Perfect fifth - Revision history

Vector at 10:24, 6 March 2026

2026-03-06T10:24:19Z

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	\|}		\|}

	{{Cat\|Intervals}}		{{Cat\|Intervals}}{{Interval regions}}

Lériendil: deploying new cat

2026-02-20T00:30:37Z

deploying new cat

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	\|22\37 = 713.5c, 3\5 = 720c, 23\38 = 726.3c		\|22\37 = 713.5c, 3\5 = 720c, 23\38 = 726.3c
	\|}		\|}

			{{Cat\|Intervals}}

Lériendil: /* Equal division */

2026-01-08T23:57:27Z

Equal division

← Older revision		Revision as of 23:57, 8 January 2026
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	As 3/2 is such a profoundly concordant and simple interval, the idea of treating 3/2 as an [[equave]] has occasionally been advanced, leading to the notion of equal divisions of the fifth (EDFs) analogous to EDOs, and even to octaveless rank-2 structures that repeat at the perfect fifth. The most well-known EDFs are 9, 11, and 20 divisions of the fifth, which have the property of accurately representing [[5/4]] and [[6/5]]; these are commonly referred to respectively as Carlos Alpha, Beta, and Gamma, though the original specifications for the Carlos scales included octaves for the purpose of transposition and slightly detuned the fifth and thus they strictly speaking are closer to [[rank-2 temperament]]s that split the fifth.		As 3/2 is such a profoundly concordant and simple interval, the idea of treating 3/2 as an [[equave]] has occasionally been advanced, leading to the notion of equal divisions of the fifth (EDFs) analogous to EDOs, and even to octaveless rank-2 structures that repeat at the perfect fifth. The most well-known EDFs are 9, 11, and 20 divisions of the fifth, which have the property of accurately representing [[5/4]] and [[6/5]]; these are commonly referred to respectively as Carlos Alpha, Beta, and Gamma, though the original specifications for the Carlos scales included octaves for the purpose of transposition and slightly detuned the fifth and thus they strictly speaking are closer to [[rank-2 temperament]]s that split the fifth.

	Regardless of the theoretical notion of fifth-equivalence, structures that divide the perfect fifth into a number of equal parts are extremely common throughout xenharmonic tuning systems - such as Rastmic (2 parts), Slendric (3 parts), Tetracot (4 parts), Miracle (6 parts), and Valentine (9 parts) - and so the divisibility of the tuning of the fifth in an EDO hence can determine a lot about the harmonies that can be found in that EDO.		Regardless of the theoretical notion of fifth-equivalence, structures that divide the perfect fifth into a number of equal parts are extremely common throughout xenharmonic tuning systems - such as Rastmic (2 parts), Slendric (3 parts), Tetracot (4 parts), Miracle (6 parts), and Valentine (9 parts, related to the aforementioned Carlos Alpha) - and so the divisibility of the tuning of the fifth in an EDO hence can determine a lot about the harmonies that can be found in that EDO.

	== Tuning range ==		== Tuning range ==

Lériendil at 23:56, 8 January 2026

2026-01-08T23:56:44Z

← Older revision		Revision as of 23:56, 8 January 2026
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	[[EDO]]s that approximate 3/2 well, in order of increasing accuracy, include [[5edo\|5]], [[7edo\|7]], [[12edo\|12]], [[29edo\|29]], [[41edo\|41]], and [[53edo]].		[[EDO]]s that approximate 3/2 well, in order of increasing accuracy, include [[5edo\|5]], [[7edo\|7]], [[12edo\|12]], [[29edo\|29]], [[41edo\|41]], and [[53edo]].

			== Equal division ==
			As 3/2 is such a profoundly concordant and simple interval, the idea of treating 3/2 as an [[equave]] has occasionally been advanced, leading to the notion of equal divisions of the fifth (EDFs) analogous to EDOs, and even to octaveless rank-2 structures that repeat at the perfect fifth. The most well-known EDFs are 9, 11, and 20 divisions of the fifth, which have the property of accurately representing [[5/4]] and [[6/5]]; these are commonly referred to respectively as Carlos Alpha, Beta, and Gamma, though the original specifications for the Carlos scales included octaves for the purpose of transposition and slightly detuned the fifth and thus they strictly speaking are closer to [[rank-2 temperament]]s that split the fifth.

