Chain of fifths - Revision history

Lériendil at 14:19, 23 February 2025

2025-02-23T14:19:22Z

← Older revision		Revision as of 14:19, 23 February 2025
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	The '''chain of fifths''' is a tool to show and measure relationships between chords or key signatures, applicable to all [[tuning system]]s generated by an [[octave]] and a [[fifth]]. The concept dates back to at least the 13th century<ref>Schulter, Margo “[https://web.archive.org/web/20120215000445/http://www.medieval.org:80/emfaq/harmony/pyth4.html Pythagorean Tuning and Medieval Polyphony]"</ref>, and was applied in [[meantone]] (including [[12edo]]), [[3-limit\|Pythagorean tuning]], and [[well temperament]]s, to help analysing chord progressions and modulations.		The '''chain of fifths''' is a tool to show and measure relationships between chords or key signatures, applicable to all [[tuning system]]s generated by an [[octave]] and a [[fifth]]. The concept dates back to at least the 13th century<ref>Schulter, Margo “[https://web.archive.org/web/20120215000445/http://www.medieval.org:80/emfaq/harmony/pyth4.html Pythagorean Tuning and Medieval Polyphony]"</ref>, and was applied in [[meantone]] (including [[12edo]]), [[3-limit\|Pythagorean tuning]], and [[well temperament]]s, to help analysing chord progressions and modulations.

	The chain of fifths starts from a [[1/1\|unison]] and then [[stacking]] fifths both downward and upward, [[octave reduction\|reducing]] the stack along the way to fit within the octave. If the fifth is Pythagorean ([[3/2]]), the most central notes of this chain form [[32/27]] - [[16/9]] - [[4/3]] - '''1/1''' - 3/2 - [[9/8]] - [[27/16]]. This chain, if extended further, is the basis of [[chain-of-fifths notation]]s; if the unison here is D, the chain proceeds in ascending order as ... D𝄫 - A𝄫 - E𝄫 - B𝄫 - F♭ - C♭ - G♭ - D♭ - A♭ - E♭ - B♭ - F - C - G - '''D''' - A - E - B - F♯ - C♯ - G♯ - D♯ - A♯ - E♯ - B♯ - F𝄪 - C𝄪 - G𝄪 - D𝄪 ....		The chain of fifths starts from a [[1/1\|unison]] and then [[stacking]] fifths both downward and upward, [[octave reduction\|reducing]] the stack along the way to fit within the octave. If the fifth is Pythagorean ([[3/2]]), the most central notes of this chain form [[32/27]] - [[16/9]] - [[4/3]] - '''1/1''' - 3/2 - [[9/8]] - [[27/16]]. This chain, if extended further, is the basis of [[chain-of-fifths notation]]s; if the unison here is D, the chain proceeds in ascending order as:
			* ... D𝄫 - A𝄫 - E𝄫 - B𝄫 - F♭ - C♭ - G♭ - D♭ - A♭ - E♭ - B♭ - F - C - G - '''D''' - A - E - B - F♯ - C♯ - G♯ - D♯ - A♯ - E♯ - B♯ - F𝄪 - C𝄪 - G𝄪 - D𝄪 ....

	For [[edo]]s in particular, this becomes a '''circle of fifths'''. If the fifth is a number of steps that is co-prime to the edo number itself, all intervals will be visited when traversing the edo by fifth-steps. See for example the intervals in [[7edo]]: (0, 4, 1, 5, 2, 6, 3)\7. Other edos have more than one circle of fifths, [[10edo]] for example has two of them: (0, 6, 2, 8, 4)\10 and (1, 7, 3, 9, 5)\10. [[15edo]] has three distinct circles of fifths: (0, 9, 3, 12, 6)\15, (1, 10, 4, 13, 7)\15, and (2, 11, 5, 14, 8)\15.		For [[edo]]s in particular, this becomes a '''circle of fifths'''. If the fifth is a number of steps that is co-prime to the edo number itself, all intervals will be visited when traversing the edo by fifth-steps. See for example the intervals in [[7edo]]: (0, 4, 1, 5, 2, 6, 3)\7. Other edos have more than one circle of fifths, [[10edo]] for example has two of them: (0, 6, 2, 8, 4)\10 and (1, 7, 3, 9, 5)\10. [[15edo]] has three distinct circles of fifths: (0, 9, 3, 12, 6)\15, (1, 10, 4, 13, 7)\15, and (2, 11, 5, 14, 8)\15.

