For my March 2021 diary, go here.

Diary — April 2021

John Baez

April 1, 2021

fragment of the face of a queen
Dynasty 18, reign of Akhenaten. c. 1353 - 1336 BC
yellow jasper
Metropolitan Museum of Art, New York

April 2, 2021

I've been reading about early music. I ran into Vicenzo Galilei, an Italian lute player, composer, and music theorist who lived during the late Renaissance and helped start the Baroque era. Of course anyone interested in physics will know Galileo Galilei. And it turns out Vicenzo was Galileo's dad!

The really interesting part is that Vincenzo did a lot of experiments — and he got Galileo interested in the experimental method!

Vicenzo started out as a lutenist, but in 1563 he met Gioseffo Zarlino, the most important music theorist of the sixteenth century, and began studying with him. Vincenzo became interested in tuning and keys, and in 1584 he anticipated Bach's Well-Tempered Clavier by composing 24 groups of dances, one for each of the 12 major and 12 minor keys.

He also studied acoustics, especially vibrating strings and columns of air. He discovered that while the frequency of sound produced by a vibrating string varies inversely with the length of string, it's also proportional to the square root of the tension applied. For example, weights suspended from strings of equal length need to be in a ratio of 9:4 to produce a perfect fifth, which is the frequency ratio 3:2.

Galileo later told a biographer that Vincenzo introduced him to the idea of systematic testing and measurement. The basement of their house was strung with lengths of lute string materials, each of different lengths, with different weights attached. Some say this drew Galileo's attention away from pure mathematics to physics!

You can see books by Vicenzo Galilei here:

Unfortunately for me they're in Italian, but the title of his Dialogo della Musica Antica et Della Moderna reminds me of his son's Dialogo sopra i Due Massimi Sistemi del Mondo (Dialog Concerning the Two Chief World Systems).

Speaking of dialogs, here's a nice lute duet by Vincenzo Galilei, played by Evangelina Mascardi and Frédéric Zigante:

It's from his book Fronimo Dialogo, an instruction manual for the lute which includes many compositions, including the 24 dances illustrating the 24 keys. "Fronimo" was an imaginary expert in the lute — in ancient Greek phronimo means sage — and the book apparently consists of dialogs with a student Eumazio (meaning "he who learns well"). So, I now suspect that Galileo also got his fondness for dialogs from his dad, too! Or maybe everyone was writing them back then?

April 3, 2021

I tend to like all sorts of complicated, disciplined music, so I'm trying to learn more about polyphonic music. I love Palestrina, and could easily bask in his music all day. I'll eventually get all of The Sixteen's albums of Palestrina, and it will take forever for me to get to the bottom of them. I also love The Sixteen's 10-CD set The Golden Age of British Polyphony, which weirdly omits Tallis and Byrd, but has a lot by Taverner and Shepphard, and a bit of Fayrfax and Mundy. And I have spent a lot of time with the Tallis Scholars' albums of music by Ockhegm, Byrd, Josquin, Brumel, etc. But I feel I'm just beginning to understand all this music. These days I've spent some evenings reading about it and listening to it.

April 11, 2021

Some great musicians of the Renaissance:

April 12, 2021

I just bought Johannes Obrecht's Missa Maria Zart, performed by the Tallis Scholars, which is only available on CD. It's great! In theory I should have started with the young Obrecht, because Wikipedia makes him sound very bold:

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 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.
But I couldn't resist getting Missa Maria Zart, because it's supposed to be his masterpiece.
This four-voice mass is based on the same Maria Zart ("Gentle Mary") hymn tune that Schlick used in his famous organ setting (the one included in Apel's Historical Anthology of Music). The tune is staunchly Phrygian in modality, and has many pungent and beautiful turns of phrase, making it an unusually fruitful source for a cyclic mass. Obrecht frequently sustains it in long note values using the cantus firmus technique that dates back to the Notre Dame school. At other times he takes its phrases as the basis for points of imitation or note-by-note elaborations in each vocal line, demonstrating his mastery of this more "modern" paraphrase technique epitomized by Ockeghem's Missa Fors Seulement. Often Obrecht combines both approaches simultaneously. Only once, at the end of the mass, does Obrecht actually quote the entire hymn straight through (starting at 8:54 of the Agnus Dei in the sopranos). But most of the rest of the work is consumed with a meticulous note-by-note parsing of the melody, spinning the contrapuntal texture out of endless local repetitions or variations of one of the hymn's 15 distinct phrases.

