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The '''3rd harmonic''', '''tritave''', '''triple''', or '''perfect twelfth''' is the [[interval]] of [[frequency ratio]] '''3/1'''. It is perhaps the most [[consonant]] interval after the [[octave]], with frequency ratio 2/1. For this reason, it is used as an [[equave]] in some [[nonoctave]] systems, such as the [[Bohlen–Pierce]] scale.

== ~~Significance~~ of prime 3 ==

It is the second [[prime harmonic]], after [[2/1]] and before [[5/1]].

== Importance of prime 3 ==

The [[octave-reduced]] 3rd harmonic is the perfect fifth [[3/2]], and the [[octave complement]] of 3/2 is the perfect fourth [[4/3]]. The perfect fifth and fourth are considered essential in western music theory, and in [[12edo]], stacking them makes the [[Circle of fifths|circle of fifths/fourths]]. The perfect fifth is often used as the base for constructing chords, such as the classical major triad [[4:5:6|1–5/4–3/2]] (4:5:6). The perfect fourth can also be used as a base in chords, such as [[6:7:8|1–7/6–4/3]] (6:7:8), which deviates from traditional harmony.

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== As an interval of equivalence ==

~~{{Main|EDT}}~~

When used as an [[interval of equivalence]], 3/1 can be called the ''tritave''. This is very xenharmonic since it assumes tritave equivalence instead of octave equivalence, so that [[1/1]], 3/1, and [[9/1]] are considered the same pitch class. Typically tritave-equivalent systems base harmony off of only [[odd harmonic]]s, for example with the [[3:5:7]] triad as analogous to 4:5:6.

When used as an interval of equivalence, 3/1 can be called the '''tritave'''. This is very xenharmonic since it ~~does not assume [[octave~~ equivalence~~]], and~~ instead ~~tritave~~ equivalence ~~is assumed~~, so that [[1/1]], 3/1, and [[9/1]] are considered the same pitch class. Typically tritave-equivalent systems base harmony off of only [[odd harmonic]]s, for example with the [[3:5:7]] triad as analogous to 4:5:6.

An example of a tritave-based ~~system~~ is the [[~~Bohlen-Pierce~~ scale]]. The [[~~Equal~~ temperament|equal-tempered]] version of the ~~Bohlen-Pierce~~ scale is [[13edt]], or 13 equal divisions of the tritave. Systems can be constructed analogously to octave-equivalent harmony, for example the 9-note [[lambda]] scale, which can be considered analogous to [[diatonic]].

An example of a system that is typically treated as tritave-based is the [[Bohlen–Pierce scale]]. The [[equal temperament|equal-tempered]] version of the Bohlen–Pierce scale is [[13edt]], or 13 equal divisions of the tritave. Systems can be constructed analogously to octave-equivalent harmony, for example the 9-note [[lambda]] scale, which can be considered analogous to [[diatonic]].

== Etymology ==

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== See also ==

* [[EDT]] (equal divisions of the tritave/twelfth)

* [[~~No-twos 31-limit~~]] – ~~non-~~octave ~~31-limit system containing neither 2 nor primes higher than 31~~

* [[3/2]] – its [[octave reduced]] form

* [[~~Tritave~~ complement]] – the analogue for [[octave complement]]

* [[Twelfth complement]] – the analogue for [[octave complement]]

== References ==

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 The '''3rd harmonic''', '''tritave''', '''triple''', or '''perfect twelfth''' is the [[interval]] of [[frequency ratio]] '''3/1'''. It is perhaps the most [[consonant]] interval after the [[octave]], with frequency ratio 2/1. For this reason, it is used as an [[equave]] in some [[nonoctave]] systems, such as the [[Bohlen–Pierce]] scale.
-== Significance of prime 3 ==
+It is the second [[prime harmonic]], after [[2/1]] and before [[5/1]].
+== Importance of prime 3 ==
 The [[octave-reduced]] 3rd harmonic is the perfect fifth [[3/2]], and the [[octave complement]] of 3/2 is the perfect fourth [[4/3]]. The perfect fifth and fourth are considered essential in western music theory, and in [[12edo]], stacking them makes the [[Circle of fifths|circle of fifths/fourths]]. The perfect fifth is often used as the base for constructing chords, such as the classical major triad [[4:5:6|1–5/4–3/2]] (4:5:6). The perfect fourth can also be used as a base in chords, such as [[6:7:8|1–7/6–4/3]] (6:7:8), which deviates from traditional harmony.
@@ Line 13: / Line 15: @@
 == As an interval of equivalence ==
-{{Main|EDT}}
+When used as an [[interval of equivalence]], 3/1 can be called the ''tritave''. This is very xenharmonic since it assumes tritave equivalence instead of octave equivalence, so that [[1/1]], 3/1, and [[9/1]] are considered the same pitch class. Typically tritave-equivalent systems base harmony off of only [[odd harmonic]]s, for example with the [[3:5:7]] triad as analogous to 4:5:6.
-When used as an interval of equivalence, 3/1 can be called the '''tritave'''. This is very xenharmonic since it does not assume [[octave equivalence]], and instead tritave equivalence is assumed, so that [[1/1]], 3/1, and [[9/1]] are considered the same pitch class. Typically tritave-equivalent systems base harmony off of only [[odd harmonic]]s, for example with the [[3:5:7]] triad as analogous to 4:5:6.
-An example of a tritave-based system is the [[Bohlen-Pierce scale]]. The [[Equal temperament|equal-tempered]] version of the Bohlen-Pierce scale is [[13edt]], or 13 equal divisions of the tritave. Systems can be constructed analogously to octave-equivalent harmony, for example the 9-note [[lambda]] scale, which can be considered analogous to [[diatonic]].
+An example of a system that is typically treated as tritave-based is the [[Bohlen–Pierce scale]]. The [[equal temperament|equal-tempered]] version of the Bohlen–Pierce scale is [[13edt]], or 13 equal divisions of the tritave. Systems can be constructed analogously to octave-equivalent harmony, for example the 9-note [[lambda]] scale, which can be considered analogous to [[diatonic]].
 == Etymology ==
@@ Line 27: / Line 28: @@
 == See also ==
 * [[EDT]] (equal divisions of the tritave/twelfth)
-* [[No-twos 31-limit]] – non-octave 31-limit system containing neither 2 nor primes higher than 31
+* [[3/2]] – its [[octave reduced]] form
-* [[Tritave complement]] – the analogue for [[octave complement]]
+* [[Twelfth complement]] – the analogue for [[octave complement]]
 == References ==