Diatonic Major Scale
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In
music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first is the "Elements of music, ...
a diatonic scale is a
heptatonic A heptatonic scale is a musical scale that has seven pitches, or tones, per octave. Examples include: * the diatonic scale; including the major scale and its modes (notably the natural minor scale, or Aeolian mode) * the melodic minor scale, l ...
(seven-note) scale that includes five
whole step In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more deta ...
s (whole tones) and two
half step A semitone, also called a minor second, half step, or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between ...
s (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 seven pitches of any diatonic scale can also be obtained by using a
chain A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A ...
of six
perfect fifth In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval f ...
s. For instance, the seven
natural Nature is an inherent character or constitution, particularly of the ecosphere or the universe as a whole. In this general sense nature refers to the laws, elements and phenomena of the physical world, including life. Although humans are part ...
pitch classes that form the C-
major scale The major scale (or Ionian mode) is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at doubl ...
can be obtained from a stack of perfect fifths starting from F: :F–C–G–D–A–E–B. Any sequence of seven successive
natural note In modern Western music notation, a natural (â™®) is a musical symbol that cancels a previous sharp or flat on a note in the written music. The natural indicates that the note is at its unaltered pitch. The natural symbol can be used as an ac ...
s, such as C–D–E–F–G–A–B, and any transposition thereof, is a diatonic scale. Modern
musical keyboard A musical keyboard is the set of adjacent depressible levers or keys on a musical instrument. Keyboards typically contain keys for playing the twelve notes of the Western musical scale, with a combination of larger, longer keys and smaller, sho ...
s 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 two
tetrachord In music theory, a tetrachord (; ) is a series of four notes separated by three interval (music), intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency proportion (approx. 498 cent (m ...
s separated by a whole tone. In
musical set theory Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonality, tonal music. Other theorists, such as Allen Forte, further devel ...
,
Allen Forte Allen Forte (December 23, 1926 – October 16, 2014) was an American music theorist and musicologist. He was Battell Professor Emeritus of the Theory of Music at Yale University and specialized in 20th-century atonal music and music analysis. ...
classifies diatonic scales as set form 7–35. The term ''
diatonic Diatonic and chromatic are terms in music theory that are used to characterize scales. The terms are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair ...
'' originally referred to the
diatonic genus In the musical system of ancient Greece, genus (Greek: γένος 'genos'' grammatical number, pl. γένη 'genē'' Latin: ''genus'', pl. ''genera'' "type, kind") is a term used to describe certain classes of Intonation (music), intonation ...
, one of the three
genera Genus (; : genera ) is a taxonomic rank above species and below family as used in the biological classification of living and fossil organisms as well as viruses. In binomial nomenclature, the genus name forms the first part of the binomial s ...
of the ancient Greeks, and comes from , of uncertain etymology. Most likely, it refers to the intervals being "stretched out" in that tuning, in contrast to the other two genera (
chromatic Diatonic and chromatic are terms in music theory that are used to characterize scales. The terms are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, es ...
and
enharmonic In music, two written notes have enharmonic equivalence if they produce the same pitch but are notated differently. Similarly, written intervals, chords, or key signatures are considered enharmonic if they represent identical pitches that ar ...
). This article does not concern alternative seven-note scales such as the
harmonic minor The harmonic minor scale (or Aeolian â™®7 scale) is a musical scale derived from the natural minor scale, with the minor seventh degree raised by one semitone to a major seventh, creating an augmented second between the sixth and seventh degree ...
or the
melodic minor In Classical_music, Western classical music theory, the minor scale refers to three Scale (music), scale patterns – the natural minor scale (or Aeolian mode), the harmonic minor scale, and the melodic minor scale (ascending or descending). ...
which, although sometimes called "diatonic", do not fulfill the condition of maximal separation of the semitones indicated above.


History

Western music from the
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the 5th to the late 15th centuries, similarly to the post-classical period of global history. It began with the fall of the Western Roman Empire and ...
until the late 19th century (see
common practice period In Western classical music, the common practice period (CPP) was the period of about 250 years during which the tonal system was regarded as the only basis for composition. It began when composers' use of the tonal system had clearly supersede ...
) is based on the diatonic scale and the unique
hierarchical A hierarchy (from Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) that are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an importan ...
relationships created by this system of organizing seven notes.