Maria Zart is rather long as hymn tunes go, so this process takes a while to unfold. Indeed Obrecht's Mass is notable for its length, probably the longest of any polyphonic Mass written before the Baroque era. It fills an entire CD, compared to typical masses from Dufay, Ockeghem, Josquin, Lasso and Palestrina which clock in at half that duration. If you aren't already interested in Renaissance sacred polyphony, then your experience listening to this work will be comparable to that of an opera hater obliged to sit through the second act of Tristan. Happily, the hymn tune is printed in the liner notes, and I encourage you to follow along using it as a kind of road map to the work's distended structure.

Here's a review on Gramophone of the Tallis Scholars' performance:
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!

April 17, 2021

I'm excited about the music by Johannes Ockeghem, 1410-1495. He was the best composer of the second generation of the Franco-Flemish School of Renaissance musicians, famous for his polyphony based on sophisticated compositional techniques. Some have called him the first real classical composer! The enthusiasm of this review by John S. Hilliard gets the point across:
Everything is related to everything else. A theory, or just a philosopher's dream? Well, in Ockeghem's Missa prolationum this dream becomes true. The previous reviewer who spoke of Bach's Art of the Fugue in the same breath as Missa prolationum had the right idea. It is simply one of the greatest works composed in the Renaissance. Ockeghem is the Bach of the Renaissance. Don't be put off by its description of almost unending canonic textures. It means each opening melody is followed in some form by other voices, e.g. "everything is related to everything else." But you need not know how it is constructed to enjoy the heavenly sheen of its melodic and harmonic flow. That is one of the breath-taking rewards of such music: pristine beauty of sound and melodic contour and modal harmony despite beneath-the-surface technical perfection the musical equivalent of Einstein's in science. The Hilliard Ensemble (no recent relation to this reviewer) do their usual fantastic job of absolute musical perfection once again. If you love a cappella vocal music of the Renaissance, please buy this disc. Keep it available for our children....they deserve to be left something of the highest aural and spiritual quality.
I just got ahold of a bunch of Ockeghem's music from a group called The Clerks. This review of one by Gio says:
The two motets on this CD — Intemerata Dei Mater & Ave Maria — are among his most accessible to modern listeners, easy to appreciate for their melodic beauty even if the intricacies of composition go unnoticed. Ockeghem's music is intricate. His polyphonic lines stretch beyond the lung capacity of most chorists, making phrasing a challenge, and their almost irrational independence of each other sounds like abstract improvisation... until everything merges in his splendid cadences. The mass Ecce ancilla domini is also one of Ockeghem's most immediate and least intellectual — though non-intellectual for Ockeghem is still astrophysics for most composers — which makes this an excellent CD choice for anyone unacquainted with the founding genius of Renaissance vocal music.

"The Clerks": it sounds like the name of a punk rock group whose members work at a grocery store! But they're a bunch of choral scholars at Oxford — also known as Academical Clerks — hence the name. Formerly called The Clerks Group, they were organized by Edward Wickham in 1992, and they the spent the 1990s performing all the sacred music of Ockeghem and recording it on six CDs, mixed with some music from some composers. They also sing the music of Josquin, Obrecht and other composers of the Franco-Flemish school.

Their CDs of Ockeghem are now out of print and quite expensive, but I got ahold of their collection, The Ockeghem Collection, in mp3 form:

It takes time for this music to sink in, and so far my best experiences have been in the later portions of the Missa prolationum, which have a soaring 'architectural' quality.

