Chain of fifths

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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 systems generated by an octave and a fifth. The concept dates back to at least the 13th century[1], and was applied in meantone (including 12edo), Pythagorean tuning, and well temperaments, to help analysing chord progressions and modulations.

For edos 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