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

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!


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 connect.com, 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.


Renaissance Polyphony: the Franco-Flemish School

29 April, 2021

Near the beginning of the pandemic I heard a conversation where Brian Eno said that he’d been listening to a radio station on the internet, based somewhere deep in rural Russia, that plays nothing but Eastern Orthodox chants 24 hours a day, with no announcements. He said that this sort of music appealed to him while locked down at home.

Somehow this led to me listening to a lot of early music on YouTube. I started with some Baroque composers I’d never paid attention to before, like Corelli and Albinoni. Previously my interest in classical music started around Beethoven and focused on the 20th century, with Bach and Vivaldi as brilliant pearls surrounded by darkness. As I began listening to more Baroque music I became very happy, like someone who’d been locked in a room for years and suddenly found a key to another room. There was so much to explore.

The Baroque period lasted from about 1580 to 1750: that’s almost two centuries of music! By getting to know it, I’ve been starting to understand the birth of common practice tonality—that is, the language of chord progressions that dominates classical music, and also, to some extent, lots of modern pop music. And I can finally start thinking clearly about what makes Bach great.

Imagine listening to just one rock group, or one jazz band. Practically nobody does that, of course, but imagine it: it would be impossible to distinguish what’s special to the artist you love from what’s common to the style as a whole. So you’d think they were more creative than they really are—but you also wouldn’t know what was truly creative about their work.

I’m still in the midst of this project, which also includes listening to Bach more thoroughly. But I got a bit distracted. I got pulled back further in time—back into Renaissance polyphony. This gave me access to another two centuries of rich, complicated music, roughly 1400 to 1600. And it let me understand the roots of Bach’s music. Bach’s music looks back to polyphony just as much as it looks forward into the future. For this he was considered a bit old-fashioned during his day; only later did people realize his greatness.

The best of Renaissance polyphony is arguably just as complex and interesting as Bach’s counterpoint: it’s just less immediately gripping. The reason is that it doesn’t follow ‘common practice’, with its familiar strategies of building and releasing tension, and its repetitive rhythmic pulse. The vocal lines are often very smooth, flowing like water. So it’s less exciting, but it’s wonderfully entrancing.

I went back even further, into the medieval—but at a certain point I ‘hit bottom’, at least when it comes to the art of harmony and juggling multiple independent melodic lines, which is what I like so much about Baroque counterpoint and Renaissance polyphony. Gregorian chant is great, but it has a single vocal line. The practice of using chords was first documented around 895: in a style called the organum, Gregorian chant was supplemented by either a supporting bass line, two parallel voices singing the melody, or both. But the ‘Big Bang’ of polyphony happened around 1170 when Léonin introduced two independent melody lines. Around 1200 his follower Pérotin went ahead and started using three or four! Imagine the dizzying sense of freedom these guys must have felt, with no precedents to guide them.

Medieval polyphony should be fascinating, and again it’s a huge territory: roughly two centuries, from 1200 to 1400. But right now I seem to be focusing on Renaissance polyphony—and especially the so-called Franco-Flemish school, which is an incredibly rich vein of music.

I’d like to say a lot about the Franco-Flemish school, but for now I just want to list a few of their best-known composers. In fact, I’ll only mention one of each ‘generation’, and give you a sample of their music.

Guillaume Dufay (1397 – 1474)

Dufay is the most famous of the first generation of the Franco-Flemish school. (This first generation is also called the Burgundian School.) He is often considered a transitional figure from the medieval to the Renaissance. His isorhythmic motets illustrate that—their tonality is dissonant and dramatic compared to typical Renaissance polyphony:

Johannes Ockeghem (1410/1425 – 1495)

Ockeghem is the most famous of the second generation of the Franco-Flemish school. His innovations firmly moved this school out of the medieval into the world of Renaissance polyphony. Some of his compositions are almost avant-garde in their structure—a mass that asks you to sing it in any of four modes, a mass where the different vocal lines are sung at different rates and drift out of synch—but they’re carried off so smoothly you might not notice. People tend to call Ockeghem “the Bach of the 1400s”, which is sort of ridiculous because there’s just one J. S. Bach, but it gives a hint of his importance. He was not a prolific composer, but he was also an honored singer, choirmaster, and teacher.

