Tonality is a system of music in which specific hierarchical pitch relationships are based on a key "center", or tonic. The term tonalité originated with Alexandre-Étienne Choron (1810) and was borrowed by François-Joseph Fétis in 1840 (Reti, 1958; Simms 1975, 119; Judd, 1998; Dahlhaus 1990). Although Fétis used it as a general term for a system of musical organization and spoke of types de tonalités rather than a single system, today the term is most often used to refer to Major-Minor tonality (also called diatonic tonality, common practice tonality, or functional tonality), the system of musical organization of the common practice period, and of Western-influenced popular music throughout much of the world today.
Characteristics and features
David Cope (1997,[page needed]) considers key, consonance and dissonance (or relaxation and tension, respectively), and hierarchical relationships to be the three most basic concepts in tonality. In describing these tenets of tonal music, several known terms are used to refer to various elements of the tonal system.
Carl Dahlhaus (1990,[page needed]) lists the characteristics schemata of tonal harmony, "typified in the compositional formulae of the 16th and early 17th centuries," as the "complete cadence" (vollständige Kadenz): I-IV-V-I; I-IV-I-V-I; or 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 equals major: iii-vi-V-I; or minor: III-VII-i-v equals major: I-V-vi-iii.
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).
Other important scales include the blues scale, the whole tone scale, the pentatonic scale, and the chromatic scale. As these are not the major or minor diatonic scales, music written exclusively with them is not tonal by the definition above.
Chords are built primarily from notes of a diatonic scale, or secondarily from chromatic notes treated as variations or embellishments of the basic scale. The identity of the scale is important, as the size of the steps between notes are used to determine the system of chord relationships.
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, with the most common being V7-I cadence. Other cadences are considered to be less powerful.
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.
Traditional tonal music is described in terms of a scale of notes, upon which are built chords. Chords in order form progressions, which 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 function by leading the music towards or away from a particular tonic chord. When the sense of which chord is the tonic 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.
History and theory
Theories of tonal music are generally said[weasel words] to have begun with Jean-Philippe Rameau's Treatise on Harmony (1722), who was the first theorist to justify harmonic practice through a coherent system based on acoustical principles (Girdlestone 1969, 520). He claimed that his work represents "the practice of the last 40 years" (Rameau 1722,[page needed]). Rameau's work was introduced to Germany by Friedrich Wilhelm Marpurg in 1757, and used Rameau's system to explain the music of Johann Sebastian Bach (Marpurg 1753–54).[vague] The vocabulary of describing notes in relationship to the tonic note, and the use of harmonic progressions and cadences, became part of Bach's practice. Essential to this version of tonal theory are the chorale harmonizations of Bach, and the method by which a church melody is given a four part harmony by first assigning cadences, then creating a natural, or most direct, thoroughbass, and finally filling in the middle voices.
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.[cite this quote] He considered tonalité moderne as "trans-tonic order" (having one established key, and allowing for modulation to other keys) and tonalité ancienne "uni-tonic order" (establishing one key and remaining in that key for the duration of the piece). He described his earliest example of tonalité moderne thus: "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 chords on the third and seventh degrees of the scale, 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" (Fétis 1844, 171).
Fétis believed that tonality, tonalité moderne, was entirely cultural, saying, "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" (Fétis 1844, 11–12). "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" (Fétis 1844, 249). In contrast, Hugo Riemann believed tonality, "affinities between tones" or Tonverwandtschaften, was entirely natural and, following Moritz Hauptmann (1853), that the major third and perfect fifth were the only "directly intelligible" intervals, and that I, IV, and V, the tonic, subdominant, and dominant were related by the perfect fifths between their root notes (Dahlhaus 1990, 101-2).
By the 1840s, the practice of harmony had expanded to include more chromatic notes and a wider chord vocabulary, particularly the more frequent use of the diminished seventh chord—a four-note chord of all minor thirds. It is in this era that the word tonality became more common. At the same time, the elaboration of both the fugue and the sonata form, in terms of key relationships, became more rigorous, and the study of harmonic progressions, voice leading, and ambiguity of key, more precise.
