The Bebop Major Scale

30 April, 2023

Though people say ‘octave’, there are only 7 different notes in the major scale. This is annoying if you’re trying to play scales in melodies with, say, 8 beats per measure. The scale keeps drifting out of synch.

One solution is to add an extra note to your scale! In this video, jazz cat Adam Maness explains the virtues of the ‘bebop major scale’, where you add a minor 6th to the major scale:

1 2 3 4 5 ♭6 6 7

Just playing this scale up and down, 8 beats per measure, already suggests some melodies. Even more so if you play it with both hands in ‘contrary motion’—up with one hand, down with the other. Listen to the video and you’ll see what I mean! This is just the start of the interesting things you can do with the bebop major scale.

Why do they call this the ‘major sixth diminished scale’? The jazz pianist and educator Barry Harris introduced this term: he said this scale is derived from a major 6th chord (1 3 5 6) and a diminished 7th chord starting at the 2 (that is, 2 4 ♭6 7).

You can get other bebop scales by putting the extra note somewhere else:

• Wikipedia, Bebop scale.

Sometimes the extra note in the bebop major scale is described as a ♯5 rather than a ♭6:

The Lydian Dominant Scale

22 April, 2023

I got excited when I learned about the Lydian dominant scale:

1 2 3 #4 5 6 ♭7

and could sing it without much effort on the first try. I seem to be getting better at this sort of thing!

More importantly, it’s a really cool scale. Rob van Hal’s video explains what it sounds like and what you can do with it:

You can think of it as a blend of the Lydian mode, with its mystical raised 4th:

1 2 3 #4 5 6 7

and the Mixolydian, with its lowered 7th:

1 2 3 4 5 6 ♭7

But why does this image from van Hal’s video call the Mixolydian mode ‘dominant’? Probably because it contains the dominant 7th chord, where we take a major triad and extend it by playing a minor 7th on top:

1 3 5 ♭7

Indeed, you can get all the notes of the Lydian dominant scale from a chord where you stack the triad 2 #4 6 on top of a dominant 7th. Since we’re playing them an octave higher, we add seven to 2, #4, and 6 and get this:

1 3 5 ♭7 9 #11 13

This is a nice jazzy chord called a ‘dominant 13th sharp 11’. And if you play it in C, musicians might call it a ‘C13#11 chord’ because the dominant 7th is so prevalent that it’s taken for granted in this notation!

But if you play this chord you might leave out the 3 and 5, which are so taken for granted that actually playing them just muddies the chord with unnecessary notes. So you might just play this:

1 ♭7 9 #11 13

There are lots of other ways to play it.

Another way to think about the Lydian dominant scale is that it’s the 4th mode of melodic minor ascending. Melodic minor is a scale you play like this when you’re going up:

1 2 ♭3 4 5 6 7

If you start it on the 4 you get a Lydian dominant scale. Think about it!

Another name for the Lydian dominant scale is the ‘acoustic scale’. This is because you can get approximately get the notes of this scale by playing all the overtones of a single tone… and quitting at the right point. Read this for the details:

• Wikipedia, Acoustic scale.

To see how people apply the Lydian dominant scale in music, start with van Hal’s video. Then take it to the next level by watching Peter Martin and Adam Maness explain how to dominate the Lydian dominant:

For me, watching these guys talk about music is like watching Ed Witten talk about physics. I get bits here and there. It all goes by too fast. Still, it’s tremendously exciting!

They describe and illustrate several things you can do in jazz with the Lydian dominant scale. Adam Maness tries to go through topics systematically, from simple to complicated. But Peter Martin—the pianist for Dianne Reeves, and a very cool cat—interrupts and talks about fancier topics, so the end result is a bit disorganized.

Nonetheless I really benefit from hearing actual jazz theory experts talk about this stuff. At the very least, it gives me a useful sense of how crappy I am at music theory, and playing the piano. But not in a depressing way. It makes me want to improve!

