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         Fractals & Music:     more detail
  1. Fractal Music, Hypercards and More Mathematical Recreations from "Scientific American" by Martin Gardner, 1991-11-30
  2. Fractals in Music: Introductory Mathematics for Musical Analysis Second Edition (Inmusic) by Charles Madden, 2007-04-23
  3. Music and Mathematics: From Pythagoras to Fractals
  4. Fractal Music, Hypercards and More. Mathematical Recreations from Scientific Ame by Martin Gardner, 1992-01-01
  5. Fractal Music, Hypercards and More : Mathematical Recreations from Scientific Am by Martin Gardner, 1991-01-01
  6. Fractal Fantasy, the Art of Mathematics by Charles Fitch, Music by Michael Angelo Strasmich, 1987
  7. Music And Mathematics: From Pythagoras To Fractals by Raymond Flood, Robin Wilson, Robin J. Wilson John Fauvel, 2003
  8. Formalised composition on the spectral and fractal trails (Skrifter fran Musikvetenskapliga institutionen, Goteborgs Universitet) by Magnus Eldenius, 1998

41. Chaffey's Fractals - Music
Chaffey's fractals on the WEB© Chaffey High School's fractals on the Web http//www.chaffey.org/fractals/. FractalMusic Links Last Updated February 28, 2003.
http://www.chaffey.org/fractals/music/
Chaffey High School's FRACTALS on the Web
http://www.chaffey.org/fractals/ Fractal Music Links
Last Updated February 28, 2003 The Fractal Music Project
http://www-ks.rus.uni-stuttgart.de/people/schulz/fmusic/

At the University of Stuttgart, Germany Fractal MIDI Music Here!
(hold SHIFT+click to download)
This is a .ZIP that contains a small collection of fractal MIDI music that was created by DT Strohbeen. Fractal Music LAB
http://members.aol.com/strohbeen/fml.html

by DT Strohbeen MIDI Piece - 'Fractal Window I'
http://sprott.physics.wisc.edu/newpc1.htm

Experiments in Fractal Music
http://www.ozemail.com.au/~davesag/snd.html ... Dave Sag in Australia Hearing the Mandelbrot Set http://www.vanderbilt.edu/VUCC/Misc/Art1/Sonify/Mandi.html by Brian Evans at the Vanderbilt University, located in Nashville, Tennessee Sonic'sFractal Sounds http://www.galivant.com/~tjustman/sonic.htm Sonic Fractal conducts aural experiments in ambience and mood alteration. Symbolic Composer: "Composition Modelling " http://www-ks.rus.uni-stuttgart.de/people/schulz/fmusic/symbolic/mainpage.html his document describes the basic features of the Symbolic Composer professional. It is one of the richest music programming languages developed on a computer. Fractal Music by bojog http://www.bali-net.com/fracmus/

42. Fractal Links
Fractal music Software from Fractovia. Fractal music Software; fractals;music FROM CHAOS index; SSEYO homesite Koan generative music
http://www.apu.ac.uk/music/rhoadley/links/links_fractals.html
links
Fractal Links

Richard Hoadley

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43. Fractal Geometry Summer Workshop
3. H. fractals in Literature and music. Literature is perceived in a serial fashion. fractalsand music form a rich ground for a variety of student projects.
http://classes.yale.edu/99-00/math190a/FracLit.html
3. H. Fractals in Literature and Music
Literature is perceived in a serial fashion. Global views are more difficult in literature than in painting, for example. Yet still the author is surrounded by the hierarchies of nature. Finding fractals in literature may be more difficult than finding fractals in painting, but as we shall see, fractals do occur in literature. Here are some examples: Dimensions of Texts Discovered Fractals in Literature Intentional Fractals in Literature Structural Fractals in Literature ... Wuorinen Return to 3. G. Complex Newton's Method and the Universality of the Mandelbrot Set Return to 3. The Mandelbrot Set and Julia Sets