Lériendil: added much-needed mention of the chain in notation

2025-02-23T14:18:50Z

added much-needed mention of the chain in notation

← Older revision		Revision as of 14:18, 23 February 2025
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	{{Wikipedia\|Circle of fifths}}		{{Wikipedia\|Circle of fifths}}
	The '''chain of fifths''' is a tool to show and measure relationships between chords or key signatures, applicable to all [[tuning system]]s generated by an octave and a fifth. The concept dates back to at least the 13th century<ref>Schulter, Margo “[https://web.archive.org/web/20120215000445/http://www.medieval.org:80/emfaq/harmony/pyth4.html Pythagorean Tuning and Medieval Polyphony]"</ref>, and was applied in [[meantone]] (including [[12edo]]), [[3-limit\|Pythagorean tuning]], and [[well temperament]]s, to help analysing chord progressions and modulations.		The '''chain of fifths''' is a tool to show and measure relationships between chords or key signatures, applicable to all [[tuning system]]s generated by an [[octave]] and a [[fifth]]. The concept dates back to at least the 13th century<ref>Schulter, Margo “[https://web.archive.org/web/20120215000445/http://www.medieval.org:80/emfaq/harmony/pyth4.html Pythagorean Tuning and Medieval Polyphony]"</ref>, and was applied in [[meantone]] (including [[12edo]]), [[3-limit\|Pythagorean tuning]], and [[well temperament]]s, to help analysing chord progressions and modulations.

			The chain of fifths starts from a [[1/1\|unison]] and then [[stacking]] fifths both downward and upward, [[octave reduction\|reducing]] the stack along the way to fit within the octave. If the fifth is Pythagorean ([[3/2]]), the most central notes of this chain form [[32/27]] - [[16/9]] - [[4/3]] - '''1/1''' - 3/2 - [[9/8]] - [[27/16]]. This chain, if extended further, is the basis of [[chain-of-fifths notation]]s; if the unison here is D, the chain proceeds in ascending order as ... D𝄫 - A𝄫 - E𝄫 - B𝄫 - F♭ - C♭ - G♭ - D♭ - A♭ - E♭ - B♭ - F - C - G - '''D''' - A - E - B - F♯ - C♯ - G♯ - D♯ - A♯ - E♯ - B♯ - F𝄪 - C𝄪 - G𝄪 - D𝄪 ....

	For [[edo]]s in particular, this becomes a '''circle of fifths'''. If the fifth is a number of steps that is co-prime to the edo number itself, all intervals will be visited when traversing the edo by fifth-steps. See for example the intervals in [[7edo]]: (0, 4, 1, 5, 2, 6, 3)\7. Other edos have more than one circle of fifths, [[10edo]] for example has two of them: (0, 6, 2, 8, 4)\10 and (1, 7, 3, 9, 5)\10. [[15edo]] has three distinct circles of fifths: (0, 9, 3, 12, 6)\15, (1, 10, 4, 13, 7)\15, and (2, 11, 5, 14, 8)\15.		For [[edo]]s in particular, this becomes a '''circle of fifths'''. If the fifth is a number of steps that is co-prime to the edo number itself, all intervals will be visited when traversing the edo by fifth-steps. See for example the intervals in [[7edo]]: (0, 4, 1, 5, 2, 6, 3)\7. Other edos have more than one circle of fifths, [[10edo]] for example has two of them: (0, 6, 2, 8, 4)\10 and (1, 7, 3, 9, 5)\10. [[15edo]] has three distinct circles of fifths: (0, 9, 3, 12, 6)\15, (1, 10, 4, 13, 7)\15, and (2, 11, 5, 14, 8)\15.

