Musical, Physical, and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy Cross Musical, Physical, Rick Miranda and Mathematical Intervals The Physics of Sound The 2010 Leonard Sulski Lecture Length (or Frequency) Ratios College of the Holy Cross Between Notes Fretting A Guitar Geometrical Approximations Rick Miranda, Colorado State University Arithmetic Approximations Vincenzo Galilei April 12, 2010
Musical, Physical, Outline and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy Cross The Physics of Sound Rick Miranda The Physics of Sound Length (or Frequency) Ratios Between Notes Length (or Frequency) Ratios Between Notes Fretting A Guitar Fretting A Guitar Geometrical Approximations Geometrical Approximations Arithmetic Approximations Vincenzo Galilei Arithmetic Approximations Vincenzo Galilei
Musical, Physical, Vibrating Strings and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy Cross When a string vibrates, its basic pitch Rick Miranda (the frequency of the sound wave generated) is determined by The Physics of Sound ◮ the composition of the string (thickness, material, etc.) Length (or Frequency) Ratios ◮ the tension of the string Between Notes Fretting A Guitar ◮ the length of the string. Geometrical Approximations Arithmetic Approximations Vincenzo Galilei
Musical, Physical, Vibrating Strings and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy Cross When a string vibrates, its basic pitch Rick Miranda (the frequency of the sound wave generated) is determined by The Physics of Sound ◮ the composition of the string (thickness, material, etc.) Length (or Frequency) Ratios ◮ the tension of the string Between Notes Fretting A Guitar ◮ the length of the string. Geometrical Ancient Scientists knew that Frequency and Length are Approximations Arithmetic inversely proportional: Approximations Vincenzo Galilei Frequency = (constant) Length (Although they didn’t really know much about Frequency...)
Musical, Physical, Pythagorean Intervals and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy The Pythagorean School refined this one step further. Cross They considered two notes together: Harmony Rick Miranda The Physics of Sound Length (or Frequency) Ratios Between Notes Fretting A Guitar Geometrical Approximations Arithmetic Approximations Vincenzo Galilei
Musical, Physical, Pythagorean Intervals and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy The Pythagorean School refined this one step further. Cross They considered two notes together: Harmony Rick Miranda They noticed that the most pleasing harmonies The Physics of were produced by Frequencies Sound (actually, they used Lengths as the measure) Length (or Frequency) Ratios which were in ratios of small integers: Between Notes Fretting A Guitar Geometrical Approximations Arithmetic Approximations Vincenzo Galilei
Musical, Physical, Pythagorean Intervals and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy The Pythagorean School refined this one step further. Cross They considered two notes together: Harmony Rick Miranda They noticed that the most pleasing harmonies The Physics of were produced by Frequencies Sound (actually, they used Lengths as the measure) Length (or Frequency) Ratios which were in ratios of small integers: Between Notes Fretting A Guitar ◮ Octave: (e.g. middle C to high C): 1 - to - 2 Geometrical Approximations ◮ Fifth: (e.g. C to G): 2 - to - 3 Arithmetic ◮ Fourth: (e.g. C to F): 3 - to - 4 Approximations Vincenzo Galilei ◮ Etc.
