5. Math in some musical structures

How is math used to compose lyrics?How can syncopation be defined?What were Bach’s secret structural codes?

We discussed the mathematics underlying Greek tunings for the lyre, but what of that math for lyrics? These calculations are called the lyric meters, the term arising naturally from the word lyre and from the Greek metron, to measure.

Most western popular and classical forms, whether a waltz, hip hop, march, stomp, stamp or bouree, use meters made of repeating patterns of stressed pulses in ongoing patterns of 2 and 3 counts. But many other styles, very much including the ancient Greeks, use composite or compound meters with alternating patterns composed from groups of 2s and 3s.

For Greek and middle eastern lyric rhythms, use the feet

The ancient Greek lyric meters were written in time divisions that the metricians called feet. Feet are formed of short syllables that last for one beat (mora), notated as ˘ (a crochet, Œ, or quarter note) and long syllables that last for two beats, written as –– (a minim, Ó or half note).

The iambic foot was one short and one long mora, (˘ ––). The Greeks tended to use an iambic trimester. The infamous iambic pentameter, used extensively in English poetry (see most poetry by William Wordsworth), uses five iambic feet.

(˘ ) –– ˘ –– ˘ ––˘ –– ˘ ––(Œ ) Ó Œ Ó Œ Ó Œ Ó Œ Ó(the) chi- ld is the fa- ther to the man

Reading this in contemporary English, we could either put more emphasis on the stressed syllables, which is known as qualitative meter, or like the ancient Greeks, read with twice the duration on the stressed syllables, known as quantitative meter. If you try the second, you might easily compose a melody for the lyric in waltz time.

The Greek metricians cataloged a wide assortment of meters that extend far beyond those we use in contemporary lyrics. Their most popular was probably the dactyl foot

–– ˘˘

arranged in dactylic hexameter. That meter consisted of 5 dactyls,

–– ˘˘ –– ˘˘ –– ˘˘ –– ˘˘ –– ˘˘ –– X

with the X on the 6th foot representing either a spondee

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–– ––

or a trochee

––˘.

The dactylic hexameter was used in virtually all of Greek epic poetry ranging from Homer to Hesiod in 700 BC, for the next 1200 years (!), including in the presumably sung lines of the Illiad.

Despite its importance, this meter is awkward in English, but you can almost sound it in this way:

–– ˘ ˘ –– ˘ ˘ –– ˘ ˘ –– ˘ ˘ –– ˘ ˘ –– ––Ó Œ Œ Ó Œ Œ Ó Œ Œ Ó Œ Œ Ó Œ Œ Ó ÓFor-ward the light bri-gade charge for the guns he said in-to the Val-ley

Why is this so hard to recite in dactylic hexameter for contemporary English? You can recite The Charge of the Light Brigade in quantitative meter as above, but to us, it is more natural to read it in qualitative meter with a count of 3, like a waltz

FOR-ward-the LIGHT-bri-gade CHARGE for the GUNS

So that the durations or each syllable are identical, but a bit louder on the capitalized words, which still suggests a waltz rhythm

While ancient lyricists tended to be quantitative and we now tend to be qualitative, our songwriters continue to use quantitative meter with longer durations on stressed syllables:

but the FOOL on the hillsees the SUN go-ing DOWNand the EYES in his headsee the WORLD spinning ROUND

Meter becomes very challenging for us if we wish to compose like the superstar poet / singer/ songwriter Sappho of Lesbos. We know that she sang her lyrics from portraits of her singing with a lyre.

Sappho’s song lyrics from about 600 B.C. survive in more than 300 stanzas and most often consists of three “hendecasyllabic” lines of 11 syllables of 5 feet (trochee, spondee, dactyl, trochee, spondee) followed by an Adonean foot, which was

Ó ŒŒ Ó Ó or Ó ŒŒ Ó Œ __˘˘ __ __ __˘˘ __ ˘

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The standard rhythm for Sappho’s lyrics is a killer for our poets and lyricists. Alfred Tennyson wanted to show that he had the technical skill to write English lyrics in Sappho’s hendecasyllabic meter and came up with

––˘ –– –– –– ˘ ˘ –– ˘ –– ˘ Ó Œ Ó Ó Ó Œ Œ Ó Œ Ó Œ Ir-re- spon-si- ble in-do- lent re- view-ers

Try your hand at the Sapphic challenge, but it’s hard.

