Tonality is a system of writing music according to certain hierarchical pitch relationships around a "center" or tonic. The term tonalité was borrowed from Castil-Blaze (1821, François Henri Joseph Blaze) by François-Joseph Fétis in 1840 (Reti, 1958; Judd, 1998; Dahlhaus). The term is often used synonymously with major/minor tonality; however, in more recent theory, the term is used more broadly to encompass a number of systems of musical organization.
Uses of the term
What is now known as tonality originated through centuries of musical practice, during which it was not known by any name, and was defined, and its features compiled, by theorists such as Heinrich Schenker in reaction to music which broke with tradition (nontonal music). Arising from sometimes disparate practices over a large area and period of time, tonality may thus be defined in various ways:
- By history and geography: The music of a specific time period and location, such as that of the common practice period of European music from after the Renaissance to before Modernism. In this context it generally means major-minor tonality plus the use of additional scales such as the semi-tone scale, pentatonic modes and others such as the Magic Scale of alternating tones and semitones.
- By characteristics: By extension, the above music and all other music which shares the characteristics of the above music (and does not display characteristics counter to those of the above music). These characteristics may include the use of the major scale or minor scale, their triadic chords and diatonic functions, and the compositional techniques, procedures, and materials used. This would include theories of tonality which focus on the thoroughbass rather than on chords, and alternate systems of tuning such as monotonic.
- By nature: As music which corresponds to or uses the characteristics of sound, organization or order, and/or perception. Thus tonality is a practice correctly based on physical or psychological constants such as the overtone series or human perception. The most current viewpoint showing tonality arises from overtones can be found at http://www.greenwych.ca/natbasis.htm and also in the 2004 book: "On the Origin of Music" by Bob Fink (Greenwich Publ., Canada). The theory is called the "trio theory," claiming that influence from the most audible overtones of the three most nearly universal intervals (found across time & cultures), namely, a tone's octave, 4th and 5th, when placed within the range of that octave, will evolve into the most widespread of scales: Pentatonic, major & minor (depending how many of the audible overtones are so placed). The unequal audibile strengths of the overtones determine the role & power of each note in a scale (tonic, dominant or subdominant) -- i.e., tonality.
- By contrast: Tonal music may simply be contrasted with atonal music, music which does not feel as if it has a center. More subtly, it may be contrasted with earlier modal music.
One may clarify between "the principle of tonality", "the requirement that all the events in a musical group should be co-ordinated by, and experienced in relation to, a central point of reference," and "tonality" as "the specific language of 'classical tonality' - the major-minor key system of the Classical and Romantic periods." (Samson 1977)
Functional tonality, or sometimes narrative tonality, is the use of chords and other features according to their functions or relationship with the tonic (so that they "go somewhere"). "Nonfunctional" tonality such as is the use of tonal characteristics in nontonal successions or without regard to their role (so that they "go nowhere"). Examples include the pandiatonicism of Aaron Copland or Steve Reich which often consists of tonal or tonal added tone chords (trouves or "finds" as Aaron Copland described some of his own nonfunctional tonality).
Extended tonality is "the incorporation of complex harmonic phenomena within a single tonal region, as in much of the music of the late nineteenth and early twentieth centuries." (Samson 1977)
Vocabulary of tonal analysis
Many of the terms and symbols necessary to analyze tonal organization follow below.
- Main article: Scale (music).
Since the mid-18th century, tonal music has been increasingly composed of a 12-note chromatic scale in a system of equal temperament. Tonal music makes reference to "scales" of notes selected as a series of steps from the chromatic scale. Most of these scales are of 5, 6 or 7 notes with the vast majority of tonal music pitches conforming to one of four specific seven-note scales: major, natural minor, melodic minor, and harmonic minor. These scales are referred to as "diatonic" because it is felt that there are two fundamental centers of attraction: the tonic note and the dominant note.
