Diatonic scale

Diatonic Scale

Qualities
Number of pitch classes	7
Maximal evenness Well formed generated collection Myhill's property Deep scale property Cardinality equals variety Structure implies multiplicity

Diatonic scale on C, equal tempered <phonos file="Diatonic scale on C.mid">Play</phonos> and just <phonos file="Just diatonic scale on C.mid">Play</phonos>.

Pythagorean diatonic scale on C <phonos file="Pythagorean diatonic scale on C.mid">Play</phonos>. A plus sign (+) indicates the syntonic comma.

In music theory, a diatonic scale (or heptatonia prima) is a scale composed of seven distinct pitch classes. The diatonic scale includes five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps, depending on their position in the scale. This pattern ensures that, in a diatonic scale spanning more than one octave, all the half steps are maximally separated from each other (i.e. separated by at least two whole steps). The word "diatonic" comes from the Greek διατονικός, meaning progressing through tones.

The seven pitches of any diatonic scale can be obtained using a chain of six perfect fifths. For instance, the seven natural pitches which form the C-major scale can be obtained from a stack of perfect fifths starting from F:

F—C—G—D—A—E—B

Called Pythagorean tuning, this property of the diatonic scales was historically relevant and possibly contributed to their worldwide diffusion because for centuries it allowed musicians to tune musical instruments easily by ear.

Any sequence of seven successive natural notes, such as C-D-E-F-G-A-B, and any transposition thereof, is a diatonic scale. Piano keyboards are designed to play natural notes, and hence diatonic scales, with their white keys. A diatonic scale can be also described as two tetrachords separated by a whole tone.

The term diatonic originally referred to the diatonic genus, one of the three genera of the ancient Greeks. In musical set theory, Allen Forte classifies diatonic scales as set form 7–35.

This article does not include alternative seven-note diatonic scales such as the harmonic minor or the melodic minor.

History

Diatonic scales are the foundation of the European musical tradition. Western harmony from the Renaissance until the late 19th century is based on the diatonic scale and the unique hierarchical relationships, or diatonic functionality, created by this system of organizing seven notes.

The modern major and minor scales are diatonic, as were all of the 'church modes'. What are now called major and minor were in reality – during the medieval and Renaissance periods – only two of eight modes ('church modes') based on the same diatonic notes (but forming different scales when the starting note was changed). Depending on which of the seven notes is used as the beginning, the positions of the intervals, the half-steps, end at different distances from the starting tone, hence obtaining seven different scales or modes that are, as already mentioned, deduced from the diatonic scale. By the end of the Baroque period, the notion of musical key was established—based on a central triad rather than a central tone. 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.

Prehistory

Main article: Music of Mesopotamia

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.

Theory

Using the twelve notes of the chromatic scale, twelve of each of the three major scales (those with a major third/triad: Ionian, Lydian, and Mixolydian), twelve of each of the three minor scales (those with a minor third/triad: Dorian, Phrygian, and Aeolian), and twelve Locrian scales can be played, totaling eighty-four diatonic scales. The modern musical keyboard, with its black keys grouped in twos and threes, is essentially diatonic; this arrangement not only helps musicians to find their bearings on the keyboard, but simplifies the system of key signatures compared with what would be necessary for a continuous alternation of black and white keys. The black (or "short") keys were an innovation that allows the adjacent positioning of most of the diatonic whole-steps (all in the case of C major), with significant physical and conceptual advantages.

Analysis

The modern piano keyboard is based on the interval patterns of the diatonic scale. Any sequence of seven successive white keys plays a diatonic scale.

Main article: heptatonic scale

In music of the broadly western classical tradition the pattern of seven intervals separating the eight notes of an octave can be represented in three ways, which are equivalent to each other. For instance, for a major scale these intervals are:

• T-T-S-T-T-T-S: where S means semitone; T means tone
• 2–2–1–2–2–2–1: where 1 means semitone; 2 means tone (2 semitones)
• whole-whole-half-whole-whole-whole-half: where half means semitone (half a tone); whole means tone.

