Palindromic Scales and Mirror Modes

Palindromic Scales and Mirror Modes

A Man. A Plan. A Canal. Panama. This is one of my favourite palindromes (and one of my favourite The Fall Of Troy songs). But there are other palindromes in music as well. In this article, we will discuss palindromic scales and mirror modes.

Try reversing or “mirroring” the order of intervals in any given scale. Reversing the order of intervals in a palindromic scale will produce the same scale. Otherwise, we will end up with a new ‘mirror scale‘ that is on the opposite side of the brightness/darkness spectrum.

This idea of the bright/dark spectrum of scales adds another layer of thinking in the way we write and improvise with these scales.

With that brief primer out of the way, let’s get into the article on palindromic scales and mirror modes!

Palindromic Scales

In the article The Dorian Brightness Quotient, we establish the Dorian mode (2nd mode of the Major Scale) as a “neutral reference point” for determining the brightness and darkness of the other heptatonic scales.

Dorian is also a palindrome! Let’s look at how:

The Dorian Mode

Dorian is made up of the following scale degrees:

1        2      ♭3         4         5         6     ♭7

w-h-w-w-w-h-w
*w=whole step // h=half step*

Mirror those intervals and we end up with the same thing:

w-h-w-w-w-h-w

Dorian is a common palindromic scale since it’s a mode of the most common scale, the Major Scale.

Another way to visualize the palindromic nature of Dorian is in the circle of fifths. Let’s look specifically at D Dorian:

Dorian Brightness Quotient

We can see that by stacking three fifths to either side of our root, we end up with Dorian. Because the root is centred, we have a palindrome!

Let’s see what happens if we push the two outer notes by a fifth in either direction:

Mixolydian ♭6

Palindromic Scale and Mirror Modes

As we can see here, D is once again in the centre and is, therefore, the root of a palindrome. This palindrome is the 5th mode of the Melodic Minor Scale and is called Mixolydian♭6. In the diagram above, it is specifically the D Mixolydian♭6 from the G Melodic Minor Scale.

D Mixolydian♭6 is made up of:

D        E        F♯    G        A        B♭    C         D

With the scale degrees:

1        2         3         4         5     ♭6     ♭7

w-w-h-w-h-w-w
*w=whole step // h=half step*

Mirror those intervals and we end up with the same thing:

w-w-h-w-h-w-w

This visual representation on the circle of fifths is the easiest way I’ve come across to visualize palindromic scales. Let’s look at one more from a heptatonic scale I really enjoy 🙂

The Double Harmonic Major Scale

Palindromic Scales and Mirror Modes

Once again, D is in the middle with pairs of notes at 1, 4, and 5 fifths away in either direction around the circle of fifths.

D Double Harmonic Major is made up of:

D        E♭     F♯    G        A        B♭    C♯    D

***note that the circle of fifths’ D♭ is enharmonic to the scale’s C♯***

With the scale degrees:

1     ♭2         3         4         5     ♭6        7

h-wh-h-w-h-wh-h
*w=whole step // h=half step // wh=whole step + half step*

Mirror those intervals and we end up with the same thing:

h-wh-h-w-h-wh-h

For a more complex and thorough analysis of musical palindromes with varying axes of symmetry, check out the book Modalogy by Jeff Brent and Schell Barkley!


Other Palindromic Scales

We could go on and on, creating palindromic scales with varying amounts of notes. The circle of fifths is a great visual tool to help us discover these scales, so long as the scale has an odd number of notes (like heptatonic and pentatonic scales).

For example, much like the second mode of the Major Scale (Dorian), the second mode of the Major Pentatonic Scale is also a palindrome:

Palindromic Scales and Mirror Modes

D        E        G        A         C        D

1        2         4         5     ♭7

w-m3-w-m3-w
*w=whole step // m3=minor third*

Mirror those intervals and we end up with the same thing:

w-m3-w-m3-w

Other axes of symmetry

So far we’ve looked at palindromic scales with a “middle point” or reflective axis that is included in the scale. For example, the note D is part of D Dorian.

Let’s restate D Dorian surrounded by its fifths linearly rather than on the circle of fifths. We’ll add is where the axis of symmetry is:

Palindromic Scales and Mirror Modes

But what if we have an axis that is not a note of the scale, or even a chromatic note at all?

