I’ve been away from my piano since September. I really miss playing it. So, I’ve been sublimating my desire to improvise on this instrument by finally learning a bunch of basic harmony theory, which I practice just by singing or whistling.
For example, I’m getting into modes. The following 7 modes are all obtained by taking the major scale and starting it at different points. But I find that’s not the good way for me to understand the individual flavor of each one.
Much better for me is to think of each mode as the major scale (= Ionian mode) with some notes raised or lowered a half-step — since I already have an intuitive sense of what that will do to the sound:
For example, anything with the third lowered a half-step (♭3) will have a minor feel. And Aeolian, which also has the 6th and 7th lowered (♭6 and ♭7), is nothing but my old friend the natural minor scale!
A more interesting mode is Dorian, which has just the 3rd and 7th notes lowered a half-step (3♭ and 7♭). Since this 6th is not lowered this is not as sad as minor. You can play happy tunes in minor, but it’s easier to play really lugubrious tear-jerkers, which I find annoying. The major 6th of Dorian changes the sound to something more emotionally subtle. Listen to a bunch of examples here:
Some argue that the Dorian mode gets a peculiarly ‘neutral’ quality by being palindromic: the pattern of whole and half steps when you go up this mode is the same as when you go down:
This may seem crazily mathematical, but Leibniz said “Music is the pleasure the human mind experiences from counting without being aware that it is counting.”
Indeed, there is a marvelous theory of how modes sound ‘bright’ or ‘dark’ depending on how many notes are sharped—that is, raised a half-tone—or flatted—that is, lowered a half-tone. I learned about it from Rob van Hal, here:
The more notes are flatted compared to the major scale, the ‘darker’ a mode sounds! The fewer are flatted, the ‘brighter’ it sounds. And one, Lydian, is even brighter than major (= Ionian), because it has no flats and one sharp!
So, let’s list them from bright to dark. Here’s a chart from Rob van Hal’s video:
You can see lots of nice patterns here, like how the flats come in ‘from top down’ as the modes get darker: that is, starting at the 7th, then the 6th and then the 5th… but also, interspersed with these, the 3rd and then the 2nd.
But here’s something even cooler, which I also learned from Rob van Hal (though he was surely not the first to discover it).
If we invert each mode—literally turn it upside down, by playing the pattern of whole and half steps from the top of the scale down instead of from bottom to top—the brighter modes become the darker modes, and vice versa!
Let’s see it! Inverting the brightest, Lydian:
we get the darkest, Locrian:
Inverting the 2nd brightest, the happy Ionian (our familiar friend the major scale):
we get the 2nd darkest, Phrygian:
Inverting the third brightest, Mixolydian:
we get the third darkest, the sad Aeolian (our friend the natural minor):
And right in the middle is the palindromic Dorian:
What a beautiful pattern!
By the way, it’s also cool how both the ultra-bright Lydian and the ultra-dark Locrian, and only these modes, have a note that’s exactly half an octave above the 1. This is a very dissonant thing for a mode to have! In music jargon we say it like this: these modes have a note that’s a tritone above the tonic.
In Lydian this note is the sharped 4th, which is a ‘brighter than usual 4th’. In Locrian it’s the flatted 5th, which is a ‘darker than usual 5th’. But these are secretly the same note, or more technically ‘enharmonic equivalents’. They differ just in the role they play—but that makes a big difference.
Why do both Lydian and Locrian have a note that’s a tritone above the tonic? It’s not a coincidence: the tritone is mapped to itself by inversion of the octave, and inversion interchanges Lydian and Locrian!
This stuff is great, especially when I combine it with actually singing in different modes and listening to how they sound. Why am I learning it all just now, after decades of loving music? Because normally when I want to think about music I don’t study theory—I go to the piano and start playing!
The mathematics of modes
We clearly have an action of the 7-element cyclic group on the set of modes I’m talking about: they’re defined by taking the major scale and cyclically permuting its notes. But as we’ve seen, inversion gives an action of on the set of modes, with Dorian as its only fixed point.
Putting these two groups together, we get an action of the 14-element dihedral group on the modes. This is the semidirect product More intuitively, it’s the symmetry group of the regular heptagon! The modes can be seen as the vertices of this heptagon.
We’ve also seen the modes have a linear ordering by ‘brightness’. However, this ordering is preserved by the symmetries I’ve described: only the identity transformation preserves this linear ordering.
All this should have been studied in neo-Riemannian music theory, but I don’t know if it has—so if you know references, please tell me! The group here is a baby version of the group often studied in neo-Riemannian theory. For more, see:
• Alissa S. Crans, Thomas M. Fiore and Ramon Satyendra, Musical actions of dihedral groups, American Mathematical Monthly 116 (2009), 479–495.
More on individual modes
For music, more important than the mathematical patterns relating different modes is learning the ‘personality’ of individual modes and how to compose or improvise well in each mode.
Here are some introductions to that! Since I’m in awe of Rob van Hal I will favor his when possible. But there are many introductions to each mode on YouTube, and it’s worth watching a lot, for different points of view.
Locrian is so unloved that I can’t find a good video on how to compose in Locrian. Instead, there’s a good one on how Björk created a top 20 hit that uses Locrian:
and also a good one about Adam Neely and friends trying to compose in Locrian:
For more, read Modes (part 2).