44. Fractal Geometry Panorama
compositions. fractals and music is another aspect of this relationship,though as some argue, more farreaching than most. Some
http://classes.yale.edu/fractals/Panorama/Music/Mus/Music.html
Music
Music and mathematics have a long and complicated relationship, from the Platonists' Music of the Spheres, through Bach's geometrical symmetries, to Cage's stochastic compositions. Fractals and music is another aspect of this relationship, though as some argue, more far-reaching than most. Some contemporary composers explicitly use fractals to guide their compositions (some student projects have pursued this direction, with varying degrees of success), others find fractal aspects in the works of great baroque and classical composers. Gardner Hsu and Hsu Ligeti 1/f Music ... Wuorinen

45. Fractals
But there is something defeatistic in denying the possibilities of actually usingfractals in music while admiring the PeitgenRichter pictures for their
http://www.notam02.no/~rolfwa/Fractalarticle.html
Rolf Wallin:
Fractal Music - Red Herring or Promised Land?
or "Just Another of those Boring Papers on Chaos"

Lecture given at the Nordic Symposium for Computer Assisted Composition
Stockholm 1989
(Oops - seems that the references to illustrations don't refer to any illustrations... sorry...)
If the Vietnam war was the first TV war, the so-called Chaos theory must be the first commercially available scientific revolution, brought to a large public step by step. Chaos, fractals, Mandelbrot Sets, strange attractors, Cantor dust - these and some dozens other magical words have totally invaded popular science magazines during the 80's, accompanied by glossy colourprints of strange, disturbingly organical computergenerated shapes.
However, this is just the shiny facade of a revolution that has left the international community of scientists of most diciplines at first totally dumbfounded by seeing central parts of their theoretical basis crumble, then completely frantic to investigate to what extent these scientific Columbi eggs can be applicable on their particular field. Fluid mechanics, geology, medicine, meteorology, in most fields many unsolvable problems have been solved by applying some part of this manyfaceted and bewildering bag of theories. Visual artists has also been attracted by the organical shapes that appear on the computer screens.
So how about Music? Will fractal mathematics come to rescue in a time of apparent stagnation, this time in the guise of simple mathematical algorithms that yield delicious patterns that can be further processed, either as succession of musical events or as complex frequency spectra? The attempts to find an answer to this has barely begun. In the field of music the answers are not as selfevident as for the visual field. But this can actually turn out to be an advantage, if not technically, then at least aesthetically.

46. Applications Of Fractals
(Landini 2512). fractals AND music. Schroeder, Manfred. fractals in music, chapter11 in CA Pickover (Ed.). Fractal Horizons The Future Use of fractals.
http://www.stanford.edu/~jje/fractals/html/applications.html
by Jonathan Effrat MEDICAL APPLICATIONS OF FRACTALS Histopathology is concerned with the study of the morphological changes in cells and tissues during disease. These changes occur at the microscopic or submicroscopic level. Histopathology helps to interpret histological images (tissue samples) to make a diagnosis and selection of treatment. The earlier the diagnosis is made, the better the chances of successful treatment. The analysis and description of such complex and irregular morphologies is difficult and, to date, has been qualitative and subjective. These difficulties may be surmountable by using fractal geometry, which provides new approaches to objective measurement and understanding of shape complexity. Fractal geometry enables us to model and understand many physical and natural process that previously were considered irregular and patternless. We can use these “pathological” geometries to quantify morphological features of cells and tissues in order to differentiate between normal and pathological changes. (Landini 251-2) FRACTALS AND MUSIC Certain number-theoretic sequences with fractal properties have been found to generate appealing melodies and rhythms when the numerical data is transformed into musical notes and rhythmic patterns. Fractal designs have been used in composition, leading to melodies that, although purely mathematical in origin, sound surprisingly baroque. The analysis of music in terms of fractal patterns may lead to new insights in the classification of music. New musical instruments with fractal design features, such as drums with fractal perimeters, await exploration. (Schroeder 207)