C major scale:
A natural minor scale:
Other scales or modes are often introduced for variety within the context of a major-minor tonal system without disturbing the diatonic nature of the work. The major scale predominates and the melodic minor contains nine pitches (seven with two alterable). The seven basic notes of a scale are notated in the key signature, and whether the piece is in the major or minor key is either stated in the title or implied in the piece (there is a major and minor key for each key signature). While other scales and modes are used in tonal music, particularly after 1890, these two scales are the reference point for most tonal music and its vocabulary.
Tonal music composed in other scale systems is referred to as microtonal, and while microtonal music draws from tonal theory, it is generally treated separately in text books and other works on music. However, within the tonal system, notes "between" the chromatic system are used in various contexts, including quarter tones and various effects such as portamento or glissando, where the instrumentalist moves between established notes of the chromatic scale. These are usually thought of as being for "colour" rather than harmonic function, and do not disturb the fundamental scale being used.
Chords are built from notes on a scale or on chromatic notes, which are supposed to be heard as variations of the basic scale. The identity of the scale is important in that the scale's steps number the system of chord relationships. At any given time one scale is heard as the most important, and the chord, almost always the major or minor triad, is heard as the most forceful closure.
- Main article: Scale degree.
- Main article: Chord (music).
These numerals also may indicate chords which are built upon the indicated degree. This degree is then known as the root of that chord. Thus "I" describes the tonic chord, the chord built on the tonic note, at a given time. These chords are generally all triads (having three notes, built from thirds, and having a diatonic function).
The degree of a scale is both the pitch (frequency) of that note and that pitch's diatonic function (role), which is why chords are named by scale degree. Thus the notes of a chord do not have to be sounded simultaneously, and one to two notes may function as, or imply, a three (or more) note chord. Thus a chord described as "V" is based on the fifth note of the prevailing tonic scale (V-VII-II). In C Major, that would be a triad based on G, and would be the G Major triad (G-B-D). To describe a chord progression, the Roman numerals of the chords are listed. Thus IV-V-I describes a chord progression of a chord based on the fourth note of a scale, then one based on the fifth note of the scale, and then one on the first note of the scale.
Chords are then further named according to their quality or makeup, determined by the scale notes which lie a third and fifth (two thirds) above the degree a chord is built upon. Capital Roman numerals refer to the major chord, and lower-case Roman numerals refer to the minor chord. Quality is generally not as important as the chord's root.
This means that in the traditional major scale, the ii, iii and vi are minor chords, where as I, IV, V are major. The chord on the seventh note is a diminished triad and is written vii with a degree sign. Numbers attached to a chord indicate additional notes, and one of the most important chords in tonal harmony is the V7 chord, which is a four note chord that includes the fourth note of the tonic scale. The "7" refers to a note seven diatonic steps up from the fundamental note of the chord, not the seventh note of the tonic scale.
- Main article: Inversion (music).
A chord's root is determined by which note establishes the chord's relationship to the tonic and not which is in the bass, or the lowest played note. Thus chords are said to be "inverted" when this root note is not sounded as the lowest. For example in C Major C-E-G is the tonic chord. If C is not the lowest note played, it is said to be in "inversion". The first inversion would be E-G-C, and the second inversion would be G-C-E. Since inverted chords are also chords in their own right, in context a chord is sometimes thought to be inverted only when voice leading implies it.
- Main article: form.
The traditional form of tonal music begins and ends on the tonic of the piece, and many tonal works move to a closely related key, such as the dominant of the main tonality (for example sonata form). Establishing a tonality is traditionally accomplished through a cadence which is two chords in succession which give a feeling of completion or rest - the most common being V7-I cadence. Other cadences are considered to be less powerful. The cadences determines the form of a tonal piece of music, and the placement of cadences, their preparation and establishment as cadences, as opposed to simply chord progressions, is central to the theory and practice of tonal music.
- Main article: Harmony.
