Chain of fifths

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Circle of fifths

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, 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

• Chain-of-fifths notation
• Fifthspan
• Ring number
• Antipodes
• Pythagorean tuning