Authors
The CD offers a chronological succession of canons, composed by 27 different Authors.
Of particular interest, the contemporary compositions of Luis Miguel Aguilar, David Manzanares Viles, Jacques Paroissien, Hans Schoutens, Melquíades Vasconcelos and Rieks Veenker, here proposed in the version for flute choir with accompaniment, in first ever recording for DaVinci Classic as the result of a close cooperation between Authors and Flöten am Bass in the search for a performance as close as possible to the Composer’s idea.
As for the remaining tracks, with the exception of 16 -26 – 27, which are presented in the original version, these are transcriptions of the original with adaptation for Flöten am Bass by Maria Teresa Carlevato and Gian Paolo Guercio.
Middle Ages and Renaissance
The CD opens with David Manzanares Viles’s “Fantasia Canon”, based on two ancient arias from Canavese, North-Western Piedmont. Following contacts to define the interpretative modalities of the “Canon per augmentationem” by the same Author, a close collaborative relationship was born, which materialized in the composition in March 2025 of a dedicated piece, inspired by the lands of origin of the ensemble.
The first theme, elaborated in the form of a canon alla terza, the Muren, can be found in the collection “Canti Popolari del Piemonte” by Costantino Nigra, while the theme of the double canon in 6/8 is still played by the fifes in Ivrea on the occasion of the “Alzata dell’Abbá” of the parish of San Grato during the Carnival.
Manzanares Viles was born in 1963 in Barcelona, Spain. He studied at Barcelona’s Conservatory and got his degree in classical guitar, harmony, counterpoint and fugue at the same place. Among others, he studied guitar with José María Olmos, Xavier Coll, José Tomás, and Armando Marrosu. He has performed in concerts as a guitarist as soloist and as a member of chamber groups. He has been a music teacher at several schools in Spain.
As a composer, he wrote more than 200 pieces for guitar, chamber music, choir, and organ. Some of them has been performed in concert.
Guillaume de Machaut (c. 1300 – April 1377), French composer and poet, was the central figure of the ars nova style in late medieval music. His dominance of the genre is such that modern musicologists use his death to separate the ars nova from the subsequent ars subtilior movement. Regarded as the most significant French composer and poet of the 14th century, he is often seen as the century’s leading European composer, stretching back to the traditions of troubadour and trouvère.
Guillaume Du Fay (5 August 1397 – 27 November 1474) was a composer and music theorist of early Renaissance music, who is variously described as French or Franco-Flemish. Considered the leading European composer of his time, his music was widely performed and reproduced. Du Fay worked throughout Europe and he has been described as leading the first generation of European musicians who were primarily considered ‘composers’ by occupation. His erratic career took him throughout Western Europe, forming a ‘cosmopolitan style’ and an extensive oeuvre which included representatives of virtually every polyphonic genre of his time.
Jacobus Clemens non Papa (c. 1510 to 1515 – 1555 or 1556) was a Netherlandish composer of the Renaissance. Unlike many of his contemporaries, Clemens seems never to have traveled to Italy, with the result that Italian influence is absent in his music. He was a prolific composer in many of the current styles, and was especially famous for his polyphonic settings of the psalms in Dutch known as the Souterliedekens.
Regarding the origin of the epithet “non Papa”, one theory is that it was jokingly added by his publisher, Susato, to distinguish him from Pope Clement VII—”Jacob Clemens—but not the Pope.” Another states that it is to distinguish him from Jacobus Papa, a poet also from Ypres.
Clemens was one of the chief representatives of the generation after Josquin and before that of Palestrina and Orlandus Lassus.
Michael Praetorius (probably 28 September 1571 – 15 February 1621) was a German composer, organist, and music theorist. Praetorius was a key figure in the transition to the High Baroque and one of the leading exponents of the German musical Renaissance. He was one of the most versatile composers of his age, being particularly significant in the development of musical forms based on Protestant hymns.
His first compositions reflect the care for music at the court of Gröningen, the motets of this collection were the first in Germany to make use of the new Italian performance practices; as a result, they established him as a proficient composer.
