Anhemitonic scale: Difference between revisions
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[[Image:Miyako-bushi scale.png|thumb|300px|''Miyako-bushi'' scale on D, equivalent to ''in'' scale on D,<ref>Titon, Jeff Todd (1996). ''Worlds of Music: An Introduction to the Music of the World's Peoples'', p. 373. ISBN 0-02-872612-X.</ref> an hemitonic scale {{audio|Miyako-bushi scale.mid|Play}}.]] |
[[Image:Miyako-bushi scale.png|thumb|300px|''Miyako-bushi'' scale on D, equivalent to ''in'' scale on D,<ref>Titon, Jeff Todd (1996). ''Worlds of Music: An Introduction to the Music of the World's Peoples'', p. 373. ISBN 0-02-872612-X.</ref> an hemitonic scale {{audio|Miyako-bushi scale.mid|Play}}.]] |
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[[Musicology]] commonly classifies note scales as either '''hemitonic''' or '''anhemitonic'''. Hemitonic scales contain one or more [[semitone]]s and anhemitonic scales do not contain semitones. For example, in Japanese music the anhemitonic [[yo scale|''yo'' scale]] is contrasted with the hemitonic [[in scale|''in'' scale]].<ref>Anon. (2001) "Ditonus", ''The New Grove Dictionary of Music and Musicians'', second edition, edited by [[Stanley Sadie]] and [[John Tyrrell (professor of music)|John Tyrrell]]. London: Macmillan Publishers; Bence Szabolcsi (1943), "Five-Tone Scales and Civilization", ''Acta Musicologica'' 15, Fasc. 1/4 (January–December): pp. 24–34, citation on p. 25.</ref> The simplest scale in most common use over the planet, the [[Pentatonic scale#Hemitonic and anhemitonic|anhemitonic pentatonic scale]], is anhemitonic, so also the [[whole tone scale]]. |
[[Musicology]] commonly classifies note scales as either '''hemitonic''' or '''anhemitonic'''. Hemitonic scales contain one or more [[semitone]]s and anhemitonic scales do not contain semitones. For example, in Japanese music the anhemitonic [[yo scale|''yo'' scale]] is contrasted with the hemitonic [[in scale|''in'' scale]].<ref>Anon. (2001) "Ditonus", ''The New Grove Dictionary of Music and Musicians'', second edition, edited by [[Stanley Sadie]] and [[John Tyrrell (professor of music)|John Tyrrell]]. London: Macmillan Publishers; Bence Szabolcsi (1943), "Five-Tone Scales and Civilization", ''Acta Musicologica'' 15, Fasc. 1/4 (January–December): pp. 24–34, citation on p. 25.</ref> The simplest scale in most common use over the planet, the [[Pentatonic scale#Hemitonic and anhemitonic|atritonic anhemitonic ("Major") pentatonic scale]], is anhemitonic, so also the [[whole tone scale]]. |
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[[Image:Hungarian minor scale on C.png|thumb|right|Hungarian minor scale on C, a cohemitonic scale.<ref name="Kahan">Kahan, Sylvia (2009). ''In Search of New Scales'', p. 39. ISBN 978-1-58046-305-8. Cites Liszt. ''Des Bohémians'', p. 301.</ref> {{audio|Hungarian minor scale on C.mid|Play}}]] |
[[Image:Hungarian minor scale on C.png|thumb|right|Hungarian minor scale on C, a cohemitonic scale.<ref name="Kahan">Kahan, Sylvia (2009). ''In Search of New Scales'', p. 39. ISBN 978-1-58046-305-8. Cites Liszt. ''Des Bohémians'', p. 301.</ref> {{audio|Hungarian minor scale on C.mid|Play}}]] |
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''Ancohemitonic'' scales, by contrast, possess either no semitones (and thus are anhemitonic), or possess semitones (being hemitonic) but ordered such that none are consecutive.<ref>Tymoczko, Dmitri (1997). ''The Consecutive-Semitone Constraint on Scalar Structure: A Link between Impressionism and Jazz'', Intégral, v.11, (1997), p. 135-179.</ref> In some uses as vary by author, only the more specific second definition is to be understood. Examples are numerous, as ancohemitonia is favored over cohemitonia in the world's musics: [[diatonic scale]], melodic major/[[Melodic_minor#Melodic_minor_scale|melodic minor]], [[Hungarian major]], [[harmonic major scale]], [[Harmonic_minor#Harmonic_minor_scale|harmonic minor scale]], and the so-called [[octatonic scale]]. |
