3/1: Difference between revisions

(24 intermediate revisions by 9 users not shown)

Line 1:

{{Infobox Interval

| Ratio = 3/1

| Name = tritave, ~~3rd harmonic~~, perfect twelfth

| Name = 3rd harmonic, tritave, triple, perfect twelfth

| Color name = w12, wa 12th

| Sound = jid_3_1_pluck_adu_dr220.mp3

}}

The '''tritave''' ~~(interval [[ratio]]~~ '''~~3/1~~''') is the [[interval]] ~~between a fundamental tone and its~~ '''~~3rd harmonic~~'''. It is perhaps the most [[~~consonance|~~consonant]] interval after the [[octave]]. For this reason, it is used as an [[equave]] in some [[nonoctave]] systems, such as the [[~~Bohlen-Pierce~~]] scale.

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.

The ~~tritave~~ is ~~one octave above~~ [[3/2]], the perfect fifth. ~~Therefore~~, in a [[~~5L 2s~~|~~diatonic~~]] ~~context~~, 3/1 is also called the ''~~'perfect twelfth'~~''.

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.

In [[just intonation]], 3/1 is the first [[prime harmonic]] that adds [[pitch class]]es besides the unison, octave, and multiples of the octave. [[Pythagorean tuning]], also known as the [[3-limit]], is the subset of just intonation containing all intervals where the only prime factors are 2 and 3. Pythagorean tuning generates the [[pentic]] and [[diatonic]] scales, and is often used as a system for interval classification in just intonation.

== As an interval of equivalence ==

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.

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 ==

The term ''tritave'' was coined by [[John Pierce]]<ref>https://www.huygens-fokker.org/bpsite/intervals.html</ref>. It was derived from the word ''octave'' by replacing the perceived prefix ''octo-'' (eight, for the eighth degree of the diatonic scale) by ''tri-'' (three, for 3/1). ~~It should be noted, however~~, ~~that~~ the ''oct'' in ''octave'' is not a prefix, but part of the single-morpheme word derived from Latin ''octavus'' (eighth).

The term ''tritave'' was coined by [[John Pierce]]<ref>[https://www.huygens-fokker.org/bpsite/intervals.html ''The Bohlen-Pierce Site: BP Interval Properties'']</ref>. It was derived from the word ''octave'' by replacing the perceived prefix ''octo-'' (eight, for the eighth degree of the diatonic scale) by ''tri-'' (three, for 3/1). However, the ''oct'' in ''octave'' is not a prefix, but part of the single-morpheme word derived from Latin [[Wiktionary:octavus #Latin|''octavus'']] ("eighth"). In this sense, ''tritave'' is more of a contraction of ''tri-'' and ''octave'' than anything else. As such, the term usually refers to 3/1 as an interval of equivalence; in other contexts, it is more often called the perfect twelfth (after the 12th degree of the diatonic scale).

''Triple'' is a proposed term which relates itself to the ancient Greek concept of [[harmonic|multiples]]. It also fixes the problem of using part of the word ''octave''.

Since the enneatonic {{mos scalesig|4L 5s<3/1>|link=1}} ("Lambda") scale is the BP substitute for the diatonic scale, the term ''decade''<ref>[https://www.youtube.com/watch?v=Ur6GOoSNGN0 12tone – How A Pair Of Microwave Engineers Broke Music]</ref> or ''decim''{{citation needed}} (tenth degree of the Lambda scale) has been proposed as an alternative to tritave, though ''decade'' almost always refers to ten times the frequency ([[10/1]]) in audio engineering.

== See also ==

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

* [[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 ==

[[Category:Tritave| ]]

~~[[Category:Terms]]~~

~~[[Category:Todo:expand]]~~

@@ Line 1: / Line 1: @@
 {{Infobox Interval
 | Ratio = 3/1
-| Name = tritave, 3rd harmonic, perfect twelfth
+| Name = 3rd harmonic, tritave, triple, perfect twelfth
 | Color name = w12, wa 12th
 | Sound = jid_3_1_pluck_adu_dr220.mp3
 }}
-The '''tritave''' (interval [[ratio]] '''3/1''') is the [[interval]] between a fundamental tone and its '''3rd harmonic'''. It is perhaps the most [[consonance|consonant]] interval after the [[octave]]. For this reason, it is used as an [[equave]] in some [[nonoctave]] systems, such as the [[Bohlen-Pierce]] scale.
+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.
-The tritave is one octave above [[3/2]], the perfect fifth. Therefore, in a [[5L 2s|diatonic]] context, 3/1 is also called the '''perfect twelfth'''.
+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.
+In [[just intonation]], 3/1 is the first [[prime harmonic]] that adds [[pitch class]]es besides the unison, octave, and multiples of the octave. [[Pythagorean tuning]], also known as the [[3-limit]], is the subset of just intonation containing all intervals where the only prime factors are 2 and 3. Pythagorean tuning generates the [[pentic]] and [[diatonic]] scales, and is often used as a system for interval classification in just intonation.
+== As an interval of equivalence ==
+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.
+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 ==
-The term ''tritave'' was coined by [[John Pierce]]<ref>https://www.huygens-fokker.org/bpsite/intervals.html</ref>. It was derived from the word ''octave'' by replacing the perceived prefix ''octo-'' (eight, for the eighth degree of the diatonic scale) by ''tri-'' (three, for 3/1). It should be noted, however, that the ''oct'' in ''octave'' is not a prefix, but part of the single-morpheme word derived from Latin ''octavus'' (eighth).
+The term ''tritave'' was coined by [[John Pierce]]<ref>[https://www.huygens-fokker.org/bpsite/intervals.html ''The Bohlen-Pierce Site: BP Interval Properties'']</ref>. It was derived from the word ''octave'' by replacing the perceived prefix ''octo-'' (eight, for the eighth degree of the diatonic scale) by ''tri-'' (three, for 3/1). However, the ''oct'' in ''octave'' is not a prefix, but part of the single-morpheme word derived from Latin [[Wiktionary:octavus #Latin|''octavus'']] ("eighth"). In this sense, ''tritave'' is more of a contraction of ''tri-'' and ''octave'' than anything else. As such, the term usually refers to 3/1 as an interval of equivalence; in other contexts, it is more often called the perfect twelfth (after the 12th degree of the diatonic scale).
+''Triple'' is a proposed term which relates itself to the ancient Greek concept of [[harmonic|multiples]]. It also fixes the problem of using part of the word ''octave''.
+Since the enneatonic {{mos scalesig|4L 5s<3/1>|link=1}} ("Lambda") scale is the BP substitute for the diatonic scale, the term ''decade''<ref>[https://www.youtube.com/watch?v=Ur6GOoSNGN0 12tone – How A Pair Of Microwave Engineers Broke Music]</ref> or ''decim''{{citation needed}} (tenth degree of the Lambda scale) has been proposed as an alternative to tritave, though ''decade'' almost always refers to ten times the frequency ([[10/1]]) in audio engineering.
 == See also ==
-* [[EDT]] (equal divisions of the tritave)
+* [[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 ==
-<references />
+<references/>
 [[Category:Tritave| ]] <!-- main article -->
-[[Category:Terms]]
-[[Category:Todo:expand]]