Baroque Period
Georg Philipp Telemann (1681–1767) within his prodigious output, cultivated to a considerable degree the genre of chamber music without continuo, as in the present Cantabile from the Canonic Sonata N°5 in A Major, TWV 40:122. The “Six Canonic Sonatas” for two flutes or two violins, are all in the tripartite form, with two fast outer movements typically frame a highly “affective” slow movement. The themes are gallant in character, which define “melodic canons”, standing at the threshold of a new era.
Henry Purcell (London1658 or 1659 – ivi 1695). Written for the feast of Saint Cecilia, this Ode by Purcell remains one of the compositions most representative of the personality of the English composer, and one of his liveliest pages, judging also by the frequency of its performances, and the number of recordings of it. There are four Odes that Henry Purcell wrote in honor of Saint Cecilia: Welcome to all the pleasures and Laudate Coeciliam are from 1683: the date of the third, Raise, raise the voice is uncertain, but it is believed to be from 1685; the last, the most famous and the one on today’s program, Hail, bright Caecilia, is from 1692. Purcell was to die three years later, on the eve of the feast of the Saint.
Following the canons of Johan Sebastian Bach (BWV 659 – 616 and Musikalisches Opfer BWV 1079, which consists of canons, fugues, ricercars and a trio sonata, all based on a tune made up by King Frederick II of Prussia the Great) the work of Hans Schoutens has been added, which constitutes a wonderful example of contemporary writing respecting the compositional rigour of the Baroque period.
Hans Schoutens grew up in Leuven, Belgium, where he also got his doctorate in mathematics, but for the longest time now, he is a professor at City University of New York. It is often said that mathematics and music go well together, and perhaps that is also true in this case. Schoutens didn’t have formal music training, and by far his most favorite composer was Bach. “As long as I can remember – he said -I was always playing his music on the piano or imitating his style; however, I only started to seriously compose music about a decade ago and posting my work on musescore.com”. Remaining true to his first musical inspiration, he writes choir works and fugues in the style of Bach, or perhaps more accurately, in a neo-baroque style.
The Choral fugue on JS Bach op.67 is based on one of Bach’s favorite hymns, used multiple times for instance in his St Matthew Passion. Schoutens only used the first two phrases, giving the two main themes, henceforth T1 and T2, and marked each time with their corresponding lyrics T1: O Haupt voll Blut und Wunden; T2: voll Schmerz und voller Hohn; any other melody line has the Latin original lyrics for the first phrase: Salve, caput cruentatum.
Started out as a modest work in which the Author wanted to use the T1 theme as basis for an organ fugue (whence still the name “choral fugue”), very soon he started adding voices, variations, incorporating also T2 and then more instruments as well as the expansion of the contrapuntal structure, with 7 final fugal passages.
Antonio Caldara (c. 1670 – 28 December 1736) was born in Venice where he became a chorister at St Mark’s and he learned several instruments, probably under the instruction of Giovanni Legrenzi. He was active in Mantua, Barcelona as chamber composer to Charles III, in Rome, becoming maestro di cappella to Francesco Maria Marescotti Ruspoli, 1st Prince of Cerveteri, at the Salzburg court and, at the end, he was appointed Vize-Kapellmeister to the Imperial Court in Vienna, where he remained until his death. Caldara composed more than 70 operas, more than 30 oratorios, and other works including motets and sonatas. Several of his compositions have libretti by Pietro Metastasio, the court poet at Vienna from 1729.
Thomas Warren (1727–1767) was an English bookseller, printer, publisher and businessman. He was also known for publishing collections of contemporary musical catches, canons, glees and rounds, more than 650 works by over 100 composers, including the two “canon à boire” of Thomas Augustine Arne (12 March 1710 – 5 March 1778) performed in the CD. Arne was a leading British theatre composer of the 18th century, working at the West End’s Drury Lane and Covent Garden. He wrote many operatic entertainments for the London theatres and pleasure gardens, as well as concertos, sinfonias, and sonatas.
