Monday, September 29, 2008
Monday, September 22, 2008
"...what do you see? Yourself"
The Creation Simulation
Why does a blockbuster video game that embraces biological evolution resemble intelligent design?
by Margaret Robertson (for Seed Magazine) • Posted September 8, 2008 11:47 AM
Official Spore Website:
Why does a blockbuster video game that embraces biological evolution resemble intelligent design?
by Margaret Robertson (for Seed Magazine) • Posted September 8, 2008 11:47 AM
Official Spore Website:
Interesting graph... but why the naked dude Economist?
On Your Bike
Sep 22nd 2008
From Economist.com
Bicycle and car production since 1950
FAT-BUSTING but not wallet-busting, the humble bicycle is an increasingly popular choice of transport. Around 130m bikes rolled off production lines in 2007 and even more are set to be made this year. Bicycle and car production grew pretty much in tandem in the two decades beginning in 1950. But since 1970 bike production has nearly quadrupled while car production has roughly doubled. Much of the recent growth has been driven by electric bikes; production has doubled since 2004, to 21m.
Sep 22nd 2008
From Economist.com
Bicycle and car production since 1950
FAT-BUSTING but not wallet-busting, the humble bicycle is an increasingly popular choice of transport. Around 130m bikes rolled off production lines in 2007 and even more are set to be made this year. Bicycle and car production grew pretty much in tandem in the two decades beginning in 1950. But since 1970 bike production has nearly quadrupled while car production has roughly doubled. Much of the recent growth has been driven by electric bikes; production has doubled since 2004, to 21m.
The Shape of Music
Chopin Nocturne Op.9 No.2 (Arthur Rubinstein)
The Shape of Music
How do harmony and melody combine to make music?
by Dmitri Tymoczko • Posted July 9, 2008 04:03 PM
Zaha Hadid/Swarovski Crystal Palace Collection
Roughly 2,500 years ago, Pythagoras observed that objects, such as the anvils he purportedly studied, produced harmonious sounds while vibrating at frequencies in simple whole-number ratios.
More complex ratios gave rise to more dissonant sounds, which indicated that human beings were unconsciously sensitive to mathematical relationships inherent in nature. By showing that the world could be described mathematically, Pythagoras not only provided an important inspiration for physics, but he also discovered a particular affinity between mathematics and music--one that Gottfried Leibniz was to invoke centuries later when he described music as the "unknowing exercise of our mathematical faculties."
For a thousand years, Western musicians have endeavored to satisfy two fundamental constraints in their compositions. The first is that melodies should, in general, move by short distances. When played on a piano, melodies typically move to nearby keys rather than take large jumps across the keyboard. The second is that music should use chords (collections of simultaneously sounded notes) that are audibly similar. Rather than leap willy-nilly between completely unrelated sonorities, musicians typically restrict themselves to small portions of the musical universe, for instance by using only major and minor chords. While the melodic constraint is nearly universal, the harmonic constraint is more particularly Western: Many non-Western styles either reject chords altogether, using only one note at a time or build entire pieces around a single unchanging harmony.
Together these constraints ensure a two-dimensional coherence in Western music analogous to that of a woven cloth. Music is a collection of simultaneously occurring melodies, parallel horizontal threads that are held together tightly by short-distance motion. But Western music also has a vertical, or harmonic, coherence. If we consider only the notes sounding at any one instant, we find that they form familiar chords related to those that sound at other instants of time. These basic requirements impose nontrivial constraints on composers--not just any sequence of chords we imagine can generate a collection of short-distance melodies. We might therefore ask, how do we combine harmony and melody to make music? In other words, what makes music sound good?
To answer these questions, we need mathematics, just as Pythagoras supposed. But as I and other music theorists have recently shown, we need a kind of mathematics that Pythagoras could not have imagined: the geometry and topology of what mathematicians call "quotient spaces" or "orbifolds." These exotic spaces contain singularities-- "unusual" points that are analogous to the black holes of Einstein's general relativity--that can be described using only very recent mathematics. Western music can ultimately be represented as a series of points and line segments on abstract shapes in higher dimensions. If we can understand their structure, then the deep principles underlying Western music will finally be revealed.