Most tonality uses "functional harmony", which is a term used to describe music where changes in the predominate scale or additional notes to chords are explainable by their place in stabilizing or destabilizing a tonality. This is a complex way of saying that it is possible to explain why a particular note was included, and what that note means in relation to the tonic chord. Harmony with a large number of notes which do not have clear structural function is called "nonfunctional" harmony, which is not to imply "dysfunctional", but merely that the additional notes are not to be played or heard as restricting or advancing the harmonic progression.
Consonance and dissonance
- Main article: Consonance and dissonance.
In the context of tonal organization a chord or a note is said to be "consonant" when it implies stability, and "dissonant" when it implies instability. This is not the same as the ordinary use of the words consonant and dissonant. A dissonant chord is in tension against the tonic, and implies that the music is distant from that tonic chord. "Resolution" is the process by which the harmonic progression moves from dissonant chords to consonant chords and follows counterpoint or voice leading. Voice leading is a description of the "horizontal" movement of the music, as opposed to chords which are considered the "vertical".
To summarize, traditional tonal music is described in terms of a scale of notes. On the notes of that scale are built chords. Chords in order form a progression. Progressions establish or deny a particular chord as being the tonic chord. The cadence is held to be the sequence of chords which establishes one chord as being the tonic chord; more powerful cadences create a greater sense of closure and a stronger sense of key. Chords have a function when it can be explained how they lead the music towards or away from a particular tonic chord. When the sense of which tonic chord is changed, the music is said to have "changed key" or "modulated". Roman numerals and numbers are used to describe the relationship of a particular chord to the tonic chord.
The techniques of accomplishing this process, are the subject of tonal music theory and compositional practice.
Carl Dahlhaus (1990) lists the characteristic schemata of tonal harmony, "typified in the compositional formulas of the 16th and early 17th centuries," as the "complete cadence" (vollstandige Kadenz), I-IV-V-I, I-IV-I-V-I, or even I-ii-V-I; the circle of fifths progression: I-IV-vii-iii-vi-ii-V-I, and the "major-minor parallelism", minor: v-i-VII-III = major: iii-vi-V-I or minor: III-VII-i-v = major: I-V-vi-iii.
Theory of tonal music
Tonality allows for a great range of musical materials, structures, meanings, and understandings. It does this through establishing a tonic, or central chord based on a pitch which is the lowest degree of a scale, and a somewhat flexible network of relations between any pitch or chord and the tonic similar to perspective in painting. This is what is meant by tonality having a hierarchical relationship, one triad, the tonic triad, is the "center of gravity" to which other chords are supposed to lead. Changing which chord is felt to be the tonic triad is referred to as "modulation". As within a musical phrase, interest and tension may be created through the move from consonance to dissonance and back, a larger piece will also create interest by moving away from and back to the tonic and tension by destabilizing and re-establishing the key. Distantly related pitches and chords may be considered dissonant in and of themselves since their resolution to the tonic is implied. Further, temporary secondary tonal centers may be established by cadences or simply passed through in a process called modulation, or simultaneous tonal centers may be established through polytonality. Additionally, the structure of these features and processes may be linear, cyclical, or both. This allows for a huge variety of relations to be expressed through dissonance and consonance, distance or proximity to the tonic, the establishment of temporary or secondary tonal centers, and/or ambiguity as to tonal center. Music notation was created to accommodate tonality and facilitates interpretation.
The majority of tonal music assumes that notes spaced over several octaves are perceived the same way as if they were played in one octave or octave equivalency. Tonal music also assumes that scales have harmonic implication or diatonic functionality. This is generally held to imply that a note which has different places in a chord will be heard differently, and that therefore there is not enharmonic equivalency. In tonal music chords which are moved to different keys, or played with different root notes are not perceived as being the same, and thus transpositional equivalency and far less still inversional equivalency are not generally held to apply.
