Diatonic scale

Antiquity

Evidence that the Sumerians and Babylonians used a version of the diatonic scale is found in cuneiform inscriptions that contain both musical compositions and a tuning system.^[1][2] Despite the conjectural nature of reconstructions of the Hurrian songs, the diatonic nature of the tuning system is demonstrated by the fact that it involves a series of six perfect fifths, which is a recipe for the construction of a diatonic scale.

The 9,000-year-old flutes found in Jiahu, China, indicate the evolution over 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.^[3]

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 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♮:

1. A–B–C–D–E–F–G–A was described as D–E–F–G–A–B♭–C–D (the modern Aeolian modes whose reference notes are A and D, respectively, corresponding to the Aeolian modes of C major and F major, respectively)
2. C–D–E–F–G–A–B–C was described as F–G–A–B♭–C–D–E–F (the modern Ionian modes whose reference notes are C and F, respectively, corresponding to the Ionian modes of C major and F major, respectively).

As a result, medieval theory described the church modes as corresponding to four diatonic scales only (two of which had the variable B♮/♭). They were the modern Dorian, Phrygian, Lydian, and Mixolydian modes of C major, plus the Aeolian and Ionian modes of F major when B♭ was substituted into the Dorian and Lydian modes of C major, respectively.

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. These were the 6 non-Locrian modes of C major and F major.

Modern

By the beginning of the Baroque period, the notion of the 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 in classical and 20th-century music, and jazz (see chord-scale system).

Major scale

The major scale or Ionian mode 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 solfège, the syllables used to name each degree of the scale are Do–Re–Mi–Fa–Sol–La–Ti–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, especially when used in a tonal context:

• 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. That is, 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 a natural minor scale, called the A natural minor scale.

The degrees of the natural minor scale, especially in a tonal context, 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) below the tonic, as is the case in the major scale.

Besides the natural minor scale, five other kinds of scales can be obtained from the notes of a major scale, by simply choosing a different note as the starting note. 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:

The first column examples shown above are formed by natural notes (i.e. neither sharps nor flats, 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. This is shown in the second column, with each mode transposed to start on C.

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]

each tetrachord being formed of two tones and a semitone, T–T–S,

and the natural minor of A would be:

[A–B–C–D] – [E–F–G–A]

formed two different tetrachords, the first consisting in a semitone between two tones, T–S–T, and the second of a semitone and two tones, S–T–T.

The medieval conception of the tetrachordal structure, however, was based on one single tetrachord, that of the D scale,

[D–E–F–G] – [A–B–C–D]

each formed of a semitone between tones, T–S–T. It viewed other diatonic scales as differently overlapping disjunct and conjunct tetrachords:

E scale: E–F–G | A–B–C–D = D–E

F scale: F–G | A–B–C–D = D–E–F

G scale: G | A–B–C–D = D–E–F–G

A scale: A–B–C–D = D–E–F–G | A

B scale: B–C–D = D–E–F–G | A–B

C scale: C–D = D–E–F–G | A–B–C

(where G | A indicates the disjunction of tetrachords, always between G and A, and D = D indicates their conjunction, always on the common note D).

Iteration of the fifth

If the scale is produced by the iteration of six perfect fifths, for instance F–C–G–D–A–E–B, the result is Pythagorean tuning:

This tuning dates to Ancient Mesopotamia^[5] (see Music of Mesopotamia § Music theory), and was done by alternating ascending fifths with descending fourths (equal to an ascending fifth followed by a descending octave), resulting in the notes of a pentatonic or heptatonic scale falling within an octave.

Six of the "fifth" intervals (C–G, D–A, E–B, F–C', G–D', A–E') are all 3⁄2 = 1.5 (701.955 cents), but B–F' is the discordant tritone, here 729⁄512 = 1.423828125 (611.73 cents). Tones are each 9⁄8 = 1.125 (203.91 cents) and diatonic semitones are 256⁄243 ≈ 1.0535 (90.225 cents).

Extending the series of fifths to eleven fifths would result into the Pythagorean chromatic scale.

Equal temperament

Equal temperament is the division of the octave in twelve equal semitones. The frequency ratio of the semitone then becomes the twelfth root of two (¹²√2 ≈ 1.059463, 100 cents). The tone is the sum of two semitone. Its ratio is the sixth root of two (⁶√2 ≈ 1.122462, 200 cents). Equal temperament can be produced by a succession of tempered fifths, each of them with the ratio of 2^7⁄12 ≈ 1.498307, 700 cents.

Meantone temperament

The fifths could be tempered more than in equal temperament, in order to produce better thirds. See quarter-comma meantone for a meantone temperament commonly used in the sixteenth and seventeenth centuries and sometimes after, which produces perfect major thirds.

Just intonation

Just intonation often is represented using Leonhard Euler's Tonnetz, with the horizontal axis showing the perfect fifths and the vertical axis the perfect major thirds. In the Tonnetz, the diatonic scale in just intonation appears as follows:

F–A, C–E and G–B, aligned vertically, are perfect major thirds; A–E–B and F–C–G–D are two series of perfect fifths. The notes of the top line, A, E and B, are lowered by the syntonic comma, 81⁄80, and the "wolf" fifth D–A is too narrow by the same amount. The tritone F–B is 45⁄32 ≈ 1.40625.

This tuning has been first described by Ptolemy and is known as Ptolemy's intense diatonic scale. It was also mentioned by Zarlino in the 16th century and has been described by theorists in the 17th and 18th centuries as the "natural" scale.

Since the frequency ratios are based on simple powers of the prime numbers 2, 3, and 5, this is also known as five-limit tuning.