To turn music into math, we begin by numbering the keys on the piano from low to high. Musicians typically number the 88 piano keys so that the lowest is 21 and the highest is 108, with middle C at 60. Mathematically, these numbers are the logarithms of the slowest frequency at which the piano string is vibrating. In principle we can assign numbers even to notes that are not found on the keyboard, with 60.5 referring to the note halfway between middle C and the next-highest key. These numbers refer to pitches.
Next we model the phenomenon of "octave equivalence": the fact that notes 12 keys apart sound similar. (As Maria teaches in The Sound of Music, "ti" brings us back to "do.") To do this mathematically, we divide our piano key numbers by 12 and keep only the remainder. In this way each of the 88 piano keys is assigned a number less than 12: the "C" keys 48, 60, and 72 are represented by 0, while the "C-sharp" (or "D-flat"), keys 49, 61, and 73 are all represented by 1, and so on. Musicians say that these numbers refer to pitch classes, representing the intrinsic "character" or "color" of the note. Geometrically, pitch classes all live on a circle divided into 12 equal parts, exactly like the face of an ordinary clock--though "12" on this clock refers to "0."
Musically, the order of a group of notes is less important than its content. The ordered sequence C-E-G, or 12-4-7, on the clock, is audibly related to E-G-C, or 4-7-12; musicians consider both to be "C major chords." A chord is therefore defined as an unordered collection of pitch classes, corresponding geometrically to an unordered set of points on a circle like hours on a clock face.
Chords that are related by rotation on the clock face all sound similar. For example, take the C major chord (12, 4, 7), and move each of the notes clockwise two places. This is the D major chord (2, 6, 9 on the clock), which sounds very much like the C major. In fact, a chord is a major chord if and only if it can be obtained by rotating the C major (so 3, 7, 10 would be another one, the E-flat major chord). The reason these chords all sound alike is that the human ear is more sensitive to the distances between notes than their absolute position on the clockface. Rotating each of the hands of a clock together doesn't change the distance between them and doesn't alter the chord's quality.
We can use this clock analogy to understand the two constraints of Western music mentioned earlier. To satisfy the harmonic constraint, composers need to use chords that are related by rotation, or at least approximately so. This ensures that the distances between the notes in each successive chord stay pretty much constant. To satisfy the melodic constraint, composers connect the notes of successive chords by short distances. For example, one could connect the C major chord (12, 4, 7) to the F major chord (5, 9, 12) by keeping the 12 fixed, moving the 4 one place clockwise to 5, and moving the 7 two clockwise places to 9. This represents a much more efficient alternative to moving each note five places clockwise. Western music is built out of a sequence of such mappings, forming a two-dimensional sonic tapestry.
The final stage in the process of translating music into math is to pass into the clock's configuration space: Rather than representing chords using multiple points on a one-dimensional circle, we construct an equivalent, higher-dimensional space in which every chord is a different point. The term "configuration space" refers to the fact that points in the higher-dimensional space represent "configurations" (or arrangements) of the points on the lower-dimensional circle. These spaces are considerably more interesting than the plain-vanilla spaces of ordinary Euclidean geometry.
Here, a complexity arises because the notes in a chord are unordered, whereas the coordinates of a geometrical point are typically ordered. Recall from high school geometry that a Cartesian plane is used to model ordered pairs of real numbers (x, y). To create the space of unordered points on a circle, we can just "fold" the familiar Cartesian spaces (representing ordered points on a line) in various ways. In two dimensions (when there are two notes in each chord), we first wrap around each axis, x and y, so that they become circles rather than lines. The resulting space is a doughnut, or in mathematical parlance, a torus. Second, we glue together all the points in the doughnut whose coordinates are related by reordering--so in two dimensions, (x, y) and (y, x) become the same point. In three dimensions (for three notes in each chord), the process is much trickier; we must glue together all six permutations of (x, y, z), and so on.
When, the dust settles, two-note chords live on a Möbius strip, three-note chords live on a solid, twisted triangular doughnut, and larger notes live on higher-dimensional analogues, whose shapes become difficult to describe nonmathematically. The boundary of each space, or shape, is geometrically unusual ("singular")--line segments appear to "bounce off" the boundary, rather like billiard balls reflecting off the edge of a pool table.