ArrowHead294 at 11:06, 21 June 2024

2024-06-21T11:06:49Z

← Older revision		Revision as of 11:06, 21 June 2024
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	{{Wikipedia\|Circle of fifths}}		{{Wikipedia\|Circle of fifths}}
	The '''chain of fifths''' is a tool to show and measure relationships between chords or key signatures, applicable to all [[tuning system]]s generated by an octave and a fifth. The concept dates back to at least the 13th century<ref>Schulter, Margo “[https://web.archive.org/web/20120215000445/http://www.medieval.org:80/emfaq/harmony/pyth4.html Pythagorean Tuning and Medieval Polyphony]"</ref>, and was applied in [[meantone]], [[~~well temperament~~]]s, and [[~~12edo~~]] to help analysing chord progressions and modulations.		The '''chain of fifths''' is a tool to show and measure relationships between chords or key signatures, applicable to all [[tuning system]]s generated by an octave and a fifth. The concept dates back to at least the 13th century<ref>Schulter, Margo “[https://web.archive.org/web/20120215000445/http://www.medieval.org:80/emfaq/harmony/pyth4.html Pythagorean Tuning and Medieval Polyphony]"</ref>, and was applied in [[meantone]] (including [[12edo]]), [[3-limit\|Pythagorean tuning]], and [[well temperament]]s, to help analysing chord progressions and modulations.

	For [[edo]]s in particular, this becomes a '''circle of fifths'''. If the fifth is a number of steps that is co-prime to the edo number itself, all intervals will be visited when traversing the edo by fifth-steps. See for example the intervals in [[7edo]]: (0, 4, 1, 5, 2, 6, 3)\7. Other edos have more than one circle of fifths, [[10edo]] for example has two of them: (0, 6, 2, 8, 4)\10 and (1, 7, 3, 9, 5)\10. [[15edo]] has three distinct circles of fifths: (0, 9, 3, 12, 6)\15, (1, 10, 4, 13, 7)\15, and (2, 11, 5, 14, 8)\15.		For [[edo]]s in particular, this becomes a '''circle of fifths'''. If the fifth is a number of steps that is co-prime to the edo number itself, all intervals will be visited when traversing the edo by fifth-steps. See for example the intervals in [[7edo]]: (0, 4, 1, 5, 2, 6, 3)\7. Other edos have more than one circle of fifths, [[10edo]] for example has two of them: (0, 6, 2, 8, 4)\10 and (1, 7, 3, 9, 5)\10. [[15edo]] has three distinct circles of fifths: (0, 9, 3, 12, 6)\15, (1, 10, 4, 13, 7)\15, and (2, 11, 5, 14, 8)\15.

ArrowHead294 at 12:40, 25 April 2024

2024-04-25T12:40:32Z

← Older revision		Revision as of 12:40, 25 April 2024
Line 1:		Line 1:
	{{Wikipedia\|Circle of fifths}}		{{Wikipedia\|Circle of fifths}}
	The '''chain of fifths''' is a tool to show and measure relationships between chords or key signatures, applicable to all [[tuning system]]s generated by an octave and a fifth. The concept dates back to at least the 13th century<ref>Schulter, Margo “[https://web.archive.org/web/20120215000445/http://www.medieval.org:80/emfaq/harmony/pyth4.html Pythagorean Tuning and Medieval Polyphony]"</ref>, and was applied in [[meantone]], [[well temperament]]s and [[12edo]] to help analysing chord progressions and modulations.		The '''chain of fifths''' is a tool to show and measure relationships between chords or key signatures, applicable to all [[tuning system]]s generated by an octave and a fifth. The concept dates back to at least the 13th century<ref>Schulter, Margo “[https://web.archive.org/web/20120215000445/http://www.medieval.org:80/emfaq/harmony/pyth4.html Pythagorean Tuning and Medieval Polyphony]"</ref>, and was applied in [[meantone]], [[well temperament]]s, and [[12edo]] to help analysing chord progressions and modulations.

