Diatonic scale

History

Western music from the Middle Ages until the late 19th century is based on the diatonic scale and the unique hierarchical relationships created by this system of organizing seven notes.

Prehistory

There is one claim that the 45,000-year-old Divje Babe Flute uses a diatonic scale, but there is no proof or consensus it is even a musical instrument.

There is evidence that the Sumerians and Babylonians used some version of the diatonic scale. This derives from surviving inscriptions that contain a tuning system and musical composition. Despite the conjectural nature of reconstructions of the piece known as the Hurrian songs from the surviving score, the evidence that it used the diatonic scale is much more soundly based. This is because instructions for tuning the scale involve tuning a chain of six fifths, so that the corresponding circle of seven major and minor thirds are all consonant-sounding, and this is a recipe for tuning a diatonic scale.

9,000-year-old flutes found in Jiahu, China indicate the evolution, over a period of 1,200 years, of flutes having 4, 5 and 6 holes to having 7 and 8 holes, the latter exhibiting striking similarity to diatonic hole spacings and sounds.

Middle Ages

The scales corresponding to the medieval Church modes were diatonic. Depending on which of the seven notes of the diatonic scale you use as the beginning, the positions of the intervals, the half-steps, fall at different distances from the starting tone (the "reference note"), producing seven different scales. One of these, the one starting on B, has no pure fifth above its reference note (B–F is a diminished fifth): it is probably for this reason that it was not used. Of the six remaining scales, two were described as corresponding to two others with a B♭ instead of a B♮: A–B–C–D–E–F–G–A was described as D–E–F–G–A–B♭C–D and C–D–E–F–G–A–B–C as F–G–A–B♭–C–D–E–F (which share the same series of intervals, T–S–T–T–S–T–T and T–T–S–T–T–T–S respectively, where T=tone and S=semitone). As a result, medieval theory described the Church modes as corresponding to four diatonic scales only (two of which with the variable B♮/♭).

Renaissance

Heinrich Glarean considered that the modal scales including a B♭ had to be the result of a transposition. In his Dodecachordon he not only described six "natural" diatonic scales (still neglecting the seventh one with a diminished fifth above the reference note), but also six "transposed" ones, each including a B♭, resulting in the total of twelve scales that justified the title of his treatise.

Modern

By the beginning of the Baroque period, the notion of musical key was established, describing additional possible transpositions of the diatonic scale. Major and minor scales came to dominate until at least the start of the 20th century, partly because their intervallic patterns are suited to the reinforcement of a central triad. Some Church modes survived into the early 18th century, as well as appearing occasionally in classical and 20th-century music, and later in modal jazz.

Theory

Glarean's six natural scales could be transposed not only to include one flat in the signature, but to all twelve notes of the chromatic scale. Of these six scales, three are major scales (those with a major third/triad: Ionian, Lydian, and Mixolydian), three are minor (those with a minor third/triad: Dorian, Phrygian, and Aeolian). To these may be added the seventh diatonic scale, with a diminished fifth above the reference note, the Locrian scale. Transposing each of these seven scales on the twelve degrees of the chromatic scale results in a total of eighty-four diatonic scales.

The modern musical keyboard originated as a diatonic keyboard with only lower (white) keys. The upper (black) keys progressively were added for several purposes:

improving the consonances, mainly the thirds, by providing a major third on each degree;

allowing all twelve transpositions described above:

and helping musicians to find their bearings on the keyboard.

The pattern of elementary intervals forming the diatonic scale can be represented either by the letters T and S, for Tone and Semitone respectively, or by their size in number of semitones, 2 for the tone and 1 for the semitone. The major scale, for instance, can be represented as

T–T–S–T–T–T–S

or

2–2–1–2–2–2–1

Major scale

The major scale or Ionian scale is one of the diatonic scales. It is made up of seven distinct notes, plus an eighth that duplicates the first an octave higher. The pattern of seven intervals separating the eight notes is T-T-S-T-T-T-S. In solfege, the syllables used to name each degree of the scale are Do–Re–Mi–Fa–Sol–La–Si–Do. A sequence of successive natural notes starting from C is an example of major scale, called C-major scale.

