I have mathematically figured out how to map the chromatic scale onto the circle of fifths for even temperaments. The equation is to multiply each interval by (n/2)+1 where n is the n TET.
For example, for 12 TET, we apply the function (n/2)+1 and get 7 ((12/2)+1=7), from there I multiply each interval by 7:
0 semitones * 7 = 0 semitones (eg: C is mapped to C);
1 semitone * 7 = 7 semitones (eg: C# is mapped to G);
2 semitone * 7 = 14 semitones (eg: D is mapped to D);
3 semitone * 7 = 21 semitones (eg: D# is mapped to A)
This will give me 12 TET circle of fifths.

However, I've noticed this equation only works for even temperaments because for odd temperaments, we have to multiply by a number with a half, giving me quartertones.
Are there no odd temperament circle of fifths equivalent?
Any guidance would be appreciated

  • As stated, this isn't possible to answer because you haven't defined what you mean by a "circle of fifths equivalent". Sep 5 '21 at 2:26

You can create a circle of any number of steps mutually prime to the total number of pitches in the octave.

  • However I'm looking for a circle with the same amount of steps as pitches within the octave, with circle of fifths capabilities; for example: a representation of key signatures. Therefore the number of steps round the circle cannot be coprime to the total number of pitches on the octave.
    – Mixtli
    Sep 4 '21 at 17:15
  • 1
    @Mixtli I'm not sure I understand. Any coprime number of steps will touch every pitch in the octave.
    – Aaron
    Sep 4 '21 at 17:18
  • 3
    @Mixtli That's because everything is mutually prime to 13. The "circle of fifths" is a special property of 12n-TET because of the harmonic (i.e., frequency ratio) relationship between the fourth/fifth and octave. for other ETs, you would have to create your own rules governing harmony.
    – Aaron
    Sep 4 '21 at 17:31
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    @Mixtli 12 is very divisible, so you cannot have a circle of whole steps, minor thirds, major thirds, tritones, minor sixths, major sixths, or minor sevenths, because none of these interval sizes (2, 3, 4, 6, 8, 9, 10) is coprime to 12. You can have a circle of fourths, of fifths, and of major sevenths (a "circle" of minor seconds is perhaps not a circle, depending on your definition; it seems like more of a straight line). I also note that the rule articulated here is independent of the equality of the temperament; it only depends on the number of divisions of the octave.
    – phoog
    Sep 4 '21 at 20:13
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    That is, the circle of fifths exists in the 12-tone system regardless of the temperament. Similarly, there is a circle of "fifths" in 22-tone systems because the fifth is 13 divisions in size, and 13 is coprime with 22. By contrast, the perfect fifth in 34-tone systems is 20 divisions wide, so the circle of fifths in that system doesn't reach every pitch class, because 34 and 20 share the common prime factor of 2.
    – phoog
    Sep 4 '21 at 20:27

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