Chain 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[1], and was applied in meantone (including 12edo), Pythagorean tuning, and well temperaments, to help analysing chord progressions and modulations.
The chain of fifths starts from a unison and then stacking fifths both downward and upward, 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 notations; 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 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
References
- ↑ Schulter, Margo “Pythagorean Tuning and Medieval Polyphony"