Anhemitonic scale

Musicology commonly classifies scales as either hemitonic or anhemitonic. Hemitonic scales contain one or more semitones, while anhemitonic scales do not contain semitones. For example, in traditional Japanese music, the anhemitonic yo scale is contrasted with the hemitonic in scale.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. The simplest and most commonly used scale in the world is the atritonic anhemitonic "major" pentatonic scale. The whole tone scale is also anhemitonic.

A special subclass of the hemitonic scales is the cohemitonic scales.Christ, William (1966). Materials and Structure of Music, v.1, p. 39. Englewood Cliffs: Prentice–Hall. LOC 66-14354. 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.Tymoczko, Dmitri (1997). "The Consecutive-Semitone Constraint on Scalar Structure: A Link between Impressionism and Jazz", Intégral, v.11, (1997), p. 135-179. 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, melodic major/melodic minor, harmonic major scale, harmonic minor scale, Hungarian major scale, Romanian major scale, and the so-called octatonic scale.

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 tritones, 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.Keith, Michael. 1991. From Polychords to Polya : Adventures in Musical Combinatorics, p. 48-49. Princeton: Vinculum Press. . 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.

Sonority Semitone counts Notes Count 0 1 2 2A 2C 3 3A 3C >=4 >=4A >=4C 1 1 1 0 0 0 0 0 0 0 0 0 0 2 6 5 1 0 0 0 0 0 0 0 0 0 3 19 10 8 1 0 1 0 0 0 0 0 0 4 43 10 21 11 4 7 1 0 1 0 0 0 5 66 3 20 30 15 15 12 0 12 1 0 1 6 80 1 5 26 16 10 34 4 30 14 0 14 7 66 0 0 3 2 1 20 4 16 43 0 43 8 43 0 0 0 0 0 0 0 0 43 1 42 9 19 0 0 0 0 0 0 0 0 19 0 19 10 6 0 0 0 0 0 0 0 0 6 0 6 11 1 0 0 0 0 0 0 0 0 1 0 1 12 1 0 0 0 0 0 0 0 0 1 0 1 TOTALS 351 30 55 71 37 34 67 8 59 128 1 127

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 whole tone scale.Hanson, Howard. (1960) Harmonic Materials of Modern Music, p.367. New York: Appleton-Century-Crofts. LOC 58-8138. Column "2A", row "7", a local minimum, refers to the diatonic scale and melodic major/melodic minor scales.Hanson, Howard. (1960) Harmonic Materials of Modern Music, p.362-363. New York: Appleton-Century-Crofts. LOC 58-8138. 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.Hanson, Howard. (1960) Harmonic Materials of Modern Music, p.363. New York: Appleton-Century-Crofts. LOC 58-8138. Column "3A", row "7", another local minimum, represents the harmonic major scale and its involution harmonic minor scale, and the Hungarian major scale and its involution Romanian major scale.Hanson, Howard. (1960) Harmonic Materials of Modern Music, p.364. New York: Appleton-Century-Crofts. LOC 58-8138. Column "3A", row "6", are the hexatonic analogs to these four familiar scales,Hanson, Howard. (1960) Harmonic Materials of Modern Music, p.369. New York: Appleton-Century-Crofts. LOC 58-8138. one of which being the Augmented scale,Hanson, Howard. (1960) Harmonic Materials of Modern Music, p.368. New York: Appleton-Century-Crofts. LOC 58-8138. and another the analog of the Octatonic scale - which itself appears, alone and solitary, at Column ">=4A". row "8".Hanson, Howard. (1960) Harmonic Materials of Modern Music, p.360. New York: Appleton-Century-Crofts. LOC 58-8138. 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;Cooper, Paul. 1973. Perspectives in Music Theory: An Historical-Analytical Approach, p. 18. New York: Dodd, Mead. . 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.

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.Hanson, Howard. (1960) Harmonic Materials of Modern Music, p.29. New York: Appleton-Century-Crofts. LOC 58-8138. "The hexad [consisting of perfect fifths] adds B, C-G-D-A-E-B, or melodically, producing C-D-E-F-G-A-B, its components being five perfect fifths, four major seconds, three minor thirds, two major thirds, and--for the first time--the dissonant minor second (or major seventh), pmnsd."

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.Hanson, Howard. (1960) Harmonic Materials of Modern Music, p.40. New York: Appleton-Century-Crofts. LOC 58-8138.

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 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 [of the perfect fifth] is 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, 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