A successful tonal piece of music, or a successful performance of one, will give the listener a feeling that a particular chord — the tonic chord — is the most stable and final. It will then use musical materials to tell the musician and the listener how far the music is from that tonal center, most commonly, though not always, to heighten the sense of movement and drama as to how the music will resolve the tonic chord. The means for doing this are described by the rules of harmony and counterpoint, though some influential theorists prefer the term "thoroughbass" instead of harmony, the concept is the same. Counterpoint is the study of linear resolutions of music, while harmony encompasses the sequences of chords which form a chord progression.
Though modulation may occur instantaneously without indication or preparation, the least ambiguous way to establish a new tonal center is through a cadence, a succession of two or more chords which ends a section and/or gives a feeling of closure or finality, or series of cadences. Traditionally cadences act both harmonically to establish tonal centers and formally to articulate the end of sections, just as the tonic triad is harmonically central, a dominant-tonic cadence will be structurally central. The more powerful the cadence, the larger the section of music it can close. The strongest cadence is the perfect authentic cadence, which moves from the dominant to the tonic, most strongly establishes tonal center, and ends the most important sections of tonal pieces, including the final section. This is the basis of the "dominant-tonic" or "tonic-dominant" relationship. Common practice placed a great deal of emphasis on the correct use of cadences to structure music, and cadences were placed precisely to define the sections of a work. However, such strict use of cadences gradually gave way to more complex procedures where whole families of chords were used to imply particular distance from the tonal center. Composers, beginning in the late 18th Century began using chords (such as the Neapolitan, French or Italian Sixth) which temporarily suspended a sense of key, and by freely changing between the major and minor voicing for the tonic chord, thereby making the listener unsure whether the music was major or minor. There was also a gradual increase in the use of notes which were not part of the basic 7 notes, called chromaticism, culminating in post-Wagnerian music such as that by Mahler and Strauss and trends such as impressionism and dodecaphony.
One area of disagreement, going back to the origin of the term tonality, is whether, and to what degree, tonality is "natural" or inherent in music, and whether, and to what degree, it is constructed by the composer, performer and listener. The arguments involved are too complex to summarize, and it is difficult to draw clear lines. There is also a disagreement as to how "natural" the practice of Western tonal harmony is versus other forms of harmony, and what grounds would prove or disprove this hypothesis. Since these arguments are often centered around what kind of music should be performed and taught, they often assume a vehemence or dogmatism which goes beyond the nominal issue of hearing and perception.
A current viewpoint among many laypersons and scholars indicates tonal scales and tonality arise from natural overtones. Most of the archaeological evidence regarding this has been found only in the last several decades, and most of it, if not all, supports many earlier claims of the universal or "natural" evolution of the scales most widely found in human music. The evidence for this now includes the recent "Neanderthal Flute," 50,ooo years old; The world's oldestn known song (Assyrian cuniform artifacts) 4,ooo years old; and the recent find of many 9,ooo year-old flutes in China, one of them fully still playable with 8 notes, including the octave. The links below discuss these archaeological finds. The finds, by independent archaeologists, reveal similarlities to present day widespread musical scales. One published theory involved to which these finds provide support, originally published in 1958, is called the trio theory, claiming that influence from the most audible overtones of the three most nearly universal intervals (found across time & cultures), namely, a tone's octave, 4th and 5th, when placed within the range of that octave, will evolve into the most widespread of scales: Pentatonic, major & minor (depending how many of the audible overtones are so placed). The unequal audibile strengths of the overtones determine the role & power of each note in a scale (tonic, dominant or subdominant) -- i.e., tonality and tonal scales.
History of the term
Theories of tonal music are generally dated from Jean-Philippe Rameau's Treatise on Harmony (1722), where he describes music written through chord progressions, cadences and structure. He claims that his work represents "the practice of the last 40 years [1682-1722]", however, this is probably not the case. Rameau's work, initially controversial, was adopted by Friedrich Wilhelm Marpurg (1718-1795) in his explanation of the music of Johann Sebastian Bach. The vocabulary of describing notes in relationship to the tonic note, and the use of harmonic progressions and cadences becomes absorbed into the practice of Bach. Essential to this version of tonal theory are the chorales harmonizations of Bach, and the method by which a church melody is given a four part harmony by assigning cadences, and then creating a "natural", meaning in this case the most direct, thoroughbass and then filling in the middle voices.
