Musicology Musicology is the academic, research-based study of music, as opposed to musical composition or performance. Musicology research combines and intersects with many fields, including psychology, sociology, acoustics, neurology, natural sciences, ...

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

semitone A semitone, also called a minor second, 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 ...

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 ethnomusic ...

, 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 (music), scale in which each Musical note, note is separated from its neighbors by the interval (music), interval of a whole tone. In twelve-tone equal temperament, there are only two Complement (music)#Ag ...

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 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 a heptatonic scale, heptatonic (seven-note) 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 eith ...

, melodic major/

melodic minor In Classical_music, Western classical music theory, the minor scale refers to three Scale (music), scale patterns – the natural minor scale (or Aeolian mode), the harmonic minor scale, and the melodic minor scale (ascending or descending). ...

harmonic major scale In music theory, the harmonic major scale is a Scale (music), musical scale found in some music from the common practice era and now used occasionally, most often in jazz. It corresponds to the Raga Sarasangi in Indian Carnatic music, or Raag Nat ...

harmonic minor scale The harmonic minor scale (or Aeolian ♮7 scale) is a Scale (music), musical scale derived from the natural minor scale, with the minor seventh degree raised by one semitone to a major seventh, creating an augmented second between the sixth and ...

, Hungarian major scale, Romanian major scale, and the so-called

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

. 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 unhemitonic 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 interval (music), musical interval spanning three adjacent Major second, 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 ...

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

and its involution

, and the Hungarian major scale and its involution Romanian major scale. Column "3A", row "6", are the hexatonic analogs to these four familiar scales, one of which being the Augmented scale, 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 ancohemitonic symmetric scale composed of alternating whole and half steps, as shown at right. In classical theory (in contrast to jazz theory), ...

- 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. 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 (sometimes also cycle of fifths) is a way of organizing pitches as a sequence of perfect fifths. Starting on a C, and using the standard system of tuning for Western music (12-tone equal temperament), the se ...

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

octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...

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. 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 interv ...

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 interv ...

, 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, popular ...

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

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

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 theory, counterpoint is the relationship of two or more simultaneous Part (music), musical lines (also called voices) that are harmonically dependent on each other, yet independent in rhythm and Pitch contour, melodic contour. The term ...

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 The double harmonic major scaleStetina, Troy (1999). ''The Ultimate Scale Book'', p. 59. . is a musical scale with a flattened second and sixth degree. This scale is enharmonic to the Mayamalavagowla raga, Bhairav raga, Byzantine scale, Arabic ...

), 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 cl ...

is based upon the assumption of a

heptatonic scale A heptatonic scale is a musical scale (music), scale that has seven pitch (music), pitches, or musical tone, tones, per octave. Examples include: * the #Diatonic scale, diatonic scale; including the major scale and its modes (notably the natural 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 a heptatonic scale, heptatonic (seven-note) 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 eith ...

* Melodic major/

Harmonic major scale In music theory, the harmonic major scale is a Scale (music), musical scale found in some music from the common practice era and now used occasionally, most often in jazz. It corresponds to the Raga Sarasangi in Indian Carnatic music, or Raag Nat ...

Harmonic minor scale The harmonic minor scale (or Aeolian ♮7 scale) is a Scale (music), musical scale derived from the natural minor scale, with the minor seventh degree raised by one semitone to a major seventh, creating an augmented second between the sixth and ...

* Hungarian major scale * Romanian major scale These cohemitonic scales are less common: * Double harmonic major scale * Neapolitan 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 ( 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 the setting fo ...

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 (; ; , ; ) is the traditional musical ensemble, ensemble music of the Javanese people, Javanese, Sundanese people, Sundanese, and Balinese people, Balinese peoples of Indonesia, made up predominantly of percussion instrument, per ...

music and some folk musical styles of

Angola Angola, officially the Republic of Angola, is a country on the west-Central Africa, central coast of Southern Africa. It is the second-largest Portuguese-speaking world, Portuguese-speaking (Lusophone) country in both total area and List of c ...

) 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 set classes This is a list of set classes, by Forte number. In music theory, a ''set class'' (an abbreviation of ''pitch-class-set class'') is an ascending collection of pitch classes, transposed to begin at zero. For a list of ordered collections, see ...

. * All octatonic scales save one ("the octatonic" or Diminished scale) 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 Complement may refer to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class collections into complementary sets * Complementary color, in the visu ...

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