music Music is the arrangement of sound to create some combination of Musical form, form, harmony, melody, rhythm, or otherwise Musical expression, expressive content. Music is generally agreed to be a cultural universal that is present in all hum ...

, 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 ...

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 consecutive series of notes that form a progression between one note and its octave", typically by order of pitch or fundamental frequency. The word "scale" originates from the Latin ''scala'', which literal ...

and a transposition of that scale, as in

modulation Signal modulation is the process of varying one or more properties of a periodic waveform in electronics and telecommunication for the purpose of transmitting information. The process encodes information in form of the modulation or message ...

. 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: four of them share all the pitches except on ...

s, though

keys Key, Keys, The Key or The Keys may refer to: Common uses * Key (cryptography), a piece of information needed to encode or decode a message * Key (instrument), a component of a musical instrument * Key (lock), a device used to operate a lock * ...

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 set theory (music), musical set theory which applies the techniques and musical analysis, analysis of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myh ...

the common tone theorem explains that scales possessing the deep scale property share a different number of common tones, not counting

enharmonic In music, two written notes have enharmonic equivalence if they produce the same pitch but are notated differently. Similarly, written intervals, chords, or key signatures are considered enharmonic if they represent identical pitches that ar ...

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 Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval f ...

) 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 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 ...

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 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 ...

(including

major Major most commonly refers to: * Major (rank), a military rank * Academic major, an academic discipline to which an undergraduate student formally commits * People named Major, including given names, surnames, nicknames * Major and minor in musi ...

natural 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). ...

, and the

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 ...

). In twelve-tone

equal temperament An equal temperament is a musical temperament or Musical tuning#Tuning systems, tuning system that approximates Just intonation, just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequency, frequencie ...

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

'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