Figure 5.1 Sappho Sappho with an 8 string lyre standing with another singing poet, Alcaeus with a seven string lyre, from a vase from 480-470 BC attributed to the Brygos painter, and redrawn on a flat surface by Valerie Woelfel, with permission by the artist.

The math of composite rhythms

Sappho’s lyrics alternate duplet and triplet phrases, something our typical western meters don’t do, and composite rhythms are required. While foreign to much of the west, composite meters are very popular in contemporary Greece, the Indian subcontinent, southern Spain, and Middle Eastern music. Far from confusing the audiences, composite rhythms often provide their most popular dances and even now provide the basis of sung epic lyrics.

The epic lyric that is still widely sung in Greece, specifically in Crete, is the Erotokritos, written by Vincezo Kornaros in the 1600s. While it is “only” 4 centuries old, it provides insight into how older lyrics were sung by Homer, Hesiod, and Sappho.

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The Erotokritos comprises 10,000 lines of 15 syllables written in a dialect of eastern Crete, and only short sections are performed at a time. The most typical melody I have heard uses a composite rhythm of 2+2+2+3+2+2+4, providing a hypnotic groove of 17 beats (!). I am confident that the rhythm, which doesn’t go stale even when repeated many times, is a strong reason that this immensely long poem remains popular.

Composite rhythms are also popular in middle eastern and north African music. One of the best known songs in the world, and surely the best known from the pre 1492- era of Arabic Andalusia in southern Spain is Lamma Bada Yatathanna, attributed to Al-Khatib of Granada (1313-1374). This lyric is in a poetic form known as muwashahat that along with the similar zajal introduced the verse–chorus structure that provided the basis for the songs of the European troubadours, and by that route is the ancestor of most songwriting in the west.

Like many lyric poet musicians of Al-Andalus, Al-Khatib was also a scientist and mathematician. He survived the bubonic plague and was the first to publish the hypothesis that diseases can be contagious.

Lamma Bada is performed in a compound rhythm known as sama’I taquil essentially constructed from trochee, spondee, and trochee feet in a “decasyllabic” line. Mideastern rhythms are today often taught on the dumbek, a hand drum that produces a low dum like a bass drum and a sharp bek that resembles a snare drum. The ten beats of sama’I taquil on the dumbek are played (with hyphens for quiet or silent beats)

Dum __ __ bek __ DUM DUM bek __ __

In contrast to Greek quantitative meter, the goal here is to arrange the stressed syllables to coincide with the strong beats on the dum as qualitative meter

Lam –

ma ba- da ya-ta than- na Lam-Ó ‰‰ ‰‰‰‰ ‰‰ ‰‰ Ó Œ Dum __ __ bek __ Dum Dum bek __ __

ma ba- da ya-ta than- na Ó ‰‰ ‰‰‰‰ ‰‰ ‰‰ Ó Œ Dum __ __ bek __ Dum Dum bek __ __

The flamenco tradition that extended the Arabic rhythms in southern Spain also uses composite rhythms, particularly for a large variety of divisions of 12 beats. The specific metrical patterns are known as palos, after the walking cane used to tap out the rhythms, and each has its own personality based on lyrics, harmonies, history, structure, and subject.

For example, the seguiriya palo is associated with blacksmiths, one of the few professions Gypsies were allowed to pursue in Spain. It is sung in a Phrygian mode, and

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the rhythm was originally hammered on the anvil: I assume the sequriya inspired Verdi’s Anvil chorus in La Traviata, although he wrote the piece in 4/4 in a major key.

