music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspe ...

, a common tone is a

pitch class In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...

that is a member of, or common to (shared by) two or more

scales Scale or scales may refer to: Mathematics * Scale (descriptive set theory), an object defined on a set of points * Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original * Scale factor, a number w ...

or sets.

Common tone theorem

A common tone is a

that is a member of, or common to, a

musical scale In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Often, especially in the ...

and a transposition of that scale, as in

modulation In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the '' carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informat ...

. Six of seven possible common tones are shared by

closely related key In music, a closely related key (or close key) is one sharing many common tones with an original key, as opposed to a distantly related key (or distant key). In music harmony, there are six of them: five share all, or all except one, pitches wi ...

s, though

keys Key or The Key may refer to: Common meanings * Key (cryptography), a piece of information that controls the operation of a cryptography algorithm * Key (lock), device used to control access to places or facilities restricted by a lock * Key (m ...

may also be thought of as more or less closely related according to their number of common tones. "Obviously, tonal distance is in some sense a function of the extent of intersection between diatonic PC collections of tonal systems". In

diatonic set theory Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and analysis of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, the ...

the common tone theorem explains that scales possessing the

deep scale property In music, a common tone is a pitch class that is a member of, or common to (shared by) two or more scales or sets. Common tone theorem A common tone is a pitch class that is a member of, or common to, a musical scale and a transposition of ...

share a different number of common tones, not counting

enharmonic In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a writte ...

equivalents (for example, C and C have no common tones with C major), for every different transposition of the scale. However many times an

interval class In musical set theory, an interval class (often abbreviated: ic), also known as unordered pitch-class interval, interval distance, undirected interval, or "(even completely incorrectly) as 'interval mod 6'" (; ), is the shortest distance in pitch ...

occurs in a diatonic scale is the number of tones common both to the original scale and a scale transposed by that particular interval class. For example, then, modulation to the dominant (transposition by a

perfect fifth In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of five ...

) includes six common tones between the keys as there are six perfect fifths in a diatonic scale, while transposition by the

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 adj ...

includes only one common tone as there is only one tritone in a diatonic scale.

Deep scale property

Deep scale property lack of the whole tone scale notation

, the deep scale property is the quality of

collections or

containing each

a unique number of times. Examples include the

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 st ...

(including

major Major ( commandant in certain jurisdictions) is a military rank of commissioned officer status, with corresponding ranks existing in many military forces throughout the world. When used unhyphenated and in conjunction with no other indicato ...

natural 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 ...

, and the

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 ...

). In twelve-tone

equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, ...

, all scales with the deep scale property can be generated with any interval coprime with twelve. For example, the diatonic scale's

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 inte ...

contains: The common tone theorem describes that scales possessing the deep scale property share a different number of common tones for every different transposition of the scale, suggesting an explanation for the use and usefulness of the diatonic collection. In contrast, 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' ...

's interval vector contains: and has only two distinct transpositions (every even transposition of the whole tone scale is identical with the original and every odd transposition has no common tones whatsoever).

Common tone theorem

Deep scale property

See also

References

Further reading