Theorists such as Hugo Riemann, and later Edward Lowinsky and others, pushed back 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, or modern and ancient, dichotomy, with Renaissance music being modal. The term modality has been criticized by Harold Powers, among others. However, it is used to describe music whose harmonic function centers on notes rather than on chords, including some of the music of Bartók, Stravinsky, Vaughan Williams, Charles Ives, and composers of minimalist music. This and other modal music is broadly tonal.
In the early 20th century, the vocabulary of tonal theory was decisively influenced by two theorists: composer Arnold Schoenberg, whose Harmonielehre (Theory of Harmony) describes in detail chords, chord progressions, vagrant chords, creation of tonal areas, and 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 was evolving a theory based on the 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 and foreground. 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[who?] 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.
Tonality may be considered generally, with no restrictions on the date or place the music was produced, and little restriction on the materials and methods used. This definition includes pre-17th century western music, as well as much non-western music. By the middle of the 20th century, it had become "evident that triadic structure does not necessarily generate a tone center, that non-triadic harmonic formations may be made to function as referential elements, and that the assumption of a twelve-tone complex does not preclude the existence of tone centers" (Perle 1991, 8). Centric is sometimes used to describe music which is not traditionally tonal, but which nevertheless has a relatively strong tonal center. Often the term common practice tonality is used specifically to refer to tonal music that utilizes the diatonic system of relationship between tonic and dominant, whereas tonal or tonality refers more broadly to describe any music or musical practice that relies on or exhibits tonal centers, modalities, or both, often with triadic organization and relatively consonant harmonies.
In the early 20th century, the tonality which had prevailed since the 17th century was seen to have reached a crisis or break down point. Because of the "increased use of the ambiguous chords, the less probable harmonic progressions, and the more unusual melodic and rhythmic inflections" (Meyer 1967, 241), the syntax of functional harmony was loosened to the point where "At best, the felt probabilities of the style system had become obscure; at worst, they were approaching a uniformity which provided few guides for either composition or listening" (Meyer 1967, 241).
Alfred Einstein wrote that in ancient China, "the development from the non-semitonal pentatonic to the seven-note scale is certainly traceable, even though the old pentatonic always remained the foundation of its music" (Einstein 1954, 7). He notes a similar development in ancient Japan and Java. Much folk and art music focuses on a pentatonic, or five-note scale, including Beijing Opera, the folk music of Hungary, and the musical traditions of Japan.
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 the lowest pitch, or degree, of a scale, and using a somewhat flexible network of relations between any pitch or chord and the tonic, similar to perspective in painting. Tonality has a hierarchical structure: one triad, the tonic triad, is the center to which other chords are supposed to lead.
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. Temporary secondary tonal centers may be established by cadences, or simply passed through in a process called modulation, while 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 consonance and dissonance, distance or proximity to the tonic, the establishment of temporary or secondary tonal centers, and ambiguity as to tonal center. Music notation was created to accommodate tonality and facilitate 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 means a note which has different places in a chord will be heard differently, thus 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; transpositional equivalency and especially inversional equivalency are not considered applicable.
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 (or throughbass) and counterpoint. 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, gives a feeling of closure or finality, or both. 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. These temporarily suspended a sense of key, and by freely changing between the major and minor voicing for the tonic chord, they made the listener unsure of 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 tonality is natural or inherent in acoustical phenomena, whether it is inherent in the human nervous system or a psychological construct, whether it is inborn or learned, and to what degree it is all these things (Meyer 1967, 236). A viewpoint held by many theorists since the third quarter of the 19th century, following the publication in 1862 of the first edition of Helmholtz's On the Sensation of Tone (Helmholtz 1877), holds that diatonic scales and tonality arise from natural overtones (Riemann 1872, 1875, 1882, 1893, 1905, 1914–15; Schenker 1906–35; Hindemith 1937–70).
