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Pythagoras to Secor: Optimal Tuning of Two-Dimensional Keyboards

  • Thursday, November 30, 2017
  • 3:40 PM–4:30 PM
  • North Hall 276

Anil Venkatesh, Ferris State University

In music theory, a temperament is a system of tuning that is generated by one or more regular pitch intervals. Today, the most common temperament in Western music is the piano tuning (12-TET), which is generated by a single pitch interval that subdivides the octave into twelve parts. While 12-TET gives a good approximation of some important harmonic intervals, it deviates sharply from others. Consequently, music played in 12-TET will occasionally sound unpleasantly discordant. One possible way of resolving this issue is to consider two-dimensional temperaments, i.e. temperaments that are generated by the octave plus a second independent interval. In 1974, George Secor discovered a two-dimensional temperament that has excellent approximation of all important harmonics. His result came to be called the miracle temperament in recognition of its quality. In this talk, we formulate the question as a linear programming problem on families of constraints, and provide exact solutions for many new keyboard dimensions. We also show that an optimal tuning for harmonic approximation can be obtained for any keyboard of given width, provided sufficiently many rows of octaves.

 Refreshments precede the talk at 3:30 p.m. in NH 282.

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