Musicology Musicology (from Greek μουσική ''mousikē'' 'music' and -λογια ''-logia'', 'domain of study') is the scholarly analysis and research-based study of music. Musicology departments traditionally belong to the humanities, although some mu ...

commonly classifies scales as either hemitonic or anhemitonic. Hemitonic scales contain one or more

semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...

s, while anhemitonic scales do not contain semitones. For example, in

traditional Japanese music Traditional Japanese music is the folk or traditional music of Japan. Japan's Ministry of Education classifies as a category separate from other traditional forms of music, such as (court music) or (Buddhist chanting), but most ethnomusicolog ...

, the anhemitonic ''yo'' scale is contrasted with the hemitonic ''in'' scale. The simplest and most commonly used scale in the world is the atritonic anhemitonic "major" pentatonic scale. The

whole tone scale In music, a whole-tone scale is a scale in which each note is separated from its neighbors by the interval of a whole tone. In twelve-tone equal temperament, there are only two complementary whole-tone scales, both six-note or ''hexatonic'' sc ...

is also anhemitonic. A special subclass of the hemitonic scales is the cohemitonic scales. Cohemitonic scales contain two or more semitones (making them hemitonic) such that two or more of the semitones appear consecutively in scale order. For example, the

Hungarian minor scale The Hungarian minor scale,Christiansen, Mike (2000). ''Guitar Scale Dictionary'', p. 14. .Stetina, Troy (2007). ''Fretboard Mastery'', p. 126. .Kent Cleland, Mary Dobrea-Grindahl (2010). ''Developing Musicianship Through Aural Skills'', p. 495. Ca ...

in C includes F, G, and A in that order, with a semitone between F and G, and then a semitone between G and A. Ancohemitonic scales, in contrast, either contain no semitones (and thus are anhemitonic), or contain semitones (being hemitonic) where none of the semitones appear consecutively in scale order. Some authors, however, do not include anhemitonic scales in their definition of ancohemitonic scales. Examples of ancohemitonic scales are numerous, as ancohemitonia is favored over cohemitonia in the world's musics:

diatonic scale In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, ...

, melodic major/

melodic minor In music theory, the minor scale is three scale patterns – the natural minor scale (or Aeolian mode), the harmonic minor scale, and the melodic minor scale (ascending or descending) – rather than just two as with the major scale, which al ...

, Hungarian major scale,

harmonic major scale In music theory, the harmonic major scale is a musical scale found in some music from the common practice era and now used occasionally, most often in jazz. In George Russell's '' Lydian Chromatic Concept'' it is the fifth mode (V) of the Lydian ...

harmonic minor scale In music theory, the minor scale is three scale patterns – the natural minor scale (or Aeolian mode), the harmonic minor scale, and the melodic minor scale (ascending or descending) – rather than just two as with the major scale, which als ...

, and the so-called

octatonic scale An octatonic scale is any eight-note musical scale. However, the term most often refers to the symmetric scale composed of alternating whole and half steps, as shown at right. In classical theory (in contrast to jazz theory), this symmetrical ...

. Hemitonia is also quantified by the number of semitones present. Unhemitonic scales have 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 a hemitonic scale, an anhemitonic scale is less dissonant than a dihemitonic scale. The qualification of cohemitonia versus ancohemitonia combines with the cardinality of semitones, giving terms like: dicohemitonic, triancohemitonic, and so forth. 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. Additional adjacency between semitones (once adjacency is present) does not necessarily increase the dissonance, the count of semitones again being equal.Keith, Michael. 1991. ''From Polychords to Polya : Adventures in Musical Combinatorics'', p. 45. Princeton: Vinculum Press. . Related to these semitone classifications are tritonic and atritonic scales. Tritonic scales contain one or more

tritone In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be decomposed into the three a ...

s, while atritonic scales do not contain tritones. A special monotonic relationship exists between semitones and tritones as scales are built by projection, q.v. below. The harmonic relationship of all these categories comes from the perception that semitones and tritones are the severest of dissonances, and that avoiding them is often desirable. The most-used scales across the planet are anhemitonic. Of the remaining hemitonic scales, the ones most used are ancohemitonic.