''Ancohemitonic'' scales, by contrast, possess either no semitones (and thus are anhemitonic), or possess semitones (being hemitonic) but ordered such that none are consecutive.<ref>Tymoczko, Dmitri (1997). ''The Consecutive-Semitone Constraint on Scalar Structure: A Link between Impressionism and Jazz'', Intégral, v.11, (1997), p. 135-179.</ref> In some uses as vary by author, only the more specific second definition is to be understood. Examples are numerous, as ancohemitonia is favored over cohemitonia in the world's musics: [[diatonic scale]], melodic major/[[Melodic_minor#Melodic_minor_scale|melodic minor]], [[Hungarian major]], [[harmonic major scale]], [[Harmonic_minor#Harmonic_minor_scale|harmonic minor scale]], and the so-called [[octatonic scale]]. |
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Hemitonia is also quantified by the number of semitones present. Unhemitonic scales have one and only one semitone; dihemitonic scales have 2 semitones; trihemitonic scales have 3 semitones, etc. In the same way that an anhemitonic scale is less dissonant than an hemitonic scale, an unhemitonic scale is less dissonant than a dihemitonic scale. |
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The qualification of cohemitonia versus ancohemitonia combines with the cardinality of semitones, giving terms like: diancohemitonic, tricohemitonic, and so forth. The importance of this lies in the fact that an ancohemitonic scale is less dissonant than a cohemitonic scale, the count of their semitones being equal. In general, the number of semitones is more important to the perception of dissonance than the adjacency (or lack thereof) of any pair of them. More adjacency between semitones does not necessarily increase the dissonance, the count of semitones again being equal.<ref>Keith, Michael. 1991. ''From Polychords to Polya : Adventures in Musical Combinatorics'', p. 45. Princeton: Vinculum Press. ISBN 978-0963009708.</ref> |
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| ⚫ | The harmonic relationship of all these categories lies in their bases of semitones and tritones being the severest of [[Consonance and dissonance|dissonances]], avoidance of which is often relatively desirable. The most used scales across the planet are anhemitonic; of the remaining hemitonic scales, the most used are ancohemitonic. The fundamental importance is confirmed that in study of these categories, the names of the commonest scales in use appear again and again. |
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== Quantification of Hemitonia and Relationship to Ancohemitonia == |
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Most of the world's music is anhemitonic, perhaps 90%. Of that other hemitonic portion, perhaps 90% is unhemitonic, predominating in chords of only 1 semitone, all of which are ancohemitonic by definition.<ref>Keith, Michael. 1991. ''From Polychords to Polya : Adventures in Musical Combinatorics'', p. 43. Princeton: Vinculum Press. ISBN 978-0963009708.</ref> Of the remaining 10%, perhaps 90% are dihemitonic, predominating in chords of no more than 2 semitones. The analog applies to chords of 3 semitones.<ref>Keith, Michael. 1991. ''From Polychords to Polya : Adventures in Musical Combinatorics'', p. 48-49. Princeton: Vinculum Press. ISBN 978-0963009708.</ref> In both later cases, however, there is a distinct preference for ancohemitonia, as the lack of adjacency of any two semitones goes a long way towards softening the dissonance. |
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== Example: hemitonia and tritonia of the [[perfect fifth]] projection == |
== Example: hemitonia and tritonia of the [[perfect fifth]] projection == |
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Revision as of 06:00, 13 July 2013
Musicology commonly classifies note scales as either hemitonic or anhemitonic. Hemitonic scales contain one or more semitones and anhemitonic scales do not contain semitones. For example, in Japanese music the anhemitonic yo scale is contrasted with the hemitonic in scale.[4] The simplest scale in most common use over the planet, the atritonic anhemitonic ("Major") pentatonic scale, is anhemitonic, so also the whole tone scale.