The seguriya uses a cycle of twelve beats formed as a composite of 2+2+3+3+2. The rhythmic pattern is sometimes taught using the name for an Andalusian blood sausage

un dos tres mor-ci-lla mor-ci-llaÓ Ó Ó Œ Œ Œ Œ Œ Œ In contrast, the buleria is presently the most popular flamenco palo in a composite rhythm, and I’ve watched teenagers in Sevilla sing and play highly complex subdivisions in a city park for hours without a break. The rhythm, nominally a composite of 3+3+2+2+2 can be divided in the many ways that it is possible to divide cycles of 12. The dancers and musicians speak of playing “in 3” or “in 6” depending on where the phrases begin and end.

There are many clapping rhythms (palmas) for the buleria, and the most common is counted and clapped as

with foot stamps underlined on 3,7,8,10, and 12. The strongest stresses are typically on the 3rd beat, where the harmonies change, and the 10th beat, where the melodic phrases often end.

The art of buleria compound meter is to make phrases that divide this long count, while never losing the pulse and ending together at the right place. For example, performers will switch to rhythms of 4 groups of 3s or two groups of 6s, or enter at unusual beats, or repeat a lopsided phrase several times but with the final phrase ending on the 10th beat. The most exciting point during a performance is when a long phrase arises in new patterns, but with all of the dancers, musicians and clappers (palmeros) realizing where to climax at the end, usually on beat 10. Sometimes the group realizes there is a syncopation and ends the phrase together on the 9th beat or on the 9th and half beat. When this is done well, the performers and audience shout encouragement to the musicians and dancers.

By far the collection with the most numerous set of composite rhythms are the tala (from the Sanskrit word for clapping) system in India, and much of the new flamenco tradition as influenced by the Indian system by the migration of the Gypsies from northern India. The north Indian Hindustani and southern Carnatic musical traditions differ in the names of the rhythms and the types of drums, but like the middle eastern dumbek, the sounds of the drum are used to define specific tals. In contrast to two sounds, the northern tabla produces seven sounds described by spoken syllables. For example, Rupak tala is a cycle of 7 made of 3+2+2, meaning the strong beats are

Tin Tin Na Di Na Di Na

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The cycles of tala meters, which were codified by Indian academics over the centuries, can run from 2 to 128 beats! Still, the long rhythms are generally divided into groups in 2s and 3s.

Medieval talea

The motet form may now be virtually extinct but motets made up most of the earliest surviving extended compositions in Europe. It provided another form of rhythmic complexity, in some ways similar to composite rhythms. While rhythms in composed European music in some ways have become less complex over the subsequent centuries, the motet evolved into polyphonic (multiple voices at once) styles of classical music including the fugues by J.S. Bach, and so its influence continues.

An ideal of the motet in the middle ages was to compose layers of melodic and rhythmic lines that were independent but work together when performed simultaneously. The recreation of their performance entails some guesswork, but early motets are thought to have been sung and instrumental parts added later in history.

The essential means to compose a motet is to first define a series of notes from a Gregorian church plainsong (cantus firmus), now often called the color. The line in which these notes were song was called the tenor (“to hold”), and the origin of the term the tenor voice. To provide the rhythm for the tenor line, a composer would use a rhythmic pattern known as an ordo or talea (which indeed is related to the Indian tala) that repeats throughout the composition.

This example of mapping two patterns of color and ordo is from a motet by Guillaume de Machaut (~1300-1377). The top line shows the rhythmic pattern of the ordo, the middle indicates the note sequence color, and the bottom how they combine to produce the tenor.

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Figure 5.2 Derivation of a tenor line in a motet.

The top line is an ordo, or sequence of note durations, consisting of 12 sounded notes and 5 rests, and the color line the order of notes, in this case 18 notes. Together, by mapping the color twice and the ordo three times, they can end together. This mapping produced the tenor line of the motet De bon espoir-Puisque la douce-Speravi by Guillaume de Machaut. Middle, the color for the motet. Bottom the resulting tenor line of the piece was derived.

These composers presumably wrote the tenor line first. The additional lines in these motets, the duplum, triplum, and sometimes quadruplum were often more freely written than the tenor but with the constraint of the rules for harmony of the period, when consonances were considered to be perfect fourths and fifths.