Antiquity

Evidence that the
Sumer Sumer () is the earliest known civilization, located in the historical region of southern Mesopotamia (now south-central Iraq), emerging during the Chalcolithic and Early Bronze Age, early Bronze Ages between the sixth and fifth millennium BC. ...
ians and
Babylonia Babylonia (; , ) was an Ancient history, ancient Akkadian language, Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Kuwait, Syria and Iran). It emerged as a ...
ns used a version of the diatonic scale is found in
cuneiform Cuneiform is a Logogram, logo-Syllabary, syllabic writing system that was used to write several languages of the Ancient Near East. The script was in active use from the early Bronze Age until the beginning of the Common Era. Cuneiform script ...
inscriptions that contain both musical compositions and a tuning system. Despite the conjectural nature of reconstructions of the
Hurrian songs The Hurrian songs (or Hurrian Hymns) are a collection of music inscribed in cuneiform on clay tablets excavated from the ancient Amorite-Canaanite city of Ugarit, a headland in northern Syria, which date to approximately 1400 BC.Marguerite Yon, ...
, 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 The flute is a member of a family of musical instruments in the woodwind group. Like all woodwinds, flutes are aerophones, producing sound with a vibrating column of air. Flutes produce sound when the player's air flows across an opening. In th ...
found in
Jiahu Jiahu () was the site of a Neolithic settlement based in the central plain of ancient China, near the Yellow River. It is located between the floodplains of the Ni River to the north, and the Sha River to the south, north of modern Wuyang in ...
, 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.


Middle Ages

The scales corresponding to the medieval
church modes A Gregorian mode (or church mode) is one of the eight systems of pitch organization used in Gregorian chant. History The name of Pope Gregory I was attached to the variety of chant that was to become the dominant variety in medieval western and ...
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 Diminished may refer to: *Diminution In Western culture, Western music and music theory, diminution (from Medieval Latin ''diminutio'', alteration of Latin ''deminutio'', decrease) has four distinct meanings. Diminution may be a form of embel ...
): 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 (the modern
Aeolian mode The Aeolian mode is a musical mode or, in modern usage, a diatonic scale also called the natural minor scale. On the piano, using only the white keys, it is the scale that starts with A and continues to the next A only striking white keys. Its a ...
s whose reference notes are A and D, respectively, corresponding to the
Aeolian mode The Aeolian mode is a musical mode or, in modern usage, a diatonic scale also called the natural minor scale. On the piano, using only the white keys, it is the scale that starts with A and continues to the next A only striking white keys. Its a ...
s of
C major C major is a major scale based on C, consisting of the pitches C, D, E, F, G, A, and B. C major is one of the most common keys used in music. Its key signature has no flats or sharps. Its relative minor is A minor and its parallel min ...
and
F major F major is a major scale based on F, with the pitches F, G, A, B, C, D, and E. Its key signature has one flat.Music Theory'. (1950). United States: Standards and Curriculum Division, Training, Bureau of Naval Personnel. 28. Its relati ...
, respectively) # C–D–E–F–G–A–B–C was described as F–G–A–B–C–D–E–F (the modern
Ionian mode The Ionian mode is a Mode (music), musical mode or, in modern usage, a diatonic scale also called the major scale. It is named after the Ionians, Ionian Greeks. It is the name assigned by Heinrich Glarean in 1547 to his new Gregorian mode#Authent ...
s whose reference notes are C and F, respectively, corresponding to the
Ionian mode The Ionian mode is a Mode (music), musical mode or, in modern usage, a diatonic scale also called the major scale. It is named after the Ionians, Ionian Greeks. It is the name assigned by Heinrich Glarean in 1547 to his new Gregorian mode#Authent ...
s of
C major C major is a major scale based on C, consisting of the pitches C, D, E, F, G, A, and B. C major is one of the most common keys used in music. Its key signature has no flats or sharps. Its relative minor is A minor and its parallel min ...
and
F major F major is a major scale based on F, with the pitches F, G, A, B, C, D, and E. Its key signature has one flat.Music Theory'. (1950). United States: Standards and Curriculum Division, Training, Bureau of Naval Personnel. 28. Its relati ...
, 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 Mixolydian mode may refer to one of three things: the name applied to one of the ancient Greek ''harmoniai'' or ''tonoi'', based on a particular octave species or scale; one of the medieval church modes; or a modern musical mode or diatonic sca ...
modes of
C major C major is a major scale based on C, consisting of the pitches C, D, E, F, G, A, and B. C major is one of the most common keys used in music. Its key signature has no flats or sharps. Its relative minor is A minor and its parallel min ...
, plus the Aeolian and Ionian modes of
F major F major is a major scale based on F, with the pitches F, G, A, B, C, D, and E. Its key signature has one flat.Music Theory'. (1950). United States: Standards and Curriculum Division, Training, Bureau of Naval Personnel. 28. Its relati ...
when B was substituted into the Dorian and Lydian modes of
C major C major is a major scale based on C, consisting of the pitches C, D, E, F, G, A, and B. C major is one of the most common keys used in music. Its key signature has no flats or sharps. Its relative minor is A minor and its parallel min ...
, respectively.


