Editing Diatonic scale

{{Short description|Class of music scales with seven notes}}
{{Use dmy dates|date=January 2022}}
{{More citations needed|date=June 2013}}

{{Image frame
|content = <score sound="1">
    \relative c' {
        \clef treble \time 7/4 \hide Staff.TimeSignature
        c4 d e f g a b c2
    }
</score>
|width=300|caption = Diatonic scale on C. The half steps are E–F (the third to the fourth note) and B–C (the seventh to the recurring first note, often called the eighth note).
}}

In [[music theory]] a '''diatonic scale''' is a [[heptatonic scale|heptatonic]] (seven-note) scale that includes five [[whole step]]s (whole tones) and two [[half step]]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 [[pitch (music)|pitches]] of any diatonic scale can also be obtained by using a [[Interval cycle|chain]] of six [[perfect fifth]]s. For instance, the seven [[natural (music)|natural]] pitch classes that form the C-[[major scale]] 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]]s, such as C–D–E–F–G–A–B, and any [[Transposition (music)|transposition]] thereof, is a diatonic scale. Modern [[musical keyboard]]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]]s separated by a whole tone. In [[set theory (music)|musical set theory]], [[Allen Forte]] classifies diatonic scales as [[set form]] 7–35.

The term ''[[diatonic]]'' originally referred to the [[diatonic genus]], one of the three [[genus (music)|genera]] of the ancient Greeks, and comes from {{langx|grc|διατονικός|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 scale|chromatic]] and [[Enharmonic scale|enharmonic]]).

This article does not concern alternative seven-note scales such as the [[harmonic minor]] or the [[melodic minor]] which, although sometimes called "diatonic", do not fulfill the condition of maximal separation of the semitones indicated above.

==History==
Western music from the [[Medieval music|Middle Ages]] until the [[Romantic music|late 19th century]] (see [[common practice period]]) is based on the diatonic scale and the unique [[hierarchy|hierarchical]] relationships created by this system of organizing seven notes.

===Antiquity===
{{main|Music of Mesopotamia}}

Evidence that the [[Sumer]]ians and [[Babylonia]]ns used a version of the diatonic scale is found in [[cuneiform]] inscriptions that contain both musical compositions and a tuning system.<ref>{{cite journal |last=Kilmer |first=Anne Draffkorn |date=1998 |title=The Musical Instruments from Ur and Ancient Mesopotamian Music |url=http://www.penn.museum/sites/expedition/?p=5425 |journal=[[Expedition (magazine)|Expedition]] |volume=40 |issue=2 |pages=12–19 |access-date=2015-12-29}}</ref><ref>{{cite book |last=Crickmore |first=Leon |title=ICONEA 2008: Proceedings of the International Conference of Near Eastern Archaeomusicology |publisher=Iconea Publications |year=2010 |editor-last1=Dumbrill |editor-first1=Richard |editor1-link=Richard Dumbrill (musicologist) |volume=24 |location=London |pages=11–22 |chapter=New Light on the Babylonian Tonal System |access-date=2015-12-29 |editor-last2=Finkel |editor-first2=Irving |editor2-link=Irving Finkel |chapter-url=http://xa.yimg.com/kq/groups/10599417/281241773/name/Crickmore-Iconea2008.pdf |archive-url=https://web.archive.org/web/20160113110832/http://xa.yimg.com/kq/groups/10599417/281241773/name/Crickmore-Iconea2008.pdf |archive-date=13 January 2016 |url-status=dead}}</ref> 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 [[Gudi (instrument)|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.<ref>{{cite journal |last1=Zhang |first1=Juzhong |last2=Harbottle |first2=Garman |last3=Wang |first3=Changsui |last4=Kong |first4=Zhaochen |date=23 September 1999 |title=Oldest playable musical instruments found at Jiahu early Neolithic site in China |journal=[[Nature (journal)|Nature]]|volume=401 |issue=6751 |pages=366–368|doi=10.1038/43865 |pmid=16862110 |bibcode=1999Natur.401..366Z |s2cid=205033636 |doi-access=free }}</ref>

===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 [[Locrian mode|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{{music|b}} instead of a B{{music|natural}}:

# A–B–C–D–E–F–G–A was described as D–E–F–G–A–B{{music|b}}–C–D (the modern [[Aeolian mode]]s whose reference notes are A and D, respectively, corresponding to the [[Aeolian mode]]s of [[C major]] and [[F major]], respectively)
# C–D–E–F–G–A–B–C was described as F–G–A–B{{music|b}}–C–D–E–F (the modern [[Ionian mode]]s whose reference notes are C and F, respectively, corresponding to the [[Ionian mode]]s 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{{music|natural}}/{{music|b}}). They were the modern [[Dorian mode|Dorian]], [[Phrygian mode|Phrygian]], [[Lydian mode|Lydian]], and [[Mixolydian mode|Mixolydian]] modes of [[C major]], plus the [[Aeolian mode|Aeolian]] and [[Ionian mode|Ionian]] modes of [[F major]] when B{{music|b}} was substituted into the Dorian and Lydian modes of [[C major]], respectively.

