Apr 7

Two Views of a Scale

by Anton Schwartz

Even non-musicians know: music has a lot to do with scales. But why? Given the amount of time we spend playing them and talking about them, I’d say it’s a fine question.

A scale has two distinct uses:

  • A scale is a kit for making melodies
  • A scale defines a particular “sound”

We can call the first use “melodic” and the second one “harmonic.” The unfortunate situation is this: all the ways we have for picturing scales are suited to their melodic use, at the expense of their harmony. This post aims to help rectify that.

How do we write a scale?

We always write a scale as a sequence of notes ordered chromatically, in increasing pitch order, starting on the tonic. And when we’re taught how to form a scale, we’re often taught in terms of the intervals between its notes… so that a C major scale is written as

CWDWEHFWGWAWBHC

where W and H are whole and half steps.

What do these ways have in common? We’re ordering the notes by pitch, focusing on are which notes are higher/lower, and by how much. And, sure enough, that’s very important when we are interested in the shape of the phrases we construct. Listen to the following melody and you’ll see what I mean:

The importance of melody

You know the song well. You probably even have a relative who sings it this way 😆. The harmony is completely, painfully wrong, but the melody is clear because of the way the notes move up & down, and how far, in relation to the rhythm. If some of the notes are a semitone off—or all of them!—it doesn’t hurt that shape very much, and the melody is still unmistakable.

But how about harmony?

If do something similar with harmony, the result is unrecognizable. First, listen to the normal chords to the song:

Now, let’s change each chord by moving just one of its notes by a semitone. Here’s the result:

So what’s going on here?

Pitch is a great way to look at notes when we’re concerned with the contour of a melody because a slight difference in “pitch space” makes for just a slight difference in the shape of a melody.

Not so when we’re interested in harmony. Harmonically, two notes a semitone apart clash strongly. So what’s a good way to look at notes that’ll tell us their harmonic similarity, if pitch is so bad?

It’s been starting us in the face

The perfect fifth is the most basic of relationships between notes. Two notes that are a fifth apart have a frequency ratio of 3:2, which is simpler than any other interval, save the octave and its 2:1 ratio. Two notes that are a fifth apart are so closely related that, if you sound them together, many people won’t realize they are hearing more than one note:

Perfect Fifth: One Note or Two?

Since notes a fifth apart are so harmonically similar, is there a way of organizing all the notes such that ones a fifth a part are right near each other? Of course. It’s our friend the circle of fifths:

CircleOfFifthsWebCFBbEbAbDbGbGDAEB

Most of us know it from talking about keys and key signatures. But for harmonic purposes, it offers a way for us to look at the notes in a scale that can’t be beat.

Look at a scale’s notes on the circle of fifths

If we’re looking to know what sound a scale conveys, let’s look at them on the circle of fifths, with the root of the scale positioned at the top. For instance, if we outline a scale’s notes in red and indicate its root with a white double-outline, a C major pentatonic scale looks like this:

C Pentatonic

We see immediately that, in contrast to chromatic ordering—in which the notes are spaced haphazardly throughout the octave—on the circle of fifths the notes are all clustered together. What’s more, we see that they are all clockwise of the root. That’s because it’s a major pentatonic scale, and all the notes on the right side of the circle are the bright “major” ones: the 5, 2, 6, 3, 7 and +4 of C, respectively. Going counterclockwise from C we encounter all the dark “minor” ones: the 4, ♭7, ♭3, ♭6, ♭2 and ♭5 respectively. (See this post on Harmonic Brightness & Darkness.) No surprise, then, that if we look at the minor pentatonic scale, we see that it is simply the major pentatonic shifted counterclockwise, in the dark direction:

C Minor Pentatonic

Returning to the major pentatonic, notice that if we add one note (B) on the bright side and one note (F) on the dark side, the result is a scale of seven notes that are all adjacent on the circle of fifths—namely, the major scale:

C Major

The act of rotating the notes of a scale on the circle, in one direction or the other, so as to put different roots at the top, is the act of taking the modes of that scale. If we rotate C major one “notch” in the bright, clockwise direction, we get the Lydian scale, the brightest of all the major modes:

F Lydian

And if we rotate in the opposite direction, we successively generate all of the darker modes of the scale, from Mixolydian and Dorian all the way through to Locrian:

B Locrian

A look at a chromatic scale (with root placed at the far left) reveals that as pitch increases, the harmonic quality of each note with respect to the root fluctuates somewhat erratically between dark (blue) and bright (green), and between consonant and dissonant:

R
m2
M2
m3
M3
P4
d5
P5
m6
M6
m7
M7

Similarly, on the circle of fifths, each of the harmonic roles has its own fixed location:

sP1P4m7m3m6m2d5A4P5M2M6M3M7

But now there’s great logic in the arrangement, with the notes growing steadily brighter/darker as one goes progressively clockwise/counterclockwise. Distance on the circle of fifths is connected with dissonance—with the notes at 0° and ±30° being the perfect intervals, and the notes at ±150° and 180° being the dissonant semitone and tritone.

