3/2 - Revision history

Furcht968 at 00:34, 2 April 2026

2026-04-02T00:34:45Z

← Older revision		Revision as of 00:34, 2 April 2026
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	\| en =		\| en =
	\| es =		\| es =
	\| ja =		\| ja =3/2
	\| ko =		\| ko =
	\| ro = 3/2 (ro)		\| ro = 3/2 (ro)

FloraC: /* Usage */ - biased opinion

2026-01-27T11:07:19Z

Usage: - biased opinion

← Older revision		Revision as of 11:07, 27 January 2026
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	Because 3/2 is a very simple and concordant interval, it is still recognizable even when heavily tempered. Often it is tempered so that an octave-reduced stack of fourths or fifths approximates some other interval. Some examples:		Because 3/2 is a very simple and concordant interval, it is still recognizable even when heavily tempered. Often it is tempered so that an octave-reduced stack of fourths or fifths approximates some other interval. Some examples:

	[[Meantone]] temperament flattens the fifth from just (to around 695–700 cents) such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor ~~3rd~~ generated by stacking three fourths is closer to 6/5.		[[Meantone]] temperament flattens the fifth from just (to around 695–700 cents) such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor third generated by stacking three fourths is closer to 6/5.

	[[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor seventh 16/9 approximates 7/4 instead of 9/5.		[[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor seventh 16/9 approximates 7/4 instead of 9/5.

	* One may choose to prioritize the accurate tuning of either the thirds or the harmonic seventh, leading to a ~710{{c}} tuning when prioritizing the thirds, or a ~715{{c}} tuning when prioritizing 7/4.

	[[Schismic]] temperament adjusts the fifth such that the ''diminished fourth'' generated by stacking eight fourths approximates 5/4. As this is already a close approximation, the tuning of the fifth can be varied around its just tuning, but is most accurately flattened by a tiny amount. Thus a triad with 5/4 is written as {{nowrap\|{{dash\|C, F♭, G}}}} (unless the notation has accidentals for [[81/80]], e.g. {{nowrap\|{{dash\|C, vE, G}}}}).		[[Schismic]] temperament adjusts the fifth such that the ''diminished fourth'' generated by stacking eight fourths approximates 5/4. As this is already a close approximation, the tuning of the fifth can be varied around its just tuning, but is most accurately flattened by a tiny amount. Thus a triad with 5/4 is written as {{nowrap\|{{dash\|C, F♭, G}}}} (unless the notation has accidentals for [[81/80]], e.g. {{nowrap\|{{dash\|C, vE, G}}}}).

FloraC: Formatting and style

2026-01-27T11:04:48Z

Formatting and style

Show changes

Overthink: nvm

2025-12-22T23:22:18Z

nvm

← Older revision		Revision as of 23:22, 22 December 2025
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	__TOC__		__TOC__
	{{todo\|improve synopsis}}		{{todo\|improve synopsis}}
	~~{{Clear}}~~
	== Properties ==		== Properties ==
	For harmonic [[timbre\|timbres]], the loudest harmonics are usually the second and third ones (2/1 and 3/1). 3/2 is the interval between these two harmonics (which incidentally makes 3/2 [[superparticular]]). Thus 3/2 is easy to tune by ear, and it's easy to hear if it's mistuned.		For harmonic [[timbre\|timbres]], the loudest harmonics are usually the second and third ones (2/1 and 3/1). 3/2 is the interval between these two harmonics (which incidentally makes 3/2 [[superparticular]]). Thus 3/2 is easy to tune by ear, and it's easy to hear if it's mistuned.

Overthink: /* In regular temperament theory */better word

2025-12-20T09:21:01Z

In regular temperament theory: better word

← Older revision		Revision as of 09:21, 20 December 2025
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	* One may choose to prioritize the accurate tuning of either the thirds or the harmonic seventh, leading to a ~710c tuning when prioritizing the thirds, or a ~715c tuning when prioritizing 7/4.		* One may choose to prioritize the accurate tuning of either the thirds or the harmonic seventh, leading to a ~710c tuning when prioritizing the thirds, or a ~715c tuning when prioritizing 7/4.