The structure of these spaces, representing all possible chords, shows us exactly how the two elemental properties of Western music can be combined. Structurally similar chords live on circles that wind through the spaces multiple times (these circles can be understood as lines that return back upon themselves like the Earth's equator). Melodic connections between chords--such as "hold 12 constant, move 4 one unit clockwise to become 5, and move 7 two units clockwise to become 9"--are represented by line segments in the space that may return back on themselves, or bounce off the space's boundaries. Our original musical question about combining harmony and melody thus becomes a geometrical question about finding circles that are "close to themselves"--that is, circles containing two points that can be connected by short line segments.
The most direct way to combine melody and harmony is to use chords that divide the 12 positions on our clockface of notes nearly (but not precisely) evenly, such as the C major chord (12, 4, 7), which divides the clockface into three roughly equal parts. These harmonies occupy the center of our musical spaces, and are thus able to take effective advantage of its non-Euclidean twists. Remarkably, in the 12-tone system of notes, these are precisely the chords that Pythagoras identified almost 2,500 years ago: the chords that sound intrinsically harmonious. Far from arbitrary or haphazard, scales and chords come close to being the unique solutions to the problem of creating two-dimensional musical coherence. Contrary to the hopes of generations of avant-garde composers, it follows that the goal of developing robust alternatives to tonality may be extremely difficult, if not impossible, to achieve.
The shapes of the space of chords we have described also reveal deep connections between a wide range of musical genres. It turns out that superficially different styles--Renaissance music, classical and Romantic music, jazz, rock, and other popular forms--all make remarkably similar use of the geometry of chord space. Traditional techniques for manipulating musical scales turn out to be closely analogous to those used to connect individual chords. And some composers have displayed a profound understanding of the higher-dimensional geometry of musical chords. In fact, one can argue that Romantic composers such as Chopin had an intuitive feel for non-Euclidean higher-dimensional spaces that exceeded the explicit understanding of their mathematical contemporaries.
The ideas I have been describing were first published in an article I wrote in Science in 2006. More recently, Clifton Callender, Ian Quinn, and I have shown that the connection between music theory and geometry is in fact much deeper and more comprehensive than even my earlier work indicated: There are in fact large families of geometrical spaces corresponding to a wide range of musical terms, some of which are considerably more exotic than those described here. (For instance, three-note chord types--such as "major chord" or "minor chord"--live on a cone containing two different flavors of singularity.) Seen in the light of this new geometrical perspective, a wide number of traditional music-theoretical questions become tractable. In some sense musicians have been doing geometry all along without ever realizing it.
The mathematician Rachel Hall and I are also exploring some interesting resemblances between music theory and economics. Similar geometrical spaces appear in both disciplines, and questions about how to measure distances between musical chords are very similar to questions about how to measure the distance between economic states. This may seem implausible until one reflects that the geometrical operations we have been discussing are very general. Ultimately, the geometry of music is a branch of the geometry of unordered collections-- and unordered collections are basic enough to have applications in a wide range of fields. Pythagoras was correct more than two and a half millennia ago: Music provides one of the clearest examples of a much deeper relation between mathematics and human experience.
The Shape of Music
How do harmony and melody combine to make music?
by Dmitri Tymoczko • Posted July 9, 2008 04:03 PM
Zaha Hadid/Swarovski Crystal Palace Collection
Roughly 2,500 years ago, Pythagoras observed that objects, such as the anvils he purportedly studied, produced harmonious sounds while vibrating at frequencies in simple whole-number ratios.
More complex ratios gave rise to more dissonant sounds, which indicated that human beings were unconsciously sensitive to mathematical relationships inherent in nature. By showing that the world could be described mathematically, Pythagoras not only provided an important inspiration for physics, but he also discovered a particular affinity between mathematics and music--one that Gottfried Leibniz was to invoke centuries later when he described music as the "unknowing exercise of our mathematical faculties."
For a thousand years, Western musicians have endeavored to satisfy two fundamental constraints in their compositions. The first is that melodies should, in general, move by short distances. When played on a piano, melodies typically move to nearby keys rather than take large jumps across the keyboard. The second is that music should use chords (collections of simultaneously sounded notes) that are audibly similar. Rather than leap willy-nilly between completely unrelated sonorities, musicians typically restrict themselves to small portions of the musical universe, for instance by using only major and minor chords. While the melodic constraint is nearly universal, the harmonic constraint is more particularly Western: Many non-Western styles either reject chords altogether, using only one note at a time or build entire pieces around a single unchanging harmony.