Let’s look at the Whole-Tone Scale. It is built entirely or whole steps, and therefore has the intervals:

w-w-w-w-w-w

Obviously, the Whole-Tone Scale is palindromic! But where’s its axis exactly? Let’s take a look:

Palindromic Scales and Mirror Modes

So for the D Whole-Tone scale, the axis of symmetry must be G? That’s the halfway point between F♯ and G♯.

Palindromic Scales and Mirror Modes

Well, the interesting thing about the Whole-Tone Scale is that it only has one mode and that any “in-between” note can be its axis of symmetry.

The Whole-Tone Scale is considered a symmetrical scale or a “mode of limited transposition” if you could ask the late Olivier Messiaen.

Briefly, a symmetrical scale is one that has a repeated pattern of intervals within its octave. Like the whole-half diminished scale, which has a repeating whole step, half step pattern:

w-h-w-h-w-h-w-h

However, this whole-half diminished scale is not palindromic…

So which scales have a palindromic mode and which do not?

There’s no correlation I’ve found between scales that do and do not have a palindrome within them. It doesn’t matter how many notes are in the scale, or whether the scale is symmetrical or not.

For example, the Harmonic Minor and Harmonic Major scales are both heptatonic but do not contain a palindromic mode within them. More on these interesting scales later!

Likewise, the symmetrical Whole-Half Diminished and Half-Whole Diminished scales are not palindromic either. But let’s take a look at them anyway, as a segue to our discussion on mirror modes.

The Diminished Octatonic Scales

Let’s take a look at a common octatonic scale, the Whole-Half Diminished Scale. As the name suggests, we alternated whole step and half step intervals to create this scale. This repeating pattern makes the Whole-Half Diminished Scale a symmetrical scale, but it is not a palindromic scale.

C Whole-Half Diminished is made up of:

C        D        E♭     F       G♭   G♯     A       B       C

With the scale degrees:

1         2       ♭3         4      ♭5    ♯5       6        7

w-h-w-h-w-h-w-h
*w=whole step // h=half step*

Mirror those intervals and we end up with the same thing:

h-w-h-w-h-w-h-w

So it’s not palindromic, but, interestingly enough, when we do mirror the Whole-Half Diminished Scale we arrive at its only other mode, the Half-Whole Diminished Scale.

This brings us to our discussion on mirror modes!

Mirror Modes

So if we mirror Dorian, we get Dorian. But what happens when we mirror the other modes of the Major Scale?

Palindromic scales are simply special cases when mirroring intervals.

Something interesting happens where the brighter a mode is, the darker its mirror mode will be.

We can say that Dorian is neutral on the brightness/darkness spectrum and therefore can assign other scales and modes a number to describe their relative brightness. This number is called the Dorian Brightness Quotient (DBQ).

  • If the DBQ is negative, the scale is darker than neutral.
  • If the DBQ is positive, the scale is brighter than neutral.

Let’s look at the modes of the Major Scale complete with their Dorian Brightness Quotients:

Dorian Brightness Quotient

Now let’s rearrange the modes from brightest to darkest (the highest DBQ to the lowest DBQ):

Dorian Brightness Quotient

So we know that Dorian mirrors Dorian (0 <-> 0). But +1 mirrors -1, as does +2 <-> -2 and +3 <-> -3.

Let’s quickly go through some proofs:

Mixolydian (+1) <-> Aeolian (-1)

Dorian Brightness Quotient

Ionian (+2) <-> Phrygian (-2)

Dorian Brightness Quotient

Lydian (+3) <-> Locrian (-3)

Dorian Brightness Quotient

It’s pretty neat that these modes mirror themselves perfectly to other modes within the same parent scale based on their matching +/- DBQ values.

This is true for the Major Scale, the Melodic Minor Scale, and the Double Harmonic Major Scale.

Let’s look at the Melodic Minor and Double Harmonic Major in terms of their modes from brightest to darkest. Let’s then mirror the modes in the same way we did with the Major Scale:

Melodic Minor Modes: Brightest to Darkest

Dorian Brightness Quotient

Mixolydian ♭6 is a palindrome and has a DBQ of 0 just like Dorian! It mirrors itself with the following intervals:

w-w-h-w-h-w-w
*w=whole step // h=half step*

Let’s look at the pairs:

Ionian Minor (+1) <-> Dorian ♭9  (-1)

Dorian Brightness Quotient

Lydian Dominant (+2) <-> Aeolian Diminished  (-2)

Dorian Brightness Quotient

Lydian Augmented (+4) <-> Altered Scale  (-4)

Dorian Brightness Quotient

Once again, the modes mirror themselves perfectly to other modes within the same parent scale based on their matching +/- DBQ values.