A feature of some motets, including Machaut’s Hocket David and a well-known anonymous motet known as In seculum is that the motetus and duplum both have rests that break the flow of the melodic line, but when played together a more clear melody emerges with the melody jumping between the voices, a technique known as a hocket, that some think has the same root as “hiccup”. Hocketing of melodic lines is an unusual feature of most classical and popular styles but is a major structural element of a great deal of vocal and instrumental music in Central Africa.

Syncopation

The construction of the tenor of the motet, for which two different patterns are overlaid, leads to the question of what happens when two different rhythm patterns overlap. This result is syncopation, a feature that underlies rhythmic structure ranging from the backbeat of most of western popular music to the extraordinary complexity of religious ceremonial music in West Africa.

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A simple syncopation can be constructed by “phasing” in which two identical rhythms are shifted apart. As a familiar example, let’s say that a bass drum plays a steady beat. What happens when hand claps or a tambourine or cymbals play the same steady pattern precisely halfway between the bass drum? This syncopation is the classic “off beat”. If the hand claps are shifted a little bit from the precise half-way point, we produce “swing”: a well-known example of that is when the clap are shifted roughly two thirds of the distance after the basic beat, a very familiar sound in jazz.

An extremely common syncopation is to play two even beats in one melodic line during the same duration as three even beats in another. This pattern, known as a hemiola, bisects the duple pattern. It generally takes a while for pianists and drummers to learn to execute it with two hands but it provides a basis for many parts of Chopin’s compositions as well as a large part of the vocabulary for jazz pianists like McCoy Tyner.

Perhaps the classic studies of syncopation are by a missionary, Reverend A.M. (Arthur Morris) Jones, who studied Yoruba and Ewe drumming in West Africa, and by the master Ewe drummer C.K. Ladzekpo, who directed the African music program at Berkeley. The drumming in the religious chants of these West African traditions provide the origin of the highly syncopated traditions of Cuban, New Orleans and “funk” drumming, and from there through them styles of contemporary dance music throughout the world.

Jones called the way these patterns were produced cross-rhythm. To determine how the individual lines sound together, he asked the musicians to play each part individually. As with hocket motets, the sum total of the music entails different voices contributing to parts of an overall structure. Also reminiscent of motets, the repeated drum phrases may be of different lengths: for instance a kadodo bell performs a more complicated version of the hemiola, in which a compound pattern of eight 8th notes are played while other drums play patterns of three 8th notes, so that the phrases together require 3*8=24 beats to line up and start together again.

Patterns that are typically played on Yoruba agogo bells are the antecedent of the clave played throughout a composition in Cuban and salsa music. There are a range of clave rhythms. For example, in New York salsa, the clave of a piece in 4/4 can start on the downbeat, which is called “three two” and is identical to the “Bo Diddley beat”, or it can start on the offbeat of the first beat of the phrase, which is called “two three”.

The agogo bell patterns, which in the Yoruba tradition often use 7 rather than 5 strikes, are called “standard patterns” by those who study west African music. Most drummers in these ensembles have specific highly syncopated parts, and a master drummer improvises over the bed of interlocking rhythms to keep things still more interesting. The singing often features solo and choir parts with still more patterns superimposed. These approaches have traveled to Cuba, Haiti, Belize where they continue to be performed in religious ceremonies. A single piece can continue for hours with a world of variety in the interlocking parts, as I have heard in New World versions in rumba ceremonies in Cuba devoted to the saints.

One attempt to develop a system to understand syncopation was by the theoretician Joszef Schillinger, celebrated in part as George Gershwin’s composition coach, who

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mapped out syncopations on graph paper from multiple rhythmic patterns that eventually line up and repeat again, i.e., “periodicities”.

