Tuning Systems & Interval Ratios

This lesson is part of the Fundamentals of Sound & Music module.

Objectives

• Students will understand the concept of tuning systems and temperaments.
• Student will understand the basis of 12-TET tuning (including definitions of cents and semitones)
• Student will know the general differences between 12-TET and just intonation.
• Students will be able to derive harmonic frequencies according to simple arithmetic calculations (using harmonics or interval ratios).
• Student will have a basic understanding of tuning discrepancies based on harmonics and interval ratios
• Student will understand the concept of beats and how they are used to tune an instrument
• Student will be exposed to tuning variations in Indian Raga, Javanese Gamelan, and Maqam
• Student will understand the relationship between diatonic intervals and frequency ratios
• Student will be able to derive intervals from frequency calculations

Resources

Readings

• Handout on Simple Intervals and the Harmonic Series
• Gann, "Information on Alternate Tunings"
• Music Crash Courses: Tuning Systems
• Lewis, "Understanding Temperaments"
• Tobey, A Feeling for Harmony:
  • Chapter 1M: Music Math Ratios Part 1
  • Chapter 2C: Consonance and Dissonance
  • Chapter 2E: Just Intonation
  • Chapter 2M: Harmonic Series
  • Chapter 2D: Pythagorean Intonation
  • Optional Material on Commas - Chapter 1N: The Pythagorean Comma, 3E: The Other Commas

teInstructor References

• Wolfe, Physclips: Tartini Tones, Consonance, Temperament, Harmony, And Scales
• Forster, Excerpts from Musical Mathematics:
  • Just Intonation
  • Origin of Length Ratios
  • Greek Classification of Ratios, Tetrachords, Scales, and Modes
• Rubinstein, "Well vs. Equal Temperament"

Listening and Videos

• Cris Forsters Instruments and Music:
  • Chrysalis (Just Intonation)
  • Just Keys (Just Intonation Keyboard)
• The Rasa: "Microtonal Star Wars Cantina Band"
• 12Tone:
  • How Many Notes Are There?
  • Can You Tell These Notes Apart? (The Syntonic Comma)
• MinutePhysics: Why It's Impossible to Tune a Piano
• Huykens-Foker Foundation: Temperament Examples

Tools

• Offtonic: Microtonal Synthesizer
• Academo: 19TET Keyboard
• Sevish: Scale Workshop
• Apple Logic Pro Manual: Tuning Settings
• Norris:
  • Harmonic Series Calculator
  • Frequency to Note Converter
• Wolfram: Pythagorean, Meantone, and Equal Temperament Scales
• Thompson, Just Intonation Interval Calculator
• Scala (software)

Class Activities

• Demonstration of different tuning systems and temperaments
• Calculation of intervals using just intonation ratios - discussion of tuning discrepancies
• Demonstration of beats