3/1 - Revision history

FloraC: /* As an interval of equivalence */ rework. Edt isn't the main article of 3/1 equivalence

2026-05-22T09:03:11Z

As an interval of equivalence: rework. Edt isn't the main article of 3/1 equivalence

← Older revision		Revision as of 09:03, 22 May 2026
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	== As an interval of equivalence ==		== 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 ==		== Etymology ==

FloraC: /* See also */

2026-05-22T08:30:33Z

Overthink: /* See also */ add 3/2

2026-03-20T15:38:49Z

Overthink: + prime harmonics

2026-03-19T05:39:20Z

+ prime harmonics

← Older revision		Revision as of 05:39, 19 March 2026
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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.		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 ==		== Importance of prime 3 ==

Overthink: change header

2026-03-16T23:54:26Z

change header

← Older revision		Revision as of 23:54, 16 March 2026
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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.		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 ==		== 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.		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.

Overthink: /* Significance of prime 3 */ remove "thus"; mention in JI

2026-03-16T22:17:26Z

Significance of prime 3: remove "thus"; mention in JI

← Older revision		Revision as of 22:17, 16 March 2026
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	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.		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 ~~thus~~ often used as a system for interval classification.		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 ==		== As an interval of equivalence ==

Overthink: Add 2 sections

2026-03-16T22:09:45Z

Add 2 sections

← Older revision		Revision as of 22:09, 16 March 2026
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	\| Sound = jid_3_1_pluck_adu_dr220.mp3		\| Sound = jid_3_1_pluck_adu_dr220.mp3
	}}		}}
	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]]. 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.

			== Significance 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 thus often used as a system for interval classification.

			== 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 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]].

	== Etymology ==		== Etymology ==

Squib at 15:31, 13 February 2026

2026-02-13T15:31:55Z

← Older revision		Revision as of 15:31, 13 February 2026
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	== Etymology ==		== Etymology ==
	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.		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''.		''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''.

FloraC: +triple

2025-03-31T10:16:58Z

+triple

← Older revision		Revision as of 10:16, 31 March 2025
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	{{Infobox Interval		{{Infobox Interval
	\| Ratio = 3/1		\| Ratio = 3/1
	\| Name = 3rd harmonic, tritave, perfect twelfth		\| Name = 3rd harmonic, tritave, triple, perfect twelfth
	\| 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 '''3rd harmonic''', '''tritave''', or '''perfect twelfth''' is the [[interval]] of [[frequency ratio]] '''3/1'''. It is perhaps the most [[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]]. For this reason, it is used as an [[equave]] in some [[nonoctave]] systems, such as the [[Bohlen–Pierce]] scale.

	== Etymology ==		== Etymology ==
	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). ~~It should be noted~~, ~~however, that~~ the ''oct'' in ''octave'' is not a prefix, but part of the single-morpheme word derived from Latin [[Wiktionary:octavus#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.

			''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.		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.

FloraC: Request citation

2025-03-11T06:00:21Z

Request citation

← Older revision		Revision as of 06:00, 11 March 2025
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	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). It should be noted, however, that the ''oct'' in ''octave'' is not a prefix, but part of the single-morpheme word derived from Latin [[Wiktionary:octavus#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). It should be noted, however, that the ''oct'' in ''octave'' is not a prefix, but part of the single-morpheme word derived from Latin [[Wiktionary:octavus#Latin\|''octavus'']] ("eighth").

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

	[[Category:Tritave\| ]] <!-- main article -->		[[Category:Tritave\| ]] <!-- main article -->

← Older revision		Revision as of 08:30, 22 May 2026
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	* [[EDT]] (equal divisions of the tritave/twelfth)		* [[EDT]] (equal divisions of the tritave/twelfth)
	* [[3/2]] – its [[octave reduced]] form		* [[3/2]] – its [[octave reduced]] form
	* [[~~Tritave~~ complement]] – the analogue for [[octave complement]]		* [[Twelfth complement]] – the analogue for [[octave complement]]
	* [[No-twos 31-limit]] – non-octave 31-limit system containing neither 2 nor primes higher than 31

	== References ==		== References ==

← Older revision		Revision as of 15:38, 20 March 2026
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	== See also ==		== See also ==
	* [[EDT]] (equal divisions of the tritave/twelfth)		* [[EDT]] (equal divisions of the tritave/twelfth)
			* [[3/2]] – its [[octave reduced]] form
			* [[Tritave complement]] – the analogue for [[octave complement]]
	* [[No-twos 31-limit]] – non-octave 31-limit system containing neither 2 nor primes higher than 31		* [[No-twos 31-limit]] – non-octave 31-limit system containing neither 2 nor primes higher than 31
	* [[Tritave complement]] – the analogue for [[octave complement]]

	== References ==		== References ==