Music theory

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Music theory is the study of the structure of music. It includes considerations of melody, rhythm, counterpoint, harmony and form; tonal systems, scales, tuning, intervals, consonance, dissonance, durational proportions and the acoustics of pitch systems; composition, performance, orchestration, ornamentation, improvisation and electronic sound production; etc.[1]

Etymologically, music theory is an act of contemplation of music, from the Greek theōreō, to inspect, look at, behold, observe, contemplate, consider. As such, it has been opposed to practice, but such practical aspects as the creation or the performance of music, or their pedagogy, are today understood as belonging to the domain of music theory.[2]

Music theory usually expresses itself in a body of theoretical literature: the Western art music, the Byzantine, Arabic, Hebrew, Chinese and Indian traditions possess large bodies of such literature. Yet, some consider that theory may be unwritten and that its sources may consist also, for instance, in Ancient instruments or in modern traditional practices of music.

Some important music theorists include Aristotle, Pythagoras, Guido of Arezzo, Rameau, Schenker and Schoenberg. For a more comprehensive list, see List of music theorists.

Fundamentals of music[edit]

Main article: Aspect of music

Music is composed of phenomena of sound and “music theory” considers how those phenomena are and can be used in music. In the most general sense, “music theory” considers elements of music including, but not limited to pitch, harmony, rhythm, melody, texture, form, performance, and style.


Main article: Pitch (music)
Middle C (261.626 Hz) About this sound Play .

Pitch is the lowness or highness of a tone, for example the difference between middle C and a higher C. The frequency of the sound waves producing a pitch can be measured precisely, but the perception of pitch is more complex because we rarely hear a single frequency or pure pitch. In music, tones, even those sounded by solo instruments or voices, are usually a complex combination of frequencies, and therefore a mix of pitches. Accordingly, theorists often describe pitch as a subjective sensation.[3] Most people appear to possess relative pitch, which means they perceive each note relative to some reference pitch, or as some interval from the previous pitch. Significantly fewer people demonstrate absolute pitch (or perfect pitch), the ability to identify pitches without comparison to another pitch. 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.[4] In general, the higher the frequency of vibration, the higher the perceived pitch. The lower the frequency, the lower the pitch.[5] However, even for tones of equal intensity, perceived pitch and measured frequency do not stand in a simple linear relationship.[6]

Intensity (loudness) can change perception of pitch. Below about 1000 Hz, perceived pitch gets lower as intensity increases. Between 1000 and 2000 Hz, pitch remains fairly constant. Above 2000 Hz, pitch rises with intensity.[7] This is due to the ear's natural sensitivity to higher pitched sound, as well as the ear's particular sensitivity to sound around the 2000–5000 Hz interval,[8] the frequency range most of the human voice occupies.[9]

The difference in frequency between two pitches is called an interval. The most basic interval is the unison, which is simply two notes of the same pitch, followed by the slightly more complex octave: pitches that are either double or half the frequency of the other. The unique characteristics of octaves gave rise to the concept of what is called pitch class, an important aspect of music theory. Pitches of the same letter name that occur in different octaves may be grouped into a single "class" by ignoring the difference in octave. For example, a high C and a low C are members of the same pitch class—that class which contains all C's. The concept of pitch class greatly aids aspects of analysis and composition.[10]

Although pitch can be identified by specific frequency, the letter names assigned to pitches are somewhat arbitrary. For example, today most orchestras assign Concert A (the A above middle C on the piano) to the specific frequency of 440 Hz, rather than, for instance, 435HZ as it was in France in 1859. In England, that A varied between 439 and 452. These differences can have noticeable affect on the timbre of instruments and other phenomena. Many cultures do not attempt to standardize pitch, often considering that it should be allowed to vary depending on genre, style, mood, etc. In historically informed performance of older music, tuning is often set to match the tuning used in the period in which it was written. A frequency of 440 Hz was recommended as the standard pitch for Concert A in 1939, and in 1955 the International Organization for Standardization affirmed the choice.[11] A440 is now widely, though not exclusively, the standard for music around the world.