Musical, Physical, Pythagorean Intervals and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy The Pythagorean School refined this one step further. Cross They considered two notes together: Harmony Rick Miranda They noticed that the most pleasing harmonies The Physics of were produced by Frequencies Sound (actually, they used Lengths as the measure) Length (or Frequency) Ratios which were in ratios of small integers: Between Notes Fretting A Guitar ◮ Octave: (e.g. middle C to high C): 1 - to - 2 Geometrical Approximations ◮ Fifth: (e.g. C to G): 2 - to - 3 Arithmetic ◮ Fourth: (e.g. C to F): 3 - to - 4 Approximations Vincenzo Galilei ◮ Etc. There is a “Resonance” reason for this: the wave form produced by adding waves with these ratios are simpler, less discordant (even visually)
Musical, Physical, Octaves: Ratio = 2 and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy Cross Rick Miranda The Physics of Sound Length (or Frequency) Ratios Between Notes Fretting A Guitar Geometrical Approximations Arithmetic Approximations Vincenzo Galilei
Musical, Physical, Dissonance: Ratio = 1.8 and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy Cross Rick Miranda The Physics of Sound Length (or Frequency) Ratios Between Notes Fretting A Guitar Geometrical Approximations Arithmetic Approximations Vincenzo Galilei
Musical, Physical, More Dissonance: Ratio = 1.6 and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy Cross Rick Miranda The Physics of Sound Length (or Frequency) Ratios Between Notes Fretting A Guitar Geometrical Approximations Arithmetic Approximations Vincenzo Galilei
Musical, Physical, Fifths: Ratio = 1.5 and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy Cross Rick Miranda The Physics of Sound Length (or Frequency) Ratios Between Notes Fretting A Guitar Geometrical Approximations Arithmetic Approximations Vincenzo Galilei
√ Musical, Physical, Fifths: Ratio = 2 and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy Cross Rick Miranda The Physics of Sound Length (or Frequency) Ratios Between Notes Fretting A Guitar Geometrical Approximations Arithmetic Approximations Vincenzo Galilei
Musical, Physical, and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy Fifths and Fourths seem consistent, at least for a while: Cross Rick Miranda F G C F G C The Physics of 3 4 3 2 1 1 Sound 2 3 4 3 2 Length (or Frequency) Ratios Between Notes Fretting A Guitar Geometrical Approximations Arithmetic Approximations Vincenzo Galilei
Musical, Physical, and Mathematical Intervals The 2010 Leonard Sulski Lecture College of the Holy Fifths and Fourths seem consistent, at least for a while: Cross Rick Miranda F G C F G C The Physics of 3 4 3 2 1 1 Sound 2 3 4 3 2 Length (or Frequency) Ratios Between Notes ◮ Octaves: F - to - F ratio = 3 4 / 3 2 = 1 2 . Fretting A Guitar ◮ Also G - to - G ratio = 2 3 / 4 3 = 1 2 . Geometrical Approximations ◮ Fifths: F - to - C ratio = 1 / 3 2 = 2 3 ; Arithmetic Approximations ◮ Also upper F - to - C ratio = 1 2 / 3 4 = 2 3 . Vincenzo Galilei ◮ Fourths: G - to - C ratio = 1 / 4 3 = 1 2 / 2 3 = 3 4 .
Musical, Physical, More Notes To The Scale and Mathematical Intervals The 2010 Leonard This Pythagorean model works well for scales that only Sulski Lecture College of the Holy involve C’s, F’s, and G’s. Cross Let’s try to add a few more notes to the scale. Rick Miranda The Physics of Sound Length (or Frequency) Ratios Between Notes Fretting A Guitar Geometrical Approximations Arithmetic Approximations Vincenzo Galilei
Musical, Physical, More Notes To The Scale and Mathematical Intervals The 2010 Leonard This Pythagorean model works well for scales that only Sulski Lecture College of the Holy involve C’s, F’s, and G’s. Cross Let’s try to add a few more notes to the scale. Rick Miranda A fifth above lower G is a D, and the Length should be The Physics of Sound 3 ∗ 4 2 3 = 8 9 . Length (or Frequency) Ratios Between Notes Fretting A Guitar Geometrical Approximations Arithmetic Approximations Vincenzo Galilei
Musical, Physical, More Notes To The Scale and Mathematical Intervals The 2010 Leonard This Pythagorean model works well for scales that only Sulski Lecture College of the Holy involve C’s, F’s, and G’s. Cross Let’s try to add a few more notes to the scale. Rick Miranda A fifth above lower G is a D, and the Length should be The Physics of Sound 3 ∗ 4 2 3 = 8 9 . Length (or Frequency) Ratios Between Notes A fifth above that D is an A, and the Length should be Fretting A Guitar Geometrical 3 ∗ 8 2 9 = 16 Approximations 27 . Arithmetic Approximations Vincenzo Galilei
Musical, Physical, More Notes To The Scale and Mathematical Intervals The 2010 Leonard This Pythagorean model works well for scales that only Sulski Lecture College of the Holy involve C’s, F’s, and G’s. Cross Let’s try to add a few more notes to the scale. Rick Miranda A fifth above lower G is a D, and the Length should be The Physics of Sound 2 3 ∗ 4 3 = 8 9 . Length (or Frequency) Ratios Between Notes A fifth above that D is an A, and the Length should be Fretting A Guitar Geometrical 2 3 ∗ 8 9 = 16 Approximations 27 . Arithmetic Approximations A fourth below that A is an E, and the Length should be Vincenzo Galilei 4 3 ∗ 16 27 = 64 81 .
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