Anhemitonic scale

Most of the world's music is anhemitonic, perhaps 90%.[9] 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 same 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 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.,[8] 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.[8]

Column 0, row 5 are the full but pleasant chords: 9th, 6/9, and 9alt5 with no 7.[11] Column "0", row "6", is the unique whole tone scale.[12] Column "2A", row "7", a local minimum, refers to the diatonic scale and melodic major/melodic minor scales.[13] 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.[14] Column "3A", row "7", another local minimum, represents the Hungarian major scale, and its involution, and the harmonic major scale and involution harmonic minor scale of the same.[15] Column "3A", row "6", are the hexatonic analogs to these four familiar scales,[16] one of which being the Augmented scale,[17] and another the analog of the Octatonic scale - which itself appears, alone and solitary, at Column ">=4A". row "8".[18] Column "2A", row "4", another minimum, represents a few frankly dissonant, yet strangely resonant harmonic combinations: mM9 with no 5, 11♭9, dom13♭9, and M7♯11.[11]

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".

The interrelationship of semitones, tritones, and increasing note count can be demonstrated by taking five consecutive pitches from the circle of fifths;[19] 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

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.[20]

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.[21]

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,[22] but cohemitonia results.

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 (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.[22]

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.[22] It is near these points where most popular scales lie.

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 of the scale on the middle note of the halfstep span. This allows a leading tone 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 convergence on the tonic. It is very common that a cohemitonic (or even hemitonic) scale (e.g.: Hungarian minor { C D E♭ F♯ G A♭ B }) be displaced preferentially to a mode where the halfstep span is split (e.cont.: Double harmonic scale { G A♭ B C D E♭ F♯ }), and by which name we more commonly know the same circular series of intervals.[23] Cohemitonic scales with multiple halfstep spans present the additional possibility of modulating between tonics each furnished with both upper and lower neighbors.

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

Western music's system of key signature is based upon the assumption of a heptatonic scale 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,[24] most of which are common in romantic music, and of which most Romantic music is composed:

• Diatonic scale
• Melodic major/melodic minor
• Hungarian major scale
• involution of Hungarian major
• Harmonic major scale
• Harmonic minor 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 each, and are suitable for use in modal mutation.[25] 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.

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♯	Lydian Augmented
Melodic	F♯, B♭	Acoustic, Lydian Dominant
Melodic	E♭	Melodic minor (ascending), Jazz minor
Melodic	B♭, A♭	Melodic Major (descending), Aeolian Dominant, Mixolydian ♭13
Melodic	B♭, E♭, D♭	Dorian ♭9
Melodic	B♭, E♭, A♭, G♭	Half Diminished, Locrian ♮2, Semilocrian
Melodic	B♭, E♭, A♭, D♭, G♭, F♭	Superlocrian, Altered
Base scale	Accidentals	Mode name
Hungarian major	F♯, G♯, E♯	Lydian Augmented ♯3
Hungarian major	F♯, D♯, B♭	Hungarian Major
Hungarian major	G♯, E♭	Jazz Minor ♯5
Hungarian major	F♯, B♭, E♭, D♭	Ukrainian Dorian ♭9
Hungarian major	E♭, A♭, G♭	Harmonic Minor ♭5
Hungarian major	B♭, E♭, D♭, G♭, F♭	Altered Dominant ♮6
Hungarian major	E♭, D♭, G♭, F♭, B, A	Ultralocrian 6
Base scale	Accidentals	Mode name
involution of Hungarian major	F♯, G♯, D♯, E♯	Super Lydian Augmented ♮6
involution of Hungarian major	F♯, G♯, E♭	Lydian Augmented ♭3
involution of Hungarian major	F♯, B♭, D♭	Involution of Hungarian Major
involution of Hungarian major	E♭, G♭	Jazz Minor ♭5
involution of Hungarian major	B♭, E♭, D♭, F♭	Dorian ♭9 ♭11
involution of Hungarian major	E♭, A♭, G♭, B	Semilocrian 7
involution of Hungarian major	B♭, E♭, D♭, G♭, F♭, A	Altered Dominant 6
Base scale	Accidentals	Mode name
Harmonic major	F♯, G♯, D♯	Lydian Augmented ♯2
Harmonic major	F♯, E♭	Lydian Diminished
Harmonic major	A♭	Harmonic Major
Harmonic major	B♭, D♭	Phrygian Dominant ♮6
Harmonic major	B♭, E♭, G♭	Diminished Dorian
Harmonic major	B♭, E♭, A♭, D♭, F♭	Superphrygian
Harmonic major	E♭, A♭, D♭, G♭, B	Locrian Diminished
Base scale	Accidentals	Mode name
Harmonic minor	F♯, D♯	Lydian ♯2
Harmonic minor	G♯	Ionian Augmented
Harmonic minor	F♯, B♭, E♭	Ukrainian Dorian
Harmonic minor	E♭, A♭	Harmonic Minor
Harmonic minor	B♭, A♭, D♭	Phrygian Dominant
Harmonic minor	B♭, E♭, D♭, G♭	Locrian ♮6
Harmonic minor	E♭, A♭, D♭, G♭, F♭, B	Ultralocrian
Base scale	Accidentals	Mode name
Hungarian minor	F♯, D♯, A♯	Lydian ♯2 ♯6
Hungarian minor	G♯, D♯	Ionian Augmented ♯2
Hungarian minor	F♯, E♭, A♭	Hungarian Minor
Hungarian minor	A♭, D♭	Double harmonic
Hungarian minor	B♭, D♭, G♭	Oriental
Hungarian minor	E♭, A♭, D♭, F♭, B	Ultraphrygian
Hungarian minor	A♭, D♭, G♭, B, E	Locrian Diminished 3
Base scale	Accidentals	Mode name
Neapolitan major	F♯, G♯, A♯	Leading Whole-Tone
Neapolitan major	F♯, G♯, B♭	Lydian Augmented Dominant
Neapolitan major	F♯, B♭, A♭	Lydian Minor
Neapolitan major	E♭, D♭	Neapolitan Major
Neapolitan major	B♭, A♭, G♭	Locrian Major
Neapolitan major	B♭, E♭, A♭, G♭, F♭	Altered ♮2
Neapolitan major	B♭, A♭, D♭, G♭, F♭, E	Altered 3
Base scale	Accidentals	Mode name
Neapolitan minor	F♯, A♯	Lydian ♯6
Neapolitan minor	D♯	Ionian ♯2
Neapolitan minor	G♯, B♭	Mixoydian Augmented
Neapolitan minor	F♯, B♭, E♭, A♭	Hungarian Gypsy
Neapolitan minor	E♭, A♭, D♭	Neapolitan Minor
Neapolitan minor	B♭, A♭, D♭, G♭	Locrian Dominant
Neapolitan minor	A♭, D♭, G♭, F♭, B, E	Ultralocrian 3
Base scale	Accidentals	Mode name
Ionian ♭5	F♯, G♯, D♯, A♯, E♯	Super Lydian Augmented
Ionian ♭5	F♯, D♭	Lydian ♭2
Ionian ♭5	G♭	Ionian ♭5
Ionian ♭5	B♭, E♭, F♭	Dorian ♭4
Ionian ♭5	E♭, A♭, B	Aeolian 7
Ionian ♭5	B♭, A♭, D♭, E	Phrygian 3
Ionian ♭5	B♭, E♭, D♭, G♭, A	Locrian 6
Base scale	Accidentals	Mode name
Persian	F♯, A♯, E♯	Lydian ♯6 ♯3
Persian	D♯, A♯	Ionian ♯2 ♯6
Persian	G♯, D♯, B♭	Mixolydian Augmented ♯2
Persian	F♯, E♭, A♭, D♭	Neapolitan Minor ♯4
Persian	A♭, D♭, G♭	Persian
Persian	A♭, D♭, G♭, B, E	Ultraphrygian 3
Persian	D♭, G♭, B, E, A	Altered Altered ♮4
Base scale	Accidentals	Mode name
Locrian ♮7	F♯, E♯	Lydian ♯3
Locrian ♮7	A♯	Ionian ♯6
Locrian ♮7	D♯, B♭	Mixolydian ♯2
Locrian ♮7	G♯, B♭, E♭	Dorian Augmented
Locrian ♮7	F♯, B♭, E♭, A♭, D♭	Phrygian ♯4
Locrian ♮7	E♭, A♭, D♭, G♭	Locrian ♮7
Locrian ♮7	D♭, G♭, F♭, B, E, A	Altered Altered

• 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.

• All heptatonic and larger scales are hemitonic (ditonic or better) and tritonic.[24] 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.
• All octatonic scales save one ("the octatonic" or Diminished scale) are cohemitonic.[24]
• All enneatonic and larger scales are cohemitonic.[24]
• All sonorities with 5 or more semitones are cohemitonic.[24]
• The set complement 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.[24]
• Tetrahemitonic and pentahemitonic scales never have more than 8 notes.[24]
• Hexahemitonic and heptahemitonic scales never have more than 9 notes.[24]
• Octahemitonic and enneahemitonic scales never have more than 10 notes.[24]
• There is no 12ET scale with exactly 11 halfsteps.[24]