Rudolph Réti differentiates between harmonic tonality of the traditional kind found in homophony, and melodic tonality, as in monophony. In the harmonic kind, tonality is produced through the V-I chord progression. 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, being more or less similar to the tonic-dominant, are "the composer's free invention." He describes melodic tonality (the term coined independently and 10 years earlier by Estonian composer Jaan Soonvald (Rais 1992, 46)) 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 as 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" to return to the tonic (Reti 1958).[page needed]
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" (Reti 1958, 23).
- Beswick, Delbert Meacham. 1951. "The Problem of Tonality in Seventeenth-Century Music." Ph.D. diss. Chapel Hill: University of North Carolina.
- Jim Samson (1977) suggests the following discussions of tonality as defined by Fétis, Helmholtz, Riemann, D'Indy, Adler, Yasser, and others:
- Beswick, Delbert M. 1950. "The Problem of Tonality in Seventeenth Century Music". Ph.D. thesis. Chapel Hill: University of North Carolina. p. 1-29. OCLC accession number 12778863.
- Shirlaw, Matthew (1917). The Theory of Harmony: An Inquiry into the Natural Principles of Harmony; with an Examination of the Chief Systems of Harmony from Rameau to the Present Day. London: Novello & Co. (Reprinted New York: Da Capo Press, 1969. ISBN 0-306-71658-5.)
- Alembert, Jean le Rond D'. 1752. Elémens de musique, theorique et pratique, suivant les principes de m. Rameau. Paris: David l'aîné. Facsimile reprint, New York: Broude Bros., 1966.
- Benjamin, Thomas. 2003. The Craft of Tonal Counterpoint, with Examples from the Music of Johann Sebastian Bach, second edition. New York: Routledge. ISBN 0415943914.
- Benjamin, Thomas, Michael M. Horvit, and Robert Nelson. 2008. Techniques and Materials of Music: From the Common Practice Period through the Twentieth Century, seventh edition. Belmont, CA: Thomson Schirmer. ISBN 9780495189770.
- Choron, Alexandre. 1810. "Sommaire de l'histoire de la musique." In vol. 1 of François Fayolle and Alexandre Choron, Dictionnaire historique de musiciens. 2 vols. Paris: Valade et Lenormant, 1810–11.
- Cope, David. 1997. Techniques of the Contemporary Composer. New York: Schirmer Books. ISBN 0-02-864737-8.
- Dahlhaus, Carl. 1990. Studies in the Origin of Harmonic Tonality. Translated by Robert O. Gjerdingen. Princeton: Princeton University Press. ISBN 0-691-09135-8.
- Castile-Blaze. 1821. Dictionnaire de musique moderne. Paris: Au magazin de musique de la Lyre moderne.
- Fétis, François-Joseph. 1844. Traité complet de la théorie et de la pratique de l'harmonie contenant la doctrine de la science et de l'art. Brussels: Conservatoire de Musique; Paris: Maurice Schlesinger.
- Hauptmann, Moritz. 1853. Die Natur der Harmonik und der Metrik. Leipzig: Breitkopf und Härtel.
- Rameau, Jean-Philippe. 1737. Génération harmonique, ou Traité de musique théorique et pratique. Paris: Prault fils.
- Riemann, Hugo; cited in Gurlitt, W. (1950). "Hugo Riemann (1849–1919)".
- Einstein, Alfred. 1954. A Short History of Music, fourth American edition, revised. New York: Vintage Books.
- Fétis, François-Joseph. 1844. Traité complet de la théorie et de la pratique de l'harmonie contenant la doctrine de la science et de l'art. Brussels: Conservatoire de Musique; Paris: Maurice Schlesinger. English edition, as Complete Treatise on the Theory and Practice of Harmony, translated by Peter M Landey. Harmonologia: Studies in Music Theory 13. Hillsdale, NY: Pendragon Press, 2008. ISBN 9781576471111.
- Girdlestone, Cuthbert. 1969. Jean-Philippe Rameau: His Life and Work, revised and enlarged edition. New York: Dover Publications, Inc.