===Renaissance===
[[Heinrich Glarean]] considered that the modal scales including a B{{music|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{{music|b}}, resulting in the total of twelve scales that justified the title of his treatise. These were the 6 non-[[Locrian mode|Locrian]] [[Mode (music)#Modern modes|modes]] of [[C major]] and [[F major]].

===Modern===
By the beginning of the [[Baroque music|Baroque]] period, the notion of the musical [[key (music)|key]] was established, describing additional possible transpositions of the diatonic scale. [[Major scale|Major]] and [[minor scale]]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 (music)|triad]]. Some church modes survived into the early 18th century, as well as appearing in [[Classical period (music)|classical]] and [[20th-century music]], and [[jazz]] (see [[chord-scale system]]).

==Theory==
[[File:Piano-keyboard.jpg|thumb|right|300px|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.]]
Of Glarean's six natural scales, three have a major third/first triad: ([[Ionian mode|Ionian]], [[Lydian mode|Lydian]], and [[Mixolydian mode|Mixolydian]]), and three have a minor one: [[Dorian mode|Dorian]], [[Phrygian mode|Phrygian]], and [[Aeolian mode|Aeolian]]). To these may be added the seventh diatonic scale, with a diminished fifth above the reference note, the [[Locrian mode|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]], resulting in a total of eighty-four diatonic scales.

The modern [[musical keyboard]] originated as a diatonic keyboard with only white keys.<ref>Meeùs, Nicolas (2001). "Keyboard." ''[[Grove Music Online]]''. {{doi|10.1093/gmo/9781561592630.article.14944}}. Retrieved 9 May 2018.</ref> 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.{{Citation needed|date=May 2018}}

The pattern of elementary intervals forming the diatonic scale can be represented either by the letters T ([[whole tone|tone]]) and S ([[semitone]]) respectively. With this abbreviation, a major scale, for instance, can be represented as

:T–T–S–T–T–T–S

===Major scale===
{{Main|Major scale}}
The [[major scale]] or [[Ionian mode]] is one of the diatonic scales. It is made up of seven distinct [[Musical note|note]]s, 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 (music)|degree]] of the scale are ''Do–Re–Mi–Fa–Sol–La–Ti–Do''. A sequence of successive [[natural note]]s starting from C is an example of major scale, called [[C-major]] scale.

{| border=0 cellspacing=0 cellpadding=3
|-
| &nbsp; || Notes in C major: || C || &nbsp; || D || &nbsp; || E || &nbsp; || F || &nbsp; || G || &nbsp; || A || &nbsp; || B || &nbsp; || C
|-
| &nbsp; || Degrees in solfège: || ''Do'' || &nbsp; || ''Re'' || &nbsp; || ''Mi'' || &nbsp; || ''Fa'' || &nbsp; || ''Sol'' || &nbsp; || ''La'' || &nbsp; || ''Ti'' || &nbsp; || ''Do''
|-
| &nbsp; || Interval sequence: || &nbsp; || T || &nbsp; || T || &nbsp; || S || &nbsp; || T || &nbsp; || T || &nbsp; || T || &nbsp; || S || &nbsp;
|}
The seen degrees of the scale are also known by traditional names, especially when used in a tonal context:
:*1st – [[Tonic (music)|Tonic]] (key note)
:*2nd – [[Supertonic]]
:*3rd – [[Mediant]]
:*4th – [[Subdominant]]
:*5th – [[Dominant (music)|Dominant]]
:*6th – [[Submediant]]
:*7th – [[Leading-tone|Leading tone]]
:*(8th – [[Tonic (music)|Tonic]])

===Natural minor scale===
{{Main|Minor scale}}
For each major scale, there is a corresponding [[minor scale#Natural minor scale|natural minor scale]], sometimes called its [[relative key|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 note]]s starting from A is an example of a natural minor scale, called the A natural minor scale.