A special subclass of the hemitonic scales is the cohemitonic scales.[6] Cohemitonic scales contain two or more semitones (making them hemitonic), in particular such that the semitones fall consecutively in scale order. For example, the Hungarian minor scale in C includes F-sharp, G, and A-flat in that order, with semitones between.
Ancohemitonic scales, by contrast, possess either no semitones (and thus are anhemitonic), or possess semitones (being hemitonic) but ordered such that none are consecutive.[7] In some uses as vary by author, only the more specific second definition is to be understood. Examples are numerous, as ancohemitonia is favored over cohemitonia in the world's musics: diatonic scale, melodic major/melodic minor, Hungarian major, harmonic major scale, harmonic minor scale, and the so-called octatonic scale.
Hemitonia is also quantified by the number of semitones present. Unhemitonic scales have one and only one semitone; dihemitonic scales have 2 semitones; trihemitonic scales have 3 semitones, etc. In the same way that an anhemitonic scale is less dissonant than an hemitonic scale, an unhemitonic scale is less dissonant than a dihemitonic scale.
The qualification of cohemitonia versus ancohemitonia combines with the cardinality of semitones, giving terms like: diancohemitonic, tricohemitonic, and so forth. The importance of this lies in the fact that an ancohemitonic scale is less dissonant than a cohemitonic scale, the count of their semitones being equal. In general, the number of semitones is more important to the perception of dissonance than the adjacency (or lack thereof) of any pair of them. More adjacency between semitones does not necessarily increase the dissonance, the count of semitones again being equal.[8]
Related to these semitone classifications are tritonic and atritonic scales. Tritonic scales contain one or more tritones and atritonic scales do not contain tritones. A special monotonic relationship obtains between semitones and tritones as scales are built by projection, q.v. below.
The harmonic relationship of all these categories lies in their bases of semitones and tritones being the severest of dissonances, avoidance of which is often relatively desirable. The most used scales across the planet are anhemitonic; of the remaining hemitonic scales, the most used are ancohemitonic. The fundamental importance is confirmed that in study of these categories, the names of the commonest scales in use appear again and again.
Quantification of Hemitonia and Relationship to Ancohemitonia
Most of the world's music is anhemitonic, perhaps 90%. Of that other hemitonic portion, perhaps 90% is unhemitonic, predominating in chords of only 1 semitone, all of which are ancohemitonic by definition.[9] Of the remaining 10%, perhaps 90% are dihemitonic, predominating in chords of no more than 2 semitones. The analog applies to chords of 3 semitones.[10] In both later cases, however, there is a distinct preference for ancohemitonia, as the lack of adjacency of any two semitones goes a long way towards softening the dissonance.
Example: hemitonia and tritonia of the perfect fifth projection
Taking five consecutive pitches from the circle of fifths;[11] starting on C, these are C, G, D, A, and E. Transposing the pitches to fit into one octave rearranges the pitches into the major pentatonic scale: C, D, E, G, A. This scale is anhemitonic, having no semitones; it is atritonic, having no tritones.
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In addition, this is the maximal number of notes taken consecutively from the circle of fifths for which is it still possible to avoid a semitone.[12]
Adding another note from the circle of fifths gives the major hexatonic scale: C D E G A B. This scale is hemitonic, having a semitone between B and C; it is atritonic, having no tritones. In addition, this is the maximal number of notes taken consecutively from the circle of fifths for which is it still possible to avoid a tritone.[13]
Adding still another note from the circle of fifths gives the major heptatonic scale: C D E F G A B. This scale is strictly ancohemitonic, having 2 semitones but not consecutively; it is tritonic, having a tritone between F and B. Past this point in the projection series, no new intervals are added to the Interval vector analysis of the scale,[14] but cohemitonia will result.