If you want do jam in Lydian dominant, you can use this backing track:

If you don’t, listening to it is still a good way to absorb the vibe of Lydian dominant. There are tons of backing tracks like this on YouTube.


5 April, 2023

Here you can listen to some cool Ethiopian scales called ‘kignits’:

Listen to Ethiopian kignits.

Twenty of them, all pentatonic! One of the most popular is Ambassel 1, shown above. If you listen to just one, please try that!

Ambassel 1 sounds good because of its charming mix of symmetry and asymmetry. If you count the octave, this scale consists of two identical blocks. But each of these blocks is very lopsided: first a half step and then four half steps!

You can think of Ambassel 1 as a subset of the Phrygian mode: it has the 1st, diminished 2nd, perfect 4th, perfect 5th and minor 6th.

Some kignits are just modes of the familiar major pentatonic scale—the one you can get by playing just the black keys on the piano. This is a fave in jazz and rock. But other kignits are new tools you can add to your toolkit.

Here are some other fun pentatonic scales. The framing is practically a textbook study in ‘exoticizing the other’, and it verges on insulting: for example calling something the ‘Hindu scale’ neglects the long list of ragas in Hindustani classical music. Some of the ‘exotic’ flavor of these scales comes from how Rob van Hal subtly bends the notes while playing them. But he explains these scales clearly and illustrates them with nice examples:

Listen to him rock out in the raga called Asavari.

Upper Structures

16 February, 2023


Here Aimee Nolte shows us some ways we can stack a triad on top of the dominant 7th chord

1 3 5 ♭7

She plays this chord in C so it’s called C7. The 5th is so obvious and bland she often leaves it out, giving what’s called a ‘shell voicing’

1 3 ♭7

Then she puts a triad on top of this, giving ‘upper structures’ like a 9th, 11th or 13th.

This is right at my level now, so I like it. Let me say a bit about it.

The first one she plays is called a ‘C dominant 13 flat 9’. This amounts to

1 3 ♭7 ♭9 10 13

10 is an octave up from 3 so she calls it a 3, and she explains why it’s not bad to double the 3 an octave up this way.

More importantly she points out that you can think of the extra notes

♭9 10 13

as a major triad (in first inversion). So we’re stacking a triad on top of dominant 7th chord.

She jokes: “Dude my chords are stacked.”

She then stacks other major triads on the dominant 7th and sees what they give.

For example, the one I just explained stacks an A major triad on the C7 chord and gives us the C dominant 13 flat 9 or C13♭9 chord.

But she rules out the triads that include the 4 or 7, or more precisely the notes an octave higher than those, the 12 and 15. The first is a minor 9th above our 3, the second a minor 9th above our ♭7. The minor 9th is dissonant!

"No bueno", she decrees.

Being a bit rebellious, I want to practice all ways of stacking a triad on top of a dominant 7th chord, including the dissonant "no bueno" ones. I might decide I like those dissonant ones… or maybe not.

Here is a chart of upper structure chords:

Upper Structure LH    RH    RH Triad Chord Name
USII E B♭ D F# A E♭ C7#11
US♭III E B♭ E♭ G B♭ E♭ C7#9
US♭V E B♭ D♭ G♭ B♭ G♭ C7♭9#11
US♭VI E B♭ E♭ A♭ C A♭ C7#9♭13
USVI E B♭ A C# E A C7♭9 13
USIm E B♭ E♭ G C Cm C7#9
US♭IIm E B♭ F♭ A♭ D♭ D♭m C7♭9♭13
US♭IIIm E B♭ E♭ G♭ B♭ E♭m C7#9#11
US#IVm E B♭ C# F# A F#m C7♭9#11
USVII E B♭ F B♭ D B♭ C7sus

These chords have standard names in jazz, which are listed under Upper Structure. For example the first is USII, meaning an upper structure chord where we stack a major II on the dominant 7th. If you know what you’re doing, these names are enough to reconstruct the chords!