With this sort of approach, the amount of syncopation or cross-rhythm can be assigned a quantity from the number of beats required for a complete pattern. For instance, the simple offbeat could be written as a pattern of two beats in one voice during the time of a single beat in another, or 2:1, while the classic swing back beat would have a value of 3: 1, the hemiola can be characterized as 3:2 and the kadodo drum pattern as 3:8. The product of these could be used to assign syncopation values of 2, 3, 6 and 24 that indicate the number of time divisions when the patterns coincide. This approach would be similar to describing the contribution of harmonics to sound in Fourier analysis.

This approach however can be confusing: the New York clave pattern or 5 strikes within a 4 beat measure possesses one note that falls on a 16th of a beat: should that be 16:4 for a syncopation value of 64? If the same clave is played against an instrument playing triplets, is that 16:12 or a pretty daunting value of 192?

An attempt to quantify syncopation further becomes confusing in the real world when portions of cycles become longer, as with the drummer Clyde Stubblefield in the James Brown group who used phrases that may be quite long on multiple different drums and occur at the end of 4 or 8 or 16 bar phrases. This is even harder with a full West African or Cuban drumming ensemble with many parts of different lengths that might each begin at different points, and impossible to contemplate in a performance by Elvin Jones, the drummer in the John Coltrane Quartet, where the patterns undergo constant shifts. But artists don’t fear challenges, right?

Canons and Riddle Canons

The most extreme tradition of using math to create a large work from a small kernel are canons, from the Greek kanon, meaning rules or law. In canons, a rule modifies an original phrase.

The most familiar rule for a canon is the round, also known as a perpetual canon. One voice plays a phrase (the leader) and while still playing, another voice or voices play the same phrase (the follower(s). The entire piece can be repeated until the singers fall into stupor. Frére Jacques, Three Blind Mice (published in 1609 as a three part canon in a minor key), Row Row Row Your Boat, Hey Ho Nobody Home, are by far the most widely sung polyphonic pieces.

The most familiar canon performed in concerts and soundtracks, Johann Pachabel’s Canon in D (1663-1706, organist, composer and teacher of the Bach family), follows similar rules, but doesn’t repeat ad libitum. He wrote the piece for 3 violins, one leader and two followers, with greater elaboration of the original simple motif as it progresses. Pachabel also wrote a bass line that is not repeated by other instruments for a total of four voices. The resulting harmonies from these four lines produces one of the most used harmonic patterns popular music, showing up in perhaps hundreds of songs since the canon became very widely popular in the 1970s.

There are many ways that parts can follow the leader beyond simply repeating from a distance. For example, the leader can be repeated backwards, so that the last note of

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the leader in repeated first (retrograde), or the follower might be what would be read if you turn the page upside down (inversion), or it might move to a higher note as the leader moves to a lower note as if reflected in a mirror (contrary motion), or one might multiply (augmentation) or divide (dimunition) the duration of the notes to make the repeats follower longer or shorter. These transformations were further elaborated in fugues composed for the keyboard, and by Arnold Schoenberg and others who used each of these manipulations to transform short themes and rhythms in 20th century music.

As an example of a canonic composition, Guillaume Machaut wrote a three-part canonic motet in the 1300s, Ma fin est ma commencement et ma commencement est ma fin (my end is my beginning, and my beginning is my end). In the tenor part, the leader runs to the middle of the piece, and then the original sequence of notes is repeated backwards so that the first note also becomes the last: this type of a retrograde manipulation is sometimes called a crab canon. When the duplum and triplum lines reach their halfway points, they repeat the retrograde version of each other, suggesting a mirror image.

A special category are riddle canons, also known as puzzle or enigmatic canons, which are intended for an extremely specialized audience of composers, who use them to communicate with each over the centuries. Here, only the leader is notated, together with clues for how the riddle can be solved to create the full piece: and one must be a detective and try to figure out how this should be done.

For example, one clue is if a clef and key signature is drawn upside down, which indicates that the music is read in the same order, but that a melodic sequence written to descend in the leader is intended to ascend in the follower, a device known as a mirror canon.

Another clue is when a sort of squiggle is placed at the end of the leader, which traditionally means that the repeat should start a step higher in the scale.

