## Modes (Part 2)

When you first learn about the major scale it’s fairly straightforward, because they tell you about just one major scale. But the minor scale is more tricky, because they tell you about three—or actually four, two of which are the same!

The most fundamental of these is the natural minor scale. The C major scale goes

C D E F G A B C

The C natural minor scale goes

C D E♭ F G A♭ B♭ C

As you can see the 3rd, 6th and 7th notes of the scale are ‘flatted’: moved down a half-tone compared to the major scale. This gives the natural minor scale a darker, even ‘sadder’ quality compared to the major scale.

I prefer to work with note numbers instead of note names, not because I’m a mathematician so I love numbers, but because then we can simultaneously talk about different keys at once, not just the key of C. In this approach we call the notes of the major scale

1 2 3 4 5 6 7 8

and then the natural minor scale is

1 2 ♭3 4 5 ♭6 ♭7 8

Don’t ask me why the flats are written in front of the numbers now instead of after them—it’s just a convention.

Now, one thing about ‘common practice’ western harmony is the 7th tone plays a special role. It’s just a half-step below the 8, and we act like that dissonance makes it want very strongly to go up to the 8. The 8 is one octave above the 1, twice the frequency. Either the 1 or 8 instantly serves as a home base: we feel like a piece or passage is done, or momentarily at peace, when we play these notes. We say the 7 wants to ‘resolve’ to the 8, and we call it the ‘leading-tone’ for this reason: it suggests that we’ve almost reached the tonic, and makes us want to get there!

There’s much more we could say here, but it all combines to make people want a scale that’s like minor but contains the 7 instead of the ♭7. And since this scale is motivated by reasons of harmony theory, it’s called the harmonic minor scale. It goes like this:

1 2 ♭3 4 5 ♭6 7 8

However, now people singing this scale find it mildly awkward to jump up from ♭6 to the 7 because the distance between them is larger. In fact it’s 3 half-tones, larger than any step in the major or natural minor scale! One way to shrink this gap is to raise the ♭6 to a 6 as well. This gives the melodic minor scale:

1 2 ♭3 4 5 6 7 8

By now we’re almost back to the major scale! The only difference is the flatted 3. However, that’s still a lot: the ♭3 is considered the true hallmark of minorness. There are reasons for this, like the massive importance of the 1 3 5 chord, which serves to pound home the message “we’re back to 1, and this is the major scale, so we are very happy”. Playing 1 ♭3 5 says “we’re back to 1, but this is minor, so we are done but we are sad”.

However, singing up the scale is different from singing down the scale. When we sing up the melodic major scale we are very happy to sing the 7 right before the 8, because it’s the leading-tone: it tells us we’re almost home. But when we sing down we don’t so much mind plunging from the 8 down to ♭7, and then it’s not so far down to ♭6: these are both steps of a whole tone. If we do this we are singing in the natural minor scale. So what I called ‘melodic minor’ is also called melodic minor ascending, while natural minor is also called melodic minor descending.

Here I should admit that while this is an oft-told pedagogical story, the actual reality is more complex. Good composers or improvisers use whatever form of minor they want at any given moment! However, most western musicians have heard some version of the story I just told, and that does affect what they do.

To listen to these various forms of the minor scale, and hear them explained more eloquently than I just did, try this:

Grazie Terzian is the patient teacher of music theory I wish I’d had much earlier. You may feel a bit impatient listening to her carefully working through various scales, but that’s because she’s giving you enough time for the information to really sink into your brain!

Anyway: we’ve seen one form of major scale and three forms of minor, one of which has two names. All these scales differ solely in whether or not we flat the 3, 6 or 7. So, we can act like mathematicians and fit them into a cube where the operations of flatting the 3, 6 or 7 are drawn as arrows:

Here to save space I’ve written flatted notes with little superscripts like $3^\flat$ instead of ♭3: it makes no difference to the meaning.

This chart shows that flatting the 3 pushes our scale into minor territory, while flatting the 6 and then the 7th are ways to further intensify the darkness of the scale. But you’ll also see that we’re just using a few of the available options!

In part 1 I showed you another way to modify the major scale, namely by starting it at various different notes to get different ‘modes’. If we list them in order of the starting note—1, 2, 3, etc.—they look like this:

For example, Ionian is just major. But we saw that it is also very nice to list the modes from the ‘brightest’ to the ‘darkest’. Rob van Hal made a nice chart showing how this works:

Skipping over Lydian, which is a bit of an exception, we start with major—that is, Ionian—and then start flatting more and more notes. When we reach the Phrygian and Locrian we flat the 2 and then the 5, which are very drastic things to do. So these modes have a downright sinister quality. But before we reach these, we pass through various modes that fit into my cube!