The column LH say what the left hand plays if you’re using the key of C. Here, unlike in Aimee Nolte’s video, the left hand always plays a ‘rootless’ voicing of the dominant 7th. That, instead of playing the dominant 7th in its original form:

1 3 5 ♭7

we leave out the 5th and also the root, so we play only

3 ♭7

which in C means the two notes E and B♭.

The column RH says what the right hand plays. It’s always a triad, which is shown under RH Triad. So RH Triad gives a compressed version of the same information, and a good musician would probably prefer this compressed form.

You may wonder why the chords are listed in the order they are. It’s not random! The chords with major triads are listed first, then the chords with minor triads, and finally one with a diminished triad: USVII.

These chords also have other names, which are listed under Chord Name. These names include the key the chord is played in: for simplicity they’re listed here in the key of C. Here USII is also called C7#11, because it’s a dominant 7th with a 9th and a sharped 11th stacked on top. The 9th is left unspoken here: we could call this chord C79#11 but people consider that notation inefficient. But the 9th were sharped or flatted, we need to mention it. For example US♭V is C7♭9#11.

I find these other names confusing, because they’re not completely logical and people seem to use different conventions. I’ve gotten the names from this page:

Upper structures.

but other names seem more common. For example, the USVI chord, which I discussed at length above, is usually called C13♭9, but this page calls it C7♭9. That makes no sense to me since it omits the 13. So was it a typo, or just a notation I don’t understand? In an effort at logical consistency I made up a name I’ve never seen elsewhere: C7♭9 13. Please don’t take this one seriously!

Do you know a really careful, detailed treatment of upper structure chords, with nice complete charts?

If you find this stuff intimidating or otherwise unpleasant, I recommend ignoring it and watching Aimee Nolte’s video, since she derives the information in this list from first principles in a friendly commonsense way.

Alternatively, you can just sit down at a piano, play some of the chords, and enjoy how they sound. I’m nowhere near able to deploy these chords in improvisation, the way a good jazz musician could—mainly because I don’t tend to improvise with dominant 7ths in the left hand. But at least I can play these chords and enjoy them! And I hope someday I’ll incorporated them in my improvisations.

The Italian Sixth

5 February, 2023

I showed my wife Lisa a nice video of Tommaso Zillio explaining a chord called the ‘Italian 6th’:

But she had a complaint: where the chords finally resolve to a C major triad, he writes the final chord as CCE—but those first two notes don’t sound like they’re an octave apart!

What do you think? Here’s the video:

Just for my own sake, let me explain how an Italian 6th works.

Zillio explains it in the key of C but I prefer numbers. One of the big motors that drives classical music is the progression


from IV, called the ‘subdominant’, to V, called the ‘dominant’, to I, called the ‘tonic’. Tension followed by release! If we write these chords as triads they are

IV = 4 6 8
V = 5 7 9
I = 1 3 5

But we’d probably invert these chords for good voice leading: we don’t want notes making big jumps.

Most importantly, the 7 wants to climb up to the 8, which is one octave higher than the 1. Also, the 9 would be happy to climb up to the 10—which is an octave higher than the 3. So let’s raise the 1 and 3 an octave and get 8 and 10:

IV = 4 6 8
V = 5 7 9
I = 5   8 10

Since we’ve raised two notes, now we say the I is in ‘second inversion’.

Next: people like to heighten the tension of the V by adding an extra note and getting a 7th chord, called V7 or the ‘dominant 7th’:

IV = 4 6 8
V7 = 5 7 9 11
I =   5   8 10

This is an incredibly common chord progression, often decorated in various ways. Indeed, the music theorist Richard Goldman wrote:

The demand of the V7 for resolution is, to our ears, almost inescapably compelling. The dominant seventh is, in fact, the central propulsive force in our music; it is unambiguous and unequivocal.