Renaissance

Heinrich Glarean Heinrich Glarean also styled Henricus Glareanus (born as Heinrich Loriti on 28 February or 3 June 1488 – 28 March 1563) was a Swiss music theorist, poet, humanist, philosopher and cartographer. He was born in Mollis (in the canton of Glarus, he ...
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 Mode ( meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * MO''D''E (magazine), a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is the setting fo ...
of
C major C major is a major scale based on C, consisting of the pitches C, D, E, F, G, A, and B. C major is one of the most common keys used in music. Its key signature has no flats or sharps. Its relative minor is A minor and its parallel min ...
and
F major F major is a major scale based on F, with the pitches F, G, A, B, C, D, and E. Its key signature has one flat.Music Theory'. (1950). United States: Standards and Curriculum Division, Training, Bureau of Naval Personnel. 28. Its relati ...
.


Modern

By the beginning of the
Baroque The Baroque ( , , ) is a Western Style (visual arts), style of Baroque architecture, architecture, Baroque music, music, Baroque dance, dance, Baroque painting, painting, Baroque sculpture, sculpture, poetry, and other arts that flourished from ...
period, the notion of the musical key was established, describing additional possible transpositions of the diatonic scale.
Major Major most commonly refers to: * Major (rank), a military rank * Academic major, an academic discipline to which an undergraduate student formally commits * People named Major, including given names, surnames, nicknames * Major and minor in musi ...
and
minor scale In Classical_music, Western classical music theory, the minor scale refers to three Scale (music), scale patterns – the natural minor scale (or Aeolian mode), the harmonic minor scale, and the melodic minor scale (ascending or descending). ...
s 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 Triad or triade may refer to: * a group of three Humanities * Trichotomy (philosophy), often called triads * Triad (sociology), a group of three people as a unit of study * Triad (relationship), or ''ménage à trois'' Music * Triad (music ...
. Some church modes survived into the early 18th century, as well as appearing in classical and
20th-century music The following Wikipedia articles deal with 20th-century music. Western art music Main articles *20th-century classical music *Contemporary classical music, covering the period Sub-topics * Aleatoric music *Electronic music *Experimental music *E ...
, and
jazz Jazz is a music genre that originated in the African-American communities of New Orleans, Louisiana, in the late 19th and early 20th centuries. Its roots are in blues, ragtime, European harmony, African rhythmic rituals, spirituals, h ...
(see
chord-scale system The chord-scale system is a method of matching, from a list of possible chord (music), chords, a list of possible scale (music), scales.Mervyn Cooke, David Horn (2003). ''Cambridge Companions to Music, The Cambridge Companion to Jazz'', p.266. . ...
).


