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Music is backed by physics and math
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Retired physics professor Jim Purcell explains how the frets on the mandolin are arranged. They are spaced according to an exacting formula which places them in the right position to create the perfect pitch each note requires. (Bulletin photo by Holly Kozelsky)
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Sunday, January 20, 2013

By HOLLY KOZELSKY - Bulletin Accent Editor

Your favorite tune has a lot more to it than a pleasing melody.

It is a combination of complicated mathematical formulas and physics concepts.

Jim Purcell on Thursday talked about those topics at the Spencer-Penn Centre’s In The Know program “Some Physics, Some Math and Some Music.” Purcell, a retired physics professor, demonstrated on several instruments how the physics of sound and the mathematics of music theory work together.

He illustrated many of his points by playing some instruments, such as the cello, and showing a “virtual keyboard” from a website projected on a screen.

He described the cello as a “large viola, in the sense of how it’s strung,” not “like a small bass” as many people seem to think.

Vibrations of the string create sound, he said.

The kind of sound an instrument makes depends upon three things: the tension in the string, how thick the string is (“mass per unit length”) and the length of the string.

The thicker the string, the lower the sound.

When a string is held down at a fret, which is on the neck (wooden bar), it shortens the part of the string that can vibrate when plucked. That means a higher frequency (higher notes).

When an instrument is built, mathematical formulas determine where frets are put. Each fret is placed at the exact same ration, Purcell said. The placement of the fret determines the sound.

Purcell pointed out the difference between two types of sound: noise and pitch. “You don’t want noise. You want pitch, something that’s pleasing to the ear.”

The concept of pitch is referred to by many names. In physics, it’s called “frequency.” Older terms for pitch are “cycles per second” or “Hertz.”

The musical scale gives order to the different frequencies (pitch, or musical sounds). There are seven notes, A through G, and they repeat in instruments, such as along a piano keyboard or the frets of a guitar.

All eight notes together are called an “octave.” (“Oct” is a root word meaning “eight.”)

Each note represents a frequency. For example, “A” has 440 Hertz, which means 440 cycles per second. A# (A sharp) has 466.16 Hz.

The notes are not evenly spaced in pitch, Purcell said. Think of the white keys in a piano as the notes. The black keys found between most of the white keys are flats and sharps.

The guitar equivalent is frets, Purcell said. Frets on many stringed instruments are spaced to make notes when used.

Notes are arranged according to a mathematical formula. The steps between notes follow the 12th root of 2. The formula for A# (A sharp) is Ax(2) 112 = 440X1.05646=466.16Hz.

The Hertz determines the sound (pitch, or frequency) of the note.

The cello does not have frets; the cello player “must use the ear to decide where to place” fingers to depress strings to create the different notes, Purcell said.

Different combinations of notes form chords.

Here are some examples of chords: The A major scale has three sharps: A, B, C#, D, E, F#. The A minor scale does not have flats: A, B, C, D, E, F, G.

Purcell’s sister, Nelda Purcell, commented with a chuckle, “I don’t know how anyone” can keep the relationships of math, music and finger movements straight.

“It’s been around for hundreds of years,” her brother laughed.

Jim Purcell played music and the group sang a few songs to hear the differences in scales: “Will The Circle Be Unbroken” in the chord of G, using all F# (F sharps, with no regular F’s), and “Wayfaring Stranger” in A minor.

Purcell bought a banjo when he was 10 or 11, he said, and then his grandmother got him a guitar. He enjoyed music when he was young, but didn’t have time for it during his working years. He was a physics professor at Georgia State University.

As he was approaching retirement, he took lessons in string instruments when he lived in Georgia. He started with violin (which his daughter was playing at the time) and went to viola, then cello.

Purcell is the president of the board of directors of the Spencer-Penn Centre.


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