Diatonic Scale - Properties

Properties

The diatonic scale as defined above has specific properties that make it unique among seven-note scales. In other words, no other kind of scale has the same properties:

• It is obtained from a chain of six successive perfect fifths. For instance, the seven natural pitches which form the C-major scale can be obtained from a chain of perfect fifths starting from F (F—C—G—D—A—E—B)
• It is either a sequence of successive natural notes (such as the C-major scale, C-D-E-F-G-A-B, or the A-minor scale, A-B-C-D-E-F-G) or a transposition thereof.
• It can be written using seven consecutive notes without accidentals on a staff with a conventional key signature, or with no signature. This is because the staff is purposely designed to represent diatonic scales.

David Rothenberg conceived of a property of scales he called propriety, and around the same time Gerald Balzano independently came up with the same definition in the more limited context of equal temperaments, calling it coherence. Rothenberg distinguished proper from a slightly stronger characteristic he called strictly proper. In this vocabulary, there are five proper seven-note scales in 12 equal temperament. None of these is strictly proper, i.e., coherent in the sense of Balzano; but in any system of meantone tuning with the fifth flatter than 700 cents, they are strictly proper. The scales are the diatonic, ascending minor, harmonic minor, harmonic major, and locrian major scales; of these, all but the last are well-known and constitute the backbone of diatonic practice when taken together.

Among these four well-known variants of the diatonic scale, the diatonic scale itself has additional properties of what has been called simplicity, because it is produced by iterations of a single generator, the meantone fifth. The scale, in the vocabulary of Erv Wilson, who may have been the first to consider the notion, is sometimes called a MOS scale.

The diatonic collection contains each interval class a unique number of times. Diatonic set theory describes the following properties, aside from propriety: maximal evenness, Myhill's property, well formedness, the deep scale property, cardinality equals variety, and structure implies multiplicity.