Theory

Of Glarean's six natural scales, three have a major third/first triad: ( Ionian, Lydian, and
Mixolydian Mixolydian mode may refer to one of three things: the name applied to one of the ancient Greek ''harmoniai'' or ''tonoi'', based on a particular octave species or scale; one of the medieval church modes; or a modern musical mode or diatonic sca ...
), and three have a minor one: Dorian, Phrygian, and Aeolian). To these may be added the seventh diatonic scale, with a diminished fifth above the reference note, the Locrian scale. These could be transposed not only to include one flat in the signature (as described by Glarean), but to all twelve notes of the
chromatic scale The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the ...
, resulting in a total of eighty-four diatonic scales. The modern
musical keyboard A musical keyboard is the set of adjacent depressible levers or keys on a musical instrument. Keyboards typically contain keys for playing the twelve notes of the Western musical scale, with a combination of larger, longer keys and smaller, sho ...
originated as a diatonic keyboard with only white keys. The black keys were progressively 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 (
tone Tone may refer to: Visual arts and color-related * Tone (color theory), a mix of tint and shade, in painting and color theory * Tone (color), the lightness or brightness (as well as darkness) of a color * Toning (coin), color change in coins * ...
) and S (
semitone A semitone, also called a minor second, half step, or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between ...
) respectively. With this abbreviation, a major scale, for instance, can be represented as :T–T–S–T–T–T–S


Major scale

The
major scale The major scale (or Ionian mode) is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at doubl ...
or
Ionian mode The Ionian mode is a Mode (music), musical mode or, in modern usage, a diatonic scale also called the major scale. It is named after the Ionians, Ionian Greeks. It is the name assigned by Heinrich Glarean in 1547 to his new Gregorian mode#Authent ...
is one of the diatonic scales. It is made up of seven distinct
note Note, notes, or NOTE may refer to: Music and entertainment * Musical note, a pitched sound (or a symbol for a sound) in music * ''Notes'' (album), a 1987 album by Paul Bley and Paul Motian * ''Notes'', a common (yet unofficial) shortened versi ...
s, plus an eighth that duplicates the first an
octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
higher. The pattern of seven intervals separating the eight notes is T–T–S–T–T–T–S. In
solfège In music, solfège (British English or American English , ) or solfeggio (; ), also called sol-fa, solfa, solfeo, among many names, is a mnemonic used in teaching aural skills, Pitch (music), pitch and sight-reading of Western classical music, W ...
, the syllables used to name each degree of the scale are ''Do–Re–Mi–Fa–Sol–La–Ti–Do''. A sequence of successive
natural note In modern Western music notation, a natural (â™®) is a musical symbol that cancels a previous sharp or flat on a note in the written music. The natural indicates that the note is at its unaltered pitch. The natural symbol can be used as an ac ...
s starting from C is an example of major scale, called
C-major C major is a major scale based on C, consisting of the pitches C, D, E, F, G, A, and B. C major is one of the most common keys used in music. Its key signature has no flats or sharps. Its relative minor is A minor and its parallel minor i ...
scale. The seven degrees of the scale are also known by traditional names, especially when used in a tonal context: :*1st – Tonic (key note) :*2nd –
Supertonic In music, the supertonic is the second degree () of a diatonic scale, one whole step above the tonic. In the movable do solfège system, the supertonic note is sung as ''re''. The triad built on the supertonic note is called the supertonic ...
:*3rd –
Mediant In music, the mediant (''Latin'': "being in the middle") is the third scale degree () of a diatonic scale, being the note halfway between the tonic and the dominant.Benward & Saker (2003), p.32. In the movable do solfège system, the mediant no ...
:*4th –
Subdominant In music, the subdominant is the fourth tonal degree () of the diatonic scale. It is so called because it is the same distance ''below'' the tonic as the dominant is ''above'' the tonicin other words, the tonic is the dominant of the subdomina ...
:*5th – Dominant :*6th –
Submediant In music, the submediant is the sixth degree () of a diatonic scale. The submediant ("lower mediant") is named thus because it is halfway between the tonic and the subdominant ("lower dominant") or because its position below the tonic is symm ...
:*7th –
Leading tone In music theory, a leading tone (also called subsemitone or leading note in the UK) is a musical note, note or pitch (music), pitch which resolution (music), resolves or "leads" to a note one semitone higher or lower, being a lower and upper lea ...
:*(8th – Tonic)