	For [[edo]]s in particular, this becomes a '''circle of fifths'''. If the fifth is a number of steps that is co-prime to the edo number itself, all intervals will be visited when traversing the edo by fifth-steps. See for example the intervals in [[7edo]]: (0, 4, 1, 5, 2, 6, 3)\7. Other edos have more than one circle of fifths, [[10edo]] for example has two of them: (0, 6, 2, 8, 4)\10 and (1, 7, 3, 9, 5)\10. [[15edo]] has three distinct circles of fifths: (0, 9, 3, 12, 6)\15, (1, 10, 4, 13, 7)\15, and (2, 11, 5, 14, 8)\15.		For [[edo]]s in particular, this becomes a '''circle of fifths'''. If the fifth is a number of steps that is co-prime to the edo number itself, all intervals will be visited when traversing the edo by fifth-steps. See for example the intervals in [[7edo]]: (0, 4, 1, 5, 2, 6, 3)\7. Other edos have more than one circle of fifths, [[10edo]] for example has two of them: (0, 6, 2, 8, 4)\10 and (1, 7, 3, 9, 5)\10. [[15edo]] has three distinct circles of fifths: (0, 9, 3, 12, 6)\15, (1, 10, 4, 13, 7)\15, and (2, 11, 5, 14, 8)\15.

Fredg999: Remove deprecated link, references section

2024-03-11T21:54:08Z

Remove deprecated link, references section

← Older revision		Revision as of 21:54, 11 March 2024
Line 1:		Line 1:
	{{Wikipedia\|Circle of fifths}}		{{Wikipedia\|Circle of fifths}}
	The '''chain of fifths''' is a tool to show and measure relationships between chords or key signatures, applicable to all [[tuning system]]s generated by an octave and a fifth. The concept dates back to at least the 13th century <ref>Schulter, Margo “[https://web.archive.org/web/20120215000445/http://www.medieval.org:80/emfaq/harmony/pyth4.html Pythagorean Tuning and Medieval Polyphony]"</ref>, and was applied in [[meantone]], [[well temperament]]s and [[12edo]] to help analysing chord progressions and modulations.		The '''chain of fifths''' is a tool to show and measure relationships between chords or key signatures, applicable to all [[tuning system]]s generated by an octave and a fifth. The concept dates back to at least the 13th century<ref>Schulter, Margo “[https://web.archive.org/web/20120215000445/http://www.medieval.org:80/emfaq/harmony/pyth4.html Pythagorean Tuning and Medieval Polyphony]"</ref>, and was applied in [[meantone]], [[well temperament]]s and [[12edo]] to help analysing chord progressions and modulations.