	[[Schismatic\|Schismic]] temperament adjusts the fifth such that the ''diminished fourth'' generated by stacking eight fourths approximates 5/4. As this is already a close approximation, the tuning of the fifth can be varied around its just tuning, but is most ~~simply~~ flattened by a tiny amount. Thus a triad with 5/4 is written as {{nowrap\|{{dash\|C, F♭, G}}}} (unless the notation has accidentals for [[81/80]], e.g. {{nowrap\|{{dash\|C, vE, G}}}}).		[[Schismatic\|Schismic]] temperament adjusts the fifth such that the ''diminished fourth'' generated by stacking eight fourths approximates 5/4. As this is already a close approximation, the tuning of the fifth can be varied around its just tuning, but is most accurately flattened by a tiny amount. Thus a triad with 5/4 is written as {{nowrap\|{{dash\|C, F♭, G}}}} (unless the notation has accidentals for [[81/80]], e.g. {{nowrap\|{{dash\|C, vE, G}}}}).

	* Garibaldi temperament is an extension of schismatic that sharpens the fifth so that the small interval between the major third and diminished fourth can also be used to create simple 7-limit intervals.		* Garibaldi temperament is an extension of schismatic that sharpens the fifth so that the small interval between the major third and diminished fourth can also be used to create simple 7-limit intervals.

Overthink: /* In regular temperament theory */better italics

2025-12-20T09:19:17Z

In regular temperament theory: better italics

← Older revision		Revision as of 09:19, 20 December 2025
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	* One may choose to prioritize the accurate tuning of either the thirds or the harmonic seventh, leading to a ~710c tuning when prioritizing the thirds, or a ~715c tuning when prioritizing 7/4.		* One may choose to prioritize the accurate tuning of either the thirds or the harmonic seventh, leading to a ~710c tuning when prioritizing the thirds, or a ~715c tuning when prioritizing 7/4.

	[[Schismatic\|Schismic]] temperament adjusts the fifth such that the ''diminished'' ~~fourth~~ generated by stacking eight fourths approximates 5/4. As this is already a close approximation, the tuning of the fifth can be varied around its just tuning, but is most simply flattened by a tiny amount. Thus a triad with 5/4 is written as {{nowrap\|{{dash\|C, F♭, G}}}} (unless the notation has accidentals for [[81/80]], e.g. {{nowrap\|{{dash\|C, vE, G}}}}).		[[Schismatic\|Schismic]] temperament adjusts the fifth such that the ''diminished fourth'' generated by stacking eight fourths approximates 5/4. As this is already a close approximation, the tuning of the fifth can be varied around its just tuning, but is most simply flattened by a tiny amount. Thus a triad with 5/4 is written as {{nowrap\|{{dash\|C, F♭, G}}}} (unless the notation has accidentals for [[81/80]], e.g. {{nowrap\|{{dash\|C, vE, G}}}}).

	* Garibaldi temperament is an extension of schismatic that sharpens the fifth so that the small interval between the major third and diminished fourth can also be used to create simple 7-limit intervals.		* Garibaldi temperament is an extension of schismatic that sharpens the fifth so that the small interval between the major third and diminished fourth can also be used to create simple 7-limit intervals.

Overthink: /* In regular temperament theory */use en dash, expand range

2025-12-20T09:17:54Z

In regular temperament theory: use en dash, expand range

← Older revision		Revision as of 09:17, 20 December 2025
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	Because 3/2 is a very simple and concordant interval, it is still recognizable even when heavily tempered. Often it is tempered so that an octave-reduced stack of fourths or fifths approximates some other interval. Some examples:		Because 3/2 is a very simple and concordant interval, it is still recognizable even when heavily tempered. Often it is tempered so that an octave-reduced stack of fourths or fifths approximates some other interval. Some examples:

	[[Meantone]] temperament flattens the fifth from just (to around ~~695-698~~ cents) such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor 3rd generated by stacking three fourths is closer to 6/5.		[[Meantone]] temperament flattens the fifth from just (to around 695–700 cents) such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor 3rd generated by stacking three fourths is closer to 6/5.

	[[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor 7th 16/9 approximates 7/4 instead of 9/5.		[[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor 7th 16/9 approximates 7/4 instead of 9/5.

Overthink: rewrite synopsis a bit

2025-12-15T03:21:52Z

rewrite synopsis a bit

← Older revision		Revision as of 03:21, 15 December 2025
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	{{Wikipedia\|Perfect fifth}}		{{Wikipedia\|Perfect fifth}}

