Chain of fifths: Difference between revisions

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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]], [[~~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.

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.

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 {{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.
+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.