Pitch is also an important consideration in tuning systems, or temperament, used to determine the intervallic distance between tones, as within a scale. Tuning systems vary widely within and between world cultures. In Western culture, there have long been several competing tuning systems, all with different qualities. Internationally, the system known as Equal Temperament is most commonly used today because it is considered the most satisfactory compromise that allows instruments of fixed tuning (e.g. the piano) to sound acceptably in tune in all keys.

Scales and modes[edit]

Main articles: Musical scale and Musical mode
Pattern of whole and half steps in the Ionian mode or major scale on C About this sound Play .

Notes can be arranged in a variety of scales and modes. Western music theory generally divides the octave into a series of twelve tones, called a chromatic scale, within which the interval between adjacent tones is called a half step or semitone. In equal temperament each semitone is equidistant from the next, but other tuning systems are also used. Selecting tones from this set of 12 and arranging them in patterns of semitones and whole tones creates other scales. The most commonly encountered scales 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-tone scale, which is common in folk music and blues. Non-Western cultures often use scales that do not correspond with an equally divided twelve-tone division of the octave. For example, classical Ottoman, Persian, Indian and Arabic musical systems often make use of multiples of quarter tones (half the size of a semitone, as the name indicates), for instance in 'neutral' seconds (three quarter tones) or 'neutral' thirds (seven quarter tones) – they do not normally use the quarter tone itself as a direct interval, however.[12]

In traditional Western notation, the scale used for a composition is usually indicated by a key signature at the beginning to designate the pitches that make up that scale. As the music progresses, the pitches used may change and introduce a different scale. Music can be transposed from one scale to another for various purposes, often to accommodate the range of a vocalist. Transposition raises or lowers the overall pitch range, but preserves the intervallic relationships of the original scale. For example, transposition from the key of C major to D major raises all pitches of the scale of C major equally by a whole tone. Since the interval relationships remain unchanged, transposition may be unnoticed by a listener, however other qualities may change noticeably because transposition changes the relationship of the overall pitch range compared to the range of the instruments or voices that perform the music. This often affects the music's overall sound of the music, as well as having technical implications for the performers.[13]

The interrelationship of the keys most commonly used in Western tonal music is conveniently shown by the circle of fifths. Unique key signatures are also sometimes devised for a particular composition. During the Baroque period, emotional associations with specific keys, known as the Doctrine of the Affections, were an important topic in music theory, but the unique tonal colorings of keys that gave rise to that Doctrine were largely erased with the adoption of Equal Tempered tuning. However, many musicians continue feel that certain keys are more appropriate to certain emotions than others. Indian classical music theory continues to strongly associate keys with emotional states, times of day, and other extra-musical concepts and notably, does not employ equal temperament.

Consonance and dissonance[edit]

Consonance and dissonance are subjective qualities of the sonority of intervals that vary widely in different cultures and over the ages.

Consonance (or concord) is the quality of an interval or chord which seems stable and complete in itself. Dissonance (or discord) is the opposite in that it feels incomplete and “wants to” resolve to a consonant interval. Dissonant intervals seem to clash. Consonant intervals seem to sound comfortable together. Commonly, perfect fourths, fifths, and octaves and all major and minor thirds and sixths are considered to be consonant. All others are dissonant to greater or lesser degree. However, context and many other aspects can affect apparent dissonance and consonance. For example, in a Debussy prelude, a major second may sound stable and consonant, while the same interval may sound dissonant in a Bach fugue. In the Common Practice era, the perfect fourth is considered dissonant when not supported by a lower third or fifth. Since the early 20th century, Arnold Schoenberg’s concept of “emancipated” dissonance, in which traditionally dissonant intervals can be treated as “higher,” more remote consonances, has become more widely accepted.[14]

Dissonance is an essential element of music and used in most every culture and genre, not only for effect, but as a fundamental structural element to create motion and tension. J.S. Bach’s music depends in great part on the effect of dissonance. The art of melody writing depends heavily upon the selection of consonant and dissonant tones.


Main article: Rhythm
Metric levels: beat level shown in middle with division levels above and multiple levels below.