Quantification of hemitonia and its relationship to ancohemitonia

Most of the world's music is anhemitonic, perhaps 90%.Keith, Michael. 1991. ''From Polychords to Polya : Adventures in Musical Combinatorics'', p. 43. Princeton: Vinculum Press. . Of that other hemitonic portion, perhaps 90% is unhemitonic, predominating in chords of only 1 semitone, all of which are ancohemitonic by definition. Of the remaining 10%, perhaps 90% are dihemitonic, predominating in chords of no more than 2 semitones. The same applies to chords of 3 semitones. 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 increasing dissonance. The following table plots sonority size (downwards on the left) against semitone count (to the right) plus the quality of ancohemitonia (denoted with letter A) versus cohemitonia (denoted with letter C). In general, ancohemitonic combinations are fewer for a given chord or scale size, but used much more frequently so that their names are well known. Column "0" represents the most commonly used chords., avoiding intervals of M7 and chromatic 9ths and such combinations of 4th, chromatic 5ths, and 6th to produce semitones. Column 1 represents chords that barely use the harmonic degrees that column "0" avoids. Column 2, however, represents sounds far more intractable. Column 0, row 5 are the full but pleasant chords: 9th, 6/9, and 9alt5 with no 7.Wilmott, Brett. (1994) ''Mel Bays Complete Book of Harmony Theory and Voicing'', p.210. Pacific, Missouri: Mel Bay. . Column "0", row "6", is the unique

. Column "2A", row "7", a local minimum, refers to the

and melodic major/

scales. Ancohemitonia, inter alii, probably makes these scales popular. Column "2C", row "7", another local minimum, refers to the

Neapolitan major scale In music, the major Neapolitan scale and the minor Neapolitan scale are two musical scales. Both scales are minor, in that they both contain the note a minor third above the root. The major and minor Neapolitan scales are instead differentiated ...

, which is cohemitonic and somewhat less common but still popular enough to bear a name. Column "3A", row "7", another local minimum, represents the Hungarian major scale, and its involution, and the

and involution

of the same. Column "3A", row "6", are the hexatonic analogs to these four familiar scales, one of which being the

Augmented scale In music and music theory, a hexatonic scale is a scale with six pitches or notes per octave. Famous examples include the whole-tone scale, C D E F G A C; the augmented scale, C D E G A B C; the Prometheus scale, C D E F A B C; and the blues s ...

, and another the analog of the

Octatonic scale An octatonic scale is any eight-note musical scale. However, the term most often refers to the symmetric scale composed of alternating whole and half steps, as shown at right. In classical theory (in contrast to jazz theory), this symmetrical ...

- which itself appears, alone and solitary, at Column ">=4A". row "8". Column "2A", row "4", another minimum, represents a few frankly dissonant, yet strangely ''resonant'' harmonic combinations: mM9 with no 5, 119, dom139, and M711. As music tends towards increasing dissonance through history, perhaps someday Column 2 will be as acceptable as even Column 1 might be, and Column 3 will finally have a place in the harmony of the world. Note, too, that in the highest cardinality row for each column before the terminal zeros begin, the sonority counts are small, except for row "7" and the "3" columns of all sorts. This explosion of hemitonic possibility associated with note cardinality 7 (and above) possibly marks the lower bound for the entity called "scale" (in contrast to "chord"). As shown in the table, anhemitonia is a property of the domain of note sets cardinality 2 through 6, while ancohemitonia is a property of the domain of note sets cardinality 4 through 8 (3 through 8 for improper ancohemitonia including unhemitonia as well). This places anhemitonia generally in the range of "chords" and ancohemitonia generally in the range of "scales".

Example: hemitonia and tritonia of the perfect-fifth projection

The interrelationship of semitones, tritones, and increasing note count can be demonstrated by taking five consecutive pitches from the

circle of fifths In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval ...