47. BUBL LINK / 5:15 Internet Resources: Fractals
Author Larry Solomon Subjects computer music, fractals, music composition, musictheory DeweyClass 781.3 ResourceType documents Location usa Last checked
http://bubl.ac.uk/link/f/fractals.htm
BUBL LINK / 5:15 Catalogue of Internet Resources Home Search Subject Menus A-Z ... About
Fractals
A-Z Index Titles Descriptions
  • Center for Polymer Studies
  • Chaos Hypertextbook
  • Complexity International
  • Fractal Music Project ...
  • WWW Virtual Library: Complex Systems Page last updated: 17 March 2003 Comments: bubl@bubl.ac.uk
    Center for Polymer Studies
    Interdisciplinary research centre for aspects of polymer, random, and fractal systems which develops experimental and computational materials for school and undergraduate education.
    Author: Boston University
    Subjects: chemistry education, fractals, organic chemistry, polymers
    DeweyClass:
    ResourceType:
    guides
    Location: usa
    Last checked:
    Chaos Hypertextbook
    An essay on chaos, fractals, and non-linear dynamics. Chapter headings include mathematical experiments, strange and complex, what is dimension and measuring chaos.
    Author: Glenn Elert
    Subjects: chaos, fractals
    DeweyClass:
    ResourceType:
    documents, images Location: usa Last checked:
    Complexity International
    A refereed journal dealing with any area of complex systems research and the generation of complex behaviour from the interaction of multiple parallel processes. Author: Charles Sturt University Subjects: chaos, complex systems, fractals, neural networks
  • 48. Abstract: Fractals In Music
    Fractal music. Abstract. This speculation will introduce a new, interestingmusical genre. It discusses the existence of fractals in music.
    http://www.sewanee.edu/Phy_Students/123_Spring01/bateyav0/PROJECT.html
    Fractal Music Abstract This speculation will introduce a new, interesting musical genre. It discusses the existence of fractals in music. It will discuss the characteristics of fractal music, how it can be generated, and why musical fractals can be interesting. This discussion will also link fractal music to the basic concepts of fractal images and chaos. What is Fractal Music? Characteristics Generating Fractal Music Similarity to Fractal Concepts ... Bibliography What is Fractal Music?
    • Fractal music is the multiplication of an algorithm that produces a recursive process. It will musically repeat the previous process.
    Characteristics
    • It includes self similar, recursive patterns that are repeated continuously. There should be slight changes in the rhythm pattern.
    • There are three different types of noises in nature. There is white noise, brown noise, and pink noise.
    • Fractal music is made mostly of pink noise.
    Generating Fractal Music
    • Altering pink noise can be used to make fractal music.
    • If a section of the pink noise above was taken and notes were translated to it, it would look like this:
    • If another pink noise is added to this picture, it will give the notes duration, or length, which is:

    49. CONTEMPORARY MUSIC By DAVID CLARK LITTLE * COMPOSITION With FRACTALS And CHAOS
    CONTEMPORARY music by DAVID CLARK LITTLE COMPOSITION with fractals andCHAOS. David Clark Little (1952, USA) studied chemistry, harpsichord
    http://www.sca.ahk.nl/david/
    CONTEMPORARY MUSIC by DAVID CLARK LITTLE
    COMPOSITION with FRACTALS and CHAOS
    This hypertext index will lead to more information about my compositions and compositional methods. This website has been selected as one of the Net's finest informative and is included in StudyWeb's listing of educational links.
    Go to StudyWeb's site
    Links2Go Key Resource Award, Computer Music Topic.
    Go to
    Links Go, Key Resource, Computer Music Topic
    Website created by DCL, updated Jan 2002