Rhythm is produced by the sequential arrangement of sounds and silences in time. Meter measures music 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 or felt as a single beat. Through increased stress, or variations in duration or articulation, particular tones may be accented. There are conventions in most musical traditions for regular and hierarchical accentuation of beats to reinforce a given meter. Syncopated rhythms contradict those conventions by accenting 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.


Main article: Chord (music)

A chord is group of tones heard or conceived as sounding simultaneously. In some instances, tones of a chord not sounding simultaneously but successively, for example in an arpeggio, in compound melody, or in the style brisé, may still be considered to form a chord. There is on-going debate about the minimum number of tones required to constitute a chord: two, or three (see chord). The exploitation of chords and their patterns of succession or progression is a prominent feature of Western music, but is also found in other cultures. The study of chords is a primary concern of music theory for many reasons, but especially their significance in, and effect upon harmony, counterpoint, form, and tension and release.

In common-practice harmony, chords are often formed of three tones in stacked thirds; such three-tone chords are called triads. They consist of a primary tone called the root, a second tone a third above the root, and another tone a third above that (that is, a fifth above the root). Triads take the name of their root tone (e.g. C, F, or E) and are further described by the quality of the intervals between their tones (e.g. A major, C minor, E diminished). Chords may be inverted by changing the vertical arrangement of tones, extended by adding tones which in the common definition of consonance and dissonance necessarily are dissonant, or altered by modifying one or several of their tones usually by a chromatic semitone. Seventh chords consist of a triad plus an additional tone a third above the fifth, i.e. a major seventh (in the case of major triads), minor seventh (in the case of diminished, minor, or major triads), or diminished seventh (only in the case of diminished triads) above the root. Clusters are built by vertical arrangements of adjacent tones.


Main article: Melody

A melody is a series of tones sounding in succession that typically move toward a climax of tension then resolve to a state of rest. Because melody is such a prominent aspect in so much music, its construction and other qualities are a primary interest of music theory.

The basic elements of melody are pitch, duration, rhythm, and tempo. The tones of a melody are usually drawn from pitch systems such as scales or modes. Melody may consist, to increasing degree, of the figure, motive, semi-phrase, antecedent and consequent phrase, and period or sentence. The period may be considered the complete melody, however some examples combine two periods, or use other combinations of constituents to create larger form melodies.[16]


Main article: Harmony
IV-V-I progression in C About this sound Play 

Harmony is the study of vertical sonorities in music. Vertical sonority is produced by the relationships of 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. How tones sound together and in succession to create what we recognize as music is a principal concern in music theory.

In Western music of the Common Practice Era, harmonies are generally tertian. This means that the intervals of which the chords are composed are a third—not that the chord is necessarily a triad (comprised of three notes). Quartal and quintal harmony are built on the interval of a fourth and fifth, respectively. In tertian harmony, 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 fourth note, a third above the top note of a triad (a seventh above the root). In 20th century classical music and jazz, many alternative types of harmony are explored: modal, quartal, etc. One way to analyze harmony is through a Roman numeral system whereby Roman numerals are used to identify chords based on their scalar roots (I through VII). in popular music and jazz a system of chord symbols is commonly used (Cmaj7, E9, etc.). post-tonal music employs a variety of approaches, most frequently set theory.


Main article: Musical texture
Introduction to Sousa's "Washington Post March," m. 1-7About this sound Play  features octave doubling (Benward & Saker 2003, 133) and a homorhythmic texture.

Musical texture is the overall sound of a passage or complete composition, 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 can also be affected by the timbre of the instruments, the number of instruments used, and the intervallic distance between each musical line, among other things. Its theoretical interest includes its effects on perception, form, and style.


Main article: Timbre

Timbre, sometimes called "color", or "tone color," is the principal phenomenon that allows us to distinguish one instrument from another when both play at the same pitch and volume, a quality of a voice or instrument often described in terms like bright, dull, shrill, etc. It is of considerable interest in music theory, especially because it is one component of music that has as yet, no standardized nomenclature. It has been called "...the psychoacoustician's multidimensional waste-basket category for everything that cannot be labeled pitch or loudness,"[17] but can be accurately described and analyzed by Fourier analysis and other methods[18] because it results from the combination of all sound frequencies, attack and release envelopes, and other qualities that comprise a tone.

