1 of 48

MATH & MUSIC A HARMONIOUS DUET

Sean Saunders – Sheridan College

2 of 48

GOOD VIBRATIONS: �THE MATH OF HARMONICS

3 of 48

UNDERSTANDING HARMONICS

  •  

Reference: Stewart, I. (2012). In Pursuit of the Unknown. pp. 137-44

4 of 48

UNDERSTANDING HARMONICS

  •  

Reference: Stewart, I. (2012). In Pursuit of the Unknown. pp. 137-44

5 of 48

STRINGED INSTRUMENTS

  • Consider a guitar string
  • Pitch is increased by shortening the length of the string by fingering the fret board
  • Pitch can be adjusted using the tuning pegs to adjust tension
  • Differently-pitched strings use differently-weighted materials/ materials of different thickness
  • Piano strings work the same way

Reference: Walker, L. (2014). Wave Equation Modelling of Vibrating String

6 of 48

WIND INSTRUMENTS – OPEN TUBES

Reference: Walker, L. (2014). Wave Equation Modelling of Vibrating String

7 of 48

WIND INSTRUMENTS – CLOSED TUBES

Reference: Walker, L. (2014). Wave Equation Modelling of Vibrating String

8 of 48

WELL TEMPERED: THE MATH OF MUSICAL TUNINGS

9 of 48

MUSIC VS NOISE: �NO DIFFERENCE (ONLY RATIOS)

10 of 48

THE HARMONIC RATIOS

  •  

11 of 48

PYTHAGOREAN TUNING (3-LIMIT)

Interval name

Short

Number of�semitones

Pythagorean tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

256/243

1.0535

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

32/27

1.1852

Major Third

M3

4

81/64

1.2656

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

1024/729

1.4047

Diminished Fifth (Tritone+)

d5

6

729/512

1.4238

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

128/81

1.5802

Major Sixth

M6

9

27/16

1.6875

Minor Seventh

m7

10

16/9

1.7778

Major Seventh

M7

11

243/128

1.8984

Perfect Octave

P8

12

2/1

2.0000

12 of 48

PYTHAGOREAN TUNING (3-LIMIT)

Interval name

Short

Number of�semitones

Pythagorean tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

256/243

1.0535

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

32/27

1.1852

Major Third

M3

4

81/64

1.2656

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

1024/729

1.4047

Diminished Fifth (Tritone+)

d5

6

729/512

1.4238

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

128/81

1.5802

Major Sixth

M6

9

27/16

1.6875

Minor Seventh

m7

10

16/9

1.7778

Major Seventh

M7

11

243/128

1.8984

Perfect Octave

P8

12

2/1

2.0000

13 of 48

PYTHAGOREAN TUNING (3-LIMIT)

Interval name

Short

Number of�semitones

Pythagorean tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

256/243

1.0535

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

32/27

1.1852

Major Third

M3

4

81/64

1.2656

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

1024/729

1.4047

Diminished Fifth (Tritone+)

d5

6

729/512

1.4238

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

128/81

1.5802

Major Sixth

M6

9

27/16

1.6875

Minor Seventh

m7

10

16/9

1.7778

Major Seventh

M7

11

243/128

1.8984

Perfect Octave

P8

12

2/1

2.0000

14 of 48

PYTHAGOREAN TUNING (3-LIMIT)

Interval name

Short

Number of�semitones

Pythagorean tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

256/243

1.0535

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

32/27

1.1852

Major Third

M3

4

81/64

1.2656

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

1024/729

1.4047

Diminished Fifth (Tritone+)

d5

6

729/512

1.4238

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

128/81

1.5802

Major Sixth

M6

9

27/16

1.6875

Minor Seventh

m7

10

16/9

1.7778

Major Seventh

M7

11

243/128

1.8984

Perfect Octave

P8

12

2/1

2.0000

15 of 48

PYTHAGOREAN TUNING (3-LIMIT)

Interval name

Short

Number of�semitones

Pythagorean tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

256/243

1.0535

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

32/27

1.1852

Major Third

M3

4

81/64

1.2656

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

1024/729

1.4047

Diminished Fifth (Tritone+)

d5

6

729/512

1.4238

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

128/81

1.5802

Major Sixth

M6

9

27/16

1.6875

Minor Seventh

m7

10

16/9

1.7778

Major Seventh

M7

11

243/128

1.8984

Perfect Octave

P8

12

2/1

2.0000

16 of 48

PYTHAGOREAN TUNING (3-LIMIT)