Adding still another note from the circle of fifths gives the major octatonic scale: C D E F F♯ G A B. This scale is cohemitonic, having 2 semitones together at E F F♯, and tritonic as well.[14]
Similar behavior is seen across all scales generally, that more notes in a scale tend cumulatively to add dissonant intervals (specifically: hemitonia and tritonia in no particular order) and cohemitonia not already present. While also true that more notes in a scale tend to allow more and varied intervals in the interval vector, there might be said to be a "point of diminishing returns", when qualified against the also increasing dissonance, hemitonia, tritonia and cohemitonia. It is near these points where most popular scales lie.[14]
Modes of the ancohemitonic heptatonic scales and the key signature system
Western music's system of key signature is based upon the assumption of an heptatonic scale of 7 notes, such that there is never more than 7 accidentals present in a valid key signature. The global preference for anhemitonic scales combines with this basis to highlight the 6 ancohemitonic heptatonic scales,[15] most of which are common in romantic music, and of which most Romantic music is composed:
- Diatonic scale
- Melodic major/melodic minor
- Hungarian major
- involution of Hungarian major
- Harmonic major scale
- harmonic minor scale.
Adhering to the definition of heptatonic scales, these all possess 7 modes each, and are suitable for use in modal mutation.[16]
Table of key signatures
The following lists the key signatures for all possible untransposed modes of the ancohemitonic heptatonic scales.
| Base scale | Accidentals | Mode name |
|---|---|---|
| Diatonic | F♯ | Lydian |
| Diatonic | Ionian | |
| Diatonic | B♭ | Mixolydian |
| Diatonic | B♭, E♭ | Dorian |
| Diatonic | B♭, E♭, A♭ | Aeolian |
| Diatonic | B♭, E♭, A♭, D♭ | Phrygian |
| Diatonic | B♭, E♭, A♭, D♭, G♭ | Locrian |
| Base scale | Accidentals | Mode name |
| Melodic | F♯, G♯ | |
| Melodic | F♯, B♭ | Acoustic |
| Melodic | E♭ | Melodic minor |
| Melodic | B♭, A♭ | Melodic major |
| Melodic | B♭, E♭, D♭ | |
| Melodic | B♭, E♭, A♭, G♭ | |
| Melodic | B♭, E♭, A♭, G♭, D♭, F♭ | Super-Locrian |
| Base scale | Accidentals | Mode name |
| Hungarian major | F♯, G♯, E♯ | |
| Hungarian major | F♯, D♯, B♭ | Hungarian major |
| Hungarian major | G♯, E♭ | |
| Hungarian major | F♯, B♭, E♭, D♭ | |
| Hungarian major | E♭, A♭, G♭ | |
| Hungarian major | B♭, E♭, D♭, G♭, F♭ | |
| Hungarian major | E♭, D♭, G♭, F♭, B , A |
|
| Base scale | Accidentals | Mode name |
| involution of Hungarian major | F♯, G♯, D♯, E♯ | |
| involution of Hungarian major | F♯, G♯, E♭ | |
| involution of Hungarian major | F♯, B♭, D♭ | involution of Hungarian major |
| involution of Hungarian major | E♭, G♭ | |
| involution of Hungarian major | B♭, E♭, D♭, F♭ | |
| involution of Hungarian major | E♭, A♭, G♭, B |
|
| involution of Hungarian major | B♭, E♭, D♭, G♭, F♭, A |
|
| Base scale | Accidentals | Mode name |
| Harmonic major | F♯, G♯, D♯ | |
| Harmonic major | F♯, E♭ | |
| Harmonic major | A♭ | Harmonic major |
| Harmonic major | B♭, D♭ | |
| Harmonic major | B♭, E♭, G♭ | |
| Harmonic major | B♭, E♭, A♭, D♭, F♭ | |
| Harmonic major | E♭, A♭, D♭, G♭, B |
|
| Base scale | Accidentals | Mode name |
| Harmonic minor | F♯, D♯ | |
| Harmonic minor | G♯ | |
| Harmonic minor | F♯, B♭, E♭ | |
| Harmonic minor | E♭, A♭ | Harmonic minor |
| Harmonic minor | B♭, A♭, D♭ | |
| Harmonic minor | B♭, E♭, D♭, G♭ | |
| Harmonic minor | E♭, A♭, D♭, G♭, F♭, B |
Common citation in theories
- Dimitri Tymoczko, in A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (ISBN 978-0195336672), includes hemitonia in calculation formulas for contrapuntal smoothness and harmonic force transfer.