- Gustin, Molly. 1969. Tonality. Philosophical Library. LCCN 68-18735.
- Harrison, Lou. 1992. "Entune." Contemporary Music Review 6 (2), 9-10.
- Helmholtz, Hermann von. 1877. Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik. Fourth edition. Braunschweig: F. Vieweg. English, as On the Sensations of Tone as a Physiological Basis for the Theory of Music. 2d English ed. translated, thoroughly rev. and corrected, rendered conformal to the 4th (and last) German ed. of 1877, with numerous additional notes and a new additional appendix bringing down information to 1885, and especially adapted to the use of music students, by Alexander J. Ellis. With a new introd. (1954) by Henry Margenau. New York, Dover Publications, 1954.
- Hindemith, Paul. Unterweisung im Tonsatz. 3 vols. Mainz, B. Schott's Söhne, 1937–70. First two volumes in English, as The Craft of Musical Composition, translated by Arthur Mendel and Otto Ortmann. New York: Associated Music Publishers; London: Schott & Co., 1941-42.
- Janata P, J. Birk, J. Van Horn J, M. Leman, B. Tillmann, and J. Bharucha. 2002. "The cortical topography of tonal structures underlying Western music." Science (Dec. 13).
- Judd, Cristle Collins. 1998. "Introduction: Analyzing Early Music", Tonal Structures of Early Music (ed. Judd). New York: Garland Publishing. ISBN 0-8153-2388-3.
- Kepler, Johannes. 1619. Harmonices mundi (Latin: The Harmony of the Worlds). Linz: Godofredo Tampechi.
- Kilmer, Anne Draffkorn, Richard L. Crocker, and Robert R. Brown. 1976. Sounds from Silence, Recent Discoveries in Ancient Near Eastern Music. LP sound recording, 33⅓ rpm, 12 inch, with bibliography (23 p. ill.) laid in container. [n.p.]: Bit Enki Records. LCC#75-750773 /R
- Marpurg, Friedrich Wilhelm. 1753–54. Abhandlung von der Fuge nach dem Grundsätzen der besten deutschen und ausländischen Meister. 2 vols. Berlin: A. Haude, und J.C. Spener.
- Marpurg, Friedrich Wilhelm. 1757. Systematische Einleitung in die musikalische Setzkunst, nach den Lehrsätzen des Herrn Rameau, Leipzig: J. G. I. Breitkopf. [translation of D'Alembert 1752]
- Meyer, Leonard B. 1967. Music, the Arts, and Ideas: Patterns and Predictions in Twentieth-Century Culture. Chicago and London: University of Chicago Press.
- Perle, George. 1978. Twelve-Tone Tonality. University of California Press. ISBN 0-520-20142-6. (reprinted 1992)
- Perle, George. 1991. Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern, sixth edition, revised. Berkeley and Los Angeles: University of California Press. ISBN 0-520-07430-0
- Pleasants, Henry. 1955 The Agony of Modern Music, Simon & Shuster, N.Y., LCC#54-12361.
- Rais, Mark. 1992. "Jaan Soonvald and His Musical System". Leonardo Music Journal 2, no. 1:45–47.
- Rameau, Jean-Phillipe. 1722. Traité de l’harmonie réduite à ses principes naturels. Paris: Ballard.
- Rameau, Jean-Phillipe. 1726. Nouveau Systême de Musique Theorique, où l'on découvre le Principe de toutes les Regles necessaires à la Pratique, Pour servir d'Introduction au Traité de l'Harmonie. Paris: L'Imprimerie de Jean-Baptiste-Christophe Ballard.
- Rameau, Jean-Phillipe. 1737. Génération harmonique, ou Traité de musique théorique et pratique. Paris: Prault fils.
- Rameau, Jean-Phillipe. 1750. Démonstration du Principe de L'Harmonie, Servant de base à tout l'Art Musical théorique et pratique. Paris: Durand et Pissot.