; starting on C, these are C, G, D, A, and E. Transposing the pitches to fit into one

octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

rearranges the pitches into the

major pentatonic scale A pentatonic scale is a musical scale with five notes per octave, in contrast to the heptatonic scale, which has seven notes per octave (such as the major scale and minor scale). Pentatonic scales were developed independently by many ancie ...

: C, D, E, G, A. This scale is anhemitonic, having no semitones; 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 semitone. 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. Adding still another note from the circle of fifths gives the major heptatonic scale: C D E F G A B (when the fifth is added from below the tonic). 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 In musical set theory, an interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. (That is, a set of pitches where octaves are disregarded.) Other names include: ic vector (or interva ...

analysis of the scale,Hanson, Howard. (1960) ''Harmonic Materials of Modern Music'', p. 33. New York: Appleton-Century-Crofts. LOC 58-8138. "When the projection f the perfect fifthis carried beyond seven tones, no ''new'' intervals can be added." "On the other hand, as sonorities are projected beyond the six-tone series they tend to lose their individuality. ''All'' seven-tone series, for example, contain ''all'' of the six basic intervals, and difference in their proportion decreases as additional tones are added....Such patterns tend to lose their identity, producing a monochromatic effect with its accompanying lack of the essential element of contrast." but cohemitonia results. Adding still another note from the circle of fifths gives the major octatonic scale: C D E F F G A B (when the fifth is added from above the top note in the series--B in this case). This scale is cohemitonic, having 3 semitones together at E F F G, and tritonic as well. 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 In musical set theory, an interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. (That is, a set of pitches where octaves are disregarded.) Other names include: ic vector (or interva ...

, 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.

Cohemitonic and hemitonic scales

Though less used than ancohemitonic scales, the cohemitonic scales have an interesting property. The sequence of two (or more) consecutive halfsteps in a scale presents the opportunity to "split" the scale by placing the

tonic note In music, the tonic is the first scale degree () of the diatonic scale (the first note of a scale) and the tonal center or final resolution tone that is commonly used in the final cadence in tonal (musical key-based) classical music, popul ...

of the scale on the middle note of the halfstep span. This allows a

leading tone In music theory, a leading-tone (also called a subsemitone, and a leading-note in the UK) is a note or pitch which resolves or "leads" to a note one semitone higher or lower, being a lower and upper leading-tone, respectively. Typically, ''the ...

from below resolving upwards, as well as a descending flat-supertonic upper neighbor, both converging on the tonic. The split turns a weakness - dissonance of cohemitonia - to a strength:

contrapuntal In music, counterpoint is the relationship between two or more musical lines (or voices) which are harmonically interdependent yet independent in rhythm and melodic contour. It has been most commonly identified in the European classical tradi ...

convergence on the tonic. It is very common that a cohemitonic (or even hemitonic) scale (e.g.: Hungarian minor ) be displaced preferentially to a mode where the halfstep span is split (e.cont.:

Double harmonic scale In music, the double harmonic major scaleStetina, Troy (1999). ''The Ultimate Scale Book'', p. 59. . is a scale whose gaps may sound unfamiliar to Western listeners. This is also known as Mayamalavagowla, Bhairav Raga, Byzantine scale, Arabic (Hi ...

), and by which name we more commonly know the same circular series of intervals. Cohemitonic scales with multiple halfstep spans present the additional possibility of modulating between tonics each furnished with both upper and lower neighbors.

Modes of heptatonic scales and the key signature system

Western music's system of

key signature In Western musical notation, a key signature is a set of sharp (), flat (), or rarely, natural () symbols placed on the staff at the beginning of a section of music. The initial key signature in a piece is placed immediately after the clef at ...

is based upon the assumption of a

heptatonic scale A heptatonic scale is a musical scale that has seven pitches, or tones, per octave. Examples include the major scale or minor scale; e.g., in C major: C D E F G A B C—and in the relative minor, A minor, natural minor: A B C D E F G A; the m ...

of 7 notes, such that there are 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,Hanson, Howard. (1960) ''Harmonic Materials of Modern Music'', p. 362ff. New York: Appleton-Century-Crofts. LOC 58-8138. most of which are common in

romantic music Romantic music is a stylistic movement in Western Classical music associated with the period of the 19th century commonly referred to as the Romantic era (or Romantic period). It is closely related to the broader concept of Romanticism—the ...