Interval name

Short

Number of�semitones

Pythagorean tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

256/243

1.0535

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

32/27

1.1852

Major Third

M3

4

81/64

1.2656

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

1024/729

1.4047

Diminished Fifth (Tritone+)

d5

6

729/512

1.4238

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

128/81

1.5802

Major Sixth

M6

9

27/16

1.6875

Minor Seventh

m7

10

16/9

1.7778

Major Seventh

M7

11

243/128

1.8984

Perfect Octave

P8

12

2/1

2.0000

17 of 48

PYTHAGOREAN TUNING (3-LIMIT)

Interval name

Short

Number of�semitones

Pythagorean tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

256/243

1.0535

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

32/27

1.1852

Major Third

M3

4

81/64

1.2656

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

1024/729

1.4047

Diminished Fifth (Tritone+)

d5

6

729/512

1.4238

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

128/81

1.5802

Major Sixth

M6

9

27/16

1.6875

Minor Seventh

m7

10

16/9

1.7778

Major Seventh

M7

11

243/128

1.8984

Perfect Octave

P8

12

2/1

2.0000

18 of 48

PYTHAGOREAN TUNING (3-LIMIT)

Interval name

Short

Number of�semitones

Pythagorean tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

256/243

1.0535

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

32/27

1.1852

Major Third

M3

4

81/64

1.2656

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

1024/729

1.4047

Diminished Fifth (Tritone+)

d5

6

729/512

1.4238

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

128/81

1.5802

Major Sixth

M6

9

27/16

1.6875

Minor Seventh

m7

10

16/9

1.7778

Major Seventh

M7

11

243/128

1.8984

Perfect Octave

P8

12

2/1

2.0000

19 of 48

PYTHAGOREAN TUNING (3-LIMIT)

Interval name

Short

Number of�semitones

Pythagorean tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

256/243

1.0535

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

32/27

1.1852

Major Third

M3

4

81/64

1.2656

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

1024/729

1.4047

Diminished Fifth (Tritone+)

d5

6

729/512

1.4238

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

128/81

1.5802

Major Sixth

M6

9

27/16

1.6875

Minor Seventh

m7

10

16/9

1.7778

Major Seventh

M7

11

243/128

1.8984

Perfect Octave

P8

12

2/1

2.0000

20 of 48

PYTHAGOREAN TUNING (3-LIMIT)

Interval name

Short

Number of�semitones

Pythagorean tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

256/243

1.0535

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

32/27

1.1852

Major Third

M3

4

81/64

1.2656

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

1024/729

1.4047

Diminished Fifth (Tritone+)

d5

6

729/512

1.4238

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

128/81

1.5802

Major Sixth

M6

9

27/16

1.6875

Minor Seventh

m7

10

16/9

1.7778

Major Seventh

M7

11

243/128

1.8984

Perfect Octave

P8

12

2/1

2.0000

21 of 48

PYTHAGOREAN TUNING (3-LIMIT)

Interval name

Short

Number of�semitones

Pythagorean tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

256/243

1.0535

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

32/27

1.1852

Major Third

M3

4

81/64

1.2656

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

1024/729

1.4047

Diminished Fifth (Tritone+)

d5

6

729/512

1.4238

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

128/81

1.5802

Major Sixth

M6

9

27/16

1.6875

Minor Seventh

m7

10

16/9

1.7778

Major Seventh

M7

11

243/128

1.8984

Perfect Octave

P8

12

2/1

2.0000

22 of 48

PYTHAGOREAN TUNING (3-LIMIT)

Interval name

Short

Number of�semitones

Pythagorean tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

256/243

1.0535

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

32/27

1.1852

Major Third

M3

4

81/64

1.2656

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

1024/729

1.4047

Diminished Fifth (Tritone+)

d5

6

729/512

1.4238

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

128/81

1.5802

Major Sixth

M6

9

27/16

1.6875

Minor Seventh

m7

10

16/9

1.7778

Major Seventh

M7

11

243/128

1.8984

Perfect Octave

P8

12

2/1

2.0000

23 of 48

PYTHAGOREAN TUNING (3-LIMIT)

