Musical Math

Music Has Its Own Geometry, Researchers Find



The figure shows how geometrical music theory represents four-note chord-types -- the collections of notes form a tetrahedron, with the colors indicating the spacing between the individual notes in a sequence. In the blue spheres, the notes are clustered, in the warmer colors, they are farther apart. The red ball at the top of the pyramid is the diminished seventh chord, a popular 19th-century chord. Near it are all the most familiar chords of Western music. (Credit: Dmitri Tymoczko, Princeton University)

ScienceDaily (Apr. 17, 2008) — The connection between music and mathematics has fascinated scholars for centuries. More than 200 years ago Pythagoras reportedly discovered that pleasing musical intervals could be described using simple ratios.

And the so-called musica universalis or "music of the spheres" emerged in the Middle Ages as the philosophical idea that the proportions in the movements of the celestial bodies -- the sun, moon and planets -- could be viewed as a form of music, inaudible but perfectly harmonious.

Now, three music professors -- Clifton Callender at Florida State University, Ian Quinn at Yale University and Dmitri Tymoczko at Princeton University -- have devised a new way of analyzing and categorizing music that takes advantage of the deep, complex mathematics they see enmeshed in its very fabric.

Writing in the April 18 issue of Science, the trio has outlined a method called "geometrical music theory" that translates the language of musical theory into that of contemporary geometry. They take sequences of notes, like chords, rhythms and scales, and categorize them so they can be grouped into "families." They have found a way to assign mathematical structure to these families, so they can then be represented by points in complex geometrical spaces, much the way "x" and "y" coordinates, in the simpler system of high school algebra, correspond to points on a two-dimensional plane.

Different types of categorization produce different geometrical spaces, and reflect the different ways in which musicians over the centuries have understood music. This achievement, they expect, will allow researchers to analyze and understand music in much deeper and more satisfying ways.

The work represents a significant departure from other attempts to quantify music, according to Rachel Wells Hall of the Department of Mathematics and Computer Science at St. Joseph's University in Philadelphia. In an accompanying essay, she writes that their effort, "stands out both for the breadth of its musical implications and the depth of its mathematical content."

The method, according to its authors, allows them to analyze and compare many kinds of Western (and perhaps some non-Western) music. (The method focuses on Western-style music because concepts like "chord" are not universal in all styles.) It also incorporates many past schemes by music theorists to render music into mathematical form.

"The music of the spheres isn't really a metaphor -- some musical spaces really are spheres," said Tymoczko, an assistant professor of music at Princeton. "The whole point of making these geometric spaces is that, at the end of the day, it helps you understand music better. Having a powerful set of tools for conceptualizing music allows you to do all sorts of things you hadn't done before."

Like what?

"You could create new kinds of musical instruments or new kinds of toys," he said. "You could create new kinds of visualization tools -- imagine going to a classical music concert where the music was being translated visually. We could change the way we educate musicians. There are lots of practical consequences that could follow from these ideas."

"But to me," Tymoczko added, "the most satisfying aspect of this research is that we can now see that there is a logical structure linking many, many different musical concepts. To some extent, we can represent the history of music as a long process of exploring different symmetries and different geometries."

Understanding music, the authors write, is a process of discarding information. For instance, suppose a musician plays middle "C" on a piano, followed by the note "E" above that and the note "G" above that. Musicians have many different terms to describe this sequence of events, such as "an ascending C major arpeggio," "a C major chord," or "a major chord." The authors provide a unified mathematical framework for relating these different descriptions of the same musical event.

The trio describes five different ways of categorizing collections of notes that are similar, but not identical. They refer to these musical resemblances as the "OPTIC symmetries," with each letter of the word "OPTIC" representing a different way of ignoring musical information -- for instance, what octave the notes are in, their order, or how many times each note is repeated. The authors show that five symmetries can be combined with each other to produce a cornucopia of different musical concepts, some of which are familiar and some of which are novel.

In this way, the musicians are able to reduce musical works to their mathematical essence.

Once notes are translated into numbers and then translated again into the language of geometry the result is a rich menagerie of geometrical spaces, each inhabited by a different species of geometrical object. After all the mathematics is done, three-note chords end up on a triangular donut while chord types perch on the surface of a cone.