Next, we could invert the IV chord by pushing the 4 up an octave to get 11:

IV =   6 8   11
V7 = 5 7 9 11
I =   5   8 10

Now the 11 just sits there for the first two chords. We could get a cooler sound by sharping that 11:

IV#° = 6 8   11#
V7 =   5 7 9 11
I =     5 8 10

The first chord is more fancy now: it’s called a ‘diminished sharp IV in first inversion’. (A IV chord is 4 6 8. If we push up the 4 by a half-tone we get 4# 6 8: now all three notes are a minor third apart, so it’s a ‘diminished’ triad, called a diminished sharp IV. To get a diminished sharp IV in first inversion we move the bottom note an octave higher and get 6 8 11#, which is what we have here.)

In this new chord progression the top voice nicely descends by half-steps from the 11# to the 11 to the 10:

IV#° = 6 8   11#
V7 =   5 7 9 11
I =     5 8 10

or in other words, subtracting an octave so it’s easier to understand, from the 4# to the 4 to the 3.

Next, we can flat the 6 in that first chord:

It6 = 6♭ 8   11#
V7 = 5 7 9 11
I =   5 8   10

Now the first chord is called the Italian 6th.

This progression is a much loved variant of our original basic IV V I. The interval from the 6♭ to the 4# is called an augmented 6th, so you may read about the Italian 6th in discussions of how to use the augmented 6th interval.

There are also two popular variants of the Italian 6th which add an extra note to the chord, called
the ‘German 6th’:

and the ‘French 6th’:

But watch the videos for more on those!

The Circle of Fifths

12 November, 2022

The circle of fifths is a beautiful thing, fundamental to music theory.

Sound is vibrations in air. Start with some note on the piano. Then play another note that vibrates 3/2 times as fast. Do this 12 times. Since

(3/2)¹² ≈ 128 = 2⁷

when you’re done your note vibrates about 2⁷ times as fast as when you started!

Notes have letter names, and two notes whose frequencies differ by a power of 2 have the same letter name. So the notes you played form a 12-pointed star:

Each time you increase the frequency by a factor of 3/2 you move around the points of this star: from C to G to D to A, and so on. Each time you move about 7/12 of the way around the star, since

log(3/2) / log(2) ≈ 7/12

This is another way of stating the approximate equation I wrote before!

It’s great! It’s called the circle of fifths, for reasons that don’t need to concern us here.

But this pattern is just approximate! In reality

(3/2)¹² = 129.746…

not 128, and

log(3/2) / log(2) = 0.58496…

not 7/12 = 0.58333… So the circle of fifths does not precisely close:

The failure of it to precisely close is called the Pythagorean comma, and you can hear the problem here:

This video plays you notes that increase in frequency by a factor of 3/2 each time, and finally two notes that differ by the Pythagorean comma: they’re somewhat out of tune.

People have dealt with this in many, many ways. No solution makes everyone happy.

For example, the equal-tempered 12-tone scale now used on most pianos doesn’t have ‘perfect fifths’—that is, frequency ratios of 3/2. It has frequency ratios of

2^{7/12} \approx 1.4983

I have tried in this blog article to be understandable by people who don’t know standard music theory terminology—basic stuff like ‘octaves’ and ‘fifths’, or the letter names for notes. But the circle of fifths is very important for people who do know this terminology. It’s a very practical thing for musicians, for example if you want to remember how many sharps or flats there are in any key. Here’s a gentle introduction to it by Gracie Terzian:

Here she explains some things you can do with it:

Here’s another version of the circle of fifths made by “Just plain Bill”>—full of information used by actual musicians:

If you watch Terzian’s videos you’ll learn what all this stuff is about.

Modes (Part 2)

6 November, 2022

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


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!

Modes (Part 1)

1 November, 2022

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:

w h w w w h w

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:

w w w h w w h

we get the darkest, Locrian:

h w w h w w w

Inverting the 2nd brightest, the happy Ionian (our familiar friend the major scale):

w w h w w w h

we get the 2nd darkest, Phrygian:

h w w w h w w

Inverting the third brightest, Mixolydian:

w w h w w h w

we get the third darkest, the sad Aeolian (our friend the natural minor):

w h w w h w w

And right in the middle is the palindromic Dorian:

w h w w w h w

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 \mathbb{Z}/7 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 \mathbb{Z}/2 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 \mathrm{D}_{14} on the modes. This is the semidirect product \mathbb{Z}/2 \ltimes \mathbb{Z}/7. 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 \mathrm{D}_{14} group here is a baby version of the \mathrm{D}_{24} 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).

Clemens non Papa

25 December, 2021

As I’ve explored more music from the Franco-Flemish school, I’ve gotten to like some of the slightly less well-known composers—though usually famous in their day—such as Jacobus Clemens non Papa, who lived in Flanders from roughly 1510 to 1555. I enjoy his clear, well-balanced counterpoint. It’s peppy, well-structured, but unromantic: no grand gestures or strong emotions, just lucid clarity. That’s quite appealing to me these days.

On a website about Flemish music I read that:

The style of his work stayed “northern”, without any Italian influences. As far as is known Clemens never ventured out of the Low Countries to pursue a career at a foreign court or institution, unlike many of his contemporaries. This is reflected in most of his religious pieces, where the style is generally reliant on counterpoint arrangements where every voice is independently formed.

Not much is known of his life. The name ‘Clemens non Papa’ may be a bit of a joke, since his last name was Clemens, but there was also a pope of that name, so it may have meant ‘Clemens — not the Pope’.

That makes it all the more funny that if you look for a picture of Clemens non Papa, you’ll quickly be led to Classical, which has a nice article about him—with this picture:

Yes, this is Pope Clement VII.

Clemens non Papa was one of the best musicians of the fourth generation of the Franco-Flemish school, along with Nicolas Gombert, Thomas Crequillon and my personal favorite, Pierre de Manchicourt. He was extremely prolific! He wrote 233 motets, 15 masses, 15 Magnificats, 159 settings of the Psalms in Dutch, and a bit over 100 secular pieces, including 89 chansons.

But unfortunately, he doesn’t seem to have inspired the tireless devotion among modern choral groups that more famous Franco-Flemish composers have. I’m talking about projects like The Clerks’ complete recordings of the sacred music of Ockeghem in five CDs, The Sixteen’s eight CDs of Palestrina, or the Tallis Scholars’ nine CDs of masses by Josquin. There’s something about early music that incites such massive projects! I think I know what it is: it’s beautiful, and a lot has been lost or forgotten, so you when you fall in love with it you start wanting to preserve and share it.

Maybe someday we’ll see complete recordings of the works of Clemens non Papa! But right now all we have are small bits—and let me list some.

A great starting-point is Clemens non Papa: Missa Pastores quidnam vidistis by the Tallis Scholars. This whole album is currently available as a YouTube playlist:

Another important album is Behold How Joyful – Clemens non Papa: Mass and Motets by the Brabant Ensemble. It too is is available as a playlist on YouTube:

The Brabant Ensemble have another album of Clemens non Papa’s music, Clemens non Papa: Missa pro defunctis, Penitential Motets. I haven’t heard it.

Next, the Egidius Kwartet has a wonderful set of twelve CDs called De Leidse Koorboeken—yet another of the massive projects I mentioned—in which they sing everything in the Leiden Choirbooks. These were six volumes of polyphonic Renaissance music of the Franco-Flemish school copied for a church in Leiden sometime in the 15th or 16th century, which somehow survived an incident in 1566 when a mob burst into that church and ransacked it.

You can currently listen to the Egidius Kwartet’s performances of the complete Leiden Choirbooks on YouTube playlists:

Volume 2 contains these pieces by Clemens non Papa—click to listen to them:

Heu mihi Domine, a4. Anima mea turbata est, a4.

Maria Magdalena, a5. Cito euntes, a5.

Jherusalem surge, a5. Leva in circuitu, a5.

Magnificat quarti thoni, a4.

Magnificat sexti thoni, a4.

Magnificat octavi toni, a4-5.

Volume 3 contains these:

Cum esset anna, a5.

Domine probasti, a5.

Advenit ignis divinus, a5.

Volume 4 contains these:

Angelus domini ad pastores, a4 – Secunda pars: Parvulus filius, a4.

Pastores loquebantur, a5 – Secunda pars: Et venerunt festinantes, a5.

Congratulamini mihi omnes, a4.

Sancti mei qui in carne – Secunda pars: Venite benedicti patris.

Pater peccavi, a4 – Secunda pars: Quanti mercenarii, a4.

Volume 5 contains this:

Ave Maria.

And finally, the group Henry’s Eight has a nice album Pierre de la Rue: Missa cum incundate, curently available as a YouTube playlist, which includes two pieces by Clemens non Papa:

Here are those pieces—click to hear them:

Ego flos campi.

Pater peccavi.

Here also is a live performance of Ego flos campi by the Choir of St James, in Winchester Cathedral:

Happy listening! And if you know a big trove of recordings of music by Clemens non Papa, let me know. I just know what’s on Discogs.

Jacob Obrecht

15 June, 2021

This is a striking portrait of the “outsider genius” Jacob Obrecht:

Obrecht, ~1457–1505, was an important composer in the third generation of the Franco-Flemish school. While he was overshadowed by the superstar Josquin, I’m currently finding him more interesting—mainly on the basis of one long piece called Missa Maria zart.

Obrecht was very bold and experimental in his younger years. He would do wild stuff like play themes backwards, or take the notes in a melody, rearrange them in order of how long they were played, and use that as a new melody. Paraphrasing Wikipedia:

Combining modern and archaic elements, Obrecht’s style is multi-dimensional. Perhaps more than those of the mature Josquin, the masses of Obrecht display a profound debt to the music of Johannes Ockeghem in the wide-arching melodies and long musical phrases that typify the latter’s music. Obrecht’s style is an example of the contrapuntal extravagance of the late 15th century. He often used a cantus firmus technique for his masses: sometimes he divided his source material up into short phrases; at other times he used retrograde (backwards) versions of complete melodies or melodic fragments. He once even extracted the component notes and ordered them by note value, long to short, constructing new melodic material from the reordered sequences of notes. Clearly to Obrecht there could not be too much variety, particularly during the musically exploratory period of his early twenties. He began to break free from conformity to formes fixes (standard forms) especially in his chansons (songs). However, he much preferred composing Masses, where he found greater freedom. Furthermore, his motets reveal a wide variety of moods and techniques.

But I haven’t heard any of these far-out pieces yet. Instead, I’ve been wallowing in his masterpiece: Missa Maria zart, an hour-long mass he wrote one year before he died of the bubonic plague. Here is the
Tallis Scholars version, with a score:

It’s harmonically sweet: it seems to avoid the pungent leading-tones that Dufay or even Ockeghem lean on. It’s highly non-repetitive: while the same themes get reused in endless variations, there’s little if any exact repetition of anything that came before. And it’s very homogeneous: nothing stands out very dramatically. So it’s a bit like a beautiful large stone with all its rough edges smoothed down by water, that’s hard to get a handle on. And I’m the sort of guy who finds this irresistibly attractive. After about a dozen listens, it reveals itself.

The booklet in the Tallis Scholars version, written by Peter Phillips, explains it better:

To describe Obrecht’s Missa Maria zart (‘Mass for gentle Mary’) as a ‘great work’ is true in two respects. It is a masterpiece of sustained and largely abstract musical thought; and it is possibly the longest polyphonic setting of the Mass Ordinary ever written, over twice the length of the more standard examples by Palestrina and Josquin. How it was possible for Obrecht to conceive something so completely outside the normal experience of his time is one of the most fascinating riddles in Renaissance music.