Let’s look at them:

We’re now tracing out a different path from top to bottom. Ionian has no notes flatted. In Mixolydian we flat the 7. In Dorian we also flat the 3. Then in Aeolian we also flat the 6.

I mentioned that the ♭3 is considered the true hallmark of minorness. Thus, in the classification of modes, those with a flatted 3 are considered ‘minor’ while those without are considered ‘major’. So in our new path from the cube’s top to its bottom, we switch from major to minor modes when we pass from Mixolydian to Dorian.

Note that Ionian is just our old friend the major scale, and Aeolian is our friend the natural minor. We can combine the two cubes I’ve showed you, and see how they fit together:

Now we can get from the top to Dorian following two paths that pass only through scales or modes we’ve seen! Similarly we can get from melodic minor ascending to the bottom following two paths through scales or modes we’ve seen. In general, moving around this cube through the course of a piece provides a lot of interesting ways to subtly change the mood.

But two corners of our cube don’t have names yet! These are more exotic! But of course they exist, and are sometimes used in music. The mode

1 2 3 4 5 ♭6 7

is called harmonic major, and it’s used in the Beatles’ ‘Blackbird’. The mode

1 2 3 4 5 ♭6 ♭7

is called the melodic major scale, or also Mixolydian flat 6 or Aeolian dominant. It’s used in the theme song of the movie The Mask of Zorro, called ‘I Want to Spend My Lifetime Loving You’.

So, let’s add these two modes to our cube:

This is the whole enchilada: a ‘commuting cube’, meaning that regardless of which path we take from any point to any other point, we get the same mode in the end. We can also strip it of all the musical names and think of it in a purely mathematical way:

We could go further and study a 5-dimensional hypercube where we also consider the results of flatting the 2 and 5. That would let us include darker and scarier modes like Phrygian, Phrygian dominant and Locrian—but it would be tougher to draw!

### 9 Responses to Modes (Part 2)

1. allenknutson says:

I’m guessing you know the book https://dmitri.mycpanel.princeton.edu/geometry-of-music.html (which I haven’t read), but I’ll bring it up anyway.

2. John Baez says:

Yes, that’s a cool book! Tymoczko sent me a copy of it, and I’ve read it, but right now I’m in Edinburgh and it’s sitting in Riverside. So it may contain some stuff I’m thinking about now.

I know he discussed how a piece by Chopin explored a certain hypercube of seventh chords, but I forget the details and I want to know. There’s a movie here but it doesn’t help enough. There it says:

ChordGeometries represents chords and voice leadings in a variety of 3D geometrical spaces. You can enter chords on a MIDI keyboard or using the Keyboard window. Voice leadings between successive chords are represented by continuous paths in the spaces. The program is meant to accompany the paper “The Geometry of Musical Chords” [Science 313 (2006): 72-74]. Further information can be found in “Generalized Voice-leading Spaces,” with Clifton Callender and Ian Quinn.

Three movies demonstrate the program, using the opening of Chopin’s E minor prelude. This movie depicts Chopin’s piece as it travels through a slice of the four-dimensional space containing seventh chords.

• Jesús López says:

It’s quite a stretch to pass from day 2 on modes to Tymoczko :-). Hope you share some of your thoughts on the book at some point.

Harmony is a big topic in itself, the vertical dimension of chords in the staff and the engine that makes the story advance by their progression, while Voice Leading is already in contrapunctal terrain, viewing chords as if sung by singers each with their own melodies, their voices being the voices that are led, in the horizontal (time) perspective. Voice melodic character is reaffirmed by using melodic intervals in preference to harmonic, wider ones (minor third or more). 2 days ago here, a user of the Music Theory Reddit group depicts a mind experiment: what happens to a voice that moves a melodic step while the piece modulates a fifth down. What does it become in the new key? The pattern (graph) ends to be 2D and topologically non-trivial, living in a Möbius strip.

As application he shows how to voice-lead an iterated secondary dominant sequence, just downing a semitone each note of the whole block each time. Wikipedia tells us that a 13th chord functions as dominant, as also the jazzy alt versions, that are defined by bearing all possible alterations that don’t denaturalize them. So the progression alternates two types of voicing of dominant chords. Harmony dictates to end at some point with a tonic chord. The more secondary dominants are stacked on the tonic, the further the modulation goes counterclock-wise from the starting tonality of the tonic chord.

3. Steven Sagaert says:

Instead of a 5D hypercube you could draw it as a 2D graph though. So we have objects (modes) en transformations between them… Next step: category theory of music :)

4. John Baez says:

Adam Neely claims to have studied all 7-note modes (i.e. 7-note subsets of the chromatic scale) that have no two half-tones right next to each other, and he made a video about them: