turn determine the hierarchical structure perceived by the listener.[citation needed] Although this theory is well developed and complete, it is by far not the only system designed to discuss music in this manner, and there is no acceptance of this theory as being the sole theory by which to discuss perception of music (see Jonathan Kramer).[citation needed]

Serial composition and set theory

Twelve-tone technique was developed by Arnold Schoenberg to order and repeat all the 12 pitches of the chromatic scale with specific order. From 1947, this technique has been alternatively designated in French and English sources by the word serialism. An ordered row of the 12 pitches is created, then all possible transformations are explored. The analytic techniques involve writing a 12 × 12 matrix of the tone row, and all of its forms (transposition, inversion, retrograde, retrograde inversion, and possibly other mappings, such as the cycle-of-fourths or M5 transformation). This technique is primarily associated with the composers of the Second Viennese School, but also has been incorporated into the languages of many other composers.[citation needed]

The term serialism does not necessarily refer only to twelve-tone technique, especially in the German language; many composers have explored serialism using fewer than 12 notes, repeating tones inside of the row, serialism of microtonal scales, permutational serialism (in which note order is not fixed), distributional serialism, and serial composition without pitches at all. Also, composers such as Pierre Boulez explored integral serialism, or the serialization of all possible musical parameters (pitch, rhythm, dynamics, etc.).[citation needed]

Set Theory is another approach to understanding atonal music that may or may not be serial. Although more akin to the mathematical field of Group Theory than mathematical Set Theory, the nomenclature has become standard inside the musical community. Set theory represents the pitch classes as numbers to allow a methodology of examining music without tonic or triadic functional harmony. This technique allows for exploration of the construction of a serial tone row as well as less strict atonal works. This technique has been extended with a great deal of mathematical rigor to both tonal and atonal systems by David Lewin in his transformational approach utilizing networks of related sets.[citation needed]

Musical semiotics

Music subjects

Notation

Musical notation is the symbolic representation of music (not to be confused with audio recording). Historically, and in the narrow sense, this is achieved with graphic symbols. Computer file formats have become important as well.[10] Spoken language and hand signs are also used to symbolically represent music, primarily in teaching.[citation needed]

In standard Western music notation, music is represented graphically by notes placed on a staff or staves with the vertical axis roughly corresponding to pitch and the horizontal axis roughly corresponding to time. Note head shapes, stems, flags, and ties are used to indicate duration. Additional symbols represent key, tempo, dynamics, accents, rests, etc.

Mathematics

Music and mathematics are strongly intertwined. As noted above, the concept of pitch and temperament are both strongly tied to mathematics, and acoustics in particular. Analysis often takes a mathematical route; musical set theory and Transformational theory are both steeped in mathematics.[citation needed]

Some methods of composition are mathematically based. Iannis Xenakis developed several methods using stochastic methods. The French school of spectral music uses mathematical analysis of sounds to develop compositional materials.[citation needed]

In music history mathematics were the foundation of the first understanding of tones, intervals, and scales developed by the Greeks between 530 and 500 BC. This discovery was based upon shortening a harp’s string by a half, creating an octave. Further, separating the same string into two-thirds or four equal parts produced intervals known as fifths and fourths, respectively. This discovery had a philosophical impact on the importance of mathematics, “It was the first consistent realization that there is a mathematical rationality in the universe and that the human mind can make sense of that rationality,” said Kitty Ferguson, the author of The Music of Pythagoras.[cite this quote] Recently, Princeton University music theorist Dmitri Tymoczko discovered that relationships between notes exist in multi-dimensional geometric forms, or orbifolds. Tymoczko made his discovery when writing down all possible two note chords in columns on a sheet of paper. After doing so Tymoczko observed the possibility that a pattern existed,

Suddenly Tymoczko realized that if he cut two triangles from the piece of paper, turned one of the triangles upside down, and reconnected the two triangles where the chords overlapped, the two-note chords on one edge of the resulting strip of paper would be the reversed versions of those on the opposite edge. If he then twisted the paper and attached the two edges, the chords would line up. "That’s when I got a tingly feeling in my fingers," he says.[cite this quote]