Jacob Obrecht (1457/8–1505) was born in Ghent and died in Ferrara. If the place of death suggests that he was yet another Franco-Flemish composer who received his training in the Low Countries and made his living in Italy, this is inaccurate. For although Obrecht was probably the most admired living composer alongside Josquin des Prés, he consistently failed to find employment in the Italian Renaissance courts. The reason for this may have been that he could not sing well enough: musicians at that time were primarily required to perform, to which composing took second place. Instead he was engaged by churches in his native land—in Utrecht, Bergen op Zoom, Cambrai, Bruges and Antwerp—before he finally decided in 1504 to take the risk and go to the d’Este court in Ferrara. Within a few months of arriving there he had contracted the plague. He died as the leading representative of Northern polyphonic style, an idiom which his Missa Maria zart explores to the full.

This Mass has inevitably attracted a fair amount of attention. The most recent writer on the subject is Rob Wegman (Born for the Muses: The Life and Masses of Jacob Obrecht by Rob C Wegman (Oxford 1994) pp.322–330. Wegman, Op.cit., p.284, is referring to H Besseler’s article ‘Von Dufay bis Josquin, ein Literaturbericht’, Zeitschrift für Musikwissenschaft, 11 (1928/9), p.18): ‘Maria zart is the sphinx among Obrecht’s Masses. It is vast. Even the sections in reduced scoring … are unusually extended. Two successive duos in the Gloria comprise over 100 bars, two successive trios in the Credo close to 120; the Benedictus alone stretches over more than 100 bars’; ‘Maria zart has to be experienced as the whole, one-hour-long sound event that it is, and it will no doubt evoke different responses in each listener … one might say that the composer retreated into a sound world all his own’; ‘Maria zart is perhaps the only Mass that truly conforms to Besseler’s description of Obrecht as the outsider genius of the Josquin period.’

The special sound world of Maria zart was not in fact created by anything unusual in its choice of voices. Many four-part Masses of the later fifteenth century were written for a similar grouping: low soprano, as here, or high alto as the top part; two roughly equal tenor lines, one of them normally carrying the chant when it is quoted in long notes; and bass. The unusual element is to a certain extent the range of the voices—they are all required to sing at extremes of their registers and to make very wide leaps—but more importantly the actual detail of the writing: the protracted sequences against the long chant notes, the instrumental-like repetitions and imitations.

It is this detail which explains the sheer length of this Mass. At thirty-two bars the melody of Maria zart is already quite long as a paraphrase model (the Western Wind melody, for example, is twenty-two bars long) and it duly becomes longer when it is stated in very protracted note-lengths. This happens repeatedly in all the movements, the most substantial augmentation being times twelve (for example, ‘Benedicimus te’ and ‘suscipe deprecationem nostram’ in the Gloria; ‘visibilium’ and ‘Et ascendit’ in the Credo). But what ultimately makes the setting so extremely elaborate is Obrecht’s technique of tirelessly playing with the many short phrases of this melody, quoting snippets of it in different voices against each other, constantly varying the extent of the augmentation even within a single statement, taking motifs from it which can then be turned into other melodies and sequences, stating the phrases in antiphony between different voices. By making a kaleidoscope of the melody in these ways he literally saturated all the voice-parts in all the sections with references to it. To identify them all would be a near impossible task. The only time that Maria zart is quoted in full from beginning to end without interruption, fittingly, is at the conclusion of the Mass, in the soprano part of the third Agnus Dei (though even here Obrecht several times introduced unscheduled octave leaps).