    50. Fractals In Music: Introductory Mathematics For Musical Analysis... By Charles M
    Buy fractals in music Introductory Mathematics for musical Analysis fractalsin music Introductory Mathematics for musical Analysis
    http://www.rbookshop.com/mathematics/t/Topology/Fractals_in_Music_Introductory_M
    AMAZON.COM
    Search All Products Electronics Books Classical Music DVD Kitchen Popular Music Software Toys VHS Video Video Games
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    Electronics Store Electronics Store ... Toy Store NOTICE : All prices, availability, and specifications are subject to verification by their respective retailers. Privacy Policy info@rbookshop.com Last Modified : 4-7-2003 Fractals in Music: Introductory Mathematics for Musical Analysis... Home Mathematics Books Topology Fractals in Music: Introductory Mathematics for Musical Analysis... by Charles Madden (Hardcover - August 1999) List Price: $39.95 At Amazon on 4-7-2003. Features
  • Hardcover: 210 pages ; Dimensions (in inches): 0.77 x 8.85 x 5.75 Publisher: High Art Pr; (August 1999)
  • ISBN: 0967172756 From the Publisher Fractals in Music is an interesting crossover between disciplines. It shows how the mathematics of chaos theory and fractal geometry can be applied to understanding music in ways that may be more applicable to the technological age than current musical analysis is. Advertised as "music theory for the new millennium," it discusses Fibonacci numbers, statistics, and Fourier analysis, as well as randomicity, correlation, and inner/outer form.
  • 51. Computer Music And The Importance Of Fractals, Chaos, And Complexity Theory
    Computer music and the Importance of fractals, Chaos, and ComplexityTheory. Dr. Mladen Milicevic, The University of South Carolina
    http://www.ircam.fr/equipes/repmus/jim96/actes/milicevic.html
    Computer Music and the Importance of Fractals, Chaos, and Complexity Theory
    Dr. Mladen Milicevic, The University of South Carolina - Media Arts, Columbia SC 29208, USA MMladen@SC.edu Abstract: The main concern addressed in this paper relates to the postmodern vs. hypermodern use of knowledge based on scientific findings as applied to philosophy and the aesthetics of computer music. 1. Possibilities of Complexity do something. So when the next falling grain hits the pile, nothing oranythingmay happen. It can result in a tiny shift of just few grains or in a catastrophic collapse of the whole pilethere is no way of predicting this non-linear result. This example displays a behavior very similar to the critical mass of plutonium, in which the chain reaction is just barely on the verge of nuclear explosion. Such behavior is very common in nature, points out Bak, and we can see the ubiquity of large impulses being rare and small ones being common. Crisis management becomes a norm for nature, pushing itself to the brink between control and chaos. There is an easy misunderstanding of emergent behavior which lies in the assumption that whatever is produced on the verge of chaos is good or creative. It is simply new, and evolution over time may show only if it is anthropocentrically positive or negative. Even when the product of emergent behavior fits into the ecological system, this does not mean that it is elegant or aesthetically pleasing; it simply means that it works within the larger context. It does not even mean that it is the best solutionit simply represents the first one that came along, hit critical mass and went into action.

    52. Links
    Fractal music Sites. There are many sites out there which explain fractals,fractal music and the approaches to generate fractal music.
    http://home.pacbell.net/bulens/Links.htm
    "If in mathematics we treat nature a an analogy of art, in the greatest of music we treat art as an analogy of nature."
    - Edward Rothstein
    Fractal Music Sites
    There are many sites out there which explain fractals, fractal music and the approaches to generate fractal music. These are my favorite sources online. I confess that when I started this project, I was convinced that I was working on something completely unique. I did not know other folks were working on fractal music. When I found the Yahoo site listed below I was initially pretty bummed out... It popped my bubble... But the truth is that there is a lot of excellent work out there, and a lot of resources to bounce ideas and music off of... Yahoo! Entertainment:Music:Computer Generated:Computer Generated Composition:Fractal Music - Many great links.
    The Fractal Music Lab
    - Lotsa information on fractals, fractal music and fractal music software.
    CNFractal_Music Forum
    - A forum specifically for fractal music. At the moment, it's almost all composers and programmers, which makes it a very valuable resource for sharing ideas and music.
    Techniques for Algorithmic Composition of Music
    - One of my favorite papers on the subject. I stole

