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Inmusic theory adiatonic scale is aheptatonic (seven-note) scale that includes fivewhole steps (whole tones) and twohalf steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps. In other words, the half steps are maximally separated from each other.
The sevenpitches of any diatonic scale can also be obtained by using achain of sixperfect fifths. For instance, the sevennatural pitch classes that form the C-major scale can be obtained from a stack of perfect fifths starting from F:
Any sequence of seven successivenatural notes, such as C–D–E–F–G–A–B, and anytransposition thereof, is a diatonic scale. Modernmusical keyboards are designed so that the white-key notes form a diatonic scale, though transpositions of this diatonic scale require one or more black keys. A diatonic scale can be also described as twotetrachords separated by a whole tone. Inmusical set theory,Allen Forte classifies diatonic scales asset form 7–35.
The termdiatonic originally referred to thediatonic genus, one of the threegenera of the ancient Greeks, and comes fromAncient Greek:διατονικός,romanized: diatonikós, of uncertain etymology. Most likely, it refers to the intervals being "stretched out" in that tuning, in contrast to the other two genera (chromatic andenharmonic).
This article does not concern alternative seven-note scales such as theharmonic minor or themelodic minor which, although sometimes called "diatonic", do not fulfill the condition of maximal separation of the semitones indicated above.
Western music from theMiddle Ages until thelate 19th century (seecommon practice period) is based on the diatonic scale and the uniquehierarchical relationships created by this system of organizing seven notes.
Evidence that theSumerians andBabylonians used a version of the diatonic scale is found incuneiform inscriptions that contain both musical compositions and a tuning system.[1][2] Despite the conjectural nature of reconstructions of theHurrian 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-oldflutes found inJiahu, 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]
The scales corresponding to the medievalchurch 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, theone starting on B, has no pure fifth above its reference note (B–F is adiminished 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♮:
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 modernDorian,Phrygian,Lydian, andMixolydian modes ofC major, plus theAeolian andIonian modes ofF major when B♭ was substituted into the Dorian and Lydian modes ofC major, respectively.
Heinrich Glarean considered that the modal scales including a B♭ had to be the result of a transposition. In hisDodecachordon, 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-Locrianmodes ofC major andF major.
By the beginning of theBaroque period, the notion of the musicalkey was established, describing additional possible transpositions of the diatonic scale.Major andminor 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 centraltriad. Some church modes survived into the early 18th century, as well as appearing inclassical and20th-century music, andjazz (seechord-scale system).
Of Glarean's six natural scales, three have a major third/first triad:Ionian,Lydian, andMixolydian, and three have a minor one:Dorian,Phrygian, andAeolian. To these may be added the seventh diatonic scale, with a diminished fifth above the reference note, theLocrian scale. These could be transposed not only to include one flat in the signature (as described by Glarean), but to all twelve notes of thechromatic scale, resulting in a total of eighty-four diatonic scales.
The modernmusical keyboard originated as a diatonic keyboard with only white keys.[4] The black keys were progressively added for several purposes:
The pattern of elementary intervals forming the diatonic scale can be represented either by the letters T (tone) and S (semitone) respectively. With this abbreviation, a major scale, for instance, can be represented as
Themajor scale orIonian mode is one of the diatonic scales. It is made up of seven distinctnotes, plus an eighth that duplicates the first anoctave higher. The pattern of seven intervals separating the eight notes is T–T–S–T–T–T–S. Insolfège, the syllables used to name eachdegree of the scale areDo–Re–Mi–Fa–Sol–La–Ti–Do. A sequence of successivenatural notes starting from C is an example of major scale, calledC-major scale.
| Notes in C major: | C | D | E | F | G | A | B | C | ||||||||
| Degrees in solfège: | Do | Re | Mi | Fa | Sol | La | Ti | Do | ||||||||
| Interval sequence: | T | T | S | T | T | T | S |
The seven degrees of the scale are also known by traditional names, especially when used in a tonal context:
For each major scale, there is a correspondingnatural minor scale, sometimes called itsrelative 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 successivenatural notes starting from A is an example of a natural minor scale, called the A natural minor scale.
| Notes in A minor: | A | B | C | D | E | F | G | A | ||||||||
| Interval sequence: | T | S | T | T | S | T | T |
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 thesubtonic because it is a whole step below the tonic. The termleading tone is generally reserved for seventh degrees that are ahalf 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.
The whole collection of diatonic scales as defined above can be divided into seven different scales.