- Brett Willmott, in Mel Bays Complete Book of Harmony Theory and Voicing (ISBN 978-1562229948), restricts the scope of his guitar chord voicing to ancohemitonic tetrads.
- Michael Keith, in From Polychords to Polya : Adventures in Musical Combinatorics (ISBN 978-0963009708), draws his list of basic harmonies as anhemitonic sonorities.
Miscellania
- All heptatonic and larger scales are hemitonic and tritonic.[14]
- All octatonic scales save one ("the octatonic" or Diminished_scale) are cohemitonic.[14]
- All enneatonic and larger scales are cohemitonic.[14]
- All sonorities with 5 or more semitones are cohemitonic.[14]
References
- ^ Susan Miyo Asai (1999). Nōmai Dance Drama, p. 126. ISBN 978-0-313-30698-3.
- ^ Minoru Miki, Marty Regan, Philip Flavin (2008). Composing for Japanese instruments, p. 2. ISBN 978-1-58046-273-0.
- ^ Titon, Jeff Todd (1996). Worlds of Music: An Introduction to the Music of the World's Peoples, p. 373. ISBN 0-02-872612-X.
- ^ Anon. (2001) "Ditonus", The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers; Bence Szabolcsi (1943), "Five-Tone Scales and Civilization", Acta Musicologica 15, Fasc. 1/4 (January–December): pp. 24–34, citation on p. 25.
- ^ Kahan, Sylvia (2009). In Search of New Scales, p. 39. ISBN 978-1-58046-305-8. Cites Liszt. Des Bohémians, p. 301.
- ^ Christ, William (1966). Materials and Structure of Music, v.1, p. 39. Englewood Cliffs: Prentice–Hall. LOC 66-14354.
- ^ Tymoczko, Dmitri (1997). The Consecutive-Semitone Constraint on Scalar Structure: A Link between Impressionism and Jazz, Intégral, v.11, (1997), p. 135-179.
- ^ Keith, Michael. 1991. From Polychords to Polya : Adventures in Musical Combinatorics, p. 45. Princeton: Vinculum Press. ISBN 978-0963009708.
- ^ Keith, Michael. 1991. From Polychords to Polya : Adventures in Musical Combinatorics, p. 43. Princeton: Vinculum Press. ISBN 978-0963009708.
- ^ Keith, Michael. 1991. From Polychords to Polya : Adventures in Musical Combinatorics, p. 48-49. Princeton: Vinculum Press. ISBN 978-0963009708.
- ^ Cooper, Paul. 1973. Perspectives in Music Theory: An Historical-Analytical Approach, p. 18. New York: Dodd, Mead. ISBN 0-396-06752-2.
- ^ Hanson, Howard. (1960) Harmonic Materials of Modern Music, p.29. New York: Appleton-Century-Crofts. LOC 58-8138.
- ^ Hanson, Howard. (1960) Harmonic Materials of Modern Music, p.40. New York: Appleton-Century-Crofts. LOC 58-8138.
- ^ a b c d e f g Hanson, Howard. (1960) Harmonic Materials of Modern Music, p. 33. New York: Appleton-Century-Crofts. LOC 58-8138. Cite error: The named reference "HansonHoward" was defined multiple times with different content (see the help page).
- ^ Hanson, Howard. (1960) Harmonic Materials of Modern Music, p. 362ff. New York: Appleton-Century-Crofts. LOC 58-8138.
- ^ Christ, William (1966). Materials and Structure of Music, v.1, p. 45. Englewood Cliffs: Prentice-Hall. LOC 66-14354.