Josquin de Prez (~1450 – 1521)

Josquin, of the third generation, is the real superstar of the Franco-Flemish school. Luther wrote that “He is the master of the notes. They must do as he wills; as for the other composers, they have to do as the notes will.” He was influenced by Ockeghem, and composed a motet in commemoration of Ockeghem’s death. But he polished Ockeghem’s ideas and in some ways simplified them, making music that was easier to understand and more popular. Despite being prolific and very influential, nothing is known of his personality, and the only known writing that may be in his own hand is a piece of graffiti on the wall of the Sistine Chapel.

Nicholas Gombert (1495 – 1460)

Gombert is probably the most famous of the fourth generation of the Franco-Flemish school. He wrote polyphonic masses and motets with as many as 6 separate melody lines, and sometimes with one voice imitating another after a very short time interval. His work marks the height of complexity of the Franco-Flemish school.

Orlande de Lassus (1530 – 1594)

Lassus, of the fifth generation of the Franco-Flemish school, is one of the composers of a style known as musica reservata—roughly, sophisticated and highly chromatic music. But he also wrote drinking songs in German, and one of his motets satirizes poor singers. He wrote over 2,000 pieces of music, all vocal—none purely instrumental! In the “The Adventure of the Bruce-Partington Plans,” Sherlock Holmes said he was working on a monograph about the polyphonic motets of Lassus.

So there you are: a microscopic overview of a musical tradition lasting over 200 years.

Maybe listing one just composer of each generation was a bad idea: I feel I’m doing an injustice to Gilles Binchois of the first generation, often considered the finest melodist of the 1400s, and Jacob Obrecht of the third generation, who was the most famous composer of masses in the late 1400s before Josquin came along. Obrecht was very adventurous: he often played melodies backwards (in retrograde), and once he even took the notes from a melody and played them in order of duration, long to short, to get a new melody.

There is also a lot going on in Renaissance polyphony outside the Franco-Flemish school! There’s the British tradition, including great composers such as William Byrd and Thomas Tallis. And I haven’t even mentioned the most famous of all polyphonists: Giovanni Pierluigi da Palestrina! Or the most dissonant and in some ways the most intriguing of the lot: Carlo Gesualdo.

But it’s easy to get lost in unfamiliar territory, so I just wanted to give a quick outline of the Franco-Flemish school. At the very least, listening to their music should be fun.


Dufay’s Isorhythmic Motets

23 April, 2021

I’ve been reading about Renaissance music. People sometimes say that it began in the early 1400s when musicians rebelled against the dry, complicated mathematical structures of late medieval music and switched to a more emotionally expressive style. For example, the New Oxford History of Music writes:

The isorhythmic motet, the highest achievement of medieval rationalism, reached its climax during Dufay’s prentice years (c. 1410-20), with works in which the quasi-mathematical construction arouses more admiration than pleasure.

But since I’m a mathematician, this actually got me interested in isorhythmic motets!

I found them hard to understand from written descriptions. Isorhythm involves a rhythmic pattern called a talea which is applied to different melodies. Often the talea lasts for a different amount of time than the melody, which leads to some interesting effects. For example, here is a melody that lasts for 28 measures divided into 7 talea that each last for 4 measures:

This is from a composition written sometime around 1360. Isorhythm gets a lot more complicated than this! The music typically has several parts, in which talea get sped up or slowed down independently. But reading about these things didn’t give me much of a feel for what isorhythmic motets actually sound like.

When I listened to some by Guillaume Dufay, they didn’t sound dry at all! For example:

It’s quite thrilling and romantic, actually! Listen to how he uses the leading-tone, the note one half a step below the tonic, to build tension. This is a big medieval thing. If you don’t know what the heck I’m talking about, wait to the very end of the piece! Here Dufay clobbers us with a loooong leading-tone, the medieval equivalent of the wail of electric guitar at the end of a classic rock song. He’s really hamming it up.

Now, Dufay is famous for being the first really major Renaissance composer and breaking from the medieval traditions, so maybe his isorhythmic motets are more exciting than average. But still, I hear they are intensely mathematical.

