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== Miscellania ==
== Miscellania ==
* All heptatonic and larger scales are hemitonic and tritonic.<ref name="HansonHoward">Hanson, Howard. (1960) ''Harmonic Materials of Modern Music'', p. 362 ff. New York: Appleton-Century-Crofts. LOC 58-8138.</ref>
* All heptatonic and larger scales are hemitonic and tritonic.<ref name="HansonHoward">Hanson, Howard. (1960) ''Harmonic Materials of Modern Music'', p. 362 ff. New York: Appleton-Century-Crofts. LOC 58-8138.</ref>

* All octatonic scales save one ("the octatonic" or [[Diminished_scale]]) are cohemitonic.<ref name="HansonHoward" />


* All enneatonic and larger scales are cohemitonic.<ref name="HansonHoward" />
* All enneatonic and larger scales are cohemitonic.<ref name="HansonHoward" />

* All sonorities with 5 or more semitones are cohemitonic.<ref name="HansonHoward" />


== References ==
== References ==

Revision as of 04:34, 13 July 2013

Minyō scale on D,[1] equivalent to yo scale on D,[2] an anhemitonic scale Playⓘ.
Miyako-bushi scale on D, equivalent to in scale on D,[3] an hemitonic scale Playⓘ.

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 anhemitonic pentatonic scale, is anhemitonic, so also the whole tone scale.

Hungarian minor scale on C, a cohemitonic scale.[5] Playⓘ

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.

Octatonic scales on C, hemitonic but ancohemitonic Playⓘ.

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.

Orthagonal to these classifications are tritonic and atritonic scales. Tritonic scales contain one or more tritones and atritonic scales do not contain tritones.

The significance of these categories lies in their bases of semitones and tritones being the severest of dissonances, avoidance of which is often desirable. The most used scales across the planet are anhemitonic; of the remaining hemitonic scales, the most used are ancohemitonic.

Example: hemitonia and tritonia of the perfect fifth projection

Taking five consecutive pitches from the circle of fifths;[8] 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.

C major pentatonic scale
C major pentatonic scale
 playⓘ

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.[9]

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.[10]

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,[11] but cohemitonia will result.

P7 projection/major heptatonic on C Playⓘ.

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.[11]

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.[11]

Modes of the ancohemitonic heptatonic scales and the key signature system

Key signature A major / F♯ minor , an ancohemitonic heptatonic scale.

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,[12] most of which are common in romantic music, and of which most Romantic music is composed:

Adhering to the definition of heptatonic scales, these all possess 7 modes each, and are suitable for use in modal mutation.[13]

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♭, Bdouble flat, Adouble flat
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♭, Bdouble flat
involution of Hungarian major B♭, E♭, D♭, G♭, F♭, Adouble flat
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♭, Bdouble flat
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♭, Bdouble flat

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.[11]
  • All enneatonic and larger scales are cohemitonic.[11]
  • All sonorities with 5 or more semitones are cohemitonic.[11]

References

  1. ^ Susan Miyo Asai (1999). Nōmai Dance Drama, p. 126. ISBN 978-0-313-30698-3.
  2. ^ Minoru Miki, Marty Regan, Philip Flavin (2008). Composing for Japanese instruments, p. 2. ISBN 978-1-58046-273-0.
  3. ^ Titon, Jeff Todd (1996). Worlds of Music: An Introduction to the Music of the World's Peoples, p. 373. ISBN 0-02-872612-X.
  4. ^ 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.
  5. ^ Kahan, Sylvia (2009). In Search of New Scales, p. 39. ISBN 978-1-58046-305-8. Cites Liszt. Des Bohémians, p. 301.
  6. ^ Christ, William (1966). Materials and Structure of Music, v.1, p. 39. Englewood Cliffs: Prentice–Hall. LOC 66-14354.
  7. ^ Tymoczko, Dmitri (1997). The Consecutive-Semitone Constraint on Scalar Structure: A Link between Impressionism and Jazz, Intégral, v.11, (1997), p. 135-179.
  8. ^ Cooper, Paul. 1973. Perspectives in Music Theory: An Historical-Analytical Approach, p. 18. New York: Dodd, Mead. ISBN 0-396-06752-2.
  9. ^ Hanson, Howard. (1960) Harmonic Materials of Modern Music, p.29. New York: Appleton-Century-Crofts. LOC 58-8138.
  10. ^ Hanson, Howard. (1960) Harmonic Materials of Modern Music, p.40. New York: Appleton-Century-Crofts. LOC 58-8138.
  11. ^ 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).
  12. ^ Hanson, Howard. (1960) Harmonic Materials of Modern Music, p. 362ff. New York: Appleton-Century-Crofts. LOC 58-8138.
  13. ^ Christ, William (1966). Materials and Structure of Music, v.1, p. 45. Englewood Cliffs: Prentice-Hall. LOC 66-14354.
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