Guerino Mazzola

Guerino Mazzola (born 1947) is a Swiss mathematician, musicologist, jazz pianist as well as book writer.

Guerino Mazzola
Guerino Mazzola
At University of Zurich, 2002
Background information
Birth nameGuerino Mazzola
BornFebruary 2, 1947 (age 72)
OriginDübendorf (Canton of Zürich), Switzerland
Occupation(s)mathematician, music theorist, and jazz pianist.


Mazzola graduated at the University of Zürich in Mathematics, Theoretical Physics and Crystallography and completed his PhD in Mathematics in 1971. In 1980, he habilitated in Algebraic Geometry and Representation Theory. In 2000, he was awarded the medal of the Mexican Mathematical Society. In 2003, he habilitated in Computational Science at the University of Zürich.

Mazzola has recorded several free jazz CDs with musicians like Mat Maneri, Heinz Geisser, Sirone, Jeff Kaiser, Scott Fields, Matt Turner and Rob Brown.

Mazzola is well known for his application of sophisticated mathematical concepts such as topos theory to music theory, described in his book The Topos of Music. The result has been somewhat controversial, drawing praise from some mathematicians and music theorists, such as Alexander Grothendieck, Yuri Manin, Yves André, François Nicolas and Thomas Noll, and dissent from others such as Dmitri Tymoczko, who said of Mazzola: "If you can't learn algebraic geometry, he sometimes seems to be saying, then you have no business trying to understand Mozart."[1]

Currently he is Professor at the School of Music at the University of Minnesota.[2] Since 2007 he is the president of the Society for Mathematics and Computation in Music.[3]


  • Gruppen und Kategorien in der Musik. Hermann (1985) ISBN 3-88538-210-5.
  • Rasterbild - Bildraster, CAD-gestützte Analyse von Raffaels "Schule von Athen". Springer (1987) ISBN 9783540172673.
  • Geometrie der Töne. Birkhäuser (1990) ISBN 3-7643-2353-1.
  • Ansichten eines Hirns. Birkhäuser (1990) ISBN 3-7643-2484-8.
  • The Topos of Music, Geometric Logic of Concepts, Theory, and Performance. Birkhäuser (2002) ISBN 3-7643-5731-2.
  • Perspectives in Mathematical Music Theory.. EpOs (2004). ISBN 978-3-923486-57-1.
  • Comprehensive Mathematics for Computer Scientists I & II here and Errata here
  • Elemente der Musikinformatik. Birkhäuser (2006) ISBN 3-7643-7745-3.
  • La vérité du beau dans la musique. Delatour/IRCAM (2007) ISBN 2-7521-0029-9.
  • Flow, Gesture, and Spaces in Free Jazz—Towards a Theory of Collaboration. Springer (2009) ISBN 978-3-540-92194-3.
  • Musical Performance. Springer (2011) ISBN 978-3-642-11837-1.
  • Musical Creativity—Strategies and Tools in Composition and Improvisation. Springer (2011) ISBN 978-3-642-24516-9.
  • Computational Musicology in Hindustani Music. Springer (2014) ISBN 978-3319114712.
  • Computational Counterpoint Worlds. Springer (2015) ISBN 978-3-319-11235-0.
  • Cool Math for Hot Music. Springer (2016) ISBN 978-3-319-42935-9.
  • All About Music. Springer (2016) ISBN 978-3-319-47334-5.
  • The Topos of Music, 2nd ed. Vol. I: Theory. Springer (2017) ISBN 978-3-319-64364-9.
  • The Topos of Music, 2nd ed. Vol. II: Performance. Springer (2017) ISBN 978-3-319-64444-8.
  • The Topos of Music, 2nd ed. Vol. III: Gestures. Springer (2017) ISBN 978-3-319-64481-3.
  • The Topos of Music, 2nd ed. Vol. IV: Roots. Springer (2017) ISBN 978-3-319-64495-0.
  • Basic Music Technology. Springer (2018) ISBN 978-3-030-00982-3.