The eight degrees of the scale are also known by traditional names:

1st – Tonic (key note)

2nd – Supertonic

3rd – Mediant

4th – Subdominant

5th – Dominant

6th – Submediant

7th – Leading tone

8th – Tonic (Octave)

Natural minor scale

For each major scale, there is a corresponding natural minor scale, sometimes called its relative minor. It uses the same sequence of notes as the corresponding major scale, but starts from a different note. Namely, it begins on the sixth degree of the major scale and proceeds step by step to the first octave of the sixth degree. A sequence of successive natural notes starting from A is an example of natural minor scale, called A-minor scale.

The degrees of the natural minor scale have the same names as those of the major scale, except the seventh degree, which is known as the subtonic because it is a whole step below the tonic. The term leading tone is generally reserved for seventh degrees that are a half step (semitone) away from the tonic, as is the case in the major scale or the harmonic minor scale and the melodic minor ascending – which are not diatonic. In solfege the scale degrees are named in two different ways: either La–Ti–Do–Re–Mi–Fa–Sol–La or Do–Re–Me–Fa–Sol–Le–Te–Do.

Besides the natural minor scale, five other kinds of scales (often improperly called modes) can be obtained from the notes of a major scale, by simply choosing a different note as the starting note or tonic. All these scales meet the definition of diatonic scale.

"Modes"

The whole collection of diatonic scales as defined above can be divided into seven different scales.

As explained above, all major scales use the same interval sequence T-T-S-T-T-T-S. This interval sequence was called the Ionian mode by Glarean. It is one of the seven modern modes. From any major scale, a new scale is obtained by taking a different degree as the tonic. With this method it is possible to generate six other scales or modes from each major scale. Another way to describe the same result would be to consider that, behind the diatonic scales, there exists an underlying "diatonic system" which is the series of diatonic notes without a reference note; assigning the reference note in turn to each of the seven notes in each octave of the system produces seven diatonic scales, each characterized by a different interval sequence:

For the sake of simplicity, the examples shown above are formed by natural notes (also called "white-notes", as they can be played using the white keys of a piano keyboard). However, any transposition of each of these scales (or of the system underlying them) is a valid example of the corresponding mode. In other words, transposition preserves mode.

The whole set of diatonic scales is commonly defined as the set composed of these seven natural-note scales, together with all of their possible transpositions. As discussed elsewhere, different definitions of this set are sometimes adopted in the literature.

Diatonic scales and tetrachords

A diatonic scale can be also described as two tetrachords separated by a whole tone. For example, under this view the two tetrachord structures of C major would be:

[C-D-E-F]-[G-A-B-C]

and the natural minor of A would be:

[A-B-C-D]-[E-F-G-A].

The set of intervals within each tetrachord comprises two tones and a semitone.

Properties

The diatonic scale as defined above has specific properties that make it unique among seven-note scales. Many of these properties arise from the fact that our Western music theory and our Western music notation were conceived with the diatonic scale in mind. These properties include:

It is obtained from a chain of six successive perfect fifths. For instance, the seven natural pitches that 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.

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. These properties may be considered to generalize properties of the diatonic scale to other scales. The generation of the diatonic scale by iterations of a single generator, the fifth, has also been generalized by Erv Wilson, in what is sometimes called a MOS scale. See also Diatonic set theory for a discussion of other properties that transcend the diatonic scale properly speaking.

Tuning

In just intonation the diatonic scale is tuned:

In Pythagorean tuning the diatonic scale is:

The differences between the two, i.e. between 5/4 and 81/64 for E, 5/3 and 27/16 for A, and 15/8 and 243/128 for B, always amount to a syntonic comma, 81/80.