Timbre is principally determined by two things: (1) the relative balance of overtones produced by a given instrument due its construction (e.g. shape, material), and (2) the envelope of the sound (including changes in the overtone structure over time). Timbre varies widely between different instruments, voices, and to lesser degree, between instruments of the same type due to variations in their construction, and significantly, the performer's technique. The timbre of most instruments can be changed by employing different techniques while playing. For example, the timbre of a trumpet changes when a mute is inserted into the bell, the player changes their embouchure, or volume. A voice can change its timbre by the way the performer manipulates their vocal apparatus, (e.g. the shape of the vocal cavity or mouth). Musical notation frequently specifies alteration in timbre by changes in sounding technique, volume, accent, and other means. These are indicated variously by symbolic and verbal instruction. For example, the word dolce (sweetly) indicates a non-specific, but commonly understood soft and “sweet” timbre. Sul tasto instructs a string player to bow near or over the fingerboard to produce a less brilliant sound. Cuivre instructs a brass player to produce a forced and stridently brassy sound. Accent symbols like marcato (^) and dynamic indications (pp) can also indicate changes in timbre.


Main article: Musical expression

Expression is created by nuances of any phenomena of sound including timbre, variation of pitch, tempo, volume, etc. Due to its great effect on perception and emotional response, it is of particular interest in music theory. Although frequently indicated in music notation verbally or by symbols, those indications are imprecise in comparison to elements like pitch, and so highly dependent on the interpretation and performance of the player. For example, although pianissimo pp and the word dolce indicate a low volume and sweet or tender feeling, precisely how quietly and with what technique it may be played is subject to the player’s interpretation. Common music notation is incapable of directing every aspect of a player's performance.


Main article: Dynamics (music)
Music hairpins.svg

In music, “dynamics” normally refers to variations of intensity or volume, as may be measured by physicists and audio engineers in decibels or phons. In music notation, however, dynamics are not treated as absolute values, but a relative ones. Because they are usually measured subjectively, there are factors besides amplitude that affect the performance or perception of intensity, such as timbre, vibrato, and articulation. The conventional indications of dynamics are abbreviations for Italian words like forte (f) for loud and piano (p) for soft. These two basic notations are modified by indications including mezzo piano (mp) for moderately soft (literally "half soft") and mezzo forte (mf) for moderately loud, sforzando or sforzato (sfz) for a surging or "pushed" attack, or fortepiano (fp) for a loud attack with a sudden decrease to a soft level. The full span of these markings usually range from a nearly inaudible pianissississimo (pppp) to a loud-as-possible fortissississimo (ffff). Greater extremes of pppppp and fffff and nuances such as p+ or più piano are sometimes found. Other systems of indicating volume are also used in both notation and analysis: dB (decibels), numerical scales, colored or different sized notes, words in languages other than Italian, and symbols such as those for progressively increasing volume (crescendo) or decreasing volume (decrescendo), often called "hairpins" when indicated with diverging or converging lines.


Main article: Articulation (music)
Examples of articulations. From left to right: staccato, staccatissimo, martellato, marcato, tenuto.

Articulation is the manner in which the performer sounds notes. For example, staccato is the shortening of duration compared to the written note value, legato performs the notes in a smoothly joined sequence with no separation. 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 to reduce the notated duration. But, for example, violin players use a variety of techniques to perform different qualities of staccato. The manner in which a performer decides to execute a given articulation is usually based on the context of the piece or phrase, but many articulation symbols and verbal instructions depend on the instrument and musical period (e.g. viol, wind; classical, baroque; etc.). There is a set of articulations that most all instruments and voices perform in common. They are, in order of long to short: legato (smooth, connected); tenuto (pressed or played to full notated duration); marcato (accented and detached); staccato ("separated", "detached"); martelé (heavily accented or "hammered"). Many of these can 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 bowed strings, where the bow bounces off the string.