Major scale

Main article: 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 which 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–Ti–Do". A sequence of successive natural notes starting from C is an example of major scale, called C-major scale.

Notes in C major:

C

D

E

F

G

A

B

C

Degrees in solfege:

Do

Re

Mi

Fa

Sol

La

Ti

Do

Interval sequence:

T

T

S

T

T

T

S

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

Main article: 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.

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 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). 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 or modes of scales 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

Main article: Mode (music)

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

As explained above, all major scales use the same interval sequence T-T-s-T-T-T-s. From the modal point of view, this interval sequence is called the Ionian mode. It is one of the seven modern modes. Taking any major scale, a new scale is obtained by taking a different degree of the major scale as the tonic. With this method, from each major scale it is possible to generate six other scales or modes, each characterized by a different interval sequence:

Mode	Also known as	Tonic relative to major scale	Interval sequence	Example
Ionian	Major scale	I	T-T-s-T-T-T-s	C-D-E-F-G-A-B-C
Dorian		II	T-s-T-T-T-s-T	D-E-F-G-A-B-C-D
Phrygian		III	s-T-T-T-s-T-T	E-F-G-A-B-C-D-E
Lydian		IV	T-T-T-s-T-T-s	F-G-A-B-C-D-E-F
Mixolydian	Dominant scale	V	T-T-s-T-T-s-T	G-A-B-C-D-E-F-G
Aeolian	Natural minor scale	VI	T-s-T-T-s-T-T	A-B-C-D-E-F-G-A
Locrian		VII	s-T-T-s-T-T-T	B-C-D-E-F-G-A-B

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 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.

Pitch constellations of the modern musical modes

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. 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.

Tuning

In just intonation the diatonic scale is tuned:

C	D	E	F	G	A	B	C
1	9/8	5/4	4/3	3/2	5/3	15/8	2

In Pythagorean tuning the diatonic scale is:

C	D	E	F	G	A	B	C
1	9/8	81/64	4/3	3/2	27/16	243/128	2

See also

• Circle of fifths text table
• Piano key frequencies
• History of music
• Prehistoric music
• Musical acoustics
• Jiahu Site of oldest still-playable flute—Neolithic
• Diatonic and chromatic

Further reading

• Balzano, Gerald J. (1980). "The Group Theoretic Description of 12-fold and Microtonal Pitch Systems", Computer Music Journal 4:66–84.
• Balzano, Gerald J. (1982). "The Pitch Set as a Level of Description for Studying Musical Pitch Perception", Music, Mind, and Brain, Manfred Clynes, ed., Plenum press.
• Clough, John (1979). "Aspects of Diatonic Sets", Journal of Music Theory 23:45–61.
• Ellen Hickmann, Anne D. Kilmer and Ricardo Eichmann, (ed.) Studies in Music Archaeology III, 2001, VML Verlag Marie Leidorf GmbH., Germany ISBN 3-89646-640-2.
• Franklin, John C. (2002). "Diatonic Music in Greece: a Reassessment of its Antiquity", Mnemosyne 56.1:669–702
• Gould, Mark (2000). "Balzano and Zweifel: Another Look at Generalised Diatonic Scales", "Perspectives of New Music" 38/2:88–105
• Johnson, Timothy (2003). Foundations Of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. ISBN 1-930190-80-8.
• Kilmer, A.D. (1971) "The Discovery of an Ancient Mesopotamian Theory of Music'". Proceedings of the American Philosophical Society 115:131–149.
• Kilmer, Crocket, Brown: Sounds from Silence 1976, Bit Enki Publications, Berkeley, Calif. LC# 76-16729.
• David Rothenberg (1978). "A Model for Pattern Perception with Musical Applications Part I: Pitch Structures as order-preserving maps", Mathematical Systems Theory 11:199–234

External links

• Diatonic Scale on Eric Weisstein's Treasure trove of Music
• The diatonic scale on the guitar