SYMMETRIC SCALES:
Symmetric scales are constructed when the octave is divided into equal parts.
The 12 notes may be divided into 2, 3, 4, 6 or 12 equal parts:
Before we go on, remember that one of the accepted conditions for a series of notes to be deemed a "scale" is that we should not have any intervals larger than a 2nd... (see: What Is A Scale ?). More on this later.
Let's take a look at those equal divisions now:
1. When the octave is divided in 12 equals parts, we get a Chromatic Scale. This is a series of minor 2nds.
Using C as Root we have: C C# D D# E F F# G G# A A# B (C)
2. When the octave is divided in 6 equals parts, we get a WholeTone Scale. This is a series of Major 2nds.
Using C as Root again: C D E F# G# Bb (C)
3. When the octave is divided in 4 equals parts, we get a Diminished 7th arpeggio. This is a series of Augmented 2nds (those happen to also be minor 3rds).
From C root: C Eb Gb A (C)
This series could actually fit our description of a scale, since all of the intervals are Augmented seconds! However, everyone will agree that it definitely makes more sense to call it an arpeggio (a sequence of minor 3rds).
Now, if we precede each one of the notes of the arpeggio with its leadingtone, we construct a Diminished Scale.
From C root: C D Eb F Gb Ab A B (C)
4. When the octave is divided in 3 equals parts, we get an Augmented Triad arpeggio, which is a series of Major 3rds. Definitely not a scale at this point, but clearly an arpeggio.
From C root: C E G# (C)
But, if we do what we just did with the Diminished 7th arpeggio and precede each note with a halfstep, we now have what we call an Augmented Scale.
From C root: C D# E G G# B (C)
5. When the octave is divided in 2 equals parts, we simply get the interval of a Tritone.
From C root: C Gb (C)
We can here as well add the halfstep below each note and get the following series:
C F Gb B (C)
Not sure what to name it though... a CsusMaj7(b5) arpeggio? Sid Jacobs, in his "Complete Book of Jazz Guitar Lines & Phrases" refers to it as a "Tetratonic Tritone Scale" and shows several lines and voicing derived from it. You may want to check it out.
Here's a Summary:
(or another way to look at the same thing...)
1. a series of m2 will create a Chromatic scale (the octave is divided equally into 12 notes).
2. a series of M2 will form a WholeTone scale (the octave is divided equally into 6 notes).
3. a series of A2 will generate... a Diminished Arpeggio (the octave is divided equally into 4 notes).
Now we can also combine 2 types of 2nds. Let's start with the more common ones:
1. M2 and m2 will give us a Diminished Scale.
2. m2 and M2 will be the "other" Diminished Scale, sometimes called AuxiliaryDiminished.
3. A2 and m2 will be referred to as an Augmented Scale.
4. m2 and A2 will be deemed the "other" Augmented Scale, which following a previous logic could be then named AuxiliaryAugmented.
We have one more pairing of 2nds to look at.
5. M2 and A2: using C as a Root, we get:
C D F G Bb (C)
We end up with the notes of a Bb Major Pentatonic Scale (or G Minor Pentatonic, its relative). However, it is not a symmetric scale, as it cannot be divided in 2 nor in 3 equal parts. It is therefore not relevant to this article  oh well...
OK, now that we had our little distraction, let's see if we can combine all 3 types of 2nds. We will again use C as the Root for each sequence.
1. A2 M2 m2
C Eb F Gb A B (C)
2. A2 m2 M2
C D# E F# A Bb (C)
3. M2 A2 m2
C D F Gb Ab B (C)
4. M2 m2 A2
C D Eb F# G# A (C)
5. m2 A2 M2
C Db E F# G Bb (C)
6. m2 M2 A2
C Db Eb F# G A (C)
All those scales are unnamed Hexatonics (6note scales). Note that they all include the b5, can be divided into 2 equal parts, and are therefore symmetrical.
Many combinations of 2nds are possible within one octave, and you can actually construct your own symmetrical series (those will be arpeggios, "groupings" or scales). The recipe is quite simple:
If we divide the octave in 2 halves, the following template will be constant:
C Gb (C)
Just insert any note between the C root and the Gb (the Tritone). You then want to insert the same interval between the Gb and the C octave.
Example: let's insert an E between C and Gb. The interval is of a M3. We then have to insert a note that is a M3 up from the Gb:
C E Gb Bb (C)
we just created a Dominant 7(b5) arpeggio
More notes may be inserted. Let's now add a chromatic step between E and Gb. The same interval duplicated will be a chromatic step between Bb and C.
Here is the new series:
C E F Gb Bb B (C)
which is another unnamed hexatonic scale
Let's try now adding a D between the C and E (a wholestep after the Root). The same interval duplicated will be a wholestep after the Tritone:
Here is our new series:
C D E F Gb Ab Bb B (C)
This particular mode has already been catalogued by composer Olivier Messiaen. He named it Mode 5 from his collection of Modes of Limited Transposition also known as "Messiaen Modes". Messiaen went through all of the possibilities, and selected 7 of them, simply referring to them as Mode 1, Mode 2, etc.
Many options, as you can see.
You can also divide the octave into 3 equal parts instead of 2 halves and use the same approach. You can start with only using 2nds in your series, but eventually try with various intervals. Whether or not you want to refer to those things as "scales" is really up to you!
More math to conclude:
There are 2 WholeTone scales (in life...) each with 6 possible roots.
There are 3 Diminished scales (in life...) each with 4 possible roots.
There are 4 Augmented scales (in life...) each with 3 possible roots.
and:
There are 6 Tritones with 2 possible roots.
Oh, and there is also only 1 Chromatic scale, with 12 possible roots...
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