As explained above, allmajor scales use the same interval sequence T–T–S–T–T–T–S. This interval sequence was called theIonian mode by Glarean. It is one of the seven modern modes. From any major scale, a new scale is obtained by taking a differentdegree 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:
| Mode | Also known as | Starting note relative to major scale | Interval sequence | Example with white keys | Example with tonic C |
|---|---|---|---|---|---|
| Ionian | Major scale | C | T–T–S–T–T–T–S | C–D–E–F–G–A–B–C | |
| Dorian | D | T–S–T–T–T–S–T | D–E–F–G–A–B–C–D | C–D–E♭–F–G–A–B♭–C | |
| Phrygian | E | S–T–T–T–S–T–T | E–F–G–A–B–C–D–E | C–D♭–E♭–F–G–A♭–B♭–C | |
| Lydian | F | T–T–T–S–T–T–S | F–G–A–B–C–D–E–F | C–D–E–F♯–G–A–B–C | |
| Mixolydian | G | T–T–S–T–T–S–T | G–A–B–C–D–E–F–G | C–D–E–F–G–A–B♭–C | |
| Aeolian | Natural minor scale | A | T–S–T–T–S–T–T | A–B–C–D–E–F–G–A | C–D–E♭–F–G–A♭–B♭–C |
| Locrian | B | S–T–T–S–T–T–T | B–C–D–E–F–G–A–B | C–D♭–E♭–F–G♭–A♭–B♭–C | |
The first column examples shown above are formed bynatural notes (i.e. neither sharps nor flats, also called "white-notes", as they can be played using the white keys of apiano keyboard). But anytransposition 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 discussedelsewhere, different definitions of this set are sometimes adopted in the literature.
A diatonic scale can be also described as twotetrachords separated by awhole tone. For example, under this view the two tetrachord structures of C major would be:
each tetrachord being formed of two tones and a semitone, T–T–S,
and the natural minor of A would be:
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,
each formed of a semitone between tones, T–S–T. It viewed other diatonic scales as differently overlapping disjunct and conjunct tetrachords:
(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).
Diatonic scales can be tuned variously, either by iteration of a perfect or tempered fifth, or by a combination of perfect fifths and perfect thirds (Just intonation), or possibly by a combination of fifths and thirds of various sizes, as inwell temperament.
If the scale is produced by the iteration of six perfect fifths, for instance F–C–G–D–A–E–B, the result isPythagorean tuning:
| note | F | C | G | D | A | E | B | |
|---|---|---|---|---|---|---|---|---|
| pitch | 2⁄3 | 1⁄1 | 3⁄2 | 9⁄4 | 10⁄3 | 5⁄1 | 15⁄2 | |
| bring into main octave | 4⁄3 | 1⁄1 | 3⁄2 | 9⁄8 | 5⁄3 | 5⁄4 | 15⁄8 | |
| sort into note order | C | D | E | F | G | A | B | C' |
| interval above C | 1⁄1 | 9⁄8 | 5⁄4 | 4⁄3 | 3⁄2 | 5⁄3 | 15⁄8 | 2⁄1 |
| interval between notes | 9⁄8 | 10⁄9 | 16⁄15 | 9⁄8 | 10⁄9 | 9⁄8 | 16⁄15 |
This tuning dates to Ancient Mesopotamia[5] (seeMusic 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 all3⁄2 = 1.5 (701.955cents), but B–F' is the discordanttritone, here729⁄512 = 1.423828125 (611.73 cents). Tones are each9⁄8 = 1.125 (203.91 cents) and diatonic semitones are256⁄243 ≈ 1.0535 (90.225 cents).
Extending the series of fifths to eleven fifths would result into the Pythagoreanchromatic scale.
Equal temperament is the division of the octave in twelve equal semitones. The frequency ratio of the semitone then becomes thetwelfth root of two (12√2 ≈ 1.059463, 100cents). The tone is the sum of two semitones. 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 27⁄12 ≈ 1.498307, 700 cents.
The fifths could be tempered more than in equal temperament, in order to produce better thirds. Seequarter-comma meantone for a meantone temperament commonly used in the sixteenth and seventeenth centuries and sometimes after, which produces perfect major thirds.
Just intonation often is represented usingLeonhard Euler'sTonnetz, 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:
| A | E | B | |
| F | C | G | D |
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 thesyntonic comma,81⁄80, and the "wolf" fifth D–A is too narrow by the same amount. The tritone F–B is45⁄32 ≈ 1.40625.
This tuning has been first described byPtolemy and is known asPtolemy'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.
| notes | C | D | E | F | G | A | B | C' |
|---|---|---|---|---|---|---|---|---|
| pitch | 1⁄1 | 9⁄8 | 5⁄4 | 4⁄3 | 3⁄2 | 5⁄3 | 15⁄8 | 2⁄1 |
| interval between notes | 9⁄8 | 10⁄9 | 16⁄15 | 9⁄8 | 10⁄9 | 9⁄8 | 16⁄15 |
Since the frequency ratios are based on simple powers of theprime numbers 2, 3, and 5, this is also known asfive-limit tuning.
