Editing Interval (music) (section)

== Interval number and quality ==
[[File:Main intervals from C.png|thumb|450px|right|Main intervals from C[[File:Intervals.mid]]]]

In Western [[music theory]], an interval is named according to its ''number'' (also called ''diatonic number, interval size''<ref>{{Cite book |last=Burstein |first=L. Poundie |title=Concise Introduction to Tonal Harmony |last2=Straus |first2=Joseph N. |publisher=W. W. Norton |year=2016 |isbn=978-0-393-26476-0 |edition=1st |location=New York |pages=55}}</ref> or ''generic interval''<ref name=":0">{{Cite book |last=Laitz |first=Steven G. |title=The Complete Musician: An Integrated Approach to Theory, Analysis, and Listening |publisher=[[Oxford University Press]] |year=2016 |isbn=9780199347094 |edition=4th |location=New York |pages=27–31}}</ref>) and ''quality''. For instance, ''major third'' (or '''M3''') is an interval name, in which the term ''major'' ('''M''') describes the quality of the interval, and ''third'' ('''3''') indicates its number.

=== Number ===
<!--[[Major second]] links directly here-->
[[Image:Staff lines and spaces SVG.svg|right|thumb|350px|[[Staff (music)|Staff]], with [[staff position]]s indicated]]
[[Image:Fifth C to G.png|right|thumb|Fifth from C to G in the A{{music|b}} [[major scale]]]]

The number of an interval is the number of letter names or [[staff position]]s (lines and spaces) it encompasses, including the positions of both notes forming the interval. For instance, the interval B–D is a third (denoted '''m3''') because the notes from B to the D above it encompass three letter names (B, C, D) and occupy three consecutive staff positions, including the positions of B and D. The [[#Main intervals|table]] and the figure above show intervals with numbers ranging from 1 (e.g., '''P1''') to 8 (e.g., '''d8'''). Intervals with larger numbers are called [[#Compound intervals|compound intervals]].

There is a [[one-to-one correspondence]] between staff positions and diatonic-scale [[Degree (music)|degrees]] (the notes of [[diatonic scale]]).{{efn|name=diatonic|
The expression [[diatonic scale]] is herein strictly defined as a [[Heptatonic scale|7-tone scale]], which is either a sequence of successive [[natural note]]s (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 any [[transposition (music)|transposition]] thereof. In other words, a scale that can be written using seven consecutive notes without [[Accidental (music)|accidentals]] on a [[Staff (music)|staff]] with a conventional [[key signature]], or with no signature. This includes, for instance, the [[Major scale|major]] and the [[Natural minor scale|natural minor]] scales, but does not include some other seven-tone scales, such as the [[Melodic minor scale|melodic minor]] and the [[Harmonic minor scale|harmonic minor]] scales (see also [[Diatonic and chromatic]]).}}
This means that interval numbers can also be determined by counting diatonic scale degrees, rather than staff positions, provided that the two notes that form the interval are drawn from a diatonic scale. Namely, B–D is a third because in any diatonic scale that contains B and D, the sequence from B to D includes three notes. For instance, in the B-[[natural minor scale|natural minor]] diatonic scale, the three notes are B–C{{music|sharp}}–D. This is not true for all kinds of scales. For instance, in a [[chromatic scale]], there are four notes from B to D: B–C–C{{music|sharp}}–D. This is the reason interval numbers are also called ''diatonic numbers'', and this convention is called ''diatonic numbering''.

If one adds any [[accidental (music)|accidental]]s to the notes that form an interval, by definition the notes do not change their staff positions. As a consequence, any interval has the same interval number as the corresponding [[Natural (music)|natural]] interval, formed by the same notes without accidentals. For instance, the intervals B–D{{music|sharp}} (spanning 4 semitones) and B–D{{music|flat}} (spanning 2 semitones) are thirds, like the corresponding natural interval B–D (3 semitones).

Notice that interval numbers represent an inclusive count of encompassed staff positions or note names, not the difference between the endpoints. In other words, one starts counting the lower pitch as one, not zero. For that reason, the interval E–E, a perfect unison, is also called a prime (meaning "1"), even though there is no difference between the endpoints. Continuing, the interval E–F{{music|sharp}} is a second, but F{{music|sharp}} is only one staff position, or diatonic-scale degree, above E. Similarly, E–G{{music|sharp}} is a third, but G{{music|sharp}} is only two staff positions above E, and so on. As a consequence, joining two intervals always yields an interval number one less than their sum. For instance, the intervals B–D and D–F{{music|sharp}} are thirds, but joined together they form a fifth (B–F{{music|sharp}}), not a sixth. Similarly, a stack of three thirds, such as B–D, D–F{{music|sharp}}, and F{{music|sharp}}–A, is a seventh (B–A), not a ninth.

