Editing Interval (music) (section)

== Compound intervals ==
[[File:Simple and compound major third.png|thumb|Simple and compound major third[[File:Simple and compound major third.mid]]]]

A compound interval is an interval spanning more than one octave.<ref name="compound" /> Conversely, intervals spanning at most one octave are called simple intervals (see [[#Main intervals|Main intervals]] below).

In general, a compound interval may be defined by a sequence or "stack" of two or more simple intervals of any kind. For instance, a major tenth (two staff positions above one octave), also called ''compound major third'', spans one octave plus one major third.

Any compound interval can be always decomposed into one or more octaves plus one simple interval. For instance, a major seventeenth can be decomposed into two octaves and one major third, and this is the reason why it is called a compound major third, even when it is built by adding up four fifths.

The diatonic number ''DN''<sub>c</sub> of a compound interval formed from ''n'' simple intervals with diatonic numbers ''DN''<sub>1</sub>, ''DN''<sub>2</sub>, ..., ''DN''<sub>n</sub>, is determined by:
:<math>DN_c = 1 + (DN_1 - 1) + (DN_2 - 1) + ... + (DN_n - 1), \ </math>
which can also be written as:
:<math>DN_c = DN_1 + DN_2 + ... + DN_n - (n - 1), \ </math>

The quality of a compound interval is determined by the quality of the simple interval on which it is based. For instance, a compound major third is a major tenth (1+(8−1)+(3−1) = 10), or a major seventeenth (1+(8−1)+(8−1)+(3−1) = 17), and a compound perfect fifth is a perfect twelfth (1+(8−1)+(5−1) = 12) or a perfect nineteenth (1+(8−1)+(8−1)+(5−1) = 19). Notice that two octaves are a fifteenth, not a sixteenth (1+(8−1)+(8−1) = 15). Similarly, three octaves are a twenty-second (1+3×(8−1) = 22), four octaves are a twenty-ninth (1+3×(8-1) = 29), and so on.

=== Main compound intervals ===
{|class="wikitable"
|-
! Number of<br/>[[semitone]]s|| [[#Quality|Minor, major,<br/>or perfect]] intervals || Short || [[#Quality|Augmented or<br/>diminished]] intervals || Short
|-
| 12 || || || Diminished [[ninth]] || d9
|-
| 13 || [[Minor ninth]] || m9 || [[Augmented octave]] || A8
|-
| 14 || [[Major ninth]] || M9 || Diminished tenth || d10
|-
| 15 || [[Minor tenth]] || m10 || [[Augmented ninth]] || A9
|-
| 16 || [[Major tenth]] || M10 || Diminished [[eleventh]] || d11
|-
| 17 || [[Perfect eleventh]] || P11 || Augmented tenth || A10 
|-
| rowspan="2"| 18 || rowspan="2"| || rowspan="2"| || Diminished twelfth || d12
|-
| [[Augmented eleventh]] || A11
|-
| 19 || [[Perfect twelfth]] or [[Tritave]] || P12 || Diminished [[thirteenth]] || d13
|-
| 20 || Minor [[thirteenth]] || m13 || Augmented twelfth || A12
|-
| 21 || Major [[thirteenth]] || M13 || Diminished fourteenth || d14
|-
| 22 || Minor fourteenth || m14 || Augmented [[thirteenth]] || A13
|-
| 23 || Major fourteenth || M14 || Diminished [[fifteenth]] || d15
|-
| 24 || Perfect [[fifteenth]] or Double octave || P15 || Augmented fourteenth || A14
|-
| 25 || || || Augmented [[fifteenth]] || A15
|}

It is also worth mentioning here the major seventeenth (28 semitones)—an interval larger than two octaves that can be considered a multiple of a perfect fifth (7 semitones) as it can be decomposed into four perfect fifths (7 × 4 = 28 semitones), or two octaves plus a major third (12 + 12 + 4 = 28 semitones). Intervals larger than a major seventeenth seldom come up, most often being referred to by their compound names, for example "two octaves plus a fifth"<ref>Aikin, Jim (2004). ''A Player's Guide to Chords and Harmony: Music Theory for Real-World Musicians'', p.&nbsp;24. {{ISBN|0-87930-798-6}}.</ref> rather than "a 19th".