- Reti, Rudolph (1958). Tonality, Atonality, Pantonality: A Study of Some Trends in Twentieth Century Music. Westport, Connecticut: Greenwood Press. ISBN 0-313-20478-0.
- Riemann, Hugo. 1872. "Über Tonalität." Neue Zeitschrift für Musik 68.
- Riemann, Hugo. 1875. “Die objective Existenz der Untertöne in der Schallwelle.” Allgemeine Musikzeitung 2:205–6, 213–15.
- Riemann, Hugo. 1882. Die Natur der Harmonik. Sammlung musikalischer Vorträge 40, ed. Paul Graf Waldersee. Leipzig: Breitkopf und Härtel.
- Riemann, Hugo. 1893. Vereinfachte Harmonielehre oder die Lehre von den tonalen Funktionen der Akkorde. London & New York: Augener & Co. (2d ed. 1903.) Translated 1895 as Harmony Simplified, or the Theory of the Tonal Functions of Chords. London: Augener & Co.
- Riemann, Hugo. 1905. "Das Problem des harmonischen Dualismus." Neue Zeitschrift für Musik 101:3–5, 23–26, 43–46, 67–70.
- Riemann, Hugo. 1914–15. "Ideen zu einer 'Lehre von den Tonvorstellungen'." Jahrbuch der Musikbibliothek Peters 1914–15: 1–26.
- Samson, Jim. 1977. Music in Transition: A Study of Tonal Expansion and Atonality, 1900–1920. New York: W.W. Norton & Company. ISBN 0-393-02193-9.
- Schellenberg, E. Glenn, and Sandra E. Trehub. 1996. "Natural Musical Intervals: Evidence from Infant Listeners" Psychological Science Vol. 7, no. 5 (September): 272-77.
- Schenker, Heinrich. 1906–35. Neue musikalische Theorien und Phantasien. 3 vols. in 4. Vienna and Leipzig: Universal Edition.
- Schenker, Heinrich. 1979. Free composition, translated and edited by Ernst Oster. New York: Longman, 1979. Translation of Neue musikalische Theorien und Phantasien, 3. Bd., Der freie Satz. ISBN 0-582-28073-7
- Schenker, Heinrich. 1987. Counterpoint, translated by John Rothgeb and Jürgen Thym; edited by John Rothgeb. 2 vols. New York: Schirmer Books; London: Collier Macmillan. Translation of Neue musikalische Theorien und Phantasien, 2. Bd., Kontrapunkt. ISBN 0-02-873220-0
- Schenker, Heinrich; ed. and annot. Oswald Jonas (1954). Harmony. trans. Elisabeth Mann-Borgese. Chicago: University of Chicago Press. OCLC 280916. Translation of Neue musikalische Theorien und Phantasien, 1. Bd., Harmonielehre. (Reprinted Cambridge, Mass.: MIT Press, 1973, ISBN 0-262-69044-6)
- Schoenberg, Arnold. 1978. Theory of Harmony, translated by Roy E. Carter. Berkeley & Los Angeles: University of California Press. ISBN 0-520-03464-3. Reprint ed. 1983, ISBN 0-520-04945-4. Pbk ed. 1983, ISBN 0-520-04944-6.
- Simms, Bryan. 1975. "Choron, Fétis, and the Theory of Tonality." Journal of Music Theory 19, no. 1 (Spring): 112–38.
- Thomson, William. 1999. Tonality in Music: A General Theory. San Marino, Calif.: Everett Books. ISBN 0-940459-19-1.
- West, M. L. 1994. "The Babylonian Musical Notation and the Hurrian Melodic Texts." Music and Letters 75, no. 2 (May): 161–79.
- Khramov M. 2008. "Project Commator and Sonantometry." Proceedings of the International Symposium FRSM-2008. Kolkata, pp. 133–140.[Full citation needed]
- 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
- The Tonal Centre "explains and demonstrates some of the key concepts of tonality"
- Algebra of Tonal Functions.
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