— 2012 — Nov 27

Reverse Engineering Our Dominant Scales

by Anton Schwartz

Let’s look at the following four scales:

  • Altered dominant
  • Lydian dominant
  • Half-whole diminished
  • Whole tone

Other than Mixolydian, these are the four most commonly used scales over dominant chords. (The term is not widespread, but let’s call a scale used over a dominant chord a “dominant scale.”) Observe that the four scales all have four notes in common: the root, major third and minor seventh (which are essential to all dominant scales) and the tritone. If we look at the following figure, in the key of C, we see that the four common notes create four intervals—two whole steps and two major thirds:

4 note skeleton

We can generate any of our four dominant scales by adding notes to this skeleton of four notes. Specifically, we add notes that subdivide the major third intervals into smaller intervals.

Each major third consists of four half steps. As we see below, we can add one note to split a major third into two whole steps (the “sparse” method)… or we can add two notes to break it into two half steps and a whole step (the “dense” method):

Subdividing Major Thirds

Going back to our four-note skeleton, if we fill our two major third intervals “sparsely,” we add just two notes, yielding a six-note scale, the whole tone scale:

Creating a Whole Tone Scale

Alternatively, if we fill in our major third intervals “desnsely,” we add four notes giving us an eight-note scale, the half-whole diminished scale:

Creating a Diminished Scale

So far, here’s what we’ve got: If we think of a dominant scale as having a bottom half and a top half, we have two ways of populating each half with notes: a sparse way (3 notes total) and a dense way (4 notes total). A whole tone scale has a sparse bottom and a sparse top, giving us a (3+3=6) six-note scale. A diminished scale has a dense bottom and a dense top, giving us a (4+4=8) eight-note scale.

So… What if we combine a dense bottom (4 notes) and a sparse top (3 notes)? That gives us the bottom-heavy 7-note altered dominant scale:

Creating an Altered Scale

And a sparse bottom (3 notes) with a dense top (4 notes) gives us the top-heavy 7-note Lydian dominant scale:

Creating a Lydian Dominant Scale

Because the dense bottom half of the altered scale is a portion of a diminished scale, and its top half is a portion of the whole tone scale, the altered scale sometimes goes by the name “diminished whole tone scale.” We can visualize this here:

Diminished Whole Tone Scale

By the same reasoning, we could call the Lydian dominant scale the “whole tone diminished scale”… thought I’ve never actually heard that:

Whole Tone Diminished Scale

In these last two diagrams I’ve included the scale going not only up to the (redundant) high C, but also to down past C to the low B, so as to show the full extent of the portions of the scale that come from a diminished scale and a whole tone scale.

Perhaps a better way to visualize this effect is to arrange the notes wrapping around in a circle. It gives us better sense of the continuity of notes in the scale as they move past the octave boundary and, in this case, of the overlap of the diminished and whole tone portions of the scale. First, here are the C diminished and whole tone scales represented as a circle:

Diminished Scale in Wheel FormationDiminished Scale in Wheel Formation

Now, here is the C altered scale, with the sections that come from the diminished and whole tone scales indicated:

Altered Scale in Wheel Formation

Likewise, the C Lydian dominant scale represented in the same way:

Lydian Dominant Scale in Wheel Formation

Note, once again, the sparse (blue) and dense (yellow) areas of the two scales.

ADDITIONAL OBSERVATIONS

The tritone substitution of altered is lydian dominant and vice versa.

You may have heard about the tritone substitution: the idea is that if you place one dominant chord over a root a tritone away, the resulting chord is also a dominant chord. That’s because the third of the new chord is the seventh of the original chord, and vice versa. Well, for the four scales we are discussing, we can take this notion further…

Any two keys a tritone apart have identical four-note skeletons because the root, third, tritone and seventh of one are, respectively, the tritone, seventh, root and third of the other:

Tritone Substituion: skeletons

The upper half of one is the lower half of the other, and vice versa. So a lydian dominant (sparse on the bottom, dense on the top) becomes altered (dense on the bottom, sparse on the top) when we take the tritone as our new root.

Tritone Substituion: Altered & Lydian Dominant

The act of swapping the top and bottom, when we look at it on the chromatic wheel, amounts to a 180° rotation. Looking at the circular representations of the altered and Lydian dominant scales, above, it’s easy to see that rotating either one a half-turn turns it into the other.