William Boyce (11 September 1711 – 7 February 1779) was an English composer and organist. Like Beethoven later on, he became deaf but continued to compose. He knew Handel, Arne, Gluck, J.C. Bach, Abel, and a very young Mozart, all of whom respected his work.
Paolo Agostini (c. 1583 – 1629) was an Italian composer and organist of the early Baroque era.
He was a highly sophisticated contrapuntist, often using strict canonic techniques; in addition, he used colorful sonorities, changes of meter between sections, and colorful chromaticism, showing an acquaintanceship with contemporary secular practice as well as the work of the Venetian School. An Agnus Dei for eight voices is especially admired and was used as an example in Padre Martini’s Saggio di contrappunto.
Classic Period
Wolfgang Amadeus Mozart – 3 Canons: Boona Nox! Bist a rechta ox; Kanon für sechs stimmen leck mich im arsch! (KV 231); Canon Inversus are proposed in successions, together with the
“Spiegel Kanonen” which however has recently been attributed to Franz Joseph Haydn.
Luigi Maria Cherubini (8 or 14 September 1760 – 15 March 1842) was an Italian Classical composer. His most significant compositions are operas and sacred music. Beethoven regarded Cherubini as the greatest living composer of his era. Cherubini’s operas were heavily praised and interpreted by Rossini.
Ludwig Von Beethoven. Beethoven’s Canons WoO 179 and WoO 185 are proposed, together with the second movement of the 9th symphony. This Scherzo does not feature a canon in a strict sense, but uses an imitation technique that resembles a canon. The central Scherzo section (Trio) features a composition in which one part of the orchestra repeats the melody of another part, creating a canon effect, although it is not a precise form of canon like that found in some compositions by other composers. While the canon form requires a precise reproduction of a melody in another voice, the Scherzo of the Ninth Symphony uses a freer imitation. The Trio, a central section of the Scherzo, often features a melody that is repeated by different voices of the orchestra at intervals of time, creating the effect of a “canon”. Difference from strict canon: The imitation in the Scherzo Trio is not as rigid as a canon, but rather a creative reworking of the original melody, with some variations.
Romantic Period
Regarding the Romantic period, of which transcriptions of canons by Schubert, Schumann, Brahms are proposed, some biographical notes of the lesser-known Authors, Salomé and Vierne, are briefly recalled.
Théodore-César Salomé (20 January 1834 – 26 July 1896) was a French organist and composer. Théodore Salomé left some very admirable pages for the organ, among which one remembers a collection of Dix Pièces pour orgue in three volumes and another of Douze Pièces Nouvelles pour orgue, published by Leduc. Volume 3 of Maîtres parisiens de l’orgue au XXe siècle (1936) contains two of Salomé’s canons (op. 21, nos. 1 and 3).
On 5 June 1875 that Salomé played the organ for the funeral mass of Georges Bizet, in the presence of 4000 people. For a prélude, Salomé improvised on themes from Bizet’s opera Les pêcheurs de perles (“The Pearl Fishers”) then during the absolution improvised on themes from Carmen. Théodore Salomé, his wife, and son share the Condrot-Gault family mausoleum at Père Lachaise Cemetery, which includes a granite sculpture of a prie-dieu with an open covered book.
Louis Victor Jules Vierne (8 October 1870 – 2 June 1937) was a French organist and composer. He was the organist of Notre-Dame de Paris from 1900 until his death. As a composer, much of his output was organ music, including six symphonies and four suites, and works for choir and organ, including a Messe solennelle for choir and two organs. He toured Europe and the United States as a concert organist. Vierne had an elegant, clean style of writing that respected form above all else. His harmonic language was romantically rich, but not as sentimental or theatrical as that of his early mentor César Franck.
His output for organ includes six organ symphonies, 24 Fantasy Pieces (which includes his famous Carillon de Westminster), and Vingt-quatre pièces en style libre, among other works. There are also several chamber works (sonatas for violin and cello, a piano quintet and a string quartet for example), vocal and choral music, and a Symphony in A minor for orchestra.