    53. Bandwidth February 2001: Music: AutoCad P.1 Of 4
    trunk of the main tree with its own branches and leaves (For additional informationon fractals see the end of this article). So how does this relate to music?
    http://www.ybfree.com/5AUTOCAD1.html
    Current Issue The Archives Our Top 5 Articles Ebay Auctions ... Subscribe BANDWIDTH A Popular Culture Electronic Magazine Click here for a printable version of this article
    AutoCad: Turning Math into Music
    Listen to AutoCad at MP3.com By: JenniferJ W hat is the sound of a tree growing? What kind of sound do you think it would make? What if all you had to do to answer this question was input the distance from the base of the tree to its branches? Well ok, you will not discover the sound of trees growing, but you can get a pretty good guess through computers and a mathematical theory known as fractals
    A VERY Brief and SIMPLE Explanation of Fractals
    A ccording to the Polish born mathematician, Benoit Mandelbrot , who formulated the Fractal Theory, a fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced size copy of the whole." The main characteristic of a fractal is that it repeats itself to into subdivided copies of the whole. These recursive or repetitive properties help those who use fractals to measure natural objects like clouds and shorelines with more accuracy. What makes fractals so hard to measure is the fact that their shapes are not entirely predictable; meaning how a shoreline takes shape is dependent on many conditions like erosion.

    54. Fractal Visions - Sites And Sounds
    There is much information about fractals and music, and algorithmicmusic generally, available on the web. fractals, Chaos, and music.
    http://www.fractalvisions.co.uk/MLinks.htm
    Sites and Sounds - Links
    This track is selected at random - use your browser's 'refresh' button to re-select. This is the fractal music links page. Please feel free to email me if you know any other good sites. There is much information about fractals and music, and algorithmic music generally, available on the web. As is the case with fractal image generating software, many of the sites where you can download the programs have extensive information about the general principles as well as the software itself. Algoritmic/Fractal Music Software Links
    MIDI and Other Software Links
    General Information on Fractal/Algorithmic Music and Composers' Galleries
    The Fractal Music Webguide
    - Links to a lot of sites on the subject. Last time I looked, some of the links were no longer applicable, but many are and very informative. Fractal Music Webguide Fractal Music Lab - Informative site on fractal theory, composition, software and more. It also has plenty of links to software (also includes software not specifically related to fractal music such as MIDI tools, Random music etc. Fractal Music Lab Discovery Online - Fractal Music: The Sound of Chaos - Interesting article on Fractal Music, the people who make it, etc.

    55. Visual Music, Fractals, Computer Graphics Fine Arts Prints In Limited Editions
    computer. What you see is a literal rendering of the tune. Musicmade manifest to the eyes. fractals The visual art of fugue. A
    http://www.lightspace.com/finearts/gallery/gallery.html
    soundscapes
    The stuff of sound. Simple waves in the air when added together can create the sound of a wailing saxophone lament or soft soprano lullaby or these images. Nutcracker snowflakes
    Melodies from Tchaikovsky's Nutcracker Ballet rendered as colorful snowflakes, or as if captured in flight on a brisk winter's night. visual hymns
    Music made visible! Simple hymn melodies are modeled and visualized using a computer. What you see is a literal rendering of the tune. Music made manifest to the eyes. fractals
    The visual art of fugue. A return, after 14 years, to creating fractal images. The tools are too powerful, the images too fun. Wait'll you see a hi-res fractal 4 feet tall! Neva Marie
    The Infinite Heart Series. Neva is a virtual artist (a Ghost Artist) who explores symmetry, and it's resonances in the design of physical and spiritual worlds. Pier Luna
    Another Ghost Artist, Pier is developing a playful series of masks. Just for fun.
    Order
    Contact Home

    56. Mathematics Archives - Topics In Mathematics - Art & Music
    MusiNum The music in the Numbers ADD. KEYWORDS fractals, Sounds, Software; NativeAmerican Art LOGO Style SOURCE Kathryn Ernie, University of Wisconsin
    http://archives.math.utk.edu/topics/artMusic.html
    Topics in Mathematics