Natural minor scale

For each major scale, there is a corresponding
natural minor scale In Western classical music theory, the minor scale refers to three scale patterns – the natural minor scale (or Aeolian mode), the harmonic minor scale, and the melodic minor scale (ascending or descending). These scales contain all th ...
, sometimes called its
relative minor In music, 'relative keys' are the major and minor scales that have the same key signatures (enharmonically equivalent), meaning that they share all of the same notes but are arranged in a different order of whole steps and half steps. A pair of ma ...
. 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 note In modern Western music notation, a natural (â™®) is a musical symbol that cancels a previous sharp or flat on a note in the written music. The natural indicates that the note is at its unaltered pitch. The natural symbol can be used as an ac ...
s 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 In music, the subtonic is the degree of a musical scale which is a major second, whole step below the tonic (music), tonic note. In a major key, it is a lowered, or flattened, seventh Degree (music), scale degree (). It appears as the seventh scal ...
because it is a whole step below the tonic. The term
leading tone In music theory, a leading tone (also called subsemitone or leading note in the UK) is a musical note, note or pitch (music), pitch which resolution (music), resolves or "leads" to a note one semitone higher or lower, being a lower and upper lea ...
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 scale The major scale (or Ionian mode) is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at doubl ...
s 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 note In modern Western music notation, a natural (â™®) is a musical symbol that cancels a previous sharp or flat on a note in the written music. The natural indicates that the note is at its unaltered pitch. The natural symbol can be used as an ac ...
s (i.e. neither sharps nor flats, also called "white-notes", as they can be played using the white keys of a
piano keyboard A musical keyboard is the set of adjacent depressible levers or keys on a musical instrument. Keyboards typically contain keys for playing the twelve notes of the Western musical scale, with a combination of larger, longer keys and smaller, sho ...
). But 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 In music theory, a tetrachord (; ) is a series of four notes separated by three interval (music), intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency proportion (approx. 498 cent (m ...
separated by a
whole tone In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more deta ...
. For example, under this view the two tetrachord structures of C major would be: : –D–E–F– –A–B–Ceach tetrachord being formed of two tones and a semitone, T–T–S, and the natural minor of A would be: : –B–C–D– –F–G–Aformed 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, : –E–F–G– –B–C–Deach 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).


Tuning

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 In music, just intonation or pure intonation is a musical tuning, tuning system in which the space between notes' frequency, frequencies (called interval (music), intervals) is a natural number, whole number ratio, ratio. Intervals spaced in thi ...
), or possibly by a combination of fifths and thirds of various sizes, as in
well temperament Well temperament (also good temperament, circular or circulating temperament) is a type of musical temperament, tempered musical tuning, tuning used for keyboard instruments of the seventeenth and eighteenth centuries. The term is modeled on the G ...
.


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 Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are determined by choosing a sequence of fifthsBruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh editi ...
: This tuning dates to Ancient Mesopotamia The book title is of 2nd edition; the 1st edition was entitled ''The Musicology and Organology of the Ancient Near East''. (see ), 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 = 1.5 (701.955 cents), but B–F' is the discordant
tritone In music theory, the tritone is defined as a interval (music), musical interval spanning three adjacent Major second, whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be ...
, here = 1.423828125 (611.73 cents). Tones are each = 1.125 (203.91 cents) and diatonic semitones are ≈ 1.0535 (90.225 cents). Extending the series of fifths to eleven fifths would result into the Pythagorean
chromatic scale The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the ...
.


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 The twelfth root of two or \sqrt 2/math> (or equivalently 2^) is an algebraic irrational number, approximately equal to 1.0594631. It is most important in Western music theory, where it represents the frequency ratio ( musical interval) of a se ...
( ≈ 1.059463, 100 cents). The tone is the sum of two semitones. Its ratio is the sixth root of two ( ≈ 1.122462, 200 cents). Equal temperament can be produced by a succession of tempered fifths, each of them with the ratio of 2 ≈ 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 Quarter-comma meantone, or -comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma with ...
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 Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
's
Tonnetz In musical tuning and harmony, the (German for 'tone net') is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. Various visual representations of the ''Tonnetz'' can be used to show traditional ...
, 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 In music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first i ...
, , and the "wolf" fifth D–A is too narrow by the same amount. The tritone F–B is ≈ 1.40625. This tuning has been first described by
Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...
and is known as
Ptolemy's intense diatonic scale Ptolemy's intense diatonic scale, also known as the Ptolemaic sequence, justly tuned major scale, Ptolemy's tense diatonic scale, or the syntonous (or syntonic) diatonic scale, is a tuning for the diatonic scale proposed by Ptolemy, and corres ...
. 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 number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s 2, 3, and 5, this is also known as
five-limit tuning Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference not ...
.


