Alternative Tuning and How We Hear

CONTENTS

• JUST TUNING
  • SIMPLE RATIOS
  • COMMON INTERVALS
• STANDARD 12-TONE SCALE
  • MAPPING THE PIANO KEYBOARD TO FREQUENCY
  • MAPPING FREQUENCY TO THE PIANO KEYBOARD
  • LOGARITHMIC VIEW
  • CENTS
  • COMMA
• HEARING INTERVALS
  • TWO VERY SIMILAR TONES
    • PHASE SHIFTING
    • COMB FILTERS
    • RELATION TO HARMONICS
  • TWO SOMEWHAT SIMILAR TONES
    • BEAT FREQUENCY
    • CRITICAL BANDWIDTH
  • TWO SOMEWHAT DIFFERENT TONES
    • INTERVALS
    • WHOLE NUMBER RATIOS
  • TWO VERY DIFFERENT TONES
• PARTIALS
  • PHYSICAL MEANING
  • FOURIER
  • HUMAN HEARING
  • HARMONY TEMPLATE
• HEARING BELL-LIKE TONES
  • AN ILLUSION
  • NO CHORDS
• RHYTHM
• APPLICATIONS
  • HARMONIC SCALE
    • MAPPING ONTO THE STANDARD 12-TONE SCALE
    • COMPOSITIONAL POSSIBILITIES
      • MELODIC
      • CHORDS
  • JUST TUNED SCALE
    • MAPPING ONTO THE STANDARD 12-TONE SCALE
    • COMPOSITIONAL POSSIBILITIES
      • MELODIC
      • CHORDS
  • DYNAMICALLY RETUNED SCALES
    • RULES FOR PREPROCESSING A SONG
    • RULES TO TRIGGER RETUNING IN REAL-TIME
      • MELODIC
      • CHORDS
    • COMPOSITIONAL POSSIBILITIES
      • MELODIC
      • CHORDS

• just tuning frequency table
• equal tempered tuning frequency table
• logarithmic frequency graph
• comb filter
• reinforcement and cancellation
• combing partials
• harmonics of a string
• Waveforms and Spectra of Different Combinations of Partials in a Sawtooth Waveform
• harmony template
• incomplete harmony template
• harmony template of a major triad
• partials of a bell tone
• harmonic scale frequency table
• harmonic scale relationship diagram
• just tuned scale frequency table
• just tuned scale relationship diagram

• Fourier: partials 1 2 3 4 of a sawtooth wave
• Fourier: partials - 2 3 4 of a sawtooth wave
• Fourier: partials 1 - 3 4 of a sawtooth wave
• Fourier: partials - - 3 4 of a sawtooth wave
• Fourier: partials 1 2 - 4 of a sawtooth wave
• Fourier: partials - 2 - 4 of a sawtooth wave
• Fourier: partials 1 - - 4 of a sawtooth wave
• Fourier: partials - - - 4 of a sawtooth wave
• Fourier: partials 1 2 3 - of a sawtooth wave
• Fourier: partials - 2 3 - of a sawtooth wave
• Fourier: partials 1 - 3 - of a sawtooth wave
• Fourier: partials - - 3 - of a sawtooth wave
• Fourier: partials 1 2 - - of a sawtooth wave
• Fourier: partials - 2 - - of a sawtooth wave
• Fourier: partials 1 - - - of a sawtooth wave
• Fourier: partials - - - - of a sawtooth wave
• harmony template, first 10 partials
• incomplete harmony template, missing 1^st and 2^nd partials
• C note composed of 4 harmonic partials
• E note composed of 3 harmonic partials
• G note composed of 3 harmonic partials
• CEG chord from the previous three notes
• harmonic series of theory 1 (two octaves down)
• harmonic series of theory 2 (one octave down)
• C bell tone
• E bell tone
• G bell tone
• CEG chord from the previous three bell tones
• bass and snare, 2x, in phase
• bass and snare, 2x, out of phase
• tambourine, explicit 4x
• out of phase, effectively 8x
• three against two
• three against two, shifted phase
• three against four
• three against four, shifted phase
• splitting 4x into 2x plus 2x shifted
• 5x 4x 3x 2x

• An excellent webpage about Tuning Barbershop Chords

• Science of Musical Sound by John R. Pierce, 1983, Scientific American Books, Inc., distributed by W. H. Freeman and Company, 41 Madison Avenue, New York, New York, 10010

• Music of the Spheres by Jamie James, 1993, Copernicus, an imprint of Springer-Verlag New York, Inc., 175 Fifth Avenue, New York NY 10010, by arrangement with Grove Press

• The Society of Mind by Marvin Minsky, 1986, Simon and Schuster, Inc., Simon & Schuster Building, Rockefeller Center, 1230 Avenue of the Americas, New York, New York, 10020

• A Tour Up The Harmonic Series - David Canright

• Rational Notation - David Canright

• Harmonet postings about building wind chimes with barbershop 7th chords

• Justonic Tuning Inc. makers of Stay Tuned™ real-time just intonation.

This document is ©2003 by Bill Grundmann.

Return to Bill Grundmann's Music Page