Canon in F minor written by Frédéric Chopin for piano around 1839 is unfinished (inacabado), and it is unclear whether the composer intended it to be used as a fragment in a larger composition. As it lacks an opus number, it is usually designated by its Brown catalog number, B. 129. In the CD is proposed in the reinterpretation of Luis Miguel Aguilar.
Luis Miguel Aguilar began his career in classical music, performing with notable institutions like the Barenboim-Said Foundation’s Orchestral Academy, the Youth Orchestra of Andalusia, and the European Union Youth Wind Orchestra. He studied with Antonio Salguero, Piotr Szymyslik, and Israel Sánchez, among others.
But his curiosity always extended beyond boundaries. Over time, his passion for music expanded into jazz and improvised music, studying at the Conservatoire Royal de Bruxelles, with Fabrice Alleman and Manu Hermia. He attended masterclasses with renowned jazz musicians such as Lage Lund, Joe Sanders, Ben Wendel, and Miguel Zenón.
He’s performed widely across Europe, both as a soloist and collaborator, with appearances at festivals like the Cádiz Spanish Music Festival and Klarafestival in Brussels, and is involved in various projects, including Barbara Wiernik’s Ellipse Quartet, Celia Rorive’s MΣAL Collectif, and Marjan Farsad’s band. In 2025, he launched his Sextet, stepping further into compositional leadership through appearances at Festival des Nocturnes, the B-Jazz International Contest, and Brussels Jazz Weekend.
1900 and Contemporary Period
From the 20th century and contemporary period, the first movement, Munter, of the Kanonische Sonatine for zwei Flöten by Paul Hindemith and the “Vecchio carillon” from the “Tre pezzi per 2 flauti” by Nino Rota are proposed in the original version.
We provide here some biographical notes of contemporary composers who have provided an important contribution, both for the pieces and for support in their interpretation in the transcription for the Flöten am Bass
Melquíades Vasconcelos – Vasconcelos trained at Federal University of Rio Grande do Norte and currently performs as a concert clarinetist and teaches at the Public University of Music of Brazil – Escola de musica UERN
Regarding Ventos, written during the Composition Workshop 2 course, taught by Professor Alexandre Reche, Vasconcelos used Joseph Schillinger’s compositional system. This movement presents some canon characteristics, as well as the use of a theme in augmentation. The Schillinger system of musical composition, named after Joseph Schillinger (1895–1943) is a method of musical composition based on mathematical processes. It comprises theories of rhythm, harmony, melody, counterpoint, form and semantics, purporting to offer a systematic and non-genre approach to music analysis and composition; a descriptive rather than prescriptive grammar of music.
While it influenced some prominent figures, such as Lawrence Berk (founder of the Berklee College of Music) and George Gershwin, it began to fall out of favor in the 1960s after receiving criticisms for being over-complicated and pseudo-scientific, and was removed from the Berklee curriculum.
Despite the technical complexity underlying the design of the piece, originally composed for Bb clarinet quartet with bass clarinet, the piece is very linear and pleasant to listen to.
Rieks Veenker – For almost thirty-five years, Veenker (1951) was a music/art history teacher at a secondary school for havo/vwo in Middelburg. In addition to his teaching, he wrote the music and songs for musicals every two years that were performed with an average of one hundred students in the Middelburg theatre.
In his hometown of Groningen, Rieks attended both secondary school and the conservatory. In addition, during his teaching, he followed the Composition Light Music course at the Rotterdam Conservatory. The Canon in swing, which we propose here in the unpublished version for flute choir, in its original vocal version, is performed by choral groups and small vocal ensembles all over the world, perhaps because as in the Netherlands people would say, in Dutch, the canon is an Oorwurm (Oor means ear and wurm is a dialect word for ‘woodworm’): a crawly woodworm who pleasantly creeps in your ear. And you can’t rid of it, even if you want to.