In 1821 Castil-Blaze used tonalité for what he called cordes tonales (today primary triads), the tonic, fourth (subdominant), and fifth (dominant). All other chords were cordes melodiques. Hugo Riemann (n.d.) defined tonality as, "the special meaning [functions] that chords receive through their relationship to a fundamental sonority, the tonic triad."
Fétis (1844) defined tonality, specifically tonalité moderne as the, "set of relationships, simultaneous or successive, among the tones of the scale," allowing for other types de tonalités among different cultures. Further he considered tonalité moderne as "trans-tonic order" and tonalité ancienne "uni-tonic order", trans-tonic meaning simply that the dominant seventh both establishes the key and allows for modulation to other keys. He described his earliest example of tonalité moderne: "In the passage quoted here from Monteverdi's madrigal [Cruda amarilli, mm.9-19 and 24-30], one sees a tonality determined by the accord parfait [root position major chord] on the tonic, by the sixth chord assigned to the third and seventh degrees, by the optional choice of the accord parfait or the sixth chord on the sixth degree, and finally, by the accord parfait and, above all, by the unprepared seventh chord (with major third) on the dominant." (p.171)
Fétis believed that tonality, tonalite moderne, was entirely cultural, "For the elements of music, nature provides nothing but a multitude of tones differing in pitch, duration, and intensity by the greater or least degree...The conception of the relationships that exist among them is awakened in the intellect, and, by the action of sensitivity on the one hand, and will on the other, the mind coordinates the tones into different series, each of which corresponds to a particular class of emotions, sentiments, and ideas. Hence these series become various types of tonalities." (p.11f) "But one will say, 'What is the principle behind these scales, and what, if not acoustic phenomena and the laws of mathematics, has set the order of their tones?' I respond that this principle is purely metaphysical [anthropological]. We conceive this order and the melodic and harmonic phenomena that spring from it out of our conformation and education." (p.249) In contrast, Hugo Riemann believed tonality, "affinites between tones" or tonverwandtshaften, was entirely natural and, following Moritz Hauptmann (1853), that the major third and perfect fifth where the only "directly intelligible" intervals, and that I, IV, and V, the tonic, subdominant, and dominant where related by the perfect fifths between their roots. (Dahlhaus 1990, p.101-2)
By the 1840's the practice of harmony had expanded to include more chromatic notes, a wider chord vocabulary, particularly the more frequent used of the diminished seventh chord - a four note chord of all minor triads which could lead to any other chord. It is in this era that the word "tonality" becomes more commonly used. At the same time the elaboration of both the fugue and the sonata form in terms of key relationships becomes more rigorous, and the study of harmonic progressions, voice leading and ambiguity of key becomes more precise.
Theorists such as Edward Lowinsky, Hugo Riemann, and others pushed the date at which modern tonality began, and the cadence began to be seen as the definitive way that a tonality is established in a work of music (Judd, 1998).
In response Bernhard Meier instead used a "tonality" and "modality", modern vs ancient, dichotomy, with Renaissance music being modal. The term modality has been criticized by Harold Powers, among others. However, it is widely used to describe music whose harmonic function centers on notes rather than on chords, including some of the music of Bartok, Stravinsky, Vaughan Williams, Charles Ives and composers of minimalist music. This and other modal music is, broadly, often considered tonal.