The broad effort follows upon earlier work by Tymoczko in which he developed geometric models for selected musical objects.

The method could help answer whether there are new scales and chords that exist but have yet to be discovered.

"Have Western composers already discovered the essential and most important musical objects?" Tymoczko asked. "If so, then Western music is more than just an arbitrary set of conventions. It may be that the basic objects of Western music are fantastically special, in which case it would be quite difficult to find alternatives to broadly traditional methods of musical organization."

The tools for analysis also offer the exciting possibility of investigating the differences between musical styles.

"Our methods are not so great at distinguishing Aerosmith from the Rolling Stones," Tymoczko said. "But they might allow you to visualize some of the differences between John Lennon and Paul McCartney. And they certainly help you understand more deeply how classical music relates to rock or is different from atonal music."

Hmmmmmm...

Simple Tools Would Enhance Experience Of Bloggers, Blog Readers

ScienceDaily (Apr. 10, 2008) — In a first-of-its-kind study, UC Irvine researchers have provided new insight into blog readers' online habits and experiences, as well as how they perceive their roles in blog-based communities.

The research, led by Eric Baumer, doctoral candidate at UCI's Donald Bren School of Information and Computer Sciences; Mark Sueyoshi, international studies and East Asian cultures undergraduate student; and Bill Tomlinson, informatics professor, is the first to focus primarily on blog reading. Previous studies about weblogs, or blogs, typically have centered on blog writers, largely overlooking those who go online to read, comment and participate.

A better understanding of the reader-blogger connection could lead to new, advanced features that would enable richer interactions between the two groups. For readers, an installed add-on could enrich their experience by tracking blog habits of which they might not be aware. For bloggers, a logging tool could help them easily distinguish between different types of readers and allow them to better connect with audiences.

The UCI study examined in-depth the blog-reading habits of 15 participants of various ages to determine how they consume content and interact with blogs and blog writers. The research found that some readers frequently post comments, while in others "lurk," or visit without commenting. Among the findings:

• Readers have diverse opinions of what makes a blog a blog. Academic definitions generally refer to blogs as frequently modified Web pages with dated entries listed in reverse chronological order. But study participants identified a wide variety of characteristics in what they considered to be blogs. These included both technical aspects like RSS (Really Simple Syndication) feeds and trackback links, as well as social aspects, including the presence of conversation or personal content.
• Regular blog reading often becomes more habitual and less content oriented. Similar to e-mail checking, blog reading can become ingrained into users' online routine. Sometimes, even the usefulness of the blog content itself can be less vital than the activity of reading or skimming the blog to fulfill a person's particular routine.
• The timing of a blog post is not nearly as relevant to readers as its position among the other entries. Readers are more likely to read the most recent posts at the top of the screen, and are generally less concerned with the exact age of a post. A vast majority of participants said they were not bothered when they were not able to read each and every blog post, challenging a common theory that users tend to feel overwhelmed by the need to remain constantly up to date.
• Blog readers feel a responsibility to make insightful contributions. While past research noted readers expect bloggers to deliver frequent, high-quality posts, the UCI study found readers also place pressure on themselves to produce coherent, worthwhile comments in response to good blog posts.

"With the increased popularity of blogs, various tools like Blogger and Movable Type have made writing a blog easy for a wide audience," said Baumer, who studies informatics -- a discipline that focuses on the use of information technology in real-world settings. "But, until the technology embraces the role of the audience, the full social potential of blogging remains untapped.

"One of the goals of this research is to stimulate the development of tools to foster that social potential in terms of both readers and bloggers."

The researchers hope their work will prompt further studies about the roles of blog readers and how features such as commenting and linking create new ways to interact with authors and text.

This potential change in research approach would be similar to a shift that occurred in literary theory in the 1960s and 1970s, when scholars began taking into account readers' responses when studying literature.

"This study is really just the beginning," said Tomlinson, an ICS professor and affiliate of the California Institute for Telecommunications and Information Technology. "With the rapid expansion of online social media such as Flickr and YouTube, understanding how people consume these media will be vital to understanding their broader social impacts."

