fun with the circle of 5ths

7/23/2019 Fun With the Circle of 5ths

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Fun with the Circle Of 5ths & Identifying Chords

In the process of teaching myself music theory, I have become very

impressed with the power of the Circle of 5ths (here I am talking about

the standard circle in 12-equal temperament). I have cataloged someinteresting properties, and I am always on the lookout for more.

1. A line through the center identifies notes which are a tri-tone

apart, and the line slices the circle into 2 keys, which are named by

moving one step clockwise from the line (the end points are in

both keys). This also sheds light on how keys are related to each

other, by being close or far apart around the circle. If they are

close, then they share a lot of notes.


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2. There is no need to write on the diagram where the relative minor

keys are, or draw a separate diagram for minor keys, because they

can easily be found (more on this later):

3. The pattern of the chromatic scale (notes most closely related in

pitch) is obtained by drawing a line through the center, say d#to a,

and then looking counter-clockwise (descending through the

chain of 5ths) to find the note below d#, and clockwise

(ascendinga 5th) to the note above d#. The pattern of the diatonic

scale is obtained by slicing the circle to isolate the particular key

like before, and then hopping around the half circle by 2 steps

each time, never straying to the other side of the tri-tone line.


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4. Likewise, if you can only remember part of the circ le and you want

to construct the rest, you need only draw a line through the center

from the note that you remember, and you can locate its sharp (by

going clockwise one step) or its flat (by going counter-clockwise):

5. Chord Resolution

It is easy to remember what dominant 7th chord resolves to what

tonic chord (major or minor) by going counter-clockwise around

the circ le of 5ths.


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6. Symmetries

The circle and the positions of the dots are invariant under the

following operations:

o 12 clockwise rotations and 12 counter-clockwise rotations

(rotate c to g, c to d, etc...) But rotating clockwise by n/12 of a

full turn has the same result as rotating (12-n)/12

counterclockwise, so there are only 12 distinct rotations in

all, not 24.

o There are 12 lines through the center for reflections (axes of

symmetry). 6 are lines through pairs of notes (tri-tones) and 6

are through the mid points between notes.

o There is one reflection through the center point, but it is

equivalent to a rotation of 180 degrees, so we won't discuss

this one further.

o

7. Rotating

an interval line or a chord shape corresponds to transposing to a

different key. Here we rotate the C major triad shape 4 positions

clockwise to get E major:


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8. If we antic ipate a litt le the chord diagrams that occur later on this

page, it is even possible to determine what are the chords within a

given key. In the key of C major, we can fit in the shape for the F

major triad, we shift it once clockwise and get the C major and

then the G major. When we try to shift it again, the 3rd hits the tri -

tone line and has nowhere else to go but up to the note f giving us

D minornow. Continuing to shift, we get Am and Em. Then we hit

the tri-tone line again, and this time the chord shape is deformed

into B-diminished. The main idea is that you can determine the

chords by what chord shape will fit. If I would have realized this

earlier, maybe I wouldn't have had to make these chord charts !

9. Intervals

o If we regard c as the lowest note in the octave, we see the

right half of the circle is comprised of major intervals (capital

M) and the left side is composed of the complementary minor

intervals (baby m). Complementary means that the number of

semi-tones of a segment and its mirror image always adds to

12. And of course we can rotate this arrangement of arrows

to eminate from any other note on the circ le, which is what is

really important.


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o If we now let the left half circle be notes in the octave

below c and the right half be notes above, we get mirror

images of exclusively major intervals.

o We can play the same game by letting the right side of the

circ le be in the octave below c and the left side above and

get mirror images of minor intervals, but since it is late at

night I won't draw that.


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10. Reflections

Reflections are very cool because they always have the property

of turning major triads into minor ones and vice versa! First

observe the symmetry between C and Cm:

o We can reflect through any symmetry line; the rule is

that the 5th of the major triad is interchanged with the root of

the minor triad . Thus we can easily find the name of any of

these reflected chords.

o This implies an equal importance of major and minor triads.

See Theory of Chordspage.

o Not only is the root triad reflected, but the entire key is

reflected. In the picture you can check that all the notes of C

major are mapped to the notes of C (natural) minor.

o As an example, consider reflecting the C major triad through

the line joining abto d. The note g is mapped to a which

means that C is mapped to Am(NOTE: this reflection

is not what is in the picture above or the table shown below).

The tri-tone line of C is b-f and that is the perpendicular

bisector of the ab to d line. All the notes in the key of C are

preserved, but C is interchanged with Am. This perhaps

sheds some light on why Am. is called therelativeminor to C!


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Reflection between C major and C minor (the parallel minor)

Roman

numberNotes

Chord

nameChord name Notes

Roman

number

I c-e-g C Cm g-eb-c i

ii d-f-a Dm Bb f-b-d VII

iii e-g-b Em A e -c-a VI

IV f-a-c F Gm d-b -g v

V g-b-d G Fm c-a -f iv

vi a-c-e Am Eb

bb

-g-eb

III

viio b-d-f B dim D dim a

b-f-d ii

o

11. Why does any major triad always reflect to a minor triad? It is

enough to show this for C major, since we can rotate to any other

major chord. From the root note c, the 5th is 1 step clockwise (we

always measure distances clockwise, so from now on I'll just say

the distance is "+1 steps") and the major 3rd is +4 steps and the

minor 3rd is -3 steps. From rotational symmetry, this is true for all

the chords.

12. Now we make an arbitrary reflection of the circ le, and so c is

mapped to c', and likewise e e', and g g'. We have to show

that this new triad is a minor chord. c' and g' are still adjacent to

each other, since reflections can't change distances, but now g' is

-1 steps from c', so g' is the root and c' is the 5th. Since in fact all

intervals are reversed in reflections (we'll prove this at the end),

the distance from c to e is +4 and thus the distance from c' to e' is

-4. This means that the distance from g' to e' is -3 and we have all

the intervals for a minor chord.

13. Lemma: In any reflection, the direction of any interval is

reversed.


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14. If a point x on the circle is counter clockwise from a

point y then the distance (measured clockwise) satisfies d(p,x)

fun with the circle of 5ths

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