Together these constraints ensure a two-dimensional coherence in Western music analogous to that of a woven cloth. Music is a collection of simultaneously occurring melodies, parallel horizontal threads that are held together tightly by short-distance motion. But Western music also has a vertical, or harmonic, coherence. If we consider only the notes sounding at any one instant, we find that they form familiar chords related to those that sound at other instants of time. These basic requirements impose nontrivial constraints on composers--not just any sequence of chords we imagine can generate a collection of short-distance melodies. We might therefore ask, how do we combine harmony and melody to make music? In other words, what makes music sound good?
To answer these questions, we need mathematics, just as Pythagoras supposed. But as I and other music theorists have recently shown, we need a kind of mathematics that Pythagoras could not have imagined: the geometry and topology of what mathematicians call "quotient spaces" or "orbifolds." These exotic spaces contain singularities-- "unusual" points that are analogous to the black holes of Einstein's general relativity--that can be described using only very recent mathematics. Western music can ultimately be represented as a series of points and line segments on abstract shapes in higher dimensions. If we can understand their structure, then the deep principles underlying Western music will finally be revealed.
To turn music into math, we begin by numbering the keys on the piano from low to high. Musicians typically number the 88 piano keys so that the lowest is 21 and the highest is 108, with middle C at 60. Mathematically, these numbers are the logarithms of the slowest frequency at which the piano string is vibrating. In principle we can assign numbers even to notes that are not found on the keyboard, with 60.5 referring to the note halfway between middle C and the next-highest key. These numbers refer to pitches.
Next we model the phenomenon of "octave equivalence": the fact that notes 12 keys apart sound similar. (As Maria teaches in The Sound of Music, "ti" brings us back to "do.") To do this mathematically, we divide our piano key numbers by 12 and keep only the remainder. In this way each of the 88 piano keys is assigned a number less than 12: the "C" keys 48, 60, and 72 are represented by 0, while the "C-sharp" (or "D-flat"), keys 49, 61, and 73 are all represented by 1, and so on. Musicians say that these numbers refer to pitch classes, representing the intrinsic "character" or "color" of the note. Geometrically, pitch classes all live on a circle divided into 12 equal parts, exactly like the face of an ordinary clock--though "12" on this clock refers to "0."
Musically, the order of a group of notes is less important than its content. The ordered sequence C-E-G, or 12-4-7, on the clock, is audibly related to E-G-C, or 4-7-12; musicians consider both to be "C major chords." A chord is therefore defined as an unordered collection of pitch classes, corresponding geometrically to an unordered set of points on a circle like hours on a clock face.
Chords that are related by rotation on the clock face all sound similar. For example, take the C major chord (12, 4, 7), and move each of the notes clockwise two places. This is the D major chord (2, 6, 9 on the clock), which sounds very much like the C major. In fact, a chord is a major chord if and only if it can be obtained by rotating the C major (so 3, 7, 10 would be another one, the E-flat major chord). The reason these chords all sound alike is that the human ear is more sensitive to the distances between notes than their absolute position on the clockface. Rotating each of the hands of a clock together doesn't change the distance between them and doesn't alter the chord's quality.
We can use this clock analogy to understand the two constraints of Western music mentioned earlier. To satisfy the harmonic constraint, composers need to use chords that are related by rotation, or at least approximately so. This ensures that the distances between the notes in each successive chord stay pretty much constant. To satisfy the melodic constraint, composers connect the notes of successive chords by short distances. For example, one could connect the C major chord (12, 4, 7) to the F major chord (5, 9, 12) by keeping the 12 fixed, moving the 4 one place clockwise to 5, and moving the 7 two clockwise places to 9. This represents a much more efficient alternative to moving each note five places clockwise. Western music is built out of a sequence of such mappings, forming a two-dimensional sonic tapestry.
The final stage in the process of translating music into math is to pass into the clock's configuration space: Rather than representing chords using multiple points on a one-dimensional circle, we construct an equivalent, higher-dimensional space in which every chord is a different point. The term "configuration space" refers to the fact that points in the higher-dimensional space represent "configurations" (or arrangements) of the points on the lower-dimensional circle. These spaces are considerably more interesting than the plain-vanilla spaces of ordinary Euclidean geometry.
