The Geometry of Music

Steve Bleiler, Mar 10, 2018

An exploration of musical scales, temperament, and algorithmic keyboard design.

Table of Contents

  1. Musical Harmony from First Principles
    1. The Basic Sine Wave
    2. Build a Sound Wave
    3. Dyadic Harmony in the Plane
    4. Visualize Two Note Harmonies with Lissajous Diagrams
    5. Visualize 3 Note Chords with Lissajous Diagrams in 3-D
    6. The Chordal Spirograph
    7. Hearing Just Intervals
    8. Hearing 3 Note Just Chords
  2. Pythagorean, Just, and Equal Tempered Scales
    1. Just Scale
    2. Pythagorean Scale
    3. Equal Tempered Scales
    4. Contrasting Chords in Different Temperaments
    5. Circle of Chords in n-ET
    6. Extended Chords
  3. Euclidean Rhythms, Even Distribution, and Bjorklund's Algorithm
    1. A Drum Machine for Euclidean Rhythms
    2. Even Distribution and Euclidean Rhythms
    3. Generating All Euclidean Rhythms
  4. Keyboards in Geogebra
    1. Generalized Equal Tempered Keyboard
    2. Euclidean Equal Tempered Keyboard
    3. An Hexagonal Just Lattice Keyboard
    4. An Hexagonal Just Lattice Keyboard
    5. A Playable 5-Limit Tonality Diamond
    6. A Playable n-limit Tonality Diamond
    7. The Virtual Tonal Plexus

1.

Musical Harmony from First Principles

  • 1. The Basic Sine Wave
  • 2. Build a Sound Wave
  • 3. Dyadic Harmony in the Plane
  • 4. Visualize Two Note Harmonies with Lissajous Diagrams
  • 5. Visualize 3 Note Chords with Lissajous Diagrams in 3-D
  • 6. The Chordal Spirograph
  • 7. Hearing Just Intervals
  • 8. Hearing 3 Note Just Chords

1.1.

The Basic Sine Wave

– Steve Bleiler

1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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1.7.

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1.8.

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2.

Pythagorean, Just, and Equal Tempered Scales

  • 1. Just Scale
  • 2. Pythagorean Scale
  • 3. Equal Tempered Scales
  • 4. Contrasting Chords in Different Temperaments
  • 5. Circle of Chords in n-ET
  • 6. Extended Chords

2.1.

Just Scale

– Ewan Kummel

2.2.

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2.3.

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2.4.

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2.5.

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2.6.

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3.

Euclidean Rhythms, Even Distribution, and Bjorklund's Algorithm

  • 1. A Drum Machine for Euclidean Rhythms
  • 2. Even Distribution and Euclidean Rhythms
  • 3. Generating All Euclidean Rhythms

3.1.

A Drum Machine for Euclidean Rhythms

– Ewan Kummel

3.2.

–

3.3.

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4.

Keyboards in Geogebra

  • 1. Generalized Equal Tempered Keyboard
  • 2. Euclidean Equal Tempered Keyboard
  • 3. An Hexagonal Just Lattice Keyboard
  • 4. An Hexagonal Just Lattice Keyboard
  • 5. A Playable 5-Limit Tonality Diamond
  • 6. A Playable n-limit Tonality Diamond
  • 7. The Virtual Tonal Plexus

4.1.

Generalized Equal Tempered Keyboard

– Ewan Kummel

4.2.

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4.3.

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4.4.

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4.5.

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4.6.

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4.7.

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