			Regardless of the theoretical notion of fifth-equivalence, structures that divide the perfect fifth into a number of equal parts are extremely common throughout xenharmonic tuning systems - such as Rastmic (2 parts), Slendric (3 parts), Tetracot (4 parts), Miracle (6 parts), and Valentine (9 parts) - and so the divisibility of the tuning of the fifth in an EDO hence can determine a lot about the harmonies that can be found in that EDO.

	== Tuning range ==		== Tuning range ==

Lériendil: /* Chains of fifths */

2026-01-08T23:20:07Z

Chains of fifths

← Older revision		Revision as of 23:20, 8 January 2026
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	== Chains of fifths ==		== Chains of fifths ==
	The purely tuned interval 3/2 is approximately 702 cents in size, and the structure generated by it against the octave is known as [[Pythagorean tuning]], or otherwise as the [[3-limit]]. Due to the perfect fifth being foundational to so many musical structures, and especially due to the prevalence of diatonic, the tuning of 3/2 is generally one of the most important structural aspects of a tuning system. A common conceptualization of the set of intervals is in terms of a [[chain of fifths]], and many notation systems designed for [[just intonation]] rely on defining "formal commas" to offset intervals of higher primes from a Pythagorean spine.		The purely tuned interval 3/2 is approximately 702 cents in size, and the structure generated by it against the octave is known as [[Pythagorean tuning]], or otherwise as the '''3-limit'''. Due to the perfect fifth being foundational to so many musical structures, and especially due to the prevalence of diatonic, the tuning of 3/2 is generally one of the most important structural aspects of a tuning system. A common conceptualization of the set of intervals is in terms of a [[chain of fifths]], and many notation systems designed for [[just intonation]] rely on defining "formal commas" to offset intervals of higher primes from a Pythagorean spine.

	== In temperaments ==		== In temperaments ==

Lériendil at 23:18, 8 January 2026

2026-01-08T23:18:34Z

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	The '''perfect fifth (P5)'''~~, represented by~~ the frequency ratio '''3/2''', is ~~a generator~~ of the ~~MOS~~ [[~~diatonic~~]] ~~scale and of [[Pythagorean tuning\|Pythagorean]] tuning. It~~ is ~~also~~ the most consonant octave-reduced interval after the octave itself. The note a perfect fifth above the root serves as an important structural anchor for scales, similarly to the [[perfect fourth]]~~. Nearly all musical cultures use~~ the perfect fifth~~. The perfect fifth contrasts with~~ the [[~~diatonic diminished fifth~~]]~~, a dissonant "tritone" interval. In purely tuned just intonation, it is approximately 702 cents~~ in ~~size, but as an interval in the abstract diatonic scale it may range from 685.7 to 720 cents, depending on the tuning~~.		The '''perfect fifth (P5)''' is an interval that generally represents the frequency ratio '''3/2'''. As 3/2 is the simplest ratio within the span of the [[octave]], it is considered the most consonant octave-reduced interval after the octave itself, and some form of a perfect fifth is used in nearly all world musical tuning systems. It is the simplest octave-reduced [[prime harmonic]], and is also a [[superparticular]] interval. The note a perfect fifth above the root serves as an important structural anchor for scales, similarly to the [[perfect fourth]], and the perfect fifth serves as the bounding interval of the most common [[Collection of chords#Triads\|triads]] in conventional systems of harmony.