Form or structure[edit]

Main article: Musical form

Form is an important area of music theory that considers the structure, both local and global, of a composition. Examples of common forms of Western music include sonata-allegro, canon, strophic, theme and variation, and rondo. Popular Music often makes use of ballad form many times in conjunction with twelve-bar blues. In Indian music, Raga is an important form. Although classification of a musical form is useful in study and appreciation, composers vary and combine forms to such a degree that the model or "the rule" is usually outnumbered by the exceptions. Fugue is often considered as a distinct form, and even though many compositions are titled as such, most fugues bear little structural resemblance to others. The aspect they have in common is often only the process by which melodies are combined and altered. Accordingly, fugue is perhaps more accurately considered to be a process, rather than a form.

Performance and Style[edit]

Main article: Music genre

Since music is generally written to be performed, consideration of performance and style is inherent in music theory. Notation is the attempt by the composer to clearly and accurately communicate how the music is intended to be performed. Theory also considers performance practices and standards from previous eras, many of which are based on style or genre, especially the problem of changing interpretation of symbols and verbal instructions over time. For examples, symbols for mordents and turns and tempo indications such as Andante were performed differently in the Baroque period than today. The violin bow was shaped in a high arc, like a hunting bow, in the Renaissance and early Baroque, with significant effect on performance technique.

Performance is an integral aspect of style, but style also includes consideration of forces (number of players in sections), interpretation of notation markings like staccato, genre, and many other aspects.

Theories of harmonization[edit]

Four-part writing [edit]

Main article: Four-part harmony
Four-voice texture in the Genevan psalter: Old 124th.[19] 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.[20] In the German musicology tradition referred to as functional harmony.[citation needed] 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.[citation needed] 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.[citation needed] 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[edit]

Further information: Music cognition, Fred Lerdahl and Ray Jackendoff

Serial composition and set theory[edit]

Tone row from Alban Berg's Lyric Suite, mov. I. About this sound Play 

Musical semiotics[edit]

Further information: music semiology and Jean-Jacques Nattiez

Music subjects[edit]


Main articles: Musical notation and Sheet music
Tibetan musical score from the 19th century.

Musical notation is the written or symbolized representation of music. This is most often achieved by the use of commonly understood graphic symbols and written verbal instructions and their abbreviations. Computer file formats have become important as well.[21] Spoken language and hand signs are also used to symbolically represent music, primarily in teaching.

In standard Western music notation, tones are represented graphically by symbols (notes) placed on a staff or staves, the vertical axis corresponding to pitch and the horizontal axis corresponding to time. Note head shapes, stems, flags, ties and dots are used to indicate duration. Additional symbols indicate keys, dynamics, accents, rests, etc. Verbal instructions are often used to indicate tempo, technique, and other aspects.

There are many systems of music notation from different cultures and different ages. Traditional Western notation evolved during the Middle Ages and continues to be an area of experimentation and innovation.[22]


Main article: Music and mathematics

Mathematics are an integral part of music theory and have been for thousands of years. Early analyses of harmony and intervals (see Pythagoras) were often done with mathematical principles (e.g. dividing a string in half to find the octave, or the pitch that is twice the frequency of the fundamental tone). Maths are also used to consider form and structure, stylistic tendencies, instrument manufacture, tuning systems, and many other aspects. Some methods of composition, especially computer-assisted composition, rely on mathematics. (see Computer music) Many electronic instruments use a mathematical system known as MIDI (musical instrument digital interface) to specify and control pitch, duration, volume, tempo and other aspects of sound.


Bass prolongation: I–IV–V–I About this sound Play  as elaboration of I–V–I About this sound Play .

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[edit]

Main article: 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 recognize a particular audio frequency as a given musical note without any prior reference.

See also[edit]



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  • 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.
  • Stone, Kurt (1980). Music Notation in the Twentieth Century. New York: W. W. Norton & Co. ISBN 978-0-393-95053-3. 
  • Touma, Habib Hassan (1996). The Music of the Arabs, new expanded edition, translated by Laurie Schwartz. Portland, OR: Amadeus Press. ISBN 0-931340-88-8.
  • Yamaguchi, Masaya (2000). The Complete Thesaurus of Musical Scales. New York: Charles Colin. ISBN 0-9676353-0-6. 

Further reading[edit]

External links[edit]