Jacques Paroissien Born in 1957 in Lorraine, he studied piano at the Metz Conservatory with Marcel Mercier, jazz at CIM in Paris, then (many tears later) composition with Etienne Rolin in Bordeaux.
He devotes himself equally to classical music and jazz. Remaining an amateur, he is a composer and arranger for several jazz orchestras in Bordeaux.
Under the pseudonym of JIB or JP, he has a considerable production of transcriptions of classical pieces (from Satie’s Gnossiennes, to Fauré’s Sicilienne, to Chopin’s waltzes, to Debussy’s Clair de Lune) and original pieces for small orchestra, jazz band or piano trios. The Canone Spumante that closes the collection of canons on the CD is a brilliant example of the wise combination of rigorous baroque canon writing techniques with the spirit of jazz, with an extremely interesting result, both from a stylistic point of view and to the listening.
DESCRIPTION
“Music is the pleasure the human mind feels when counting without being aware of counting ”
This is how the philosopher and mathematician Gottfried Wilhelm von Leibniz defined music.
I completely agree with this definition. A mathematical formula, a law of physics or a graph are not just mere symbols, but vivid expressions that, if you observe them carefully and look further than their outward appearance, describe you the world. They speak to you! If you then translate them into music, they gift you with emotions!
Since the time of Pythagoras and perhaps even before, music has been considered part of mathematics; it is one of the liberal arts of the quadrivium together with arithmetic, geometry and astronomy
“…, the Pythagoreans were of the opinion of dividing the entire mathematical science into four parts: […] arithmetic considers the number in itself, music considers it in relation to another number; geometry considers the extension as immobile, astronomy considers it mobile on itself.” reports Proclus in his “Commentary on the First Book of Euclid’s Elements”
The tight link between music and mathematics, true for music in general, is even more evident in Canons.
This CD is born from the study and mathematical analysis of the main Canons and the belief that, as the ethnomusicologist Maurizio Giani wrote in his essay “The question of the Canon”, “to give space today to Bach, Beethoven, Shubert, etcetera, simply means doing something to ensure that an invaluable artistic heritage is still brought to the young people’s attention”.
This is a work dedicated to our friend John Lunn and all musicians and mathematicians that highlights one of the most artistic and emotional aspects of mathematics, showing that is far less strict than it might seem.
CANON
The word “canon” comes from the Sumeric term “ganeh” (rod, measuring cane, rule).
We can also find a variation in the Greek term kanṓn -ónos that means (ruler, reed) and in the Latin terms canon -ŏnis (rule)
It has different meanings: norm, rule, criteria, principle, but also group of authors or handiwork considered models, legal norm of universal value or amount of money to be given periodically.
In music, canon is a contrapuntal composition based on the repetition of a theme by one or more mimicking voices that start to play, overlapping, at different times or pitches respect to the main melody.
The main voice is named dux (or antecedent), while the second is named comes (or consequent).
If there are more than two voices, the comes becomes the dux of the following voice, this is repeated for all the available voices.
There are canons with a very simple structure, such as in the Michael Praetorius canons of the CD, where there is a single melody and the voices enter at different times creating a chasing effect, as well as extremely complex ones.
One can think of the Canon as a kind of game to showcase one’s talent. And it’s precisely in this game that mathematics comes into play; canons are nothing but geometric transformations of the main melody.
MATHEMATICS AND MUSIC
If a musician, while reading a score, can “hear” the melody in its rising and falling movement, in the mind of a mathematician this is generally translated into a cartesian chart of a function that follows the progress of the melody.
If on the cartesian plane we use on the x-axis the time (t, measured in bars, like the beats of a metronome) and on the y-axis the pitch of the sound (A, in ascending order from deep to high), a melody can be represented as a function A=f(t), whose graph is a line with ascending and descending segments that follow the melody.
By applying one or more transformations to the main melody, represented by the function A=f(t), we can generate the other voices (obtaining new functions) and build a CANON.
THE SOFTWARE
Canons can be represented by graphs, made using a simple spreadsheet.
On the x-axis is the time using a bar as a measuring unit; on the y-axis is the tone.