See also

* Circle of fifths text table *
Diatonic and chromatic Diatonic and chromatic are terms in music theory that are used to characterize Scale (music), scales. The terms are also applied to musical instruments, Interval (music), intervals, Chord (music), chords, Musical note, notes, musical styles, ...
*
History of music Although definitions of music vary wildly throughout the world, every known culture partakes in it, and it is thus considered a cultural universal. The origins of music remain highly contentious; commentators often relate it to the origin of la ...
*
Musical acoustics Musical acoustics or music acoustics is a multidisciplinary field that combines knowledge from physics, psychophysics, organology (classification of the instruments), physiology, music theory, ethnomusicology, signal processing and instrument buil ...
*
Piano key frequencies This is a list of the fundamental frequencies in hertz (cycles per second) of the keys of a modern 88-key standard or 108-key extended piano in twelve-tone equal temperament, with the 49th key, the fifth A (called A4), tuned to 440 Hz (refe ...
*
Prehistoric music Prehistoric music (previously called primitive music) is a term in the history of music for all music produced in preliterate cultures (prehistory), beginning somewhere in very late geological history. Prehistoric music is followed by ancient mu ...


References


Further reading

*Clough, John (1979). "Aspects of Diatonic Sets", ''
Journal of Music Theory The ''Journal of Music Theory'' is a peer-reviewed academic journal specializing in music theory and analysis. It was established by David Kraehenbuehl (Yale University) in 1957. According to its website, " e ''Journal of Music Theory'' fosters co ...
'' 23:45–61. *Franklin, John C. (2002).
Diatonic Music in Greece: a Reassessment of its Antiquity
, ''
Mnemosyne In Greek mythology and ancient Greek religion, Mnemosyne (; , ) is the goddess of memory and the mother of the nine Muses by her nephew Zeus. In the Greek tradition, Mnemosyne is one of the Titans, the twelve divine children of the earth-godde ...
'' 56.1:669–702 *Gould, Mark (2000). "Balzano and Zweifel: Another Look at Generalised Diatonic Scales", ''
Perspectives of New Music ''Perspectives of New Music'' (PNM) is a peer-reviewed academic journal specializing in music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Musi ...
'' 38/2:88–105 *Ellen Hickmann, Anne D. Kilmer and Ricardo Eichmann, (ed.) ''Studies in Music Archaeology III'', 2001, VML Verlag Marie Leidorf, Germany . *Johnson, Timothy (2003). ''Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals''. Key College Publishing. . *Kilmer, A. D. (1971) "The Discovery of an Ancient Mesopotamian Theory of Music'". ''
Proceedings of the American Philosophical Society In academia and librarianship, conference proceedings are a collection of academic papers published in the context of an academic conference or workshop. Conference proceedings typically contain the contributions made by researchers at the confer ...
'' 115:131–149. *Kilmer, Anne Draffkorn; Crocker, Richard L.; Brown, Robert R.: ''Sounds from Silence: Recent Discoveries in Ancient Near Eastern Music'', Volume 1. 1976, Bit Enki Publications, Berkeley, California. *
David Rothenberg David Rothenberg (born 1962) is a professor of philosophy and music at the New Jersey Institute of Technology, with a special interest in animal sounds as music. He is also a composer and jazz musician whose books and recordings reflect a long ...
(1978).
A Model for Pattern Perception with Musical Applications Part I: Pitch Structures as order-preserving maps
, ''
Mathematical Systems Theory ''Theory of Computing Systems'' is a peer-reviewed scientific journal published by Springer Verlag. Published since 1967 as ''Mathematical Systems Theory'' and since volume 30 in 1997 under its current title, it is devoted to publishing orig ...
'' 11:199–234


External links


Diatonic Scale
, Eric Weisstein's Treasure Trove of Music

{{DEFAULTSORT:Diatonic Scale Heptatonic scales Diatonic set theory