	For [[edo]]s in particular, this becomes a '''circle of fifths'''. If the fifth is a number of steps that is co-prime to the edo number itself, all intervals will be visited when traversing the edo by fifth-steps. See for example the intervals in [[7edo]]: (0, 4, 1, 5, 2, 6, 3)\7. Other edos have more than one circle of fifths, [[10edo]] for example has two of them: (0, 6, 2, 8, 4)\10 and (1, 7, 3, 9, 5)\10. [[15edo]] has three distinct circles of fifths: (0, 9, 3, 12, 6)\15, (1, 10, 4, 13, 7)\15, and (2, 11, 5, 14, 8)\15.		For [[edo]]s in particular, this becomes a '''circle of fifths'''. If the fifth is a number of steps that is co-prime to the edo number itself, all intervals will be visited when traversing the edo by fifth-steps. See for example the intervals in [[7edo]]: (0, 4, 1, 5, 2, 6, 3)\7. Other edos have more than one circle of fifths, [[10edo]] for example has two of them: (0, 6, 2, 8, 4)\10 and (1, 7, 3, 9, 5)\10. [[15edo]] has three distinct circles of fifths: (0, 9, 3, 12, 6)\15, (1, 10, 4, 13, 7)\15, and (2, 11, 5, 14, 8)\15.
Line 10:		Line 10:
	* [[Antipodes]]		* [[Antipodes]]
	* [[Pythagorean tuning]]		* [[Pythagorean tuning]]
	* [https:/~~/en.xen.wiki/w/Generalized_Pythagorean_tuning#Chain_of_fifths_.2F_fourths Generalized Pythagorean Tuning]~~
			== References ==
			<references/>

	[[Category:Pitch space]]		[[Category:Pitch space]]

Fastaro: /* See also */ added pythagorean tuning

2024-01-07T04:32:28Z

Fastaro: /* See also */ I added a see also section for generalized pythagorean tuning linking to the chain of fifths and fourths in set notation

2024-01-07T02:09:48Z

See also: I added a see also section for generalized pythagorean tuning linking to the chain of fifths and fourths in set notation

← Older revision		Revision as of 02:09, 7 January 2024
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	* [[Ring number]]		* [[Ring number]]
	* [[Antipodes]]		* [[Antipodes]]
			* [https://en.xen.wiki/w/Generalized_Pythagorean_tuning#Chain_of_fifths_.2F_fourths Generalized Pythagorean Tuning]

	[[Category:Pitch space]]		[[Category:Pitch space]]

Fastaro: I added a source to the earliest written account of the chain of fifths

2024-01-04T21:08:31Z

I added a source to the earliest written account of the chain of fifths

← Older revision		Revision as of 21:08, 4 January 2024
Line 1:		Line 1:
	{{Wikipedia\|Circle of fifths}}		{{Wikipedia\|Circle of fifths}}
	The '''chain of fifths''' is a tool to show and measure relationships between chords or key signatures, applicable to all [[tuning system]]s generated by an octave and a fifth. The concept dates back to ~~Baroque era~~, and was applied in [[meantone]], [[well temperament]]s and [[12edo]] to help analysing chord progressions and modulations.		The '''chain of fifths''' is a tool to show and measure relationships between chords or key signatures, applicable to all [[tuning system]]s generated by an octave and a fifth. The concept dates back to at least the 13th century <ref>Schulter, Margo “[https://web.archive.org/web/20120215000445/http://www.medieval.org:80/emfaq/harmony/pyth4.html Pythagorean Tuning and Medieval Polyphony]"</ref>, and was applied in [[meantone]], [[well temperament]]s and [[12edo]] to help analysing chord progressions and modulations.

	For [[edo]]s in particular, this becomes a '''circle of fifths'''. If the fifth is a number of steps that is co-prime to the edo number itself, all intervals will be visited when traversing the edo by fifth-steps. See for example the intervals in [[7edo]]: (0, 4, 1, 5, 2, 6, 3)\7. Other edos have more than one circle of fifths, [[10edo]] for example has two of them: (0, 6, 2, 8, 4)\10 and (1, 7, 3, 9, 5)\10. [[15edo]] has three distinct circles of fifths: (0, 9, 3, 12, 6)\15, (1, 10, 4, 13, 7)\15, and (2, 11, 5, 14, 8)\15.		For [[edo]]s in particular, this becomes a '''circle of fifths'''. If the fifth is a number of steps that is co-prime to the edo number itself, all intervals will be visited when traversing the edo by fifth-steps. See for example the intervals in [[7edo]]: (0, 4, 1, 5, 2, 6, 3)\7. Other edos have more than one circle of fifths, [[10edo]] for example has two of them: (0, 6, 2, 8, 4)\10 and (1, 7, 3, 9, 5)\10. [[15edo]] has three distinct circles of fifths: (0, 9, 3, 12, 6)\15, (1, 10, 4, 13, 7)\15, and (2, 11, 5, 14, 8)\15.

FloraC: FloraC moved page Circle of fifths to Chain of fifths over redirect: Circle -> chain since since chains are the more fundamental idea and circles are special cases

2023-12-18T10:57:43Z

FloraC moved page Circle of fifths to Chain of fifths over redirect: Circle -> chain since since chains are the more fundamental idea and circles are special cases

← Older revision	Revision as of 10:57, 18 December 2023
(No difference)

FloraC: Rework: circle -> chain since chains are the more fundamental idea and circles are special cases

2023-12-18T10:57:14Z

Rework: circle -> chain since chains are the more fundamental idea and circles are special cases

← Older revision		Revision as of 10:57, 18 December 2023
Line 1:		Line 1:
	{{Wikipedia\|Circle of fifths}}		{{Wikipedia\|Circle of fifths}}
	The '''~~circle~~ of fifths''' is a tool to show and measure relationships between chords or key signatures. The concept dates back to ~~Baroqe~~ era, and was applied in [[meantone]], [[well temperament]]s and [[12edo]] to help analysing chord progressions and modulations. ~~In fact, it is applicable to all tuning systems generated by an octave and a fifth.~~ For ~~edos~~ in particular, this ~~means those which map~~ the fifth to a number of steps that is co-prime to the edo number itself~~; so~~ when traversing ~~these edos~~ by fifth-steps~~, all intervals will be visited~~. See for example the intervals in [[7edo]]: (0, 4, 1, 5, 2, 6, 3)\7. Other edos have more than one circle of fifths, [[10edo]] for example has two of them: (0, 6, 2, 8, 4)\10 and (1, 7, 3, 9, 5)\10. [[15edo]] has three distinct circles of fifths: (0, 9, 3, 12, 6)\15, (1, 10, 4, 13, 7)\15, and (2, 11, 5, 14, 8)\15.		The '''chain of fifths''' is a tool to show and measure relationships between chords or key signatures, applicable to all [[tuning system]]s generated by an octave and a fifth. The concept dates back to Baroque era, and was applied in [[meantone]], [[well temperament]]s and [[12edo]] to help analysing chord progressions and modulations.

			For [[edo]]s in particular, this becomes a '''circle of fifths'''. If the fifth is a number of steps that is co-prime to the edo number itself, all intervals will be visited when traversing the edo by fifth-steps. See for example the intervals in [[7edo]]: (0, 4, 1, 5, 2, 6, 3)\7. Other edos have more than one circle of fifths, [[10edo]] for example has two of them: (0, 6, 2, 8, 4)\10 and (1, 7, 3, 9, 5)\10. [[15edo]] has three distinct circles of fifths: (0, 9, 3, 12, 6)\15, (1, 10, 4, 13, 7)\15, and (2, 11, 5, 14, 8)\15.

	== See also ==		== See also ==
	* [[~~Circle~~-of-fifths notation]]		* [[Chain-of-fifths notation]]
	* [[Fifthspan]]		* [[Fifthspan]]
			* [[Ring number]]
	* [[Antipodes]]		* [[Antipodes]]

← Older revision		Revision as of 04:32, 7 January 2024
Line 9:		Line 9:
	* [[Ring number]]		* [[Ring number]]
	* [[Antipodes]]		* [[Antipodes]]
			* [[Pythagorean tuning]]
	* [https://en.xen.wiki/w/Generalized_Pythagorean_tuning#Chain_of_fifths_.2F_fourths Generalized Pythagorean Tuning]		* [https://en.xen.wiki/w/Generalized_Pythagorean_tuning#Chain_of_fifths_.2F_fourths Generalized Pythagorean Tuning]