{| border=0 cellspacing=0 cellpadding=3
|-
| &nbsp; || Notes in A minor: || A || &nbsp; || B || &nbsp; || C || &nbsp; || D || &nbsp; || E || &nbsp; || F || &nbsp; || G || &nbsp; || A
|-
| &nbsp; || Interval sequence: || &nbsp; || T || &nbsp; || S || &nbsp; || T || &nbsp; || T || &nbsp; || S || &nbsp; || T || &nbsp; || T || &nbsp;
|}
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===
{{Main|Mode (music)}}
The whole collection of diatonic scales as defined above can be divided into seven different scales.

As explained above, all [[major scale]]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 (music)|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:

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none; text-align: center;"
! Mode
! Also known as
! Starting note relative<br/>to major scale
! Interval sequence
! Example with white keys
! Example with tonic C
|-
| [[Ionian mode|Ionian]] || Major scale
| align="center" | C
| align="center" | T–T–S–T–T–T–S
| colspan=2 | C–D–E–F–G–A–B–C
|-
| [[Dorian mode|Dorian]] ||
| align="center" | D
| align="center" | T–S–T–T–T–S–T
| D–E–F–G–A–B–C–D
| C–D–E♭–F–G–A–B♭–C
|-
| [[Phrygian mode|Phrygian]] ||
| align="center" | E
| align="center" | S–T–T–T–S–T–T
| E–F–G–A–B–C–D–E
| C–D♭–E♭–F–G–A♭–B♭–C
|-
| [[Lydian mode|Lydian]] ||
| align="center" | F
| align="center" | T–T–T–S–T–T–S
| F–G–A–B–C–D–E–F
| C–D–E–F♯–G–A–B–C
|-
| [[Mixolydian mode|Mixolydian]] ||
| align="center" | G
| align="center" | T–T–S–T–T–S–T
| G–A–B–C–D–E–F–G
| C–D–E–F–G–A–B♭–C
|-
| [[Aeolian mode|Aeolian]] || [[Minor scale#Natural minor scale|Natural minor scale]]
| align="center" | A
| align="center" | T–S–T–T–S–T–T
| A–B–C–D–E–F–G–A
| C–D–E♭–F–G–A♭–B♭–C
|-
| [[Locrian mode|Locrian]] ||
| align="center" | B
| align="center" | 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 by [[natural note]]s (i.e. neither sharps nor flats, also called "white-notes", as they can be played using the white keys of a [[piano keyboard]]). But any [[transposition (music)|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 [[diatonic and chromatic#Diatonic scales|elsewhere]], different definitions of this set are sometimes adopted in the literature.

[[File:Modes.svg|thumb|center|700px|[[Pitch constellation]]s of the modern musical modes]]

===Diatonic scales and tetrachords===
A diatonic scale can be also described as two [[tetrachord#Romantic era|tetrachords]] separated by a [[whole tone]]. For example, under this view the two tetrachord structures of C&nbsp;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).

==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]]), or possibly by a combination of fifths and thirds of various sizes, as in [[well temperament]].

===Iteration of the fifth===
{{main|Pythagorean tuning}}
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]]:

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;text-align:center"
|-
! style="text-align:left;" | note
| style="width: 60px;" | F
| style="width: 60px;" | C
| style="width: 60px;" | G
| style="width: 60px;" | D
| style="width: 60px;" | A
| style="width: 60px;" | E
| style="width: 60px;" | B ||
|-
! style="text-align:left;" | pitch
| {{frac|2|3}} || {{frac|1|1}} || {{frac|3|2}} || {{frac|9|4}} || {{frac|10|3}} || {{frac|5|1}} || {{frac|15|2}} ||
|-
! style="text-align:left;" | bring into main octave
| {{frac|4|3}} || {{frac|1|1}} || {{frac|3|2}} || {{frac|9|8}} || {{frac|5|3}} || {{frac|5|4}} || {{frac|15|8}} ||
|-
|
|
|
|
|
|
|
|
|
|-
! style="text-align:left;" | sort into note order
|C||D||E||F||G||A||B|| style="width:6em;" |C'
|-
! style="text-align:left;" | interval above C
| {{frac|1|1}} || {{frac|9|8}} || {{frac|5|4}} || {{frac|4|3}} || {{frac|3|2}} || {{frac|5|3}} || {{frac|15|8}} || {{frac|2|1}}
|-
! style="text-align:left;" | interval between notes
| || {{frac|9|8}} || {{frac|10|9}} || {{frac|16|15}} || {{frac|9|8}} || {{frac|10|9}} || {{frac|9|8}} || {{frac|16|15}}
|}