Here, a complexity arises because the notes in a chord are unordered, whereas the coordinates of a geometrical point are typically ordered. Recall from high school geometry that a Cartesian plane is used to model ordered pairs of real numbers (x, y). To create the space of unordered points on a circle, we can just "fold" the familiar Cartesian spaces (representing ordered points on a line) in various ways. In two dimensions (when there are two notes in each chord), we first wrap around each axis, x and y, so that they become circles rather than lines. The resulting space is a doughnut, or in mathematical parlance, a torus. Second, we glue together all the points in the doughnut whose coordinates are related by reordering--so in two dimensions, (x, y) and (y, x) become the same point. In three dimensions (for three notes in each chord), the process is much trickier; we must glue together all six permutations of (x, y, z), and so on.
When, the dust settles, two-note chords live on a Möbius strip, three-note chords live on a solid, twisted triangular doughnut, and larger notes live on higher-dimensional analogues, whose shapes become difficult to describe nonmathematically. The boundary of each space, or shape, is geometrically unusual ("singular")--line segments appear to "bounce off" the boundary, rather like billiard balls reflecting off the edge of a pool table.
The structure of these spaces, representing all possible chords, shows us exactly how the two elemental properties of Western music can be combined. Structurally similar chords live on circles that wind through the spaces multiple times (these circles can be understood as lines that return back upon themselves like the Earth's equator). Melodic connections between chords--such as "hold 12 constant, move 4 one unit clockwise to become 5, and move 7 two units clockwise to become 9"--are represented by line segments in the space that may return back on themselves, or bounce off the space's boundaries. Our original musical question about combining harmony and melody thus becomes a geometrical question about finding circles that are "close to themselves"--that is, circles containing two points that can be connected by short line segments.
The most direct way to combine melody and harmony is to use chords that divide the 12 positions on our clockface of notes nearly (but not precisely) evenly, such as the C major chord (12, 4, 7), which divides the clockface into three roughly equal parts. These harmonies occupy the center of our musical spaces, and are thus able to take effective advantage of its non-Euclidean twists. Remarkably, in the 12-tone system of notes, these are precisely the chords that Pythagoras identified almost 2,500 years ago: the chords that sound intrinsically harmonious. Far from arbitrary or haphazard, scales and chords come close to being the unique solutions to the problem of creating two-dimensional musical coherence. Contrary to the hopes of generations of avant-garde composers, it follows that the goal of developing robust alternatives to tonality may be extremely difficult, if not impossible, to achieve.
The shapes of the space of chords we have described also reveal deep connections between a wide range of musical genres. It turns out that superficially different styles--Renaissance music, classical and Romantic music, jazz, rock, and other popular forms--all make remarkably similar use of the geometry of chord space. Traditional techniques for manipulating musical scales turn out to be closely analogous to those used to connect individual chords. And some composers have displayed a profound understanding of the higher-dimensional geometry of musical chords. In fact, one can argue that Romantic composers such as Chopin had an intuitive feel for non-Euclidean higher-dimensional spaces that exceeded the explicit understanding of their mathematical contemporaries.
The ideas I have been describing were first published in an article I wrote in Science in 2006. More recently, Clifton Callender, Ian Quinn, and I have shown that the connection between music theory and geometry is in fact much deeper and more comprehensive than even my earlier work indicated: There are in fact large families of geometrical spaces corresponding to a wide range of musical terms, some of which are considerably more exotic than those described here. (For instance, three-note chord types--such as "major chord" or "minor chord"--live on a cone containing two different flavors of singularity.) Seen in the light of this new geometrical perspective, a wide number of traditional music-theoretical questions become tractable. In some sense musicians have been doing geometry all along without ever realizing it.
The mathematician Rachel Hall and I are also exploring some interesting resemblances between music theory and economics. Similar geometrical spaces appear in both disciplines, and questions about how to measure distances between musical chords are very similar to questions about how to measure the distance between economic states. This may seem implausible until one reflects that the geometrical operations we have been discussing are very general. Ultimately, the geometry of music is a branch of the geometry of unordered collections-- and unordered collections are basic enough to have applications in a wide range of fields. Pythagoras was correct more than two and a half millennia ago: Music provides one of the clearest examples of a much deeper relation between mathematics and human experience.