    57. Mathematics Archives - Topics In Mathematics - Fractals
    KEYWORDS Lesson Plan, Tutorial, Bibliography; Fantastic fractals ADD. KEYWORDSTutorials, Gallery, Fractal music, Bibliography, Software, Newsgroups; FractED
    http://archives.math.utk.edu/topics/fractals.html
    Topics in Mathematics Fractals

    58. The Uranian Chronicles 3D-pictures, Fractals Music-galleries
    Get it here! FREE ADS Directories Galore! All pictures webdesign copy-right© Jörgen Sangsta -99 -00. free mp3-music for the taking!
    http://zangsta.koolhost.com/
    Jörgen Sangsta
    presents: Click on picture below to enter my homepage! You can also click on buttons below to go directly to those pages Search for:
    All the words Any of the words The exact phrase The page title The person Links to this URL The Boolean phrase All forms of the words Tools and tactics to EXPLODE TRAFFIC to your website before your competition
    does! Extremely powerful! $195.00 Bonus Included. Get it here! All pictures web-design copy-right Jörgen Sangsta -99 -00

    59. Beauty Of Fractals
    Computer Generated music. In the 70s inspired by the music of IannisXenakis - I tried to convert the n-effect-free alternatives
    http://www.gwfa.de/fractal/frmus/frmusic.html
    Computer Generated Music
    In the 70s - inspired by the music of Iannis Xenakis - I tried to convert the n-effect-free alternatives of Karl Popper (which in the limit tend to a uniform distribution of 0's and 1's) into music, what seemed to be a hopeless endeavour whitout the tools we have available today.
    I'm still trying to get familiar with some very nice pieces of software, among them ArtSong by David Strohbeen or the Musical Generator of Arnold Reinders:
    Theme 1 (5'36")

    Theme 2 (3'24")

    Theme 3 (4'17")

    Theme 4 (6'8")
    ...
    Theme 14 (37'30")

    These themes are essentially based on the following two bitmaps
    ( ifs-pick of the 800x600 resolution or a complete smaller selection):
    Theme 15 (4'54")
    Theme 16 (4'04") Theme 17 (4'04") Theme 18 (3'34") ... Theme 26 (7'49") You may expect some additions to this collection in the future. Back to Gallery Index Back to Fractal Page Site Map About ... Entry Page

    60. Music, Architecture, Chaos, And Fractals
    music, Architecture, Chaos, and fractals. music provides a good exampleto use when considering the study of chaos and fractals in art.
    http://www.faegre.org/Book6.html
    Music, Architecture, Chaos, and Fractals Music provides a good example to use when considering the study of chaos and fractals in art. Much of J. S. Bach's music when deconstructed is found to have fractal forms throughout it. The music establishes certain phrases or forms, and then chronologically reverses them, or inverts them, or changes their time signature or key. Likewise, the bass line and the treble line are playful with each other, with a kind of complexity that relates to what we call fractals in today's science. Some historians have expounded on observations that in Baroque times the decoration of the architecture has a quality to it which is similar to that of the music. Some would say that the attractiveness of "jazz" music is for similar reasons. There is a playfulness and spontaneity in the taking of a figure improvising on it that results in complexity which is astoundingly beautiful, if not quite understandable or repeatable. There are composers today who are exploring the creation of modern music using mathematical fractal generators. Samples of this music can be found at one internet site www-ks.rus.uni-stuttgart.de/people/schulz/fmusic/index.html. They do capture some of the baroque quality of Bach. In 1973 I visited Frank Lloyd Wright's Zimmerman House in Manchester, New Hampshire, one of the best know of his Usonian Houses. The owners had invited our string quartet to come to their house and play some music. They said that many people had commented on how Frank Lloyd Wright's best houses (their own included) were like a piece of Bach music. It is this sense of initiating a simple form and repeating it and varying it in scale, texture, and size, which Frank Lloyd Wright's work is such a genius at accomplishing. For the owners who lived in the house, our playing of music just carried their love affair with the house to an even higher level, where the form and music interpenetrated each other.

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