Interval name

Short

Number of�semitones

Pythagorean tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

256/243

1.0535

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

32/27

1.1852

Major Third

M3

4

81/64

1.2656

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

1024/729

1.4047

Diminished Fifth (Tritone+)

d5

6

729/512

1.4238

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

128/81

1.5802

Major Sixth

M6

9

27/16

1.6875

Minor Seventh

m7

10

16/9

1.7778

Major Seventh

M7

11

243/128

1.8984

Perfect Octave

P8

12

2/1

2.0000

24 of 48

PYTHAGOREAN TUNING (3-LIMIT)

Interval name

Short

Number of�semitones

Pythagorean tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

256/243

1.0535

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

32/27

1.1852

Major Third

M3

4

81/64

1.2656

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

1024/729

1.4047

Diminished Fifth (Tritone+)

d5

6

729/512

1.4238

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

128/81

1.5802

Major Sixth

M6

9

27/16

1.6875

Minor Seventh

m7

10

16/9

1.7778

Major Seventh

M7

11

243/128

1.8984

Perfect Octave

P8

12

2/1

2.0000

25 of 48

JUST INTONATION (5-LIMIT)

Interval name

Short

Number of�semitones

Standard 5-limit tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

16/15

1.0667

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

6/5

1.2000

Major Third

M3

4

5/4

1.2500

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

45/32

1.4063

Diminished Fifth (Tritone+)

d5

6

64/45

1.4222

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

8/5

1.6000

Major Sixth

M6

9

5/3

1.6667

Minor Seventh

m7

10

9/5

1.8000

Major Seventh

M7

11

15/8

1.8750

Perfect Octave

P8

12

2/1

2.0000

26 of 48

JUST INTONATION (5-LIMIT)

Interval name

Short

Number of�semitones

Standard 5-limit tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

16/15

1.0667

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

6/5

1.2000

Major Third

M3

4

5/4

1.2500

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

45/32

1.4063

Diminished Fifth (Tritone+)

d5

6

64/45

1.4222

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

8/5

1.6000

Major Sixth

M6

9

5/3

1.6667

Minor Seventh

m7

10

9/5

1.8000

Major Seventh

M7

11

15/8

1.8750

Perfect Octave

P8

12

2/1

2.0000

27 of 48

JUST INTONATION (5-LIMIT)

Interval name

Short

Number of�semitones

Standard 5-limit tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

16/15

1.0667

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

6/5

1.2000

Major Third

M3

4

5/4

1.2500

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

45/32

1.4063

Diminished Fifth (Tritone+)

d5

6

64/45

1.4222

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

8/5

1.6000

Major Sixth

M6

9

5/3

1.6667

Minor Seventh

m7

10

9/5

1.8000

Major Seventh

M7

11

15/8

1.8750

Perfect Octave

P8

12

2/1

2.0000

28 of 48

JUST INTONATION (5-LIMIT)

Interval name

Short

Number of�semitones

Standard 5-limit tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

16/15

1.0667

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

6/5

1.2000

Major Third

M3

4

5/4

1.2500

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

45/32

1.4063

Diminished Fifth (Tritone+)

d5

6

64/45

1.4222

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

8/5

1.6000

Major Sixth

M6

9

5/3

1.6667

Minor Seventh

m7

10

9/5

1.8000

Major Seventh

M7

11

15/8

1.8750

Perfect Octave

P8

12

2/1

2.0000

29 of 48

JUST INTONATION (5-LIMIT)

Interval name

Short

Number of�semitones

Standard 5-limit tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

16/15

1.0667

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

6/5

1.2000

Major Third

M3

4

5/4

1.2500

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

45/32

1.4063

Diminished Fifth (Tritone+)

d5

6

64/45

1.4222

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

8/5

1.6000

Major Sixth

M6

9

5/3

1.6667

Minor Seventh

m7

10

9/5

1.8000

Major Seventh

M7

11

15/8

1.8750

Perfect Octave

P8

12

2/1

2.0000

30 of 48

JUST INTONATION (5-LIMIT)