The diminished scale is its own tritone substitution. And the whole tone scale is its own tritone substitution.

Since the top and bottom halves are the same for each of the two scales (for diminished both are dense; for whole tone both are sparse), reversing the two halves yields no change for either. Equivalently, spinning the wheel representation of either the diminished scale or the whole-tone scale 180° has no effect on the scale; they are “radially symmetrical.” (Which is why people use the terms “symmetric diminished” and occasionally “symmetric augmented.”)

Since lydian dominant and altered scales are modes of the melodic minor scale…

…they are not the only scales made up a diminished part and a whole tone; all the melodic minor modes share that property:

Melodic Minor in Wheel FormationHalf-Diminished in Wheel Formation
Lydian Augmented in Wheel FormationPhrygian Raised 6 in Wheel Formation

11 Responses

  1. Adam Spiers says:

    Fantastic post! I’ve been aware of these relationships for quite a while, but I’ve never seen them expressed so clearly before. How did you draw those diagrams?

  2. Jeff Taylor says:

    Yes I love the graphics as well……how did you create them?

  3. Ah, the magic of Adobe Illustrator…

  4. Michael Ray says:

    Thamk you, thank you…………………………………

  5. Narciso (nj) Jaramillo says:

    Nice illustration of the relationships!

    There’s another symmetry worth mentioning here, I think. In the bottom half of the scale, the “sparser” sound seems normal, and the “denser” sound seems altered, because the “sparser” sound matches up with the actual (ordinary) dominant scale. In the top half, it’s the opposite: the “sparser” sound seems altered, and the “denser” sound seems normal, because the “denser” sound matches up with the ordinary dominant scale. And then, of course, these relationships flip again if you tritone-sub the root (while still starting the scale at the same original note).

    In practice, though, I find that I’m thinking less of these kinds of structural symmetries in the scale, and thinking more about which color tones I want to use: do I want the b13 or the natural 13, and do I want the b9 or the natural 9. Which of those I pick tends to dictate what other notes I play around them (i.e., if I play the b9, I don’t really want to play the natural 9 except as a passing tone, so the next available scale note will be the #9; if I play the b13, I don’t really want to play the natural 13, but depending on what I’m doing I might sometimes play the natural fifth even though that doesn’t really work in any of the standard dominant scales).

    • I agree completely with your point about choosing your sound by individual notes rather than always signing up wholesale for one scale. One example: long stretches of Dorian where it’d be a crime to rule out also using major sevenths and minor sixths as needed. Still, I think there are plenty of times when, except for chromatic passing notes and such, you want to stick pretty firmly within the sound of one scale.
      As for your first point, It seems to me tantamount to saying that Lydian Dominant sounds “ordinary” or “normal” (or sounds close to Mixolydian, if that’s what we consider “normal”) and Altered sounds altered. No?

  6. Ben Hester says:

    Very intersting and so beautifully depicted! Love your music!

  7. hongyijig says:

    I am going to do my best to not to make this sound like a math lecture so I will jump directly to the punch line. If you build a dominant 7 chord (R,3,5,b7) on both the ♭5 and the #5 of the dominant chord that is written, you will be using only notes from the altered scale.

    • Indeed, Hongyijig! It’s easier to see if you look at the tritone substitution. In that “dual space”, your statement translates to this:

      If you build a dominant 7 chord (R,3,5,b7) on both the root and the second degree of the dominant chord that is written, you will be using only notes from the Lydian Dominant scale.

      It’s trivial to see that the Lydian Dominant scale contains the notes of the dom 7 chord built on the root (R,3,5,b7); as for the dom 7 chord built on the second degree, that consists of the upper structure triad (9, #11, 13… all of which belong to Lydian Dominant) and the root itself.
      A useful application: “triad pairs”. You can generate nice modern altered lines by alternating between arpeggios of the major triad built on the ♭5 and arpeggios of the major triad built on the #5.

  8. Alexander Olsen says:

    This lesson is fantastic and I really like the language you chose to describe the density of the scale, I’ve never really thought about it that way before. Very enlightening. Combining all of these on the fly, navigating densities and sparsities, rearranging them sporadically with your ear tempered by your mind based on this lesson would make even Nicolas Slonimsky jealous of the combinations that could be created.

  9. king solomon says:

    what a great teacher, keep it up.

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