Another approach is to use more than one clef for the leader. In this way, a note in the middle line of the stave can read as a B in the treble clef, a C in an alto clef, and a D in the bass clef, which can lead to dense parallel slabs of sound. This approach, I have read, was used by Ornette Coleman in his enormous orchestra piece Skies of America, which he is said to have written as a single melody played by each instrument without transposition.

The all-time master of riddle canons is the all time master of everything else, J.S. Bach. The most famous are the ten riddle canons in The Musical Offering, These pieces were written using a theme given to Bach during a visit by Frederick the Great of Prussia.

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Figure 5.3 The first riddle canon

from J.S. Bach’s A Musical Offering in the original. The title at top Canones diversi super Theme Regium (different canons on a royal theme). The label Canon Ia2 indicates that this first canon is for two voices.

In the first of these riddle canons, the leader is written in alto clef indicating that C is the first line and that we are in the key of C minor. At the end, there is the same clef and key signature facing backwards. The answer to the riddle is to play the piece in the usual direction and simultaneously backwards (retrograde), an example of a crab canon structure used centuries after Machaut.

Other riddle canons in Bach’s collection are rounds or followers formed from contrary motion. One canon climbs a whole step six times until the piece ends in the original key, reminiscent of Chapter 2 where we discussed how the interval of seconds repeated 6 times in the tunings demonstrated the Pythagorean comma.

Canons may be ancient, but new rules continue to be devised. Steve Reich’s Clapping Music is a canon that follows a simple rule. The leader is a 12 beats African drum rhythm clapped in unison by two performers. After 8 repeats, one player pauses for an eighth note rest before starting again, and this eighth rest delay pattern continues until, by advancing 12 times, the performers return to the original unison. It’s a challenge to perform, and that listeners can understand its simple and clear design is one reason it works.

At an opposite extreme, in which canons are used to produce music of extraordinary complexity, are the player piano studies by Conlon Nancarrow. Conlon wrote canons of such difficulty that they were unplayable by musicians, and he manually punched holes in rolls so that they could be performed by two player pianos at his house in Mexico City. For example, his Study 21, Canon X, has one line that begins at a speed of about four notes per second and another at 39 notes per second. The faster line slows down while the slower speeds up, so that by the end of the piece, the tempi are traded, so that the original slow line is playing a surreal 120 notes per second. His Study 33 uses tempo changes in the follower that are multiplied by an irrational number based on the square root of 2 (remember how this drove Pythagoreans to distraction in Chapter 2?).

The mathematical transformation of a leader in a canon has been limited to addition / subtraction and multiplication / division and mirroring, which makes historical sense as this form was developed before calculus described by Leibniz and Newton around 1700. It’s however not hard to transform a leader as a derivative (an operation in calculus), which measures the distance between the frequency of each successive note, which abolishes changes in melodic direction over a long phrase, and so decreases the overall range of melody. But for integration (another operation in calculus) of the leader, in which the distances are added, the frequencies quickly generally extend above our range of hearing!

The modern concept of fractal patterns provides another new approach for canonic augmentation and diminution. Fractals, named and described by the mathematician Benoit Mandelbrot (1924-2010), are geometric patterns in which the same patterns are observed when an object is viewed at different scales. A classic example a coastline,

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where the same amount of jaggedness is seen when viewed from a vantage of miles or centimeters. Mandelbrot’s book The Fractal Geometry of Nature shows how fractal shapes can resemble trees, the flow of blood in the body, and landscapes, so much so that fractal transformations are used to create landscapes of other worlds in science fiction films.

The best known simple fractal is the Koch snowflake, in which the sides of equilateral triangle undergoes repetitive division to produce another equilateral triangle on each side, first producing a star of David and then snowflake patterns.

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Figure 5.4 a Koch Snowflake

Should be redrawn, taken from internet

Applying a fractal approach to a leader, the followers can follow the same patterns of changing notes at different time scales. If this is repeated multiple times, a strange phenomenon occurs, as the music sounds similar when it is performed at different tempos, and the concept of musical time has a different, less clear meaning.