This scheme applies to intervals up to an octave (12 semitones). For larger intervals, see {{format link|#Compound intervals}} below.

=== Quality <span class="anchor" id="major_minor_perfect_anchor"></span>===
[[File:Diatonic intervals (with colors).png|right|frame|Intervals formed by the notes of a C major [[diatonic scale]]]]

The name of any interval is further qualified using the terms '''perfect''' ('''P'''), [[Major and minor|major]] ('''M'''), [[Major and minor|minor]] ('''m'''), [[Augmentation (music)|augmented]] ('''A'''), and [[Diminution|diminished]] ('''d'''). This is called its ''interval quality'' (or ''modifier''<ref name=":1">{{Cite book |last=Kostka |first=Stefan |title=Tonal Harmony, with an Introduction to Post-Tonal Music |last2=Payne |first2=Dorothy |last3=Almén |first3=Byron |publisher=[[McGraw Hill]] |year=2018 |isbn=9781259447099 |edition=8th |location=New York |pages=16–18}}</ref><ref name=":0" />). It is possible to have doubly diminished and doubly augmented intervals, but these are quite rare, as they occur only in [[chromatic]] contexts. The combination of number (or generic interval) and quality (or modifier) is called the ''specific interval'',<ref name=":0" /> ''diatonic interval'' (sometimes used only for intervals appearing in the diatonic scale), or simply ''interval''.<ref name=":1" />

The quality of a [[#Compound intervals|compound interval]] is the quality of the simple interval on which it is based. Some other qualifiers like ''neutral'', ''subminor'', and ''supermajor'' are used for [[#Non-diatonic intervals|non-diatonic intervals]].

==== Perfect ====
[[Image:Perfect intervals on C.png|thumb|Perfect intervals on C: {{audio|Unison on C.mid|PU}}, {{audio|Perfect fourth on C.mid|P4}}, {{audio|Perfect fifth on C.mid|P5}}, {{audio|Perfect octave on C.mid|P8}}]]
Perfect intervals are so-called because they were traditionally considered perfectly consonant,<ref name=Weber>
Definition of ''perfect consonance'' from
{{cite book |url=https://archive.org/details/bub_gb_20oBAAAAYAAJ |quote=perfect concord |first=Godfrey |last=Weber |title=General Music Teacher |year=1841}}
</ref>
although in Western classical music the perfect fourth was sometimes regarded as a less than perfect consonance, when its function was [[Counterpoint|contrapuntal]].{{Vague|date=February 2013}} Conversely, minor, major, augmented, or diminished intervals are typically considered less consonant, and were traditionally classified as mediocre consonances, imperfect consonances, or near-dissonances.<ref name=Weber/>

Within a [[diatonic scale]]{{efn|name=diatonic}} all unisons ('''P1''') and octaves ('''P8''') are perfect. Most fourths and fifths are also perfect ('''P4''' and '''P5'''), with five and seven semitones respectively. One occurrence of a fourth is augmented ('''A4''') and one fifth is diminished ('''d5'''), both spanning six semitones. For instance, in an E-major scale, the '''A4''' is between A and D{{music|sharp}}, and the '''d5''' is between D{{music|sharp}} and A.

The [[Inversion (music)#Intervals|inversion]] of a perfect interval is also perfect. Since the inversion does not change the [[pitch class]] of the two notes, it hardly affects their level of consonance (matching of their [[Harmonic series (music)|harmonics]]). Conversely, other kinds of intervals have the opposite quality with respect to their inversion. The inversion of a major interval is a minor interval, and the inversion of an augmented interval is a diminished interval.