Interval name

Short

Number of�semitones

Standard 5-limit tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

16/15

1.0667

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

6/5

1.2000

Major Third

M3

4

5/4

1.2500

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

45/32

1.4063

Diminished Fifth (Tritone+)

d5

6

64/45

1.4222

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

8/5

1.6000

Major Sixth

M6

9

5/3

1.6667

Minor Seventh

m7

10

9/5

1.8000

Major Seventh

M7

11

15/8

1.8750

Perfect Octave

P8

12

2/1

2.0000

31 of 48

JUST INTONATION (5-LIMIT)

Interval name

Short

Number of�semitones

Standard 5-limit tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

16/15

1.0667

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

6/5

1.2000

Major Third

M3

4

5/4

1.2500

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

45/32

1.4063

Diminished Fifth (Tritone+)

d5

6

64/45

1.4222

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

8/5

1.6000

Major Sixth

M6

9

5/3

1.6667

Minor Seventh

m7

10

9/5

1.8000

Major Seventh

M7

11

15/8

1.8750

Perfect Octave

P8

12

2/1

2.0000

32 of 48

JUST INTONATION (5-LIMIT)

Interval name

Short

Number of�semitones

Standard 5-limit tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

16/15

1.0667

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

6/5

1.2000

Major Third

M3

4

5/4

1.2500

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

45/32

1.4063

Diminished Fifth (Tritone+)

d5

6

64/45

1.4222

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

8/5

1.6000

Major Sixth

M6

9

5/3

1.6667

Minor Seventh

m7

10

9/5

1.8000

Major Seventh

M7

11

15/8

1.8750

Perfect Octave

P8

12

2/1

2.0000

33 of 48

JUST INTONATION (5-LIMIT)

Interval name

Short

Number of�semitones

Standard 5-limit tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

16/15

1.0667

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

6/5

1.2000

Major Third

M3

4

5/4

1.2500

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

45/32

1.4063

Diminished Fifth (Tritone+)

d5

6

64/45

1.4222

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

8/5

1.6000

Major Sixth

M6

9

5/3

1.6667

Minor Seventh

m7

10

9/5

1.8000

Major Seventh

M7

11

15/8

1.8750

Perfect Octave

P8

12

2/1

2.0000

34 of 48

JUST INTONATION (5-LIMIT)

Interval name

Short

Number of�semitones

Standard 5-limit tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

16/15

1.0667

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

6/5

1.2000

Major Third

M3

4

5/4

1.2500

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

45/32

1.4063

Diminished Fifth (Tritone+)

d5

6

64/45

1.4222

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

8/5

1.6000

Major Sixth

M6

9

5/3

1.6667

Minor Seventh

m7

10

9/5

1.8000

Major Seventh

M7

11

15/8

1.8750

Perfect Octave

P8

12

2/1

2.0000

35 of 48

JUST INTONATION (5-LIMIT)

Interval name

Short

Number of�semitones

Standard 5-limit tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

16/15

1.0667

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

6/5

1.2000

Major Third

M3

4

5/4

1.2500

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

45/32

1.4063

Diminished Fifth (Tritone+)

d5

6

64/45

1.4222

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

8/5

1.6000

Major Sixth

M6

9

5/3

1.6667

Minor Seventh

m7

10

9/5

1.8000

Major Seventh

M7

11

15/8

1.8750

Perfect Octave

P8

12

2/1

2.0000

36 of 48

JUST INTONATION (5-LIMIT)

Interval name

Short

Number of�semitones

Standard 5-limit tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

16/15

1.0667

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

6/5

1.2000

Major Third

M3

4

5/4

1.2500

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

45/32

1.4063

Diminished Fifth (Tritone+)

d5

6

64/45

1.4222

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

8/5

1.6000

Major Sixth

M6

9

5/3

1.6667

Minor Seventh

m7

10

9/5

1.8000

Major Seventh

M7

11

15/8

1.8750

Perfect Octave

P8

12

2/1

2.0000

37 of 48

JUST INTONATION (5-LIMIT)

Interval name

Short

Number of�semitones

Standard 5-limit tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

16/15

1.0667

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

6/5

1.2000

Major Third

M3

4

5/4

1.2500

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

45/32

1.4063

Diminished Fifth (Tritone+)

d5

6

64/45

1.4222

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

8/5

1.6000

Major Sixth

M6

9

5/3

1.6667

Minor Seventh

m7

10

9/5

1.8000

Major Seventh

M7

11

15/8

1.8750

Perfect Octave

P8

12

2/1

2.0000

38 of 48

JUST INTONATION (5-LIMIT)