How did the Missa prolationum get its name? The reason is that it has singers singing the same melody at different pitches at different speeds. This is called a mensuration canon or prolation canon. The Wikipedia article on the Missa prolationum gives some more details about this amazing piece:

The Missa prolationum is a musical setting of the Ordinary of the Mass by Johannes Ockeghem, dating from the second half of the 15th century. Based on freely written material probably composed by Ockeghem himself, and consisting entirely of mensuration canons, it has been called "perhaps the most extraordinary contrapuntal achievement of the fifteenth century", and was possibly the first multi-part work written with a unifying canonic principle for all its movements.

The mass is for four voices, and is in the usual parts:

A typical performance takes 30 to 35 minutes.

Like Palestrina's Missa Repleatur os meum (Third Book of Masses, 1570) and the canons of J. S. Bach's Goldberg Variations more than two centuries later, the Missa prolationum uses progressive canon in all its movements. Most of the movements feature pairs of mensuration canons. The interval separating the two voices in each canon grows successively in each consecutive movement, beginning at the unison, proceeding to the second, then the third, and so forth, reaching the octave at the "Osanna" section in the Sanctus. The four voices each sing in a different mensuration. For instance, in the first "Kyrie", the four voices sing in the meters 2/2, 3/2, 6/4, and 9/4 respectively (in modern notation). Thus, the second voice, in 3/2, sings the same tune as the first voice, in 2/2, but half again as slowly, so the voices gradually pull apart. The same occurs between the second pair of voices, in 6/4 and 9/4 respectively. In the score, only one voice was written out for each canon, with the mensuration marks (approximately equivalent to a modern time signature) given alongside, so the singers would understand that they are to sing in those proportions, and thus at different speeds; in addition, the intervals between the voices are given in the score by the positions of the C clefs. What has so astonished musicians and listeners from Ockeghem's age to the present day is that he was able to accomplish this extraordinarily difficult feat.

Ockeghem was the first composer of canons at the second, third, sixth, and seventh (the "imperfect" intervals), and the Missa prolationum may have been the first work to employ them. Its format, with the interval of imitation expanding from the unison up to the octave, was used by Bach in the Goldberg Variations, but it is not known whether Bach knew Ockeghem's work (which was generally unavailable in the 18th century). Another unusual feature of this mass is that the melodies used for its canons were all apparently freely composed; none have been identified as from other sources. In Ockeghem's time, composers usually built masses on preexisting tunes such as Gregorian chant or even popular songs.

April 18, 2021

Searching around for good recordings of Ockeghem I bumped into the Boston group Blue Heron, who had a project called Ockeghem@600:
Ockeghem@600 is Blue Heron’s multi-year project to perform the complete works of Johannes Ockeghem, one of the very greatest composers of the Western tradition, in thirteen programs over the course of seven seasons. Inaugurated in the spring of 2015, Ockeghem@600 will wind up in 2021, more or less commemorating the 600th anniversary of Ockeghem’s birth in circa 1420.

Besides concerts, the project demands and includes a significant component of research into the many questions of fifteenth-century performance practice which remain unsolved puzzles – questions as basic as pitch level, voice types, and scoring. By the end we expect to have a better understanding of such issues. We will also have created a new complete edition of the music of Ockeghem, scrupulously based on the original sources and rigorously tested in practice.

Along the way we will also explore music of Ockeghem’s predecessors, contemporaries, and followers, developing and sharing with our audiences a sense of the entire fifteenth-century repertoire. Succeeding our series of recordings of Music from the Peterhouse Partbooks, the fifth and final volume of which was released in March 2017, a new series of CDs is underway, including a 2-CD set of all of Ockeghem’s songs: Volume I was released in November 2019.

Joining Blue Heron as adviser for Ockeghem@600 is Professor Sean Gallagher of the New England Conservatory, one of the world’s leading experts on Ockeghem and his contemporaries.

