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 theIn such a physically natural temperament, directly related to the mathematical theory ofPythagorean Comma[Wikipedia]. One can begin the sequence starting with any frequency.

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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Patterns, Transformations and Groups in Musical Composition

Evolving Dodecaphony

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Pitch Sets in Composition

Email me, Bill Hammel at

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The URL for this document is:

http://graham.main.nc.us/~bhammel/MUSIC/5ths.html

Created: September 1997

Last Updated: May 28, 2000

Last Updated: March 1, 2011

Last Updated: May 20, 2011