  • Mazzola/Piano Solo Kelvin Null OMP Records 1001 LP
  • Mazzola/Piano Solo Akroasis Wergo SM 1024 LP
  • Mazzola/Moor/Sollberger Aus dem Hinterhalt OMP Records 1002 LP
  • Q4 Orchestra Lyons' Brood Creative Works CW 1018 CD
  • Guerino Mazzola Synthesis SToA music ST-71.1001 CD
  • Jan Beran Immaculate Concept SToA music ST-71.1002 CD
  • Q4 Orchestra Yavapai Creative Works CW 1028 CD
  • Rissi-Mazzola-Geisser Fuego Creative Works CW 1029 CD
  • Brown-Mazzola-Geisser Orbit Music & Arts CD-1015 CD
  • Mazzola-Geisser Toni's Delight Cadence Jazz Records 1090 CD
  • Mazzola/Geisser/Fields/Turner Maze Quixotic Records 5002 CD
  • Mazzola/Geisser /Fields/Maneri Heliopolis Cadence Jazz Records 1122 CD
  • Mazzola/Geisser Folia Silkheart Records SHCD 153 CD
  • Mazzola/Geisser/Rissi Tierra Cadence Jazz Records 1130 CD
  • Mazzola/Geisser/Rissi Agua Cadence Jazz Records, 1150 CD
  • Mazzola/Geisser Someday Silkheart Records 154 CD
  • Mazzola/Geisser/Fields/Maneri Chronotomy BlackSaint 120173-2 CD
  • Mazzola/Geisser/Kato/Saga Live at Airegin Ayler Records aylDL-056 CD
  • Mazzola/Geisser/Rissi Herakleitos Ayler Records aylDL-069 CD
  • Mazzola/Geisser/Rissi Aire Cadence Jazz Records 1130 CD
  • Mazzola/Geisser/Kaiser/Sirone Liquid Bridges CD in Springer book Flow, Gesture, and Spaces
  • Mazzola/Geisser/Onuma Dancing the Body of Time Cadence Jazz Records 1239 CD
  • Mazzola/Park Passionate Message Silkheart Records 159 CD
  • Mazzola/Geisser/Kita Ma pfMentum PFMCD116
  • Mazzola/Lubet Deep State pfMentum PFMCD119
  • Mazzola/Geisser Live at Le Classique pfMentum PFMCD126


  1. ^ Dmitri Tymoczko. "Mazzola's Counterpoint Theory" (PDF). Retrieved May 17, 2018.
  2. ^ "Faculty". Retrieved May 17, 2018.
  3. ^ "SMCM: Society for Mathematics and Computation in Music". Retrieved May 17, 2018.

External links

Cadence Jazz Records

Cadence Jazz is an American record company and label specializing in noncommercial modern jazz. It is associated with Cadence Magazine.

Cadence Jazz was founded by Bob Rusch in Redwood, New York in 1980.

By 2000 the label had issued more than 100 albums. Its catalogue includes Marilyn Crispell, Beaver Harris, and Frank Lowe. This label is different from the Cadence that produced pop music in the 1950s and 1960s.

Category theory

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms). A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.

Several terms used in category theory, including the term "morphism", are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself.

Samuel Eilenberg and Saunders Mac Lane introduced the concepts of categories, functors, and natural transformations in 1942–45 in their study of algebraic topology, with the goal of understanding the processes that preserve mathematical structure.

Category theory has practical applications in programming language theory, for example the usage of monads in functional programming. It may also be used as an axiomatic foundation for mathematics, as an alternative to set theory and other proposed foundations.


In music, counterpoint is the relationship between voices that are harmonically interdependent (polyphony) yet independent in rhythm and contour. It has been most commonly identified in the European classical tradition, strongly developing during the Renaissance and in much of the common practice period, especially in the Baroque. The term originates from the Latin punctus contra punctum meaning "point against point".


Dübendorf is a fast growing municipality in the district of Uster in the canton of Zürich in Switzerland.

It is a suburb of Zürich in Switzerland with a population of about 28,000 (2018). It is the fourth largest city in the canton, after Zürich, Winterthur, and Uster.

Mat Maneri

Mat Maneri (born October 4, 1969) is an American composer, violin, and viola player. He is the son of the saxophonist Joe Maneri. and his wife Sonja Maneri.

Mazzola (surname)

Mazzola is an Italian surname. Notable people with the surname include:

Alessandro Mazzola (footballer born 1969), Italian footballer

Caterino Mazzolà, Italian poet and librettist

Denia Mazzola, Italian operatic soprano

Ferruccio Mazzola, Italian footballer

Frank Mazzola, American film editor

Girolamo Mazzola Bedoli (1500–1569), Italian painter

Guerino Mazzola, Swiss mathematician

Joey Mazzola, American guitarist

José Altafini, Brazilian footballer who played under the name Mazzola

Marissa Mazzola-McMahon, American film producer

Rose Mazzola, American musician

Sandro Mazzola, Italian footballer

Valentino Mazzola, Italian footballer

Musical gesture

In music, gesture is any movement, either physical (bodily) or mental (imaginary). As such "gesture" includes both categories of movements required to produce sound and categories of perceptual moves associated with those gestures. The concept of musical gestures has received much attention in various musicological disciplines (e.g. music analysis, music therapy, music psychology, NIME) in recent years.