Double Harmonic Major Modes: Brightest to Darkest

Dorian Brightness Quotient

The Double Harmonic Major Scale is a palindrome and has a DBQ of 0 just like Dorian! It mirrors itself with the following intervals:

h-wh-h-w-h-wh-h
*w=whole step // h=half step*

The Double Harmonic Major Scale is a very interesting scale, having two consecutive half steps and two minor third intervals (whole-half steps) in its scale. The common names for its modes don’t even follow the typical modal names.

Let’s dig into the modes and their mirrors:

Hungarian Minor (+1) <-> Oriental  (-1)

Dorian Brightness Quotient

Ionian Augmented ♯9 (+4) <-> Ultraphrygian  (-4)

Lydian♯2♯6 (+5) <-> Locrian ♭♭3 ♭♭7 (-5)

Dorian Brightness Quotient

mirroring cross relationship between the harmonic minor/major scales

So we’ve checked out the Major, Melodic Minor, and Double Harmonic Major scales. They’re all heptatonic and have modes that mirror their opposites within their respective parent scales.

Each of these parent scales has a palindromic mode within them.

  • Dorian (from the Major Scale)
  • Mixolydian ♭6 (from the Melodic Minor Scale)
  • Double Harmonic Major (from the Double Harmonic Major Scale)

This makes sense since we can’t evenly split up 7 modes into pairs. The one mode that is not paired with a mirror mode is a reflection of itself!

But the Harmonic Minor and Major scales don’t contain any palindromic modes.

We can’t evenly pair up 7, but we can evenly pair up 14.

And that’s how we do it with the Harmonic Major and Minor scales. The mirror of a mode from the Harmonic Major Scale will always be a mode from the Harmonic Minor Scale.

Let’s take a look:

Harmonic Minor Modes: Brightest to Darkest

Dorian Brightness Quotient

Harmonic Major Modes: Brightest to Darkest

Dorian Brightness Quotient

The first thing to point out by looking at the modes of these two scales is that although Aeolian ♮7 and Mixolydian ♭9 have DBQs of 0, they are not palindromic. Up until this point, all out modes that have had DBQ=0 have been palindromic. This is a good time to point out that a DBQ value of zero does not necessarily mean a palindromic scale/mode!

However, the Aeolian ♮7 and Mixolydian ♭9 are mirrors of each other. Let’s get into our proofs:

Mixolydian ♭9 (0) <-> Aeolian ♮7  (0)

Dorian Brightness Quotient

Ionian ♭6 (+1) <-> Phrygian Dominant  (-1)

Dorian Brightness Quotient

Dorian ♯4 (+1) <-> Dorian Diminished  (-1)

Dorian Brightness Quotient

Lydian Minor (+2) <-> Locrian ♮6 (-2)

Dorian Brightness Quotient

Ionian Augmented (+3) <-> Phrygian ♭4 (-3)

Dorian Brightness Quotient

Lydian ♯9 (+4) <-> Locrian ♭♭7 (-4)

Dorian Brightness Quotient

Lydian Augmented ♯9 (+5) <-> Super Locrian ♭♭7 (-5)

Dorian Brightness Quotient

ReCap

It’s interesting to think of scales in terms of palindromes, brightness and darkness.

The Dorian Brightness Quotient is a neat tool for mirroring modes.

This all adds an extra layer of thinking to the relationships between musical scales that we may not think of when first learning them.

In Closing,

I hope this article provided some insight into the modes. Although not necessarily a compositional device, mirroring can provide another level of thinking and understanding to scales and modes.

I believe that the more ways we have of thinking about a subject, the more creative we can get with it. And thinking of brightness/darkness, palindromes, and mirror scales has definitely added another layer to my understanding of modes. I’m glad to have shared this with you all, and hope you’ve gotten as much out of it as I have!

As always, thanks for reading and for your support,

-Art