A parallel situation occurs in fractal geometric patterns, which are drawn on a flat surface and so are in two dimensions: but the more the patterns are repeated at different scales, the closer the drawings are fit by a fractional dimensions, for the Koch snowflake a dimension between 1 and 2, and for complex patterns like the so-called Mandelbrot or Julia sets, between the second and third dimensions.

With this bit of weirdness, in which the very dimensions of art and music are bent, one might glimpse possibilities for the future.

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Listening

A lovely attempt at recreation of all of the surviving ancient Athenian Greek scores is recorded by the Spanish monk Gregorio Paniagua and his Atrium Musicae de Madrid on the album Musique del Grece Antique.

Here are some recordings to explore compound meters.

The Arabic Andalusian song Lama Bada Yatathanna, often credited to Lisan Al-Din ibn Al-Khatb from Granada in the 1300s, is an exemplar of composite rhythms as well as the poetic form, the muwashahat, the ancestor of the verse and chorus structure of most western popular music. It has been recorded hundreds of times. A particularly outstanding singer with fantastic musicians in her orchestras is Fairouz, who is essentially the national singer of Lebanon.

The flamenco seguriya, while in principle a composite cycle of 2+2+3+3+2 can be challenging to follow in real life due to tempo changes and the freedom of expression consistent with its mournful subject matter: classic singers of the seguriya include Manuel Aguejatas and Manuel Torre.

All flamenco performers record bulerias, and any video of an Andalusian Gypsy wedding or juerga (flamenco fiesta) will demonstrate how it is performed as a series of improvised solos. The rhythm is nominally 3+3+2+2+2, but is subdivided and syncopated in myriad ways. The possible ways to combine and subdivide the counts may be clearest to hear with guitarists, including Paco de Lucia, who was the preeminent innovator and master technician of the contemporary style, as well as Tomatito and Moraito, who are masters of syncopation within the rhythmic cycle.

Of the incredible variety of Indian rhythmic cycles, some of the wildest talas were released under Ravi Shankar’s name and featured his percussionist Alla Rahka on table. These include the 14 beat tala Farodast, which is 3+4+3+4 on the album Homage to Mahatma Gandhi.

The Dagar Brothers are members of a famous family expert in an ancient style known as dhrupad. Some of their pieces are in a style known as Dhamar which is associated with the Holi spring festival and is in a 14 beat tala in subdivisions of 5+2+3+4.

For a pinnacle of syncopation, listen to music from the Ewe in West Africa. The classic recordings from AM Jones and others are great, and they have a contemporary styles of choral religious music.

To hear how the Yoruba and Ewe styles were transformed in New Orleans, you might listen to Ziggy Modeliste of the band the Meters and the drummers of the contemporary brass bands starting with the Dirty Dozen. Another extension is through the drummers with James Brown, Clyde Stubblefield and Jabo Starks around the period of The Jungle Groove.

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I would guess that the most virtuosic of complex syncopation may be music by Elvin Jones with the John Coltrane quartet when they played in rhythms that can be counted in several ways, especially in their live recordings such as India, which was likely influenced by listening to Hindustani tabla drumming. In some ways, this approach was taken even further by the drummers Sunny Murray and Dennis Charles who played with Cecil Taylor, and the drummer William Hooker.

To explore the isorhythms of the motet, listen to recordings of Guillaume de Machaut hocket motet, Hocket David.

Here are some recordings to explore mathematical transformations of music, as exemplified by the canon.

Steve Reich’s Clapping Music is a canon that follows an extremely simple and effective rule that is almost a round.

Brahms wrote a gorgeous canon for four sopranos, Mir Lächelt Kein Frühling, in which the key changes in a subtle way so that most listeners do not notice it.

Timothy Dwight Edwards maintains a website on “puzzle canons” by many composers throughout the ages, including Albert Roussel, William Byrd, and John Dowland.