==== Major and minor ====
[[Image:Major and minor intervals on C.png|thumb|Major and minor intervals on C: {{audio|Minor second on C.mid|m2}}, {{audio|Major second on C.mid|M2}}, {{audio|Minor third on C.mid|m3}}, {{audio|Major third on C.mid|M3}}, {{audio|Minor sixth on C.mid|m6}}, {{audio|Dim seventh on C.mid|M6}}, {{audio|Minor seventh on C.mid|m7}}, {{audio|Major seventh on C.mid|M7}}]]
As shown in the table, a [[diatonic scale]]{{efn|name=diatonic}} defines seven intervals for each interval number, each starting from a different note (seven unisons, seven seconds, etc.). The intervals formed by the notes of a diatonic scale are called diatonic. Except for unisons and octaves, the diatonic intervals with a given interval number always occur in two sizes, which differ by one semitone. For example, six of the fifths span seven semitones. The other one spans six semitones. Four of the thirds span three semitones, the others four. If one of the two versions is a perfect interval, the other is called either diminished (i.e. narrowed by one semitone) or augmented (i.e. widened by one semitone). Otherwise, the larger version is called major, the smaller one minor. For instance, since a 7-semitone fifth is a perfect interval ('''P5'''), the 6-semitone fifth is called "diminished fifth" ('''d5'''). Conversely, since neither kind of third is perfect, the larger one is called "major third" ('''M3'''), the smaller one "minor third" ('''m3''').

Within a diatonic scale,{{efn|name=diatonic}} unisons and octaves are always qualified as perfect, fourths as either perfect or augmented, fifths as perfect or diminished, and all the other intervals (seconds, thirds, sixths, sevenths) as major or minor.

==== Augmented and diminished ====
{{further|Augmented interval|Diminished interval}}
[[Image:Augmented and diminished intervals on C.png|thumb|300px|Augmented and diminished intervals on C: {{audio|Unison on C.mid|d2}}, {{audio|Minor third on C.mid|A2}}, {{audio|Major second on C.mid|d3}}, {{audio|Perfect fourth on C.mid|A3}}, {{audio|Major third on C.mid|d4}}, {{audio|Tritone on C.mid|A4}}, {{audio|Tritone on C.mid|d5}}, {{audio|Minor sixth on C.mid|A5}}, {{audio|Perfect fifth on C.mid|d6}}, {{audio|Minor seventh on C.mid|A6}}, {{audio|Dim seventh on C.mid|d7}}, {{audio|Perfect octave on C.mid|A7}}, {{audio|Major seventh on C.mid|d8}}, {{audio|Minor ninth on C.mid|A8}}]]
Augmented intervals are wider by one semitone than perfect or major intervals, while having the same interval number (i.e., encompassing the same number of staff positions): they are wider by a [[chromatic semitone]]. Diminished intervals, on the other hand, are narrower by one semitone than perfect or minor intervals of the same interval number: they are narrower by a chromatic semitone. For instance, an augmented sixth such as E{{music|flat}}–C{{music|sharp}} spans ten semitones, exceeding a major sixth (E{{music|flat}}–C) by one semitone, while a diminished sixth such as E{{music|sharp}}–C spans seven semitones, falling short of a minor sixth (E{{music|sharp}}–C{{music|sharp}}) by one semitone.

The augmented fourth ('''A4''') and the diminished fifth ('''d5''') are the only augmented and diminished intervals that appear in diatonic scales{{efn|name=diatonic}} (see table).

=== Example ===
Neither the number, nor the quality of an interval can be determined by counting [[semitones]] alone. As explained above, the number of staff positions must be taken into account as well.

For example, as shown in the table below, there are six semitones between C and F{{music|sharp}}, C and G{{music|flat}}, and C{{music|flat}} and E{{music|sharp}}, but
* C–F{{music|sharp}} is a fourth, as it encompasses four staff positions (C, D, E, F), and it is augmented, as it exceeds a perfect fourth (such as C–F) by one semitone.
* C–G{{music|flat}} is a fifth, as it encompasses five staff positions (C, D, E, F, G), and it is diminished, as it falls short of a perfect fifth (such as C-G) by one semitone.
* C{{music|flat}}–E{{music|sharp}} is a third, as it encompasses three staff positions (C, D, E), and it is doubly augmented, as it exceeds a major third (such as C–E) by two semitones.

{|class="wikitable"
! rowspan="2" ; style="text-align: left" | Number<br/>of semitones || rowspan="2" ; style="text-align: left" | Interval name || colspan="5"| [[Staff position]]s
|-
! 1 || 2 || 3 || 4 || 5
|-
| 6 || [[augmented fourth]] ('''A4''') || C || &nbsp; || &nbsp; || F{{music|sharp}} ||
|-
| 6 || [[diminished fifth]] ('''d5''') || C || &nbsp; || &nbsp; || &nbsp; || G{{music|flat}}
|-
| 6 || doubly [[augmented third]] ('''AA3''') || C{{music|flat}} || &nbsp; || E{{music|sharp}} || &nbsp; ||
|}