In the early 20th century the vocabulary of tonal theory is decisively influenced by two theorists: composer Arnold Schoenberg whose work "Harmonielehre" describes in detail chords, chord progressions, vagrant chords, creation of tonal areas, voice leading in terms of harmony. To Schoenberg, every note has "structural function" to assert or deny a tonality, based on its tendency to establish or undermine a single tonic triad as central. At the same time Heinrich Schenker is evolving a theory based on expansion of horizontal relationships. To Schenker the background of every successful tonal piece is based on a simple cadence, which is then elaborated and elongated in the middle ground and the background. Though adherents of the two theorists argued back and forth, in the mid-century a synthesis of their ideas was widely taught as "tonal theory", most particularly Schenker's use of graphical analysis, and Schoenberg's emphasis on tonal distance.
The practice of jazz developed its own theory of tonality, stating that while the cadence is not central to establishing a tonality - the presence of the I and V chords and either the IV or ii chord in progression is. This theory emphasized the play of modal elements against tonal elements, in an effort to allow improvisation, and inflection of standard melodies. Among theorists influenced by this view are Meier, Schillinger and the be-bop school of Jazz.
While many regard the works of Schoenberg post 1911 as "atonal", see atonality, one influential school of thought, to which Schoenberg himself belonged, argued that chromatic composition lead to a "new tonality", this view is argued by George Perle in his works on "post diatonic tonality". The central idea of this theory is that music is always perceived as having a center, and even in a fully chromatic work, composers establish and disintegrate centers in a manner analogous to traditional harmony. This view is highly controversial, and remains a topic of intense debate.
However, tonality may be considered generally with no restrictions as to the date or place at which the music was produced, or (very little) restriction as to the materials and methods used. This definition includes much non-western music and western music before 17th century. In fact, many people, including Anton Webern, consider all music to be tonal in that music is always perceived as having a center. Centric is sometimes used to describe music which is not traditionally tonal in that it used triads of a diatonic scale but which nevertheless has relatively strong tonal center. Other terms which have been used in an attempt to clarify are tonical and tonicality, as in "possessing a tonic," and Igor Stravinsky used the term polar. See: pitch center.
In the early 20th century, the definition of tonality which was held to have prevailed since the 1600's was felt to reach a crisis or break down point. The belief was that tonality had "snapped" because of expansion of vocabulary, decreased functionality, increased use of leading tones, alterations, modulations, tonicization, the increased importance of subsidiary key areas, use of non-diatonic hierarchical methods, and/or symmetry in interval cycles. This "crisis" lead to a series of responses, many of which were considered irreconcilable with tonal theory or tonality at all. At the same time, other composers and theorists maintained that tonality had been stretched but not broken. This led to more technical vocabularies to describe tonality, including pitch classes, pitch sets, graphical analysis, and describing works in terms, not of their notes, but of their dominant intervals.
While tonality is the most common form of organizing Western Music, it is not universal, nor is the seven note scale universal, much folk music and the art music of many cultures focus on a pentatonic, or five note scale, including Beijing Opera, the folk music of Hungary, and the musical traditions of Japan. (But see another perspective on this: The 7-Note Solution -- Why so many 5 and 7 scales are consistently unearthed from the ancient world). Pre-classical concert music was largely modal, as is much folk and some popular music. In the early 20th century many techniques were developed and applied to tonal music, such as non-tertian secundal or quartal music. Some, such as Benjamin Boretz, consider tonal theory a specific part of atonal theory or musical set theory, which is in turn part of a more general theory of music. Many composers such as Darius Milhaud and Philip Glass have been interested in bitonality. While at one point in the middle of the 20th century classical composers interested in the twelve tone technique and serialism declared tonality dead, many composers have since returned to tonality, including many minimalists and older composers such as George Rochberg. Other composers never abandoned tonality entirely such as Lou Harrison who says he has "always composed both modally and chromatically." (Harrison, 1992) Much music today that is described as tonal is nonfunctional tonality such as in that of Claude Debussy, Steve Reich, Aaron Copland and many others.
