Mathematics, Music, and the Guitar Martin Flashman

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Mathematics, Music, and the GuitarMartin
Flashman

• Professor of MathematicsHumboldt State
  UniversityOctober 21,2006

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Mathematics, Music, and the Guitar

• General Guitar Overview
• The Problem of Scales
• Pythagorean / Ptolemaic Proportional Scales
• Even (Well) Tempered Scales
• Fretting and Scales on the Guitar
• Some Guitar Intonation Problems
• Where and how to play a note.
• The Bridge and the Saddle.

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The Guitar Parts

• Head
• Nut
• Neck
• Body
• Bridge and Saddle

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The Head

• The strings pass over the nut and attach to
  tuning heads, which allow the player to increase
  or decrease the tension on the strings to tune
  them.
• In almost all tuning heads, a tuning knob turns
  a worm gear that turns a string post.
• Between the neck and the head is a piece called
  the nut, which is grooved to accept the strings

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The Neck

• The face of the neck, containing the frets, is
  called the fingerboard. The frets are metal
  pieces cut into the fingerboard at specific
  intervals. By pressing a string down onto a fret,
  you change the length of the string and therefore
  the tone it produces when it vibrates

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The body

• The body of most acoustic guitars has a "waist,"
  or a narrowing. This narrowing happens to make it
  easy to rest the guitar on your knee.
• The most important piece of the body is the
  soundboard. This is the wooden piece mounted on
  the front of the guitar's body, and its job is to
  make the guitar's sound loud enough for us to
  hear.
• The two widenings are called bouts. The upper
  bout is where the neck connects, and the lower
  bout is where the bridge attaches.
• In the soundboard is a large hole called the
  sound hole.

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The Bridge

• Attached to the soundboard is a piece called the
  bridge, which acts as the anchor for one end of
  the six strings. The bridge has a thin, hard
  piece embedded in it called the saddle, which is
  the part that the strings rest against.

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Building Scales(with Audacity)

• Choose one tone
• A frequency 440 cycles/sec (Hertz)
• Double the frequency cut the string length by
  1/2
• A2 frequency 2 440 880 (Octave)
• Triple the frequency cut the string length by
  1/3Then divide by 2 to bring the frequency
  between A and A2 double the string length to 2/3
• E frequency 3440/2 1320/2 660
• Divide A2 frequency by 3 then double.
• D frequency 2880/3 4/3 440 586.666

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MORE SCALE TONES

• A440 D 586.66 E 660 A2880
• Continue to multiply frequencies by 3/2, 4/3
• Multiply A by 9/4 then divide by 2
• B 4409/4990 990/2 495
• Multiply A by 16/9
• G 44016/9 782.22
• Pentatonic ScaleABDEGA (Play This)

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The round of Perfect Fifths

• FCGDAEB FCGDA FCGDAEB
• This gives a total of 12 distinct chromatic
  tones.
• The intervals between these tones in the same
  octave are roughly the same ratio.
• HOWEVER The scales are not the same if you
  start with a different tonic.

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A Pythagorean Scale based on 32
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Pythagorean A Major Scale
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Just Intonation Scale (Ptolemy)Based on triad
456
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A major Scale with Just Intonation (Ptolemy)
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Even Tempered ScaleBased on Equal step
R?1.05946
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A Major Even Tempered ScaleBased on Equal step
R?1.05946
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Comparison Just vs Even Tempered
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Frets and scales
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Scales, Frets, and logarithms
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Multiply with the guitar!
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Some Guitar Intonation Issues

• Where and how to play a note.
• At the fret.
• Vibrato and Bending.
• String qualities- multiple positions.
• The Bridge and the Saddle.
• Varying string length proportions from bridge to
  nut.
• Added tension sharper on higher frets.

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ThanksThe End of Lecture!Questions??