The circle of fifths,
C F G Bb D Eb A Ab E Db B Gb = F#Again, as far as a modern keyboard with well tempering goes:
C# = Db, D# = Eb, F# = Gb, G# = Ab, A# = Bb B = Cb, E = Fb, C = B#, F = E#,which are called enharmonic equivalences.
One can follow the circle of fifths either clockwise or counterclockwise. The more distant the tonic note of a tonality is from another on the circle of fifths, the more harmonically distant the tonalities are; thus F# Major (Gb Major) is the most harmonically distant from C Major.
Musically speaking, the notion of harmonic distance between tonailties is directly related to the amount of musical machinations one must perform in order to modulate between them, using the rules of "classical harmony". Traditional Harmony
The above is in the context of equal temperament.
If one uses perfect fifths, whose tones have frequencies in the ratio of 1:(3/2), the sequence of 12 fifths, from "middle C" is named:
C G D A E B F# C# G# D# A# E# B# and B# is *not* an integral multiple of C in frequency, bringing the B# down by 7 octave equivalences (dividing the frequency by two for each octave descent) gives a discepancy which is called the Pythagorean Comma [Wikipedia]. One can begin the sequence starting with any frequency.In such a physically natural temperament, directly related to the mathematical theory of Fourier Series [Wikipedia] the circle of fifths is clearly not closed and is not a circle.
One can also see by a little mathematical reasoning that thus tuned, the fifths, beginning on any frequency will never close.
The above explanation is a quick and dirty explanation of complicated facts of musical and mathematical life, and does not even begin to do justice to the details. For a rather extensive and properly mathematical exposition of the problems posed and compromises involved in the concept of temperament see Pianos and Continued Fractions. This is not for the dilletante, and seems to be as good as reading Continued Fractions [Wikipedia] [Helmholtz 1877] on the subject; in fact, Helmholtz may be even more accesible. Available also on the net is MTP: Algorithms for Mapping Diatonic Tunings and Temperaments
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