	~~Due~~ to the fifth's ~~role in~~ diatonic, ~~Pythagorean intervals are usually conceptualized in terms of~~ a ~~chain~~ of fifths. ~~Many temperaments~~, ~~called monocot temperaments~~, ~~use~~ the ~~perfect~~ fifth ~~as a generator~~. ~~Additionally~~, ~~it is the bounding interval of most common~~ [[~~Collection of chords#Triads\|triads~~]].		== Diatonic scales ==
			Most commonly, the term "perfect fifth" refers to the diatonic perfect fifth, that is, a generator of the MOS [[diatonic]] scale (5L 2s). As an interval in the abstract diatonic scale, it may range from 685.7 to 720 [[cent]]s, depending on the tuning; it contrasts with the [[diatonic diminished fifth]], a dissonant "tritone" interval, and with a broader melodic category of "fifths". Specifically, the diatonic scale contains six perfect fifths; in the Ionian mode, perfect fifths are found on all but the 7th degree of the scale, which instead has a diminished fifth. Note that not all diatonic perfect fifths necessarily represent 3/2, especially in larger [[EDO]]s where multiple diatonic fifths can be found.

	~~Edos that approximate~~ the perfect fifth ~~well~~, in ~~order~~ of ~~increasing accuracy, include 5, 12, 29, 41~~, and ~~53edo~~.		== Chains of fifths ==
			The purely tuned interval 3/2 is approximately 702 cents in size, and the structure generated by it against the octave is known as [[Pythagorean tuning]], or otherwise as the [[3-limit]]. Due to the perfect fifth being foundational to so many musical structures, and especially due to the prevalence of diatonic, the tuning of 3/2 is generally one of the most important structural aspects of a tuning system. A common conceptualization of the set of intervals is in terms of a [[chain of fifths]], and many notation systems designed for [[just intonation]] rely on defining "formal commas" to offset intervals of higher primes from a Pythagorean spine.

	== ~~Scale info~~ ==		== In temperaments ==
	~~The diatonic scale contains six perfect fifths. In~~ the ~~Ionian mode, perfect fifths are found~~ on ~~all but~~ the ~~7th degree~~ of the ~~scale~~, which ~~instead has a diminished fifth~~.		Further, the [[ploidacot]] system is a method of classifying temperaments and generated structures based on the position of the perfect fifth in that structure. Monocot temperaments are those which use the perfect fifth as a generator, of which there are many important ones, such as [[Meantone]], [[Archy\|Superpyth]], and [[Schismic]].

	~~== Just intonation ==~~		[[EDO]]s that approximate 3/2 well, in order of increasing accuracy, include [[5edo\|5]], [[7edo\|7]], [[12edo\|12]], [[29edo\|29]], [[41edo\|41]], and [[53edo]].
	~~The perfect fifth is a~~ [[~~superparticular~~]] ~~interval~~.

	== Tuning range ==		== Tuning range ==

Lériendil at 13:57, 15 December 2025

2025-12-15T13:57:37Z

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	== Just intonation ==		== Just intonation ==
	The perfect fifth is a superparticular interval.		The perfect fifth is a [[superparticular]] interval.

	== Tuning range ==		== Tuning range ==

Vector at 00:54, 15 December 2025

2025-12-15T00:54:00Z

← Older revision		Revision as of 00:54, 15 December 2025
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	The '''perfect fifth (P5)''', represented by the frequency ratio '''3/2''', is a generator of the MOS [[diatonic]] scale and of [[Pythagorean tuning\|Pythagorean]] tuning. It is also the most consonant octave-reduced interval after the octave itself. The note a perfect fifth above the root serves as an important structural anchor for scales, similarly to the [[perfect fourth]]. Nearly all musical cultures use the perfect fifth. The perfect fifth contrasts with the [[diatonic diminished fifth]], a dissonant "tritone" interval. It is approximately 702 cents in size, but as an interval in the abstract diatonic scale it may range from 685.7 to 720 cents, depending on the tuning.		The '''perfect fifth (P5)''', represented by the frequency ratio '''3/2''', is a generator of the MOS [[diatonic]] scale and of [[Pythagorean tuning\|Pythagorean]] tuning. It is also the most consonant octave-reduced interval after the octave itself. The note a perfect fifth above the root serves as an important structural anchor for scales, similarly to the [[perfect fourth]]. Nearly all musical cultures use the perfect fifth. The perfect fifth contrasts with the [[diatonic diminished fifth]], a dissonant "tritone" interval. In purely tuned just intonation, it is approximately 702 cents in size, but as an interval in the abstract diatonic scale it may range from 685.7 to 720 cents, depending on the tuning.