, and of which most Romantic music is composed: *

Diatonic scale In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, ...

* Melodic major/

* Hungarian major scale * involution of Hungarian major *

Harmonic major scale In music theory, the harmonic major scale is a musical scale found in some music from the common practice era and now used occasionally, most often in jazz. In George Russell's '' Lydian Chromatic Concept'' it is the fifth mode (V) of the Lydian ...

Harmonic minor scale In music theory, the minor scale is three scale patterns – the natural minor scale (or Aeolian mode), the harmonic minor scale, and the melodic minor scale (ascending or descending) – rather than just two as with the major scale, which als ...

. These cohemitonic scales are less common: * Double harmonic major scale *

* Neapolitan minor scale * Ionian 5 scale * Persian scale * Locrian 7 scale. Adhering to the definition of heptatonic scales, these all possess 7

modes Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is ...

each, and are suitable for use in modal mutation. They appear in the table above in Row "7", Columns "2A" and "3A".

Table of key signatures

The following lists the key signatures for all possible untransposed modes of the aforementioned heptatonic scales using the note C as the tonic.

Common citation in theories

* Dimitri Tymoczko, in ''A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice '' (), includes hemitonia in calculation formulas for contrapuntal smoothness and harmonic force transfer. * Brett Willmott, in ''Mel Bays Complete Book of Harmony Theory and Voicing'' (), restricts the scope of his guitar chord voicing to ancohemitonic tetrads. * Michael Keith, in ''From Polychords to Polya : Adventures in Musical Combinatorics'' (), draws his list of basic harmonies as anhemitonic sonorities.

Miscellanea

* All heptatonic (except for 7EDO temperament, so called "neutral scale", used in

Gamelan Gamelan () ( jv, ꦒꦩꦼꦭꦤ꧀, su, ᮌᮙᮨᮜᮔ᮪, ban, ᬕᬫᭂᬮᬦ᭄) is the traditional ensemble music of the Javanese, Sundanese, and Balinese peoples of Indonesia, made up predominantly of percussive instruments. T ...

music and some folk musical styles of

Angola , national_anthem = " Angola Avante"() , image_map = , map_caption = , capital = Luanda , religion = , religion_year = 2020 , religion_ref = , coordina ...

) and larger scales are hemitonic (ditonic or better) and tritonic. All pitch class sets of seven notes contain 1-3 tritones and 3-6 semitones, as can be seen in their interval vectors on

List of pitch-class sets This is a list of set classes by Forte number. For a list of ordered collections, see: list of tone rows and series. Sets are listed next to their complements. Inversions are marked "B" (sets not marked "A" or "B" are symmetrical). "T" and ...

. * All octatonic scales save one ("the octatonic" or

Diminished scale An octatonic scale is any eight- note musical scale. However, the term most often refers to the symmetric scale composed of alternating whole and half steps, as shown at right. In classical theory (in contrast to jazz theory), this symmetric ...

) are cohemitonic. * All enneatonic and larger scales are cohemitonic. * All sonorities with 5 or more semitones are cohemitonic. * The

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...

of a cohemitonic scale is often an ancohemitonic scale, and vice versa. * Unhemitonic scales never have more than 6 notes, and are always ancohemitonic. * Dihemitonic and trihemitonic scales never have more than 7 notes. * Tetrahemitonic and pentahemitonic scales never have more than 8 notes. * Hexahemitonic and heptahemitonic scales never have more than 9 notes. * Octahemitonic and enneahemitonic scales never have more than 10 notes. * There is no 12ET scale with exactly 11 halfsteps.

References

{{Reflist Musical scales Musical analysis Musicology