The Derivation of
Radially Symmetrical
Altered Scales



Central Point of Reference
D


1st Level
The Quartal Consonants of D [A and G]
4th          4th
A        D        G
(figure 1)



2nd Level
Add quartal consonants of A and G [E and C]
4th         4th        4th        4th
E        A        D        G        C
(figure 2)

the minor pentatonic scale
A         C       D       E         G
(figure 3)



3rd Level
Add quartal consonants of E and C [B and F] 4th         4th        4th        4th        4th        4th
B        E        A        D        G        C        F
(figure 4)

the minor scale
A       B        C        D        E        F        G
(figure 5)

The Major Scale
C        D        E        F       G        A       B
(figure 7)


 


Level 4

Taking this law of 4ths (as in figure 2 and 4 above) another step further,
we now have the following 9-tone row:


4th Level
The Quartal Consonants of B and F [F# and Bb]
4th      4th        4th         4th          4th          4th          4th           4th
F#       B        E        A        D        G        C        F        Bb
(figure 8)

 

By substituting both of the two new notes for their natural namesakes
(B becomes Bb and F becomes F#),
we arrive at a this radially symmetrical 7-note structure:


A       Bb        C        D        E        F#        G
(figure 9)


Rearranged similarly to figure 7 above,
this yields what is commonly known as
the G melodic minor scale


melodic minor scale
G       A        Bb        C        D        E        F#
(figure 10)


This scale's modes are quite common and very useful
(Lydian Augmented Scale, Lydian Dominant Scale, Altered Scale, etc)




Level 5

Taking the logic of outward-going 4ths the next step further,
we now have the following 11-tone row:


5th Level
Quartal Consonants of F# and Bb [C# and Eb]
  4th         4th         4th         4th          4th         4th        4th        4th      4th          4th   
C#       F#       B        E        A        D        G        C        F        Bb        Eb
(figure 11)

 

Level 5a

Preserving Level 4's Bb and F#,
substitute both of the two new notes C# and Eb for their natural namesakes,
(C becomes C#, E becomes Eb)
and we arrive at this eastern 7-note scale:


the double harmonic scale
A       Bb        C#        D      Eb      F#        G
(figure 12)



Level 5b

Preserving Level 4's Bb and F#,
substitute both the enharmonic notes Db and D# for the D,
and we arrive at the most common of all 8-note scales:


the diminished scale
A       Bb        C        Db      D#        E        F#        G
(figure 13)

In the diminished scale, both Db and D# are one half-step equidistant from D.



Level 5c

Neutralizing Level 4's Bb and F# alterations, (Bb reverts to B, F# reverts to F)
substitute the new notes C#/Db and the D#/Eb
for all of their natural namesakes,
(both C and D become C#/Db, and both D and E become D#/Eb )
and we arrive at this 6-note scale:


the wholetone scale
A       B        C#/Db      D#/Eb        F        G
(figure 14)

 



Level 6

Taking the final quartal step,
we now have the following 13(12) tone row:


6th Level
Quartal Consonant of C# and Eb [G#/Ab]
          4th        4th        4th        4th        4th         4th          4th           4th         4th          4th           4th           4th     
G#       C#       F#       B        E        A        D        G        C        F        Bb        Eb        Ab
(figure 15)

THE CHROMATIC SCALE



Level 6a

Preserving Level 4's Bb and F#,
and also using Level 5b's alterations (Db and D#),
substitute the new Ab/G# for both the A and the G. That yields this 8(7) note scale:


Ab       Bb        C        Db      D#        E        F#        G#
(figure 16)

 

rearranged and enharmonically spelled:


 C#/Db melodic minor scale
Db        Eb        Fb      Gb       Ab         Bb       C  
C#       D#        E        F#        G#        A#        B#
(figure 17)




Level 6b

Neutralizing Level 5b's Db and D# alterations
(Db reverts to D, and D# reverts to D)
while preserving Level 4's alterations (Bb and F#),
yields this 7(6) note tone row


the wholetone scale
Ab       Bb        C        D        E        F#        G#
(figure 18)



Level 6c

Preserving Level 4's Bb and F#,
and using Level 5c's alterations (C#/Db and D#/Eb for C, D, and E),
we arrive this 6(5) note scale:


Ab       Bb        C#/Db       D#/Eb      F#        G#
(figure 19)


rearranged and enharmonically spelled:


D#/Eb minor pentatonic
Eb        Gb        Ab       Bb        Db
 D#        F#        G#        A#        C#  
(figure 20)

 


Level 6d

As in 6c above, preserving Level 4's Bb and F#,
and using Level 5c's alterations (C#/Db and D#/Eb for C, D, and E),
but also allowing in the natural B and natural F from the third level,
we arrive this 8(7) note scale:


Ab       Bb        B        C#/Db      D#/Eb       F        F#        G#
(figure 21)

 

enharmonically spelled:


Ab/G# dorian mode
Ab       Bb        Cb        Db      Eb       F        Gb        Ab
   G#        A#         B         C#      D#      E#       F#        G#    
(figure 22)



Level 6e

Neutralizing all of Level 4's and Level 5's alterations,
and allowing in a natural A and a natural G from the first level,
we arrive at this 9(8) note scale:



 G#       A       B        C        D        E        F        G        Ab
(figure 23)


rearranged:


the major bebop scale
C        D        E        F        G        Ab/G#       A       B
(figure 24)

 


Conclusion

Within all of the above radially symmetrical scales
(minor and major pentatonic scales; aeolian/ionian/dorian etc modes;
double harmonic scale; the melodic minor modes; wholetone scales;
the diminished scales and the major bebop scale)
are found every practical chord possibility!


These results are all obtained by
using extremely basic laws of physics and
strict logic in only six short steps.



As a footnote to this discussion of radially symmetrical scales, it is interesting to note that whenever symmetrical scales are mentioned in theory books, that only the wholetone and diminished are the ones to get any press. Check out all these other common scales that are also symmetrical!

The minor pentatonic is radially symmetrical in two ways:

A minor pentatonic
A C D E G
m3 W | W m3

D#/Eb minor pentatonic
Ab Bb Db | D# F# G#
W m3 W m3 W

The "modes of the major scale" are also radially symmetrical in two ways:

A Aeolian
A B C D E F G
W h W | W h W

Ab Dorian
Ab Bb Cb Db | Eb F Gb Ab
W h W W W h W

And the Jazz Melodic Minor is radially symmetrical in two ways:

G Jazz Minor
A Bb C D E F# G
h W W | W W h

C#/Db Jazz Minor
Ab Bb C Db | D# E F# G#
W W h W h WW


Ockham's Razor:
'All things being equal, the simplest solution tends to be the best one'



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