	Due to the fifth's role in diatonic, Pythagorean intervals are usually conceptualized in terms of a chain of fifths. Many temperaments, called monocot temperaments, use the perfect fifth as a generator. Additionally, it is the bounding interval of most common [[Collection of chords#Triads\|triads]].		Due to the fifth's role in diatonic, Pythagorean intervals are usually conceptualized in terms of a chain of fifths. Many temperaments, called monocot temperaments, use the perfect fifth as a generator. Additionally, it is the bounding interval of most common [[Collection of chords#Triads\|triads]].

Vector at 00:52, 15 December 2025

2025-12-15T00:52:03Z

← Older revision		Revision as of 00:52, 15 December 2025
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	== Tuning range ==		== Tuning range ==
	''~~TODO: Insert info about different~~ tuning ~~ranges~~ of ~~the perfect fifth here~~. ~~Discussion~~ of ~~the~~ gentle ~~region should go here and be mirrored in~~ [[Neogothic interval]]. ~~Also discuss edo tunings~~ of ~~the perfect fifth, and the 685~~.7-~~720 cent diatonic range~~.''		{\| class="wikitable"
			\|+Fifth tuning ranges
			!Range (in cents)
			!Description
			!Notable edos
			\|-
			\|'''685.7-720'''
			\|'''Range of fifths capable of generating a diatonic scale.'''
			\|'''4\7 = 685.7c, 7\12 = 700c, 3\5 = 720c'''
			\|-
			\|666.7-685.7
			\|Range of intervals that generate a soft [[antidiatonic]] scale and may be considered fifths.
			\|9\16 = 675c
			\|-
			\|~678-693
			\|Range of intervals that generate an [[equiheptatonic]] diatonic or antidiatonic scale.
			\|13\23 = 678.3c, 4\7 = 685.7c, 15\26 = 692.3c
			\|-
			\|691.5-694.7
			\|Range of viable generators of [[List of regular temperaments\|Flattone]] temperament.
			\|26\45 = 693.3c, 15\26 = 692.3c
			\|-
			\|694.7-700
			\|Range of viable generators of [[Meantone\|Septimal Meantone]] temperament.
			\|11\19 = 694.7c, 18\31 = 696.8c, 29\50 = 696c, 7\12 = 700c
			\|-
			\|698.5-705.5
			\|Pythagorean tuning; fifths within a [[just noticeable difference]] of just intonation. Contains schismic tunings.
			\|7\12 = 700c; 24\41 = 702.4c; 31\53 = 701.9c
			\|-
			\|703.4-705.9
			\|Range of "gentle" fifths, which generate [[Neogothic interval\|neogothic intervals]].
			\|17\29 = 703.4c; 10\17 = 705.9c
			\|-
			\|~709-711
			\|Range of viable generators of [[Superpyth]] temperament.
			\|13\22 = 709.1c; 16\27 = 711.1c
			\|-
			\|~711-715
			\|Range of viable generators of other [[Archy]] temperaments.
			\|22\37 = 713.5c
			\|-
			\|~709-730
			\|Range of intervals that generate an [[equipentatonic]] pentic or antipentic scale.
			\|22\37 = 713.5c, 3\5 = 720c, 23\38 = 726.3c
			\|}

Vector at 00:33, 15 December 2025

2025-12-15T00:33:08Z

← Older revision		Revision as of 00:33, 15 December 2025
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	== Tuning range ==		== Tuning range ==
			''TODO: Insert info about different tuning ranges of the perfect fifth here. Discussion of the gentle region should go here and be mirrored in [[Neogothic interval]]. Also discuss edo tunings of the perfect fifth, and the 685.7-720 cent diatonic range.''