Circle of fifths question

Discussion in 'Tab, Tips, Theory and Technique' started by monfoodoo, Jan 11, 2010.

  1. EJG1

    EJG1 Tele-Meister

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    Thanks Goluphi! I've tried unsuccessfully to come up with a way to remember this stuff. I think your way will work for me... I copied your post and printed it out.
     
  2. nvilletele

    nvilletele Friend of Leo's

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    I fixed the diagram, reversing those two relative minors. I used some basic photo editor I had on my computer (not photoshop - PhotoStudio by ArcSoft), for the very first time, and it seems to have come out ok . . . . Also enhanced the pic a bit in iPhoto first (before I figured out iPhoto wouldnt work to edit the pic to fix the errors).

    Photo editing is not something I do very often . . . .
     

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  3. Larry F

    Larry F Doctor of Teleocity Vendor Member

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    C C# D Eb E F F# G G# A Bb B = 0 1 2 3 4 5 6 7 8 9 t e (where t = 10 and e = 11)
    C B Bb A G# G F# F E Eb D C# = 0 e t 9 8 7 6 5 4 3 2 1
    C F Bb Eb G# C# F# B E A D G = 0 5 t 3 8 1 6 e 4 8 2 7
    C G D A E B F# C# G# Eb Bb F = 0 7 2 8 4 e 6 1 8 3 t 5

    The integers 0, 1, 2,..., 11 modulo 12 can be represented as a mathematical cyclic group of order 12, denoted Z12. The term "modulo 12" provides that any number 12 or greater can be expressed as n - 12. This is like clock arithmetic. The notes of the chromatic scale, irrespective of octave, are isomorphic to Z12.

    The ascending chromatic scale is created by the successive addition of 1 semitone, mod 12.
    The descending chromatic scale is created by the successive addition of 11 semitones, mod 12.
    The descending circle of fifths is created by the successive addition of 5 semitones, mod 12.
    The ascending circle of fifths is created by the successive addition of 7 semitones, mod 12.

    The group Z12 is generated by the successive addition of 1, mod 12.
    It is also generated by the successive addition of 11 mod, 12.
    And generated by the successive addition of 5, mod 12.
    And generated by the successive addition of 7, mod 12.

    No other interval successively added will generate the chromatic scale. For example:
    The successive addition of 2 = 0 2 4 6 8 = C D E F# G# Bb, the whole-tone scale.
    The successive addition of 3 = 0 3 6 9 = C Eb F# A, the diminished 7th chord.
    The successive addition of 4 = 0 4 8 = C E G#, the augmented triad.
    The successive addition of 6 = 0 6 = C F#, the tritone.
    The other intervals are inversions of the above.

    The group Z12 is generated by 1, 11, 5, 7. Since these integers generate Z12, they are said to form a group known as an automorphism group.

    I hope all of this might help open the door to the field of mathematical groups in music. Music theorists are finding different kinds of groups in many forms of musical structures and musical compositions. I believe that a comprehensive study of pitch in music must recognize the presence of groups.

    The guitarist Pat Martino first opened this door for me in an early 70s issue of Guitar Player. I do not follow his theories, since I have gone into mathematical group theory instead, but he seems to be pursuing his studies by using 2-d geometrical patterns. If anyone runs across these, consider that some might be geometrical representations of groups or subsets of groups.

    A group is a set of elements where a * b = c, where * denotes combination (like addition or multiplication) and a, b, c are members of the set. A group, then, is closed under composition.

    A group contains an identity element, e, where a + e = a, for any element a of the set.

    An element of a of the group has an inverse a-1 which is also a member of the set. Note that a * a-1 = e.

    There is one other axiom for groups, associativity, that I won't show here, since it doesn't seem to have any relevance for music.

    Finally, groups can be combined into larger groups. The study of subgroups is where the interest lies in mathematical group theory and its musical representations.

    Some of the math-oriented members might be interested in all of this. But non-math people might be equally interested. It takes a little time for a musician to learn group theory, but wow, do things emerge.
     
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