This tuning dates to Ancient Mesopotamia<ref>{{cite book|last=Dumbrill|first=Richard J.|author-link=Richard Dumbrill (musicologist)|title=The Archaeomusicology of the Ancient Near East|page=18|url=https://www.academia.edu/875113|publisher=Tadema Press|location=London|date=1998}} The book title is of 2nd edition; the 1st edition was entitled ''The Musicology and Organology of the Ancient Near East''.</ref> (see {{slink|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 {{frac|3|2}} = 1.5 (701.955 [[Cent (music)|cents]]), but B–F' is the discordant [[tritone]], here {{frac|729|512}} = 1.423828125 (611.73 cents). Tones are each {{frac|9|8}} = 1.125 (203.91 cents) and diatonic semitones are {{frac|256|243}} ≈ 1.0535 (90.225 cents).

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

===Equal temperament===
{{main|Equal temperament#Twelve-tone 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]] ({{radic|2|12}} ≈ 1.059463, 100 [[Cent (music)|cents]]). The tone is the sum of two semitones. Its ratio is the sixth root of two ({{radic|2|6}} ≈ 1.122462, 200 cents). Equal temperament can be produced by a succession of tempered fifths, each of them with the ratio of 2<sup>{{frac|7|12}}</sup> ≈ 1.498307, 700 cents.

===Meantone temperament===
{{main|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===
{{main|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:

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;text-align:center"
| A || E || B ||
|-
| style="width:2em;" | F
| style="width:2em;" | C
| style="width:2em;" | G
| style="width:2em;" | 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 the [[syntonic comma]], {{frac|81|80}}, and the "wolf" fifth D–A is too narrow by the same amount. The tritone F–B is {{frac|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.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;text-align:center" cellspacing="0" border="1"
! style="text-align:left" | notes
| style="width: 60px;" | C
| style="width: 60px;" | D
| style="width: 60px;" | E
| style="width: 60px;" | F
| style="width: 60px;" | G
| style="width: 60px;" | A
| style="width: 60px;" | B
| style="width: 60px;" | C'
|-
! style="text-align:left" | pitch
| {{frac|1|1|}} || {{frac|9|8}} || {{frac|5|4}} || {{frac|4|3}} || {{frac|3|2}} || {{frac|5|3}} || {{frac|15|8}} || {{frac|2|1|}}
|-
! style="text-align:left" | interval between notes
| || {{frac|9|8}} || {{frac|10|9}} || {{frac|16|15}} || {{frac|9|8}} || {{frac|10|9}} || {{frac|9|8}} || {{frac|16|15}}
|}
Since the frequency ratios are based on simple powers of the [[prime number]]s 2, 3, and 5, this is also known as [[five-limit tuning]].

==See also==
* [[Circle of fifths text table]]
* [[Diatonic and chromatic]]
* [[History of music]]
* [[Musical acoustics]]
* [[Piano key frequencies]]
* [[Prehistoric music]]

==References==

{{Reflist}}

==Further reading==
*Clough, John (1979). "Aspects of Diatonic Sets", ''[[Journal of Music Theory]]'' '''23''':45–61.
*Franklin, John C. (2002). "[https://web.archive.org/web/20060624040706/http://www.kingmixers.com/Franklin%20PDF%20files%20copy/DiatonicmusicinGreece.pdf Diatonic Music in Greece: a Reassessment of its Antiquity]", ''[[Mnemosyne (journal)|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
*Ellen Hickmann, Anne D. Kilmer and Ricardo Eichmann, (ed.) ''Studies in Music Archaeology III'', 2001, VML Verlag Marie Leidorf, Germany {{ISBN|3-89646-640-2}}.
*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, 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. {{OCLC|463314183}}
*[[David Rothenberg]] (1978). "[http://lumma.org/tuning/rothenberg/AModelForPatternPerception.pdf A Model for Pattern Perception with Musical Applications Part I: Pitch Structures as order-preserving maps]", ''[[Mathematical Systems Theory]]'' '''11''':199–234

==External links==
* [http://www.ericweisstein.com/encyclopedias/music/DiatonicScale.html Diatonic Scale] {{Webarchive|url=https://web.archive.org/web/20151209011658/http://www.ericweisstein.com/encyclopedias/music/DiatonicScale.html |date=9 December 2015 }}, Eric Weisstein's Treasure Trove of Music
* [http://www.guitar-tutorial.net/diatonic.html The diatonic scale on the guitar]

{{Circle of fifths}}
{{Set theory (music)}}
{{Scales}}
{{Tonality}}
{{Authority control}}

{{DEFAULTSORT:Diatonic Scale}}
[[Category:Heptatonic scales]]
[[Category:Diatonic set theory]]