At the same time as constantly quoting from the Maria zart melody Obrecht developed some idiosyncratic ways of adorning it. Perhaps the first thing to strike the ear is that the texture of the music is remarkably homogeneous. There are none of the quick bursts of vocal virtuosity one may find in Ockeghem, or the equally quick bursts of triple-time metre in duple beloved of Dufay and others. The calmer, more consistent world of Josquin is suggested (though it is worth remembering that Josquin may well have learnt this technique in the first place from Obrecht). This sound is partly achieved by use of motifs, often derived from the tune, which keep the rhythmic stability of the original but go on to acquire a life of their own. Most famously these motifs become sequences—an Obrecht special—some of them with a dazzling number of repetitions (nine at ‘miserere’ in the middle of Agnus Dei I; six of the much more substantial phrase at ‘qui ex Patre’ in the Credo; nine in the soprano part alone at ‘Benedicimus te’ in the Gloria. This number is greatly increased by imitation in the other non-chant parts). Perhaps this method is at its most beautiful at the beginning of the Sanctus. In addition the motifs are used in imitation between the voices, sometimes so presented that the singers have to describe leaps of anything up to a twelfth to take their place in the scheme (as in the passage beginning ‘Benedicimus te’ in the Gloria mentioned above). It is the impression which Obrecht gives of having had an inexhaustible supply of these motifs and melodic ideas, free or derived, that gives this piece so much of its vitality. The mesmerizing effect of these musical snippets unceasingly passing back and forth around the long notes of the central melody is at the heart of the particular sound world of this great work.

When Obrecht wrote his Missa Maria zart is not certain. Wegman concludes that it is a late work—possibly his last surviving Mass setting—on the suggestion that Obrecht was in Innsbruck, on his way to Italy, at about the time that some other settings of the Maria zart melody are known to have been written. These, by Ludwig Senfl and others, appeared between 1500 and 1504–6; the melody itself, a devotional monophonic song, was probably written in the Tyrol in the late fifteenth century. The idea that this Mass, stylistically at odds with much of Obrecht’s other known late works and anyway set apart from all his other compositions, was something of a swansong is particularly appealing. We shall never know exactly what Obrecht was hoping to prove in it, but by going to the extremes he did he set his contemporaries a challenge in a certain kind of technique which they proved unable or unwilling to rival.

This Gramophone review of the Tallis Scholars performance, by David Fallows, is also helpful:

This is a bizarre and fascinating piece: and the disc is long-awaited, because The Tallis Scholars have been planning it for some years. It may be the greatest challenge they have faced so far. Normally a Renaissance Mass cycle lasts from 20 to 30 minutes; in the present performance, this one lasts 69 minutes. No ‘liturgical reconstruction’ with chants or anything to flesh out the disc: just solid polyphony the whole way. It seems, in fact, to be the longest known Renaissance Mass.

It is a work that has long held the attention of musicologists: Marcus van Crevel’s famous edition was preceded by 160 pages of introduction discussing its design and numerology. And nobody has ever explained why it survives in only a single source—a funny print by a publisher who produced no other known music book. However, most critics agree that this is one of Obrecht’s last and most glorious works, even if it leaves them tongue-tied. Rob C. Wegman’s recent masterly study of Obrecht’s Masses put it in a nutshell: “Forget the imitation, it seems to tell us, be still, and listen”.

There is room for wondering whether all of it needs to be quite so slow: an earlier record, by the Prague Madrigal Singers (Supraphon, 6/72 – nla), got through it in far less time. Moreover, Obrecht is in any case a very strange composer, treating his dissonances far more freely than most of his contemporaries, sometimes running sequential patterns beyond their limit, making extraordinary demands of the singers in terms of range and phrase-length. That is, there may be ways of making the music run a little more fluidly, so that the irrational dissonances do not come across as clearly as they do here. But in most ways it is hard to fault Peter Phillips’s reading of this massive work.

With only eight singers on the four voices, he takes every detail seriously. And they sing with such conviction and skill that there is hardly a moment when the ear is inclined to wander. As we have come to expect, The Tallis Scholars are technically flawless and constantly alive. Briefly, the disc is a triumph. But, more than that, it is a major contribution to the catalogue, unflinchingly presenting both the beauties and the apparent flaws of this extraordinary work. Phew!

My ear must be too jaded by modern music to notice the dissonances.