April (meteor) Showers

Star Gazing? The Moon Meets The Pleiades, And Saturn Will Be Beautiful In April



Saturn will be high in the southeast as darkness falls on April evenings, one of the first "stars" to appear and a fine sight in any telescope. (Credit: Voyager 2, NASA)

ScienceDaily (Apr. 1, 2008) — The Pleiades star cluster will have a beautiful encounter with the slender moon in the western sky after sunset on April 8. Usually the moon's brightness overpowers nearby stars, but not when it's such a thin crescent. Binoculars will reveal the spectacle as the moon passes just below the famous Seven Sisters.

The Pleiades (pronounced PLEE-ah-deez) are lovely by themselves, and on a clear night they can be seen with the unaided eye in the constellation Taurus the Bull. Known prehistorically, the cluster is identified as a group of women in many cultures around the world, from Australian Aborigine to Native American.

Saturn will be high in the southeast as darkness falls on April evenings, one of the first "stars" to appear and a fine sight in any telescope. It will be visible much of the night in the constellation Leo the Lion, near Leo's brightest star, Regulus. Saturn's rings will open slightly during April, but they will be nearly edge-on and invisible to us by the end of the year. Titan, Saturn's largest moon, will be due south of the planet on April 2 and 18 and due north on April 10 and 26.

Jupiter will rise around 3 a.m. local daylight time at the beginning of April and two hours earlier by month's end. Wait until the brilliant white object is high in the southern sky and then use binoculars to see its four largest moons, which Galileo discovered with one of the first telescopes. These tiny bits of white light are strung out in a straight line on both sides of Jupiter, and they seem to slide back and forth along the line from one night to the next as they orbit the planet. You may need to steady your binoculars by resting your elbows on a surface to keep the image from wobbling. The only times you won't see all four moons are when one or more are crossing in front of the planet or behind it.

Mars will be conspicuous during April, glowing bright orange high in the west after dark as it passes through the stars of the constellation Gemini the Twins. This will be a good chance to compare the similar colors of Mars and the bright star Pollux as they draw closer. Binoculars will make the colors stand out.

Mercury will be lost in the sun's glare during most of April for observers in the Northern Hemisphere. By month's end, it will appear below the Pleiades star cluster low in the west-northwest half an hour after sunset. Those in the Southern Hemisphere will be able to see the small planet in early morning twilight for the first half of the month.

Starting in April, Venus will be hidden in the glare of the sun for an unusually long time for observers in the Northern Hemisphere. The brilliant planet will be on the far side of the sun from us as it moves from morning to evening twilight. Not until July will Venus again be visible near the western horizon. Seen from the Southern Hemisphere, Venus will rise 90 minutes before the sun and be conspicuous in the morning sky.

Light pollution

A great deal of energy and money is wasted on inefficient, improperly directed outdoor lighting. To help call attention to this problem of light pollution, the International Dark-Sky Association is encouraging people in the United States to turn off unnecessary outside lighting during National Dark-Sky Week, which lasts until April 4.

Meteor shower

The Lyrid meteor shower will peak on the night of April 21-22. About 20 Lyrids per hour are normally visible after midnight, but this year the full moon's glare will hide the fainter ones. To minimize this effect, try watching from a place where the moon is blocked by a building or trees. The meteors can appear anywhere in the sky, but they will seem to come from a point called the radiant in the constellation Lyra the Harp, which gives the shower its name. Lyra's bright white star Vega will be almost at the radiant, and the meteor count should be highest when Vega is well above the eastern horizon, beginning around midnight local time in the Northern Hemisphere and lasting the rest of the night. The later the hour, the more meteors there will be.

Summer Triangle

Spring weather can be fickle, but a reliable sign that the seasons are changing is the appearance of a trio of bright white stars called the Summer Triangle. Vega will be the first to rise each night in the east, leading the way high across the sky. The other stars are Altair in the constellation Aquila the Eagle to the south and Deneb in the constellation Cygnus the Swan to the north. When you are looking from Deneb to Altair, you are also looking at the Milky Way, though it's usually obscured by city lights. If you find a place dark enough for you to see the Milky Way, keep the location in mind -- you'll also see a lot more meteors and other celestial objects from there.

Moon phases

The moon will be new on April 5, at first quarter on April 12, full on April 20 and at third quarter on April 28.