Wednesday, September 17, 2008
Tuesday, September 16, 2008
Saturday, September 13, 2008
Friday, September 12, 2008
Ecuador Constitution Would Grant Inalienable Rights To Nature | CommonDreams.org
Published on Thursday, September 4, 2008 by The Christian Science Monitor
Ecuador Constitution Would Grant Inalienable Rights To Nature
Ecuador's proposed constitution includes an article that grants nature the right to "exist, persist, maintain and regenerate its vital cycles, structure, functions and its processes in evolution" and will grant legal standing to any person to defend those rights in court.
![[A sea lion pup on Santiago Island, one of Ecuador's famed Galapagos Islands. (MELANIE STETSON FREEMAN/STAFF/FILE)]](http://www.commondreams.org/files/article_images/seal.jpg "seal.jpg")
A sea lion pup on Santiago Island, one of Ecuador's famed Galapagos Islands. (MELANIE STETSON FREEMAN/STAFF/FILE)The blog Green Change quotes the five articles [2] that acknowledge rights said to be possessed by nature, or "Pachamama," a goddess revered by indigenous Andean peoples whose name roughly translates into "Mother Earth."
Chapter: Rights for Nature
Art. 1. Nature or Pachamama, where life is reproduced and exists, has the right to exist, persist, maintain and regenerate its vital cycles, structure, functions and its processes in evolution.
Every person, people, community or nationality, will be able to demand the recognitions of rights for nature before the public organisms. The application and interpretation of these rights will follow the related principles established in the Constitution.
Art. 2. Nature has the right to an integral restoration. This integral restoration is independent of the obligation on natural and juridical persons or the State to indemnify the people and the collectives that depend on the natural systems.
In the cases of severe or permanent environmental impact, including the ones caused by the exploitation on non renewable natural resources, the State will establish the most efficient mechanisms for the restoration, and will adopt the adequate measures to eliminate or mitigate the harmful environmental consequences.
Art. 3. The State will motivate natural and juridical persons as well as collectives to protect nature; it will promote respect towards all the elements that form an ecosystem.
Art. 4. The State will apply precaution and restriction measures in all the activities that can lead to the extinction of species, the destruction of the ecosystems or the permanent alteration of the natural cycles.
The introduction of organisms and organic and inorganic material that can alter in a definitive way the national genetic patrimony is prohibited.
Art. 5. The persons, people, communities and nationalities will have the right to benefit from the environment and form natural wealth that will allow wellbeing.
The environmental services are cannot be appropriated; its production, provision, use and exploitation, will be regulated by the State.
The concept that nature itself can possess rights runs counter to the classical liberal theories of government that hold sway throughout much of the West, which view rights as possessed only by individual human beings. But Ecuador is not the first country to propose granting rights to nonhuman entities: Many countries, including the United States, have long held that corporations possess many of the same rights - such as the rights to free expression and to due process - that human beings have. And in June, Spain's parliament approved a measure to extend some human rights to nonhuman apes [3].
But, as an editorial in the Los Angeles Times observes, Ecuador's extension of rights to nature may represent a larger shift [4] in how humans view their place in the world:
No other country has gone as far as Ecuador in proposing to give trees their day in court, but it certainly is not alone in its recalibration of natural rights. Religious leaders, including the Archbishop of Canterbury, the Dalai Lama and the Archbishop of Constantinople, have declared that caring for the environment is a spiritual duty. And earlier this year, the Catholic Church updated its list of deadly sins to include polluting the environment.
Ecuador is codifying this shift in sensibility. In some ways, this makes sense for a country whose cultural identity is almost indistinguishable from its regional geography - the Galapagos, the Amazon, the Sierra. How this new area of constitutional law will work, however, is another question. We aren't ready to endorse such a step at home, or even abroad. But it's intriguing. We'll be watching Ecuador's example.
[via Grist [5]]
© 2008 The Christian Science Monitor
Article printed from www.CommonDreams.org
URL to article: http://www.commondreams.org/headline/2008/09/04-7
Tuesday, September 9, 2008
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