Effect of tonality
Rudolph Réti differentiates between harmonic tonality, of the traditional homophonic kind, and melodic tonality, as in monophonic. He argues that in the progression I-x-V-I (and all progressions), V-I is the only step "which as such produces the effect of tonality," and that all other chord successions, diatonic or not, though being closer or farther from the tonic-dominant, are "the composer's free invention." He describes melodic tonality as being "entirely different from the classical type," wherein, "the whole line is to be understood as a musical unit mainly through its relationship to this basic note [the tonic]," this note not always being the tonic that would be interpreted according to harmonic tonality. His examples are ancient Jewish and Gregorian chant and other Eastern music, and he points out how these melodies often may be interrupted at any point and returned to the tonic, yet harmonically tonal melodies, such as that from Mozart's The Magic Flute below, are actually "strict harmonic-rhythmic pattern[s]," and include many points "from which it is impossible, that is, illogical, unless we want to destroy the innermost sense of the whole line." (Reti, 1958)
- x = return to tonic near inevitable
- circled x = possible but not inevitable
- circle = impossible
- (Reti, 1958)
Consequently, he argues, melodically tonal melodies resist harmonization and only reemerge in western music after, "harmonic tonality was abandoned," as in the music of Claude Debussy: "melodic tonality plus modulation is [Debussy's] modern tonality." (page 23)
- Music Fundamentals: Tonal Music PDF
- Tonal Harmony Reference Materials for the Undergraduate Theory Student
- Tonality, Modality, and Atonality by Larry Solomon
- Basic guide to tonal theory
- Guide to Schenkerian analysis
- The false "science" of atonal theories
- Babies react: tonality & dissonance/consonance -- News clipping
- The development of tonal scales
- Fink, Bob On the Origin of Music -- Essays & Readings
- Gustin, Molly Malcolmson Tonality
- Helmholtz, Hermann Sensations of Tone
- Perle, George Twelve-Tone Tonality
- Rameau Phillipe Treatise on Harmony
- Role of the drone in the evolution of harmony and tonality
- Schenker, Heinrich Free Composition
- Schoenberg, Arnold The Law of Harmony
Jim Samson (1977) suggests the following discussions of tonality as defined by Fétis, Helmhotz, Riemann, D'Indy, Adler, Yasser, and others:
- Beswick, Delbert M. The Problem of Tonality in Seventeenth Century Music, p.1-29.
- Shirlaw, Matthew Theory of Harmony (1955)
- Reti, Rudolph (1958). Tonality, Atonality, Pantonality: A study of some trends in twentieth century music. Westport, Connecticut: Greenwood Press. ISBN 0313204780.
- Harrison, Lou (1992). Entune. Contemporary Music Review 6 (2), 9-10
- Cope, David (1997). Techniques of the Contemporary Composer, p.12. New York, New York: Schirmer Books. ISBN 0028647378.
- Judd, Cristle Collins (1998). "Introduction:Analyzing Early Music", Tonal Structures of Early Music (ed. Judd). New York: Garland Publishing. ISBN 0815323883.
- Dahlhaus, Carl. Gjerdingen, Robert O. trans. (1990). Studies in the Origin of Harmonic Tonality. Princeton University Press. ISBN 0691091358.
- Joseph Fétis, (1722). Traité complet de la théorie et de la pratique de l'harmonie contenant la doctrine de la science et de l'art, 2d ed., p.166. Brussels and Paris.
- Castile-Blaze, (1821). Dictionnaire de musique moderne. Paris: Au magazin de musique de la Lyre moderne.
- Hugo Riemann, cited in Gurlitt, W. (1950). "Hugo Riemann (1849-1919)".
- Hauptmann, Moritz (1853). Die Natur der Harmonik und der Metrik, p.21. Leipzig.
- Rameau, Jean-Philippe (1737). Génération harmonique, ou Traité de musique théorique et pratique. Paris.
- Samson, Jim (1977). Music in Transition: A Study of Tonal Expansion and Atonality, 1900-1920. New York: W.W. Norton & Company. ISBN 0393021939.
- Delbert M. Beswick (1950). The Problem of Tonality in Seventeenth Century Music, Ph.D. dissertation. University of North Carolinacs:Tonalita