15/8: Difference between revisions

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{{Infobox Interval

| Name = classic~~/just~~ major seventh

| Name = just major seventh, classic(al) major seventh, ptolemaic major seventh

| Color name = y7, yo 7th

| Sound = jid_15_8_pluck_adu_dr220.mp3

}}

In [[5-limit]] [[just intonation]], '''15/8''' is the '''classic major seventh''' of about 1088.3¢. It is also the ~~15th~~ [[~~harmonic~~]] ([[~~octave-reduced~~]]), and appears as a complex consonance in chords such as 8:10:12:15, a just version of a major seventh chord. Since 15 ~~is 3×5~~, it can be seen as a perfect fifth above a major third or vice versa, and this understanding ~~is compatible with~~ the ~~1100¢ interval~~ of [[~~12edo~~]].

In [[5-limit]] [[just intonation]], '''15/8''' is the '''just major seventh''', '''classic(al) major seventh''', or '''ptolemaic major seventh'''<ref>For reference, see [[5-limit]]. </ref> of about 1088.3¢. It is also the [[octave-reduced]] 15th [[harmonic]], and appears as a complex consonance in chords such as [[8:10:12:15]], a just version of a major seventh chord. Since 15/8 = [[3/2]] × [[5/4]], it can be seen as a perfect fifth above a major third or vice versa, and this understanding works in [[12edo]], as the sum of [[~]]3/2 and ~5/4 is 700{{c}} + 400{{c}} = 1100{{c}}, which 15/8 is mapped to.

Since 15 is a perfect fifth above 10 (15/10 = [[3/2]]), ~~[[List of root-3rd-P5 triads in JI|root-3rd-P5 triads]]~~ can be formed with the 10th harmonic as ~~root~~ and 15th harmonic as ~~perfect fifth~~. The simplest and most familiar example is the ~~classic minor triad~~ 10:12:15 ~~– a [[~~6/5~~]] with a [[~~5/4~~]] stacked on top of it~~. Another is the ~~Barbados triad~~, 10:13:15 ~~– a [[~~13/10~~]] on bottom~~ and a [[15~~/13~~]] ~~on top~~. ~~And a~~ particularly uncommon but mentionable example is the [[23-limit]] ~~inframinor triad~~ 20:23:30.

Since 15 is a perfect fifth above 10 (15/10 = [[3/2]]), seventh chords can be formed with the 10th harmonic as major third and 15th harmonic as major seventh. The simplest and most familiar example is the classical major seventh chord 8:10:12:15 with steps 5/4, 6/5 and 5/4. Another example replaces the 12 with 13, which leads to [[8:10:13:15]] with steps 5/4, 13/10 and 15/13, and contains the [[10:13:15]] barbados triad. A particularly uncommon but mentionable example is the [[23-limit]] seventh chord [[16:20:23:30]].

== Approximation ==

== See also ==

* [[16/15]] – its [[octave complement]]

* [[8/5]] – its [[twelfth complement]]

* [[Ed15/8]]

* [[Gallery of just intervals]]

== Notes ==

[[Category:Seventh]]

[[Category:Major seventh]]

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 {{Infobox Interval
-| Name = classic/just major seventh
+| Name = just major seventh, classic(al) major seventh, ptolemaic major seventh
 | Color name = y7, yo 7th
 | Sound = jid_15_8_pluck_adu_dr220.mp3
 }}
 {{Wikipedia|Major seventh}}
-In [[5-limit]] [[just intonation]], '''15/8''' is the '''classic major seventh''' of about 1088.3¢. It is also the 15th [[harmonic]] ([[octave-reduced]]), and appears as a complex consonance in chords such as 8:10:12:15, a just version of a major seventh chord. Since 15 is 3×5, it can be seen as a perfect fifth above a major third or vice versa, and this understanding is compatible with the 1100¢ interval of [[12edo]].
+In [[5-limit]] [[just intonation]], '''15/8''' is the '''just major seventh''', '''classic(al) major seventh''', or '''ptolemaic major seventh'''<ref>For reference, see [[5-limit]]. </ref> of about 1088.3¢. It is also the [[octave-reduced]] 15th [[harmonic]], and appears as a complex consonance in chords such as [[8:10:12:15]], a just version of a major seventh chord. Since 15/8 = [[3/2]] × [[5/4]], it can be seen as a perfect fifth above a major third or vice versa, and this understanding works in [[12edo]], as the sum of [[~]]3/2 and ~5/4 is 700{{c}} + 400{{c}} = 1100{{c}}, which 15/8 is mapped to.
-Since 15 is a perfect fifth above 10 (15/10 = [[3/2]]), [[List of root-3rd-P5 triads in JI|root-3rd-P5 triads]] can be formed with the 10th harmonic as root and 15th harmonic as perfect fifth. The simplest and most familiar example is the classic minor triad 10:12:15 – a [[6/5]] with a [[5/4]] stacked on top of it. Another is the Barbados triad, 10:13:15 – a [[13/10]] on bottom and a [[15/13]] on top. And a particularly uncommon but mentionable example is the [[23-limit]] inframinor triad 20:23:30.
+Since 15 is a perfect fifth above 10 (15/10 = [[3/2]]), seventh chords can be formed with the 10th harmonic as major third and 15th harmonic as major seventh. The simplest and most familiar example is the classical major seventh chord 8:10:12:15 with steps 5/4, 6/5 and 5/4. Another example replaces the 12 with 13, which leads to [[8:10:13:15]] with steps 5/4, 13/10 and 15/13, and contains the [[10:13:15]] barbados triad. A particularly uncommon but mentionable example is the [[23-limit]] seventh chord [[16:20:23:30]].
+== Approximation ==
+{{Interval edo approximation|15/8}}
 == See also ==
 * [[16/15]] – its [[octave complement]]
+* [[8/5]] – its [[twelfth complement]]
+* [[Ed15/8]]
 * [[Gallery of just intervals]]
+== Notes ==
+<references/>
 [[Category:Seventh]]
 [[Category:Major seventh]]