Two-note chords, the minimalist form of a chord (as a chord is any combination any number of notes played simultaneously) is represented graphically by a Mobius strip, a two-dimensional surface embedded in a three-dimensional space. As the chords are composed of increasing numbers of notes, the geometric form they take on becomes increasingly sophisticated. Three-note chords are represented by twisted three-dimensional shapes, and four-note chords, four-dimensional shapes. These principles on tonal relationships apply to every genre of music and have been unintentionally practiced by theorists and composers since medieval times.[citation needed] The significance of this finding is rooted in teaching and applying music theory. The simplistic relationships of tones as geometric shapes allow students of music to understand the composition of complex musical scores. In doing so, students can apply the relationships used in preeminent examples of melodic composition more easily into their own writing. Tymoczko explains this simplicity as being

the "amazing and mysterious" thing about music … Three singers can go from a pleasing C-major chord to the complementary and more plaintive A-minor chord by moving just one note: changing from CEG to CEA. Someone playing Hey Jude on the piano can move his or her fingers very little while moving from one sonorous chord to another. Miraculously, the chords that sound good together and the ones that produce efficient voice leading are the same.[cite this quote]

It was this idea that lead to the study of mathematical music theory and explains his discovery of orbifolds and representations of these relationships. Tymoczko used these tools to as facilitators for his own compositions and an analysis of Western music in his book A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice.[11]

Analysis

Analysis is the effort to describe and explain music. Analysis at once is a catch-all term describing the process of describing any portion of the music, as well as a specific field of formal analysis or the field of stylistic analysis. Formal analysis attempts to answer questions of hierarchy and form, and stylistic analysis attempts to describe the style of the piece. These two distinct sub-fields often coincide.

Analysis of harmonic structures is typically presented through a roman numeral analysis. However, over the years, as music and the theory of music have both grown, a multitude of methods of analyzing music have presented themselves. Two very popular methods, Schenkerian analysis and Neo-Riemannian analysis, have dominated much of the field. Schenkerian analysis attempts to "reduce" music through layers of foreground, middleground, and, eventually and importantly, the background. Neo-Riemannian (or Transformational) analysis began as an extension of Hugo Riemann's theories of music, and then expanding Riemann's concepts of pitch and transformation into a mathematically rich language of analysis. While both theories originated as methods of analysis for tonal music, both have been extended to use in non-tonal music as well.

Ear training

Aural skills – the ability to identify musical patterns by ear, as opposed to by the reading of notation – form a key part of a musician's craft and are usually taught alongside music theory. Most aural skills courses train the perception of relative pitch (the ability to determine pitch in an established context) and rhythm. Sight-singing – the ability to sing unfamiliar music without assistance – is generally an important component of aural skills courses. Absolute pitch or perfect pitch describes the ability to recognise a particular audio frequency as a given musical note without any prior reference.

See also

Notes

  1. ^ Boretz 1995,[page needed].
  2. ^ Lloyd and Boyle 1978, 142.
  3. ^ Benade 1960, 31.
  4. ^ Stevens, Volkmann, and Newman 1937, 185; Josephs 1967, 53–54.
  5. ^ Olson 1967, 248–51.
  6. ^ Cavanagh (1999).
  7. ^ Harnsberger 1997.
  8. ^ Benward and Saker 2009, 179.
  9. ^ Kostka and Payne 2004,[page needed].
  10. ^ Castan 2009.
  11. ^ Olson, Steve. "A Grand Unified Theory of Music". Princeton Alumni Weekly. http://paw.princeton.edu/issues/2011/02/09/pages/6550/index.xml. Retrieved 27 April 2011. 

