Blurred Lines: Is the Number of Pop Tunes Finite?

An infinite number of musical combinations is narrowed by listenability.

Where does the line between inspiration stop and plagiarism begin?

A federal jury last week ruled that the 2013 mega hit "Blurred Lines" by Robin Thicke and Pharrell Williams was a rip-off of "Got to Give It Up," and awarded the family of the late soul singer Marvin Gaye $7.4 million in damages. The jury didn't hear the difference between the two songs, rather the judge told the jury to review sheet music only.

The trial featured testimony by musicologists comparing individual combinations of notes, and the chances that the songs were arrived at independently. Aside from legal issues, the case raises interesting questions about the mathematical limits of pop music, and whether there is a finite number of songs out there.

"It's a really important question and has inspired some really good mathematics back to ancient India," said Rachel Hall, professor of music at St. Johns University. "Once you break it down with assumptions, the number of songs is definitely finite. But it's really large."

To figure out this number, Hall assumes that all pop songs have six notes, and each beat can either be a a note or a rest. A typical pop song lasts three or four minutes and contains about one minute of original musical material, with the rest being repeated melody and chords.

Suppose the tempo is 160 beats per minute (like Pharrell's "Happy") and each beat can be subdivided once (again, like "Happy"). That means a one-minute melody can have at most 320 notes. For each of these 320 beat subdivisions, there are seven choices. The total number of one-minute sequences of notes is then 7^320=2.7 x 10^(270)

That is, approximately 27 followed by 269 zeros. (A googol is 10^(100), so this is more than a googol squared) And of course, we've only chosen one note at a time, but in reality you'd have chords, drum sounds, etc.

Hall says the number of choices that would sound convincing musically is much, much smaller, because there are many musical expectations that we've built up over a lifetime of listening.

Still, the number is large. Even if we want to use only 16 beats, the number of possibilities is 7^16=3.3 x 10^(13), which is more than 30 trillion, according to Hall.

That figure represents the number of combinations, but not the number of songs. Richard Cohn, professor of musicology at the University of Sydney, says that some tones go well together for American music listeners, which are different than those in other cultures.

"They require judgments to be made about what the limitations of human physiology, memory, and processing and psychologists don't have robust answers to those questions," Cohn said via e-mail. "So I think the only reasonable answer is: the number of combinations is so large as to be infinite for any practical purpose."

St. Johns Hall says she listened to "Blurred Lines" and she believes it the groove, rather than the melody, that was similar.

"They took the groove and made a melody that wasn't that close," Hall said. "This is something musicians are doing a lot. That's how human culture, fashion and art work.

"There's always a balance between originality and not being so original that people can't relate what to what you are doing."

Pharrell Williams (L), Robin Thicke and T.I. perform in Beverly Hills on Jan. 25, 2014. Williams and Thicke were found liable for copyright infringement in a lawsuit accusing them of plagiarizing the late Marvin Gaye in their hit single, "Blurred Lines."

German photographer Martin Klimas' latest exhibition, a series of images he calls "Sonic Sculptures," is so explosive and colorful, it just may change the way you look -- yes, look -- at music.

For the project, Klimas put vibrantly colored paint on a diaphragm over a speaker, turned up the volume on selected music and snapped photos of what the New York Times Magazine described as "a 3-D take on Jackson Pollack."

"I use an ordinary speaker with a funnel-shaped protective membrane on top of it," he told the Smithsonian. "I pour paint colors onto the rubber membrane, and then I withdraw from the setup."

The above photo shows Prince's "Sign 'O' The Times."

Klimas' project was inspired by the research of Hans Jenny, a German physician, scientist and father of cymatics, which is the study of wave phenomena. Jenny photographed his experiments of the effects sound vibrations had on various materials such as fluids, powders and liquid paste. Jenny placed these substances on a rubber drum head and, as it vibrated, he found different tones produced different patterns in the materials. Low tones made powders assemble in straight lines, while deeper tones made for more complex patterns.

The above photo reflects Phillip Glass' "Music With Changing Parts."

Klimas used a variety of music -- everyone from Prince to James Brown and Charlie Parker to Phillip Glass. He says he leaves the "creation of the picture to the sound itself" and, after cranking the volume, steps back. Once the paint starts jumping, a sound-trigger device that detects noise spikes automatically takes photos.

"I mostly selected works that were particularly dynamic, and percussive," Klimas said. Though he used songs from a variety of music styles and eras, many of the tracks chosen were by musicians who had ties to the visual art world, such as the Velvet Underground and John Cage.

Before they struck gold with "Get Lucky," Daft Punk got dance floors thumping with "Around the World" shown here.

Klimas spent six months completing the project in his Dusseldorf studio and took about 1,000 shots to get his final 212 images. He went through 18.5 gallons of paint, on average of 6 ounces per shot, and blew two speakers while cranking the tunes. He used a Hasselblad camera with a shutter speed of 1/7000th a second.

The above image is a photo of Ornette Coleman's "Free Jazz, A Collective Improvisation."

Blown speakers and exactitudes aside, Klimas said "the most annoying thing was cleaning up the set thoroughly after every single shot." Check out more of Klimas' work on his website (www.martin-klimas.de), or better yet, if you're in New York City, stop by the Foley Gallery on the Lower East Side. There you can find his new exhibition, "SONIC," which opened earlier this month.

The above photo illustrates Pink Floyd's "On the Run."