The origin of the axes corresponds to the start of the piece and the note G on the staff.
All notes are represented by adding or subtracting 1 (the value of a semitone), starting from the middle G.
The melody is represented in a table where columns indicate the length of the note and the rows represent the notes of the melody: the base and all the ones derived through transformations that form the canon taken into consideration.
THE CANONS – EXAMPLES OF MATHEMATIC / GRAPHIC DESCRIPTION
SIMPLE or EXACT CANON
In this canon the same melody is played by multiple voices at the same time; they come to play the same melody but start at different times.
This type of canon is obtained via a horizontal translation: the pitch remains the same while time shifts for all voices as described below
t’ = t +a
A’ = A
where a is a multiple or submultiple of a beat
In this case the melody played by the instruments are identical in pitch, but shifted in time.
In case the melody of every voice, once reached the end of the piece, can start from the beginning in a loop, the canon becomes a PERPETUAL or INFINITE CANON, like in the universally known “Frère Jacques”; this is also called circular canon, round, rota or in latin canon perpetuus.
CANON at INTERVAL
The same melody is performed at the same time by multiple voices; the comes play the melody at the same time as the dux, but at different pitches (e.g., a third or a fifth above or below), as in the third canon of the Mozart “three canons”
The melody of the comes is obtained by vertically translating the melody of the dux while the time stays the same.
t’ = t
A’ = A +b
where b is a multiple or submultiple of a tone
CANON by TRANSPOSITION
As in the previous cases the melody is interpreted by multiple voices at the same time: the comes play the same melody as the dux, but at different times and pitches. This type of canoni s the combination of the previous two and is obtained by the composition of two translations: a horizontal of magnitude a and a vertical one of magnitude b; this is called oblique translation, as in Melquiades Vasconcelos’s Ventos II movement.
t’ = t +a
A’ = A +b
Where a is a multiple or submultiple of a beat and b is a multiple or submultiple of a tone
INVERTED CANON or by CONTRARY MOTION
In this type of canon the comes moves in a contrary direction respect to the dux, if the dux has an ascending motion, the comes has a descending motion and vice versa; this is obtained via an axial symmetry respect to a horizontal axis, as in the Der Spiegel Duet proposed in the CD
t’ = t
A’ = 2Ao -A
The axis of symmetry is y = Ao La whose position corresponds to the pitch of A
RETROGRADE CANON
In this type of canon, the comes starts from the last note of the dux and proceeds backwards, ending on the first note. This is obtained by applying a vertical symmetry respect to a vertical axis that corresponds to the central bar of the whole melody
t’ = 2to – t
A’ = A
The axis of symmetry is the straight line, x = to, entire melody.
CRAB CANON
In this type of canon, the comes doesn’t just start from the last note and proceeds backwards, but all note progressions are reversed. This is achieved via a central symmetry positioned on the central bar of the melody for the x-axis and the central tone for the y-axis. (see Musikalische Opfer by JS Bach)
t’ = 2to – t
A’ = 2Ao -A
This type of canon can be executed ad libitum as if it was written on a Möbius strip.
MENSURAL or PROPORTIONAL CANON
In this kind of canon the comes plays the same melody as the dux, but increasing or decreasing the rhythmic values of the notes (canon per augmentationem – as in David Manzanares Viles Canon per Augmentationem- or canon per diminutionem). This is achieved via a scaling transformation (homothety) on the x-axis.
t’ = at
A’ = A
The effect is the same as playing the same melody with a double metronome speed.
In the example the scaling a is ½: Flute 2 plays the same melody as Flute 1, but doubling the speed.
REPETITION: a melody, or just a fragment, repeated multiple times one after the other (an example is the Ostinato Bass of euphonium in Luigi Cherubini Canone a 3 voci) is comparable to a periodic function that repeats identical to itself after a fixed time (e.g. a sinusoid function, a wave). A periodic function f(t) = f(t+nT) is achieved with an horizontal translation of the original fragment. This can be considered as a repeated horizontal translation.
Fabrizia Barbin