Sources

  • Benade, Arthur H. (1960). Horns, Strings, and Harmony. Science Study Series S 11. Garden City, New York: Doubleday & Company, Inc.
  • Boretz, Benjamin (1995). Meta-Variations: Studies in the Foundations of Musical Thought. Red Hook, New York: Open Space.
  • Benward, Bruce, and Marilyn Nadine Saker (2009). Music in Theory and Practice, eighth edition, vol. 2. Boston: McGraw-Hill. ISBN 978-0-07-310188-0.
  • Bent, Ian D., and Anthony Pople (2001). "Analysis." The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.
  • Castan, Gerd (2009). "Musical Notation Codes". Music-Notation.info (Accessed 1 May 2010).
  • Cavanagh, Lynn ([1999]). "A Brief History of the Establishment of International Standard Pitch A=440 Hertz" (PDF). http://web.archive.org/web/20090325223741/http://www.wam.hr/Arhiva/US/Cavanagh_440Hz.pdf.  (Accessed 1 May 2010)
  • Harnsberger, Lindsey C. (1997). "Articulation". Essential Dictionary of Music: Definitions, Composers, Theory, Instrument and Vocal Ranges, second edition. The Essential Dictionary Series. Los Angeles: Alfred Publishing Co. ISBN 0882847287.
  • Jackendoff, Ray and Fred Lerdahl (1981). "Generative Music Theory and Its Relation to Psychology." Journal of Music Theory.[page needed]
  • Josephs, Jess L. (1967). The Physics of Musical Sound. Princeton, Toronto, London: D. Van Nostrand Company, Inc.
  • Kostka, Stefan, and Dorothy Payne (2004). Tonal Harmony, fifth edition. New York: McGraw-Hill.
  • Kramer, Jonathan (1988). The Time of Music. New York: Schirmer Books.
  • Lerdahl, Fred (2001). Tonal Pitch Space. Oxford: Oxford University Press.
  • Lewin, David (1987). Generalized Musical Intervals and Transformations. New Haven: Yale University Press.
  • Lloyd, Llewellyn S., and Hugh Boyle (1978). Intervals, Scales and Temperaments. New York: St. Martin's Press. ISBN 0-312-42533-3
  • Olson, Harry F. (1967). Music, Physics and Engineering. New York: Dover Publications. ISBN 0486217698. http://books.google.com/?id=RUDTFBbb7jAC. 
  • Stevens, S. S., J. Volkmann, and E. B. Newman (1937). "A Scale for the Measurement of the Psychological Magnitude Pitch". Journal of the Acoustical Society of America 8, no. 3:185–90.
  • Yamaguchi, Masaya (2000). The Complete Thesaurus of Musical Scales. New York: Charles Colin. ISBN 0967635306. 

Further reading

  • Apel, Willi, and Ralph T. Daniel (1960). The Harvard Brief Dictionary of Music. New York: Simon & Schuster Inc. ISBN 0-671-73747-3
  • Benward, Bruce, Barbara Garvey Jackson, and Bruce R. Jackson. (2000). Practical Beginning Theory: A Fundamentals Worktext, 8th edition, Boston: McGraw-Hill. ISBN 0697343979. [First edition 1963]
  • Brown, James Murray (1967). A Handbook of Musical Knowledge‎, 2 vols. London: Trinity College of Music.
  • Chase, Wayne (2006). How Music REALLY Works!, second edition. Vancouver, Canada: Roedy Black Publishing. ISBN 1-897311-55-9 (book)
  • Hewitt, Michael (2008). Music Theory for Computer Musicians. USA: Cengage Learning. ISBN 13-978-1-59863-503-4.
  • Lawn, Richard J., and Jeffrey L. Hellmer (1996). Jazz Theory and Practice. [N.p.]: Alfred Publishing Co. ISBN 0-882-84722-8.
  • Owen, Harold (2000). Music Theory Resource Book. Oxford University Press. ISBN 0-19-511539-2.
  • Seashore, Carl (1933). Approaches to the Science of Music and Speech. Iowa City: The University.
  • Seashore, Carl (1938). Psychology of Music, New York, London, McGraw-Hill Book Company, Inc.
  • Sorce, Richard (1995). Music Theory for the Music Professional. [N.p.]: Ardsley House. ISBN 1-880-15720-9.
  • Taylor, Eric (1989). AB Guide to Music, vol 1. England: Associated Board of the Royal Schools of Music. ISBN 1-854-72446-0
  • Taylor, Eric (1991). AB Guide to Music, vol 2. England: Associated Board of the Royal Schools of Music. ISBN 1-854-72447-9
  • Yamaguchi, Masaya (2006). The Complete Thesaurus of Musical Scales, revised edition. New York: Masaya Music Services. ISBN 0967635306.

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