Common tone (scale)

[1] Six of seven possible common tones are shared by closely related keys, though keys may also be thought of as more or less closely related according to their number of common tones.

[2] In diatonic set theory the common tone theorem explains that scales possessing the deep scale property share a different number of common tones, not counting enharmonic equivalents (for example, C♯ and C♭ have no common tones with C major), for every different transposition of the scale.

For example, then, modulation to the dominant (transposition by a perfect fifth) includes six common tones between the keys as there are six perfect fifths in a diatonic scale, while transposition by the tritone includes only one common tone as there is only one tritone in a diatonic scale.

[1] In diatonic set theory, the deep scale property is the quality of pitch class collections or scales containing each interval class a unique number of times.

[4] 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.

C is a common tone between the C and G major scales, as are D, E, G, A, and B.
Common tones between G major and C major and between C major and F ♯ major, 6 and 1 common tones respectively.
Diatonic scale transposed a perfect fifth: since it contains six perfect fifths the two scales a perfect fifth apart have six common tones.
Diatonic scale in the chromatic circle with each interval class a different color, each occurs a unique number of times
C major scale with interval classes labelled
Whole tone scale on C with interval classes labelled