While the above speaks of the "complete works" of Ockeghem, it looks like Blue Heron wants to record the complete songs of Ockeghem. This is good because The Clerks recorded the complete sacred music of Ockeghem, a distinct group of pieces, which together (I believe) covers all of Ockeghem's output.

Here is Blue Heron's description of their album Johannes Ockeghem: Complete Songs Volume 1:

This is the first disc of a 2-CD set that will include all of Ockeghem’s songs; the second is planned for release in 2022. The songs have not been recorded complete since the early 1980s.

Johannes Ockeghem (c. 1420-1497) was one of the most celebrated musicians of the fifteenth century and is one of the greatest composers of all time, every bit the equal of J.S. Bach in contrapuntal technique and profound expressivity, and like Bach able to combine the most rigorous intellectual structure with a beguiling sensuality. His two dozen songs set French lyric poetry in the courtly forms of the fifteenth century — rondeau, virelai, and ballade — to exquisitely crafted polyphony in which all voices are granted equally beautiful and compelling melodies.

Besides eleven of Ockeghem’s songs, Volume 1 includes two related works: the anonymous En atendant vostre venue from the recently-discovered Leuven Chansonnier (probably copied c. 1475 in the Loire Valley, where Ockeghem lived and worked), whose text borrows the first line of Ockeghem’s Quant de vous seul, and Au travail suis by the composer Barbingant, which quotes both text and music from the opening of Ockeghem’s Ma maistresse. The CD booklet contains complete texts, translations, and notes on the music and performance practice by Sean Gallagher (musicological adviser for Ockeghem@600, Blue Heron’s project to perform the complete works of Johannes Ockeghem in a series of thirteen concert programs) and Blue Heron’s artistic director Scott Metcalfe.

Here's a great "ad" for this album, which conveys some of the excitement people feel for Ockeghem:

And here's the track listing:

  1. Aultre Venus estes sans faille
  2. Quant de vous seul je pers la veue
  3. Anonymous En atendant vostre venue
  4. Ma maistresse et ma plus grant amye
  5. Barbingant Au travail suis que peu de gens croiroient
  6. S’elle m’amera / Petite camusecte
  7. Mort tu as navré de ton dart / Miserere
  8. O rosa bella (instrumental)
  9. D’un autre amer mon cueur s’abesseroit
  10. Fors seullement l’actente que je meure
  11. Fors seullement contre ce qu’ay promis / Fors seullement l’actente
  12. Se vostre cuer eslongne moy a tort
  13. Permanent vierge / Pulcra es / Sancta dei genitrix
For more information, including lyrics of all the songs and translations, and a discussion of Ockeghem's songs, get their booklet.

Here you can see a couple of live performances of Ockeghem by Blue Heron:

You can see more music by Blue Heron, including more Ockeghem, here.

April 19, 2021

I've got to try to find this music by Pycard, a musician from around 1400, perhaps a distant relative of Jean-Luc Picard:
A 15th-century Mass by a composer known only as Pycard found in the Old Hall manuscript (named for the English town in which it was eventually discovered), demonstrates the high sophistication and complexity of panisorhythmic techniques. The lower parts have a recurring color and talea that unite the composition. The upper parts have four different talea, one for each major section of the composition. The rhythmic relationship between upper and lower parts changes as the music progresses. Each quarter note in the lower part equals 4½ quarter notes in the upper parts, creating an uneven ratio of 4:9 that causes the parts to lose synchronization. The lower part then steadily contracts in a series of Pythagorean proportions (12:9:8:6) until the parts come back into alignment.

Those last two sentences are awesome. A ratio of 4½ is something you'd expect from math rock like King Crimson. The stuff about "color" and "talea" is terminology from "isorhythmic composition", an old technique I don't understand.

I don't know if anyone has recorded this mass by Pycard.

April 23, 2021

People sometimes say that Renaissance music 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 had 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.

For my May 2021 diary, go here.


© 2021 John Baez
baez@math.removethis.ucr.andthis.edu

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