Go learn on your own.

There’s much more to be said on the subject, but I suggest you explore the terrain yourself. Have a look at these scales on the circle of fifths:

  • The modes of the major scale. See how their ordering on the circle of fifths makes much more sense than the arbitrary “Ionian, Dorian, Phrygian, Lydian…” ordering-by-starting-pitch we’re used to.
  • The melodic minor and its modes. Look for the two tritone pairs contained in the scale. (The major scale modes have just one.) Look at how the gaps in the scale correspond to less tonal focus (concentration together) and more dissonance than the major scale.
  • The diminished and whole tone scales – notice their perfect rotational symmetry. And how they have no tonal focus at all—their “center of gravity” is not on one side of the circle, like major and minor scales, but smack in the middle.

An app helps you explore

If you’re an iPhone/iPad user, I’ve created an interactive app called ScaleMate that shows scales in both linear and circle of fifths arrangement and lets you modify them and hear the results—played both as scales and as random melodies. It also lets you learn about dozens of scales—from built-in explanations that I’ve written, which serve as an extensive, interactive theory reference.

Your thoughts?

I hope you enjoy looking as scales this way. The nice thing is that harmony doesn’t presuppose that the notes have to be listed in any order… you can just take them in all at once by looking at the diagram. And I’ve found no better way of looking at scales for thinking about their “sound” than the circle of fifths view. Not by a long shot. Let me know what you think—I’d love to hear your comments!

Further reading

Interested in more? In addition to checking out the ScaleMate App, these posts discuss similar subjects. You might find them interesting:

7 Responses

  1. David Espinosa says:

    Fantastic !
    Now I want a circle-of-fifths clock.
    (like, a physical time-telling clock)
    Labelled exactly like you drew it…

    • Sounds like a great etsy side-hustle!
      Dinner at half past tritone?

  2. Your analysis makes perfect sense as you laid out your case. It has made me reconsider how I think about the overtone series as the fundamental building block of western harmony. The question for me is will this make me play any differently? I might find out tonight…

    • I think the relation between the circle of fifths and the overtone series is one of the stickier wickets in music theory. Heck, you have to deal with it much more as a string player than I do – hats off to you. There’s something undeniably beautiful about the sound of a just-tempered 5:4 major third, which comes from the overtone series, as opposed to the 81:64 major third that comes from the circle of fifths. And yet it’s hard for me to imagine jazz without equal temperament, which wrecks the sound of the overtone-based intervals but which lets all this stuff about the circle of fifths actually work. Giant Steps without equal temperament? Or even Body & Soul? I can’t begin to contemplate.

      Let me know if you play any differently tonight!

  3. david says:

    Hey Anton, please make your app available for android if at all possible. It looks great!

    • Thank you, David! I’d love to make it available for android! It’s simply whether I can afford to – if I gain enough traction with the iPhone version then I can justify it. Same with Random Roots. Apps take a lot of time and I have to pay the mortgage somehow.

      • David says:

        Of course totally understandable.

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ScaleMate App

ScaleMate!

ScaleMate is the interactive app for visualizing scales. It shows you scales in ways that make their harmonic qualities apparent, while playing you the sounds you see. Choose from its library of scales or use it as a playground to create on your own.

Average App Store rating:
★ ★ ★ ★ ★  5.0 out of 5

Watch the short video… or visit ScaleMate.app.

—Anton

Random Roots App

RANDOM ROOTS!

Random Roots is a groundbreaking practice app I've created for players looking to deepen their musicianship and increase the efficiency and effectiveness of their practice. Guaranteed that you've never experienced anything like it.

Average App Store rating:
★ ★ ★ ★ ★  4.8 out of 5  (100+ reviews)

To learn more and download it for free, visit randomroots.app.

—Anton