For example, the "musical" movement from a close-position tonic C major chord to a close-position dominant G major chord requires on the piano the physical movement from each white key of the first chord to the right (in space, upwards in pitch) four white keys or steps. Thus gesture includes both characteristic physical movements by performers and characteristic melodies, phrases, chord progressions, and arpeggiations produced by (or producing) those movements.


In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold. It is a topological space (called the underlying space) with an orbifold structure (see below).

The underlying space locally looks like the quotient space of a Euclidean space under the linear action of a finite group. Definitions of orbifold have been given several times: by Satake in the context of automorphic forms in the 1950s under the name V-manifold; by Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the name orbifold, after a vote by his students; and by Haefliger in the 1980s in the context of Gromov's programme on CAT(k) spaces under the name orbihedron. The definition of Thurston will be described here: it is the most widely used and is applicable in all cases.

Mathematically, orbifolds arose first as surfaces with singular points long before they were formally defined. One of the first classical examples arose in the theory of modular forms with the action of the modular group SL(2,Z) on the upper half-plane: a version of the Riemann–Roch theorem holds after the quotient is compactified by the addition of two orbifold cusp points. In 3-manifold theory, the theory of Seifert fiber spaces, initiated by Seifert, can be phrased in terms of 2-dimensional orbifolds. In geometric group theory, post-Gromov, discrete groups have been studied in terms of the local curvature properties of orbihedra and their covering spaces.

In string theory, the word "orbifold" has a slightly different meaning, discussed in detail below. In two-dimensional conformal field theory, it refers to the theory attached to the fixed point subalgebra of a vertex algebra under the action of a finite group of automorphisms.

The main example of underlying space is a quotient space of a manifold under the properly discontinuous action of a possibly infinite group of diffeomorphisms with finite isotropy subgroups. In particular this applies to any action of a finite group; thus a manifold with boundary carries a natural orbifold structure, since it is the quotient of its double by an action of Z2. Similarly the quotient space of a manifold by a smooth proper action of S1 carries the structure of an orbifold.

Orbifold structure gives a natural stratification by open manifolds on its underlying space, where one stratum corresponds to a set of singular points of the same type.

One topological space can carry many different orbifold structures. For example, consider the orbifold O associated with a factor space of the 2-sphere along a rotation by ; it is homeomorphic to the 2-sphere, but the natural orbifold structure is different. It is possible to adopt most of the characteristics of manifolds to orbifolds and these characteristics are usually different from correspondent characteristics of underlying space. In the above example, the orbifold fundamental group of O is Z2 and its orbifold Euler characteristic is 1.

Orbit (Rob Brown, Guerino Mazzola and Heinz Geisser album)

Orbit is a collaborative album by American jazz saxophonist Rob Brown and the Swiss duo composed of pianist Guerino Mazzola and percussionist Heinz Geisser. It was recorded in 1996 and released on the Music & Arts label. Mazzola and Geisser worked together since 1994, before this album they played as a trio of similar instrumentation with Swiss saxophonist Mathias Rissi instead of Brown.

Pit Inn (jazz club)

The Pit Inn (ピットイン) is a jazz club in Shinjuku, Tokyo. The original opened in 1966 and was forced by demolition to close in 1992. It re-opened at a different site in Shinjuku later that year. DownBeat wrote in 2019 that the Pit Inn "is almost universally regarded as Japan's most important jazz club".