	'''3/2''', the '''just perfect fifth''', is a very [[consonance\|consonant]] interval, due to the numerator and denominator of its ratio being very small numbers~~. Only~~ the [[2/1\|octave]] and the [[3/1\|tritave]] ~~have~~ smaller numbers.		'''3/2''', the '''just perfect fifth''', is a very [[consonance\|consonant]] interval, due to the numerator and denominator of its ratio being very small numbers, with only the [[2/1\|octave]] and the [[3/1\|tritave]] having smaller numbers. As such, it is very important in western music and many musical traditions, and approximating it is key in systems like [[12edo]] and other [[EDO]]s.
			__TOC__
			{{todo\|improve synopsis}}
			{{Clear}}
	== Properties ==		== Properties ==
	For harmonic [[timbre\|timbres]], the loudest harmonics are usually the second and third ones (2/1 and 3/1). 3/2 is the interval between these two harmonics (which incidentally makes 3/2 [[superparticular]]). Thus 3/2 is easy to tune by ear, and it's easy to hear if it's mistuned.		For harmonic [[timbre\|timbres]], the loudest harmonics are usually the second and third ones (2/1 and 3/1). 3/2 is the interval between these two harmonics (which incidentally makes 3/2 [[superparticular]]). Thus 3/2 is easy to tune by ear, and it's easy to hear if it's mistuned.
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	Variations of the perfect fifth (whether [[just]] or tempered) appear in most [[Approaches to musical tuning\|music of the world]]. [[Historical temperaments\|Historically]], European music treated the perfect fifth as consonant long before it treated the major third—specifically [[5/4]]—as consonant. In the present day, the dominant tuning [[12edo]] approximates 3/2 very accurately.		Variations of the perfect fifth (whether [[just]] or tempered) appear in most [[Approaches to musical tuning\|music of the world]]. [[Historical temperaments\|Historically]], European music treated the perfect fifth as consonant long before it treated the major third—specifically [[5/4]]—as consonant. In the present day, the dominant tuning [[12edo]] approximates 3/2 very accurately.

	A [[Chain of fifths\|chain of just perfect fifths]] generates [[Pythagorean tuning]]. The chain continues indefinitely and theoretically never returns to the starting note. A chain that ends at seven notes generates the historically important [[Wikipedia:Diatonic scale #Iteration of the fifth\|Pythagorean diatonic scale]]. This scale is also the 7 natural notes of all "pyth-spine" notations, in which all uninflected notes are ~~pythogorean~~, such as [[HEJI]], [[Sagittal notation\|Sagittal]], [[Ups and downs notation\|ups and downs]], [[FJS]] and [[color notation]].		A [[Chain of fifths\|chain of just perfect fifths]] generates [[Pythagorean tuning]]. The chain continues indefinitely and theoretically never returns to the starting note. A chain that ends at seven notes generates the historically important [[Wikipedia:Diatonic scale #Iteration of the fifth\|Pythagorean diatonic scale]]. This scale is also the 7 natural notes of all "pyth-spine" notations, in which all uninflected notes are Pythagorean, such as [[HEJI]], [[Sagittal notation\|Sagittal]], [[Ups and downs notation\|ups and downs]], [[FJS]] and [[color notation]].

	Music using unusual intervals can be very disorienting. The presence of perfect fifths can provide a "ground" that make it less so. Some composers deliberately use tunings that lack fifths, to make their music sound more [[xenharmonic]].		Music using unusual intervals can be very disorienting. The presence of perfect fifths can provide a "ground" that make it less so. Some composers deliberately use tunings that lack fifths, to make their music sound more [[xenharmonic]].
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	Because 3/2 is a very simple and concordant interval, it is still recognizable even when heavily tempered. Often it is tempered so that an octave-reduced stack of fourths or fifths approximates some other interval. Some examples:		Because 3/2 is a very simple and concordant interval, it is still recognizable even when heavily tempered. Often it is tempered so that an octave-reduced stack of fourths or fifths approximates some other interval. Some examples:

	[[Meantone]] temperament flattens the fifth from just (to around 695 cents) such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor 3rd generated by stacking three fourths is closer to 6/5.		[[Meantone]] temperament flattens the fifth from just (to around 695-698 cents) such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor 3rd generated by stacking three fourths is closer to 6/5.

	[[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor 7th 16/9 approximates 7/4 instead of 9/5.		[[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor 7th 16/9 approximates 7/4 instead of 9/5.

Overthink: /* As a dyad */ + Template:Clear

2025-12-15T03:10:43Z

As a dyad: + Template:Clear

← Older revision		Revision as of 03:10, 15 December 2025
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	\|2:3:4 melodically inverted		\|2:3:4 melodically inverted
	\|}		\|}
			{{Clear}}

	== See also ==		== See also ==

TallKite at 09:16, 24 November 2025

2025-11-24T09:16:52Z

← Older revision		Revision as of 09:16, 24 November 2025
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	\|+		\|+
	!Voices		!Voices
	!EFR		![[EFR]]
	![[Kite's thoughts on hi-lo notation\|Hi-lo name]]		![[Kite's thoughts on hi-lo notation\|Hi-lo name]]
	!Special properties		!Special properties