In fact there’s an album of Dufay’s isorhythmic motets with the great title Quadrivium which features a booklet by a mathematical physicist named Guido Magnano, an expert on general relativity at the University of Turin. On a Russian website I read:

This album by the Italian vocal-instrumental group Cantica Symphonia takes off from the proportional aspect of large Dufay works like the motet Nuper rosarum flores, long thought to have been based on the proportions of the great cathedral of Florence but recently discovered to have probably been modeled on another building. The album was actually sponsored by the mathematics department of the University of Turin, and an essay by professor Guido Magnano explores the mathematical bases of the musical system Dufay knew. For the average listener the musical manifestations of these principles are going to be hard to hear sitting in front of your stereo; the chief interpretive decision made by Cantica Symphonia is to strive for a transparent texture, judiciously using a small instrumental ensemble to bring out structural details. Save for the fact that the voice parts are sung solo, it’s a Renaissance performance in the classic “pure” mold. For the numerologically inclined or for the serious student of the Renaissance era, the disc is an interdisciplinary goldmine. Recorded in an Italian church, the disc matches its engineering to its aims, and the packaging by the Spanish label Glossa is uncommonly attractive.

I decided I needed to get my hands on that booklet. The album is on YouTube:

Unfortunately the actual CD costs $47.53 on Amazon. Luckily I was able to get it for much less online from Barnes and Noble… thereby procuring the booklet, which is what I really want. This should arrive in a week or so, so with luck I’ll tell you more later. I’m also quite fascinated by Dufay and the whole Franco-Flemish school of Renaissance music that he helped start, and—as usual when I’m just starting to learn about something—I have dreams of blogging about it.

In the meantime, I found out a bit from an interview with Guido Magnano, where he says this:

Your first disc for Glossa, Quadrivium, placed special emphasis on the question of mathematical proportions in Dufay’s motets. If these considerations apply also to the works recorded here, can you provide some examples of how this came through in practice?

Guido Magnano: Mathematical proportions do not occur in medieval and Renaissance music as occasional, accessory stylistic elements: the Pythagorean-Platonic paradigm states that music itself is nothing but “auditory perception of numbers”. The hypothetical relationship between the mensural proportions of the motet Nuper rosarum flores and the proportions of Brunelleschi’s Duomo, although fascinating and quite plausible, should not obscure that in other motets, particularly in the later isorythmic motets (Fulgens iubar and Moribus et genere), Dufay attains an even higher degree of formal complexity.

The motet Magnam me gentes, (12:4:2:3) also included in this CD, has a mensural structure very close to Nuper (6:4:2:3). Worth noting is that the 15th century humanist Marsilio Ficino introduced a “Platonic-Hermetic” movement, attributing occult significance to numerical relations. Did Dufay himself share these ideas? Do the numerical ratios in his motets hide a symbolic content? Some modern scholars have claimed so, even though the pieces for which numerological interpretations have been proposed were written some thirty years before Ficino’s works, and it is impossible to obtain a conclusive proof that such interpretations reflect Dufay’s intentions.

More concretely, one could ask to what extent mathematical proportions can be perceived by the listener. The “mensural proportions” (which are but one example of numerical ratios in this music) are merely changes of meter: in an isorhythmic motet, for instance, the basic sequence of note values (talea) is repeated with all durations multiplied by a fixed ratio (e.g. 2:1, 1:2 or 2:3). Whenever the change occurs simultaneously in all voices, it can be clearly heard; in other cases, it remains hidden in the polyphonic texture. The mensural proportions also determine the ratio of the lengths of the various sections of the piece, and the choice of appropriate proportions was considered to be essential to the overall structure of the piece, much as in the Pythagorean scale where such ratios (1:2, 2:3, …) determine the consonance of a chord. As Leibniz states three centuries later: “Music is the pleasure the human mind experiences from counting without being aware that it is counting.”

Here is Nuper rosarum flores as played by Cantica Symphonia:

This is the isorhythmic motet that Dufay wrote for the consecration of the cathedral in Florence in 1436, the one with Brunelleschi’s famous dome. He was 35 then, living in Florence and working for the Pope. Later he would return to Cambrai, in what is now Northern France.

I’ve visited this cathedral and taken the terrifying tour that lets you climb up to the dome, go above it into the rafters between the dome and roof, and then out onto the roof. It’s an amazing structure:

For sheer joy, so far my favorite performance of Dufay’s isorhythmic motets is the album O gemma lux by the Huelgas Ensemble:

Here’s the man himself—Guillaume Dufay:

To be honest, nobody is completely sure whether this is Guillaume Dufay or another famous composer of the early 1400s, Gilles Binchois.