3/1: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Ratio = 3/1 | | Ratio = 3/1 | ||
| Name = 3rd harmonic, tritave, triple, perfect twelfth | |||
| Name = tritave, | |||
| Color name = w12, wa 12th | | Color name = w12, wa 12th | ||
| Sound = jid_3_1_pluck_adu_dr220.mp3 | | Sound = jid_3_1_pluck_adu_dr220.mp3 | ||
}} | }} | ||
The '''tritave''' | 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 | 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 == | == 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 prefix ''octo-'' (eight, for | 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 == | == See also == | ||
* [[EDT]] (equal divisions of the tritave) | * [[EDT]] (equal divisions of the tritave/twelfth) | ||
* [[ | * [[3/2]] – its [[octave reduced]] form | ||
* [[ | * [[Twelfth complement]] – the analogue for [[octave complement]] | ||
== References == | == References == | ||
<references /> | <references/> | ||
[[Category:Tritave| ]] <!-- main article --> | [[Category:Tritave| ]] <!-- main article --> | ||
Latest revision as of 09:03, 22 May 2026
| Interval information |
tritave,
triple,
perfect twelfth
prime harmonic
[sound info]
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.
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/fourths. The perfect fifth is often used as the base for constructing chords, such as the classical major triad 1–5/4–3/2 (4:5:6). The perfect fourth can also be used as a base in chords, such as 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 classes 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 harmonics, 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-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[1]. 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 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 multiples. It also fixes the problem of using part of the word octave.
Since the enneatonic 4L 5s⟨3/1⟩ ("Lambda") scale is the BP substitute for the diatonic scale, the term decade[2] 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/twelfth)
- 3/2 – its octave reduced form
- Twelfth complement – the analogue for octave complement