As mentioned, most stunningly complex canons are by Conlon Nancarrow. Of the player piano etudes, I am a particular fan of Study 12, which uses the Phrygian scale of flamenco and somewhat imitates the guitar. Study 21 is deservedly popular. Canon for Ursula is actually written for a human, Ursula Oppens, to perform. I adore chamber group versions of these pieces orchestrated by Yvar Mikhashoff that were recorded by the Ensemble Modern.

A popular approach for a canon, as computation has made it much easier, is simply to vastly augment music, in effect slowing it down., The software engineer Paul Nasca developed software to slow recordings by running a Fourier analysis, smoothing the gaps in the waves, and maintaining the original pitch information. Perhaps the classic use is a manipulation Justin Bieber song, U Smile, by Shamantis, who slowed it by 8 fold.

9 Beet Stretch is Beethoven’s 9th Symphony slowed to 24 hours by Leif Inge using software developed by Bill Schottstaed.

I wrote a literal fractal canon of the first five notes of the second movement of Arnold Schoenberg’s second quartet, for my String Quartet #3, The Essential. The movement, Benoit Meets Arnold in 5 Dimensions consists of the five note intervals at four speeds, with the pitch intervals superimposed on each other like a Koch snowflake. The notes follow the pattern precisely but don’t repeat and catch up at the end, which in the recording by the PubliQuartet takes about two minutes. In theory the patterns can be played at multiple tempos and still sound the same, Other transformations in this quartet include a first derivative, a Fourier transform, and the integral, in which the pitches rapidly exceed the range of hearing.

Two freer fractal transformations, as the number of notes can become dense quickly, are my piano pieces Fractals on the Names of Bach and Haydn. This is a tradition of using

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someone’s name to spell a melody (Bach is spelled in notes as B-A-C-Bb) to provide a theme. My version of the Haydn piece for organ, Olivia porphyria, is named for a sea snail with fractal patterns on its shell.

The musician Adam Neely developed a clever way to convert musical phrases to fractals apparent during playback. He chooses a short phrase from a song and repeats it at a very fast speed in which the phrase produces a pitch. If the phrase is repeated at 440 Hz it will produce the note A44 (440 Hz), and if repeated at 220 Hz, the A3 an octave below. In this way, a melodic phrase can be heard, and if the tempo is slowed in playback the phrase seems to disappear, but when played back at still slower speed, the phrase eventually reappears.

Fractals don’t only arise from notes on the grid, but also from the playback of the same sounds at different speeds. This can produce new musical features absent in the original material. My podcast for WFMU radio, Timeless Music goes through this with a bit of Frank Sinatra vocals.

I wrote Variations on Chopin’s “Minute Waltz” based on an old joke about a bad piano player “”it takes him half an hour to play the Minute Waltz.” The idea is to use newer math to produce variations. The variations include 1) the average pitch of the notes: if there is a C and a C#, we play a C quarter tone sharp, 2) playing every pitch except the notes in the original score 3) no rhythm, with every note played simultaneously, 4) a Fourier transformation where the amount of each pitch in the piece is played without regard to rhythm, 5) no pitch information, the rhythm is played on one note, 6) the piece compressed into 6 seconds, 7) the derivative, where the distance between successive pitches is played, 8) the integral, in which each pitch is added to the previous one, so that in a just a few notes the notes are above the range of human hearing, and 8) the Minute Waltz played over half an hour, recorded live in concert with the notes time stretched using a system developed by Sean Haggerty (pity the audience at the live premiere at Greenwich Village’s Le Poisson Rouge concert hall).

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The Sense of Hearing

Figure X The Sense of Hearing by Jan Breughel

1618, Prado Museum. The woman may be Euterpe, the muse for music (certainly not St. Cecilia). We see instruments including a two manual harpsichord, drum, trumpet, trombone, cornetto, lysard, lute, shawm and several flutes, various viols and and gambas. On a table on the right are a collection of musical clocks and small instruments, and underneath hunting horns. There are several singing birds, a cockatoo and two parrots. There are several musical scores, including a four part piece under a viol. One of score is a six-part madrigal dedicated to Prince Pietro Philippi Albert and Princess Isabella.

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