Interval name

Short

Number of�semitones

Standard 5-limit tuning

Perfect Unison (Fundamental)

P1

0

1/1

1.0000

Minor Second

m2

1

16/15

1.0667

Major Second

M2

2

9/8

1.1250

Minor Third

m3

3

6/5

1.2000

Major Third

M3

4

5/4

1.2500

Perfect Fourth

P4

5

4/3

1.3333

Augmented Fourth (Tritone-)

A4

6

45/32

1.4063

Diminished Fifth (Tritone+)

d5

6

64/45

1.4222

Perfect Fifth

P5

7

3/2

1.5000

Minor Sixth

m6

8

8/5

1.6000

Major Sixth

M6

9

5/3

1.6667

Minor Seventh

m7

10

9/5

1.8000

Major Seventh

M7

11

15/8

1.8750

Perfect Octave

P8

12

2/1

2.0000

39 of 48

EQUALITY FOR ALL KEYS!

  • PROBLEM: Just (5-Limit) & Pythagorean (3-Limit) tunings require different tunings for each key
    • This leads to some very bad karaoke…

​

​

​

​

​

​

  • SOLUTION: An equal-tempered tuning system

40 of 48

EQUAL-TEMPERAMENT TUNING

Interval name

Short

Number of�semitones

12-Tone�Equal-Tempered

Perfect Unison (Fundamental)

P1

0

​

1.0000

Minor Second

m2

1

​

​

Major Second

M2

2

​

​

Minor Third

m3

3

​

​

Major Third

M3

4

​

​

Perfect Fourth

P4

5

​

​

Aug. Fourth/Dim. Fifth (Tritone)

A4/d5

6

​

​

Perfect Fifth

P5

7

​

​

Minor Sixth

m6

8

​

​

Major Sixth

M6

9

​

​

Minor Seventh

m7

10

​

​

Major Seventh

M7

11

​

​

Perfect Octave

P8

12

​

2.0000

41 of 48

EQUAL-TEMPERAMENT TUNING

Interval name

Short

Number of�semitones

12-Tone�Equal-Tempered

Perfect Unison (Fundamental)

P1

0

​

1.0000

Minor Second

m2

1

​

​

Major Second

M2

2

​

​

Minor Third

m3

3

​

​

Major Third

M3

4

​

​

Perfect Fourth

P4

5

​

​

Aug. Fourth/Dim. Fifth (Tritone)

A4/d5

6

​

​

Perfect Fifth

P5

7

​

​

Minor Sixth

m6

8

​

​

Major Sixth

M6

9

​

​

Minor Seventh

m7

10

​

​

Major Seventh

M7

11

​

​

Perfect Octave

P8

12

​

2.0000

42 of 48

EQUAL-TEMPERAMENT TUNING

Interval name

Short

Number of�semitones

12-Tone�Equal-Tempered

Perfect Unison (Fundamental)

P1

0

​

1.0000

Minor Second

m2

1

​

​

Major Second

M2

2

​

​

Minor Third

m3

3

​

​

Major Third

M3

4

​

​

Perfect Fourth

P4

5

​

​

Aug. Fourth/Dim. Fifth (Tritone)

A4/d5

6

​

​

Perfect Fifth

P5

7

​

​

Minor Sixth

m6

8

​

​

Major Sixth

M6

9

​

​

Minor Seventh

m7

10

​

​

Major Seventh

M7

11

​

​

Perfect Octave

P8

12

​

2.0000

43 of 48

EQUAL-TEMPERAMENT TUNING

Interval name

Short

Number of�semitones

12-Tone�Equal-Tempered

Perfect Unison (Fundamental)

P1

0

​

1.0000

Minor Second

m2

1

​

1.0595

Major Second

M2

2

​

1.1225

Minor Third

m3

3

​

1.1892

Major Third

M3

4

​

1.2599

Perfect Fourth

P4

5

​

1.3348

Aug. Fourth/Dim. Fifth (Tritone)

A4/d5

6

​

1.4142

Perfect Fifth

P5

7

​

1.4983

Minor Sixth

m6

8

​

1.5874

Major Sixth

M6

9

​

1.6818

Minor Seventh

m7

10

​

1.7818

Major Seventh

M7

11

​

1.8877

Perfect Octave

P8

12

​

2.0000

44 of 48

SONG SYMMETRY: MUSIC AND GROUP THEORY

45 of 48

MUSIC AND GROUP THEORY

  • Circle of Fifths
    • Progressing through all keys by “Perfect Fifths” (7 semitones)
    • Cycles through all 12 �(or 15) possible keys and key signatures
  • Chord Progressions
    • “Go out” clockwise
    • “Return home” by going counter-clockwise
    • Include major & minor
    • Avoid notes outside of the key signature

Reference: http://commons.wikimedia.org/wiki/File:Circle_of_fifths_deluxe_4.svg

46 of 48

POP SONG CHORD ANALYSIS

  •  

47 of 48

JUST A SMALL TOWN GIRL, LIVING IN A LONELY WORLD�SHE TOOK THE MIDNIGHT TRAIN GOING ANYWHERE�JUST A CITY BOY, BORN AND RAISED IN SOUTH DETROIT�HE TOOK THE MIDNIGHT TRAIN GOING ANYWHERE��WHEN I FIND MYSELF IN TIMES OF TROUBLE, MOTHER MARY COMES TO ME�SPEAKING WORDS OF WISDOM, “LET IT BE”�AND IN MY HOUR OF DARKNESS, SHE IS STANDING RIGHT IN FRONT OF ME�SPEAKING WORDS OF WISDOM, “LET IT BE”��SEE THE STONE SET IN YOUR EYES, SEE THE THORN TWIST IN YOUR SIDE, I’LL WAIT FOR YOU�SLEIGHT OF HAND AND TWIST OF FATE, ON A BED OF NAILS SHE MAKES ME WAIT, AND I WAIT WITHOUT YOU�WITH OR WITHOUT YOU, WITH OR WITHOUT YOU��BABY YOU A SONG, YOU MAKE ME WANNA ROLL MY WINDOWS DOWN AND CRUISE�DOWN A BACK ROAD, BLOWIN’ STOP SIGNS, THROUGH THE MIDDLE EVERY LITTLE FARM TOWN WITH YOU�AND THIS BRAND NEW CHEVY WITH A LIFT KIT�WOULD LOOK A HELL OF A LOT BETTER WITH YOU UP IN IT�SO BABY YOU A SONG, YOU MAKE ME WANNA ROLL MY WINDOWS DOWN AND CRUISE��COUNTRY ROADS, TAKE ME HOME TO THE PLACE I BELONG�WEST VIRGINIA, MOUNTAIN MOMMA, TAKE ME HOME, COUNTRY ROADS��MY LIFE IS BRILLIANT, MY LOVE IS PURE, I SAW AN ANGEL, OF THAT I’M SURE�SHE SMILED AT ME ON THE SUBWAY, SHE WAS WITH ANOTHER MAN�BUT I WON’T LOSE NO SLEEP ON THAT, CAUSE I’VE GOT A PLAN��I’M ON THE EDGE OF GLORY, AND I’M HANGING ON A MOMENT OF TRUTH�OUT ON THE EDGE OF GLORY, AND I’M HANGING ON A MOMENT WITH YOU�I’M ON THE EDGE, THE EDGE, …�I’M ON THE EDGE OF GLORY, AND I’M HANGING ON A MOMENT WITH YOU; I’M ON THE EDGE WITH U��BABY, I'M DANCING IN THE DARK WITH YOU BETWEEN MY ARMS �BAREFOOT ON THE GRASS, LISTENING TO OUR FAVOURITE SONG �WHEN YOU SAID YOU LOOKED A MESS, I WHISPERED UNDERNEATH MY BREATH �BUT YOU HEARD IT, DARLING, YOU LOOK PERFECT TONIGHT��GOOD FRIENDS WE HAVE HAD OH, GOOD FRIENDS WE'VE LOST ALONG THE WAY �IN THIS GREAT FUTURE, YOU CAN'T FORGET YOUR PAST SO DRY YOUR TEARS, I SAY �AND NO, WOMAN, NO CRY NO, WOMAN, NO CRY �LITTLE DARLING, DON'T SHED NO TEARS NO, WOMAN, NO CRY

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THANK YOU!