A raga or raag (IAST: rāga; also raaga or ragam ; literally "coloring, tingeing, dyeing") is a melodic framework for improvisation akin to a melodic mode in Indian classical music. While the rāga is a remarkable and central feature of the classical music tradition, it has no direct translation to concepts in the classical European music tradition. Each rāga is an array of melodic structures with musical motifs, considered in the Indian tradition to have the ability to "colour the mind" and affect the emotions of the audience.A rāga consists of at least five notes, and each rāga provides the musician with a musical framework within which to improvise. The specific notes within a rāga can be reordered and improvised by the musician. Rāgas range from small rāgas like Bahar and Shahana that are not much more than songs to big rāgas like Malkauns, Darbari and Yaman, which have great scope for improvisation and for which performances can last over an hour. Rāgas may change over time, with an example being Marwa, the primary development of which has gone down to the lower octave compared to the traditionally middle octave. Each rāga traditionally has an emotional significance and symbolic associations such as with season, time and mood. The rāga is considered a means in Indian musical tradition to evoke certain feelings in an audience. Hundreds of rāga are recognized in the classical tradition, of which about 30 are common. Each rāga, state Dorothea E. Hast and others, has its "own unique melodic personality".There are two main classical music traditions, Hindustani (North Indian) and Carnatic (South Indian), and the concept of rāga is shared by both. Rāga are also found in Sikh traditions such as in Guru Granth Sahib, the primary scripture of Sikhism. Similarly it is a part of the qawwali tradition found in Sufi Islamic communities of South Asia. Some popular Indian film songs and ghazals use rāgas in their compositions.

Reza Negarestani

Reza Negarestani is an Iranian philosopher and writer, known for "pioneering the genre of 'theory-fiction' with his book" Cyclonopedia which was published in 2008. it was listed in Artforum as one of the best books of 2009.


Rhythm (from Greek ῥυθμός, rhythmos, "any regular recurring motion, symmetry" (Liddell and Scott 1996)) generally means a "movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions" (Anon. 1971, 2537). This general meaning of regular recurrence or pattern in time can apply to a wide variety of cyclical natural phenomena having a periodicity or frequency of anything from microseconds to several seconds (as with the riff in a rock music song); to several minutes or hours, or, at the most extreme, even over many years.

In the performance arts, rhythm is the timing of events on a human scale; of musical sounds and silences that occur over time, of the steps of a dance, or the meter of spoken language and poetry. In some performing arts, such as hip hop music, the rhythmic delivery of the lyrics is one of the most important elements of the style. Rhythm may also refer to visual presentation, as "timed movement through space" (Jirousek 1995) and a common language of pattern unites rhythm with geometry. In recent years, rhythm and meter have become an important area of research among music scholars. Recent work in these areas includes books by Maury Yeston (1976), Fred Lerdahl and Ray Jackendoff (Lerdahl and Jackendoff 1983), Jonathan Kramer, Christopher Hasty (1997), Godfried Toussaint (2005), William Rothstein (1989), Joel Lester (Lester 1986), and Guerino Mazzola.

Rob Brown (saxophonist)

Rob Brown (born February 27, 1962) is an American free jazz saxophonist and composer.

Rubato Composer

Rubato Composer is free (GPL) software that allows one to compose music or transform existing music with the help of mathematical Category Theory and Topos Theory. It is currently the only software for music composition based on Category Theory. It is being developed by Gérard Milmeister and Guerino Mazzola, both of whom are currently working on a book about it, The RUBATO Bible (working title). The book is currently in preparation, to appear during 2008.

The only other similar software, Presto, is no longer developed and runs only on Atari computers.

Silkheart Records

Silkheart Records is a Swedish record company and label dedicated to improvised music and free jazz.

Lars-Olof Gustavsson and Keith Knox founded Silkheart in 1985. In 1991, Jimmy Johnson of Forced Exposure suggested that Silkheart "could easily be considered for the new ESP-Disk throne." The Penguin Guide to Jazz describes the four albums that Dennis González recorded for the label as "part of a determined effort to wrest creative initiative back from New York and the West Coast". Charles Brackeen (whom Silkheart's management and González had coaxed out of retirement) recorded three albums for Silkheart.

Society for Mathematics and Computation in Music

The Society for Mathematics and Computation in Music (SMCM) was founded in 2006 as an International Forum for researchers and musicians working in the trans-disciplinary field at the intersection of music, mathematics and computation. The SMCM is registered in the USA. At its inaugural meeting in Berlin on May 20, 2007, 13 board members were elected. The board later elected the officers for the society.

The School of Athens

The School of Athens (Italian: Scuola di Atene) is a fresco by the Italian Renaissance artist Raphael. It was painted between 1509 and 1511 as a part of Raphael's commission to decorate the rooms now known as the Stanze di Raffaello, in the Apostolic Palace in the Vatican. The Stanza della Segnatura was the first of the rooms to be decorated, and The School of Athens, representing Philosophy, was probably the third painting to be finished there, after La Disputa (Theology) on the opposite wall, and the Parnassus (Literature). The picture has long been seen as "Raphael's masterpiece and the perfect embodiment of the classical spirit of the Renaissance".

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