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The interested reader is referred to "Mathematics and Music" by David Wright, published by the American Mathematical Society. The realization that the circle of

by ngneer 10y ago

The interested reader is referred to "Mathematics and Music" by David Wright, published by the American Mathematical Society. The realization that the circle of fifths is really there because 5 and 7 (aka -5) are co-prime to 12 was worth the entire thing! Together with 1 and 11 (aka -1) they generate the group Z_{n} for n = 12...

ktRolster 10y ago
That explains why the circle of fifths is a circle that goes through all keys, but the really interesting thing about the circle of fifths is that a fifth sounds relatively close to the key next to it, and it doesn't really explain that, I guess.
- djaychela 10y ago
  Any key a fifth away only changes one note - if you go from C Major (CDEFGABC) to G Major (GABCDEF#G) then only the F# has changed, and as a result there are chords which are common between the two keys - any chord without an F in it in C will be common to G as well. Works the other way if you go 'down' a fifth (to F), but has a flat instead. (Bb)
  - ngneer 10y ago
    More specifically, going up a fifth augments the fourth by a semitone to become the seventh and going down a fifth diminishes the seventh by a semitone to become the fourth. In your example, F the fourth became F# the seventh and B the seventh became Bb the fourth. A fifth away _IS_ a semitone away. Defining the following eight functions: SEMITONE = +1 mod 12; ANTI_SEMITONE = SEVENTH; FOURTH = +5 mod 12; ANTI_FOURTH = FIFTH; FIFTH = +7 mod 12; ANTI_FIFTH = FOURTH; SEVENTH = +11 mod 12; ANTI_SEVENTH = SEMITONE; We ask ourselves why the following holds: FOURTH(X) = ANTI_SEMITONE(SEVENTH(FIFTH(X))) SEVENTH(X) = SEMITONE(FOURTH(ANTI_FIFTH(X))) Substituting, the answer is clear: X + 5 = X + 7 + 11 - 1 X + 11 = X - 7 + 5 + 1 Keys one fifth apart differ by one note because: 1 = 11 + 7 - 5 mod 12 Algebraically it seems to work out...

by ngneer 10y ago

ktRolster 10y ago

That explains why the circle of fifths is a circle that goes through all keys, but the really interesting thing about the circle of fifths is that a fifth sounds relatively close to the key next to it, and it doesn't really explain that, I guess.

djaychela 10y ago
Any key a fifth away only changes one note - if you go from C Major (CDEFGABC) to G Major (GABCDEF#G) then only the F# has changed, and as a result there are chords which are common between the two keys - any chord without an F in it in C will be common to G as well. Works the other way if you go 'down' a fifth (to F), but has a flat instead. (Bb)
- ngneer 10y ago
  More specifically, going up a fifth augments the fourth by a semitone to become the seventh and going down a fifth diminishes the seventh by a semitone to become the fourth. In your example, F the fourth became F# the seventh and B the seventh became Bb the fourth. A fifth away _IS_ a semitone away. Defining the following eight functions: SEMITONE = +1 mod 12; ANTI_SEMITONE = SEVENTH; FOURTH = +5 mod 12; ANTI_FOURTH = FIFTH; FIFTH = +7 mod 12; ANTI_FIFTH = FOURTH; SEVENTH = +11 mod 12; ANTI_SEVENTH = SEMITONE; We ask ourselves why the following holds: FOURTH(X) = ANTI_SEMITONE(SEVENTH(FIFTH(X))) SEVENTH(X) = SEMITONE(FOURTH(ANTI_FIFTH(X))) Substituting, the answer is clear: X + 5 = X + 7 + 11 - 1 X + 11 = X - 7 + 5 + 1 Keys one fifth apart differ by one note because: 1 = 11 + 7 - 5 mod 12 Algebraically it seems to work out...