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February 2013 Issue

Auto-Tuning Mother Nature: Waves in Water and Music

*By Hunter C. Brown*Specialists, even in the marine and ocean industries, tend to focus their energies narrowly on immediate problems, while neglecting new and exciting applications of their work outside of their fields. An exception is the work of engineer Harold 'Andy' Hildebrand, the creator of Auto-Tune.

In 1998, the musician Cher released a song titled 'Believe,' which relied heavily on the Auto-Tune technique to produce unique vocal sounds. After the release, musicians around the world were clamoring for the new technology that could correct a singer's voice to perfect pitch or transform a smooth pitch change into a stepped pitch output. Megastars, including Madonna, Cher, Britney Spears, Tim McGraw and Kanye West, heavily use Auto-Tune not only to ensure perfect pitch but often to produce unique sounds. Perhaps unknown to these singers, wave experts in the geophysical industry have been using exactly the same mathematical functions to identify ocean-wave characteristics for more than 50 years.

The company behind Auto-Tune, Antares Audio Technologies, was the brainchild of Hildebrand, a former Exxon Mobil Corp. (Irving, Texas) seismic engineer. He realized that by applying the same principles for studying seismic waves in the Earth's crust to acoustic waves, new sound effects could be achieved.

According to Hildebrand, the inspiration for Auto-Tune came at a dinner party when one of the guests learned he was an engineer and asked if he could design a gadget to improve her singing. Since his work involved analyzing acoustic signals sent through the ground to identify possible drilling locations (essentially geological sonar), he had knowledge of the mathematical treatments of acoustic signals. The realization that these could be applied to music led to a music industry revolution—and allowed him to retire at 40.

The magic behind both Auto-Tune and recovering ocean-wave properties owes its existence to two famous mathematicians from the mid-1700s and early 1800s. Leonhard Euler's formula showed that exponential functions can comprise combinations of trigonometric functions. This laid the foundation on which Joseph Fourier would erect an additional masterpiece, later named the Fourier transform, which can be used to identify dominant components in a composite signal formed of many individual components.

Just as sound waves can be digitized via microphones, ocean waves are often measured and digitized with highly sensitive inertial instruments mounted on large buoys. These measurements are used to reconstruct the surface-displacement time series. Ocean scientists can then determine wave characteristics, such as significant wave height and dominant period, by application of the Fourier transform—just like Auto-Tune.

A perfect spar buoy for measuring waves moves vertically in the water with no pitch or roll. If an accelerometer is mounted on this buoy, it will measure the buoy's vertical acceleration, including gravity. Using elementary calculus, this time series of accelerations may be twice-integrated to estimate the sea-surface displacement. There are some difficulties, however, that arise from this scheme, such as the computational complexity of repeated integration and added low-frequency noise.

If, instead, the time series were transformed into the frequency domain, then the Fourier transform's unique properties could alleviate some of these difficulties. The dominant period of the ocean-wave time series is identified by finding the peak in the frequency spectrum, calculated with the Fourier transform. This peak represents the frequency with the most energy. Often there will be several peaks, indicating waves comprising several frequencies.

Significant wave height, defined as the average of the highest one-third of the waves as reported by a trained observer, is only slightly more difficult to calculate. Michel Ochi explains in his book 'Ocean Waves: The Stochastic Approach' that taking the square root of the spectral density function and multiplying by roughly four obtains a very close approximation to the significant wave height.

In music, the Fourier transform can be used on the sound of an instrument or a singer's voice, which can be recorded through a microphone, digitized by an analog-to-digital converter and stored as a series of digital measurements. Once converted to a digital form, a process known as the discrete Fourier transform (DFT) can be applied to the sound data to identify the dominant frequencies within that signal.

A 440-hertz tuning fork when struck, for example, produces air-pressure fluctuations at 440 times a second. When this sound is digitized and run through a DFT, the result will show a large spike at 440 hertz. If, however, the tuning fork were slightly warped, it might produce a 430-hertz tone. This would show up as a spike at 430 hertz in the DFT results. One would instantly know that the tuning fork was damaged and should be replaced. Similarly, if a singer attempts to sing the note middle C at around 261 hertz, the DFT would show the actual frequency produced. If the singer were slightly flat, say 256 hertz, one could identify the problem and adjust the frequency in the recording back to 261 hertz for perfect tuning.

While it is possible to construct a real-time Auto-Tune system using just the Fourier transform, there are less computationally costly and faster, more recent analysis techniques for music, although traditional Fourier analysis remains the foundation of Auto-Tune.

Scientists, mathematicians and engineers tend to associate a technique with a particular domain and rarely consider its significance outside its natural habitat. But, as Hildebrand proved, cross-discipline collaborations can often lead to scientific and commercial revolutions.

*Hunter C. Brown is the lead engineer for the Delaware Environmental Observing System (DEOS) at the University of Delaware, which maintains a region-wide network of automated real-time environmental monitoring stations. Brown's work focuses on large-scale environmental data management systems, underwater technologies involving augmented reality and autonomous sensing platforms, and sustainable environmental technologies.*

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