Musical Systems

• Facts about musical systems

• Musical cultures make use of variation in pitch

• Use tones of low to high frequency, and combine them in various ways

• Pitch and frequency are continuous scales

• Yet musical cultures use discrete pitches

• Use of discrete pitches, as opposed to continuously varying pitches, a universal

• Although there is potentially a large set, we don’t actually use the entire set

• Octave equivalence – repeat “notes” with 2:1 frequency ratio

• Collapse across octaves, have 12 distinct tones – called chromatic set

Musical Scales

C C# D D# E F F# G G# A A# B C

Db Eb Gb Ab Bb

Note Names:

“C” “D” “E” “F” “G” “A” “B”

“C Sharp” “D Sharp” “F Sharp” “G Sharp” “A Sharp”

“D Flat” “E Flat” “G Flat” “A Flat” “B Flat”

The Chromatic Scale

C C# D D# E F F# G G# A A# B C

Difference: 1 Semitone┌─┐┌─┐

└───┘└───┘Difference: 2 Semitones

Musical Systems

• Chromatic Set

• Octave equivalence

• Tones with 2:1 frequency ratio have the same note name

• Twelve equally divided logarithmic intervals

• Produces 12 equal steps within the octave

• Calculated by multiplying each frequency by 21/12, or 1.059

Intervals and Frequency RatiosInterval Note Frequency RatioName Name Equal Unison C 1.000

Minor Second C# 1.059 Db 1.059 

Major Second D 1.122

Minor Third D# 1.189 Eb 1.189

Major Third E 1.260 

Perfect Fourth F 1.335 

Tritone F# 1.414 Gb 1.414 

Perfect Fifth G 1.498 

Minor Sixth G# 1.587 Ab 1.587 

Major Sixth A 1.682 

Minor Seventh A# 1.782 Bb 1.782 

Major Seventh B 1.888 

Octave C 2.000

Musical Systems

• Is the division of the octave into 12 steps a norm?

• The use of quartertones (24 steps to the octave)

• First proposed in West in 19th century, uses freq ratio of 21/24

• http://www.youtube.com/watch?v=Nxrfoar3HfQ

• Karl Stockhausen

• Works using 7 – 60 steps per octave

• Classical Indian music

• 22 notes per octave

• Basic structure same as 12 tone Western system, though

• Arab Persian music

• 15-24 steps per octave

• Scales not played microtonally, though

Tuning Systems

• Consonance vs. Dissonance

• Roughly defined by freq ratio between notes

• Smaller frequency ratios are more consonant

• How well do two notes go together?

• What are some consonant frequency ratios?

• 2:1 – Octave

• 3:2 – Musical fifth

Intervals and Frequency RatiosInterval Note Frequency RatioName Name Equal JustUnison C 1.000 1.000

Minor Second C# 1.059 1.067Db 1.059  1.067

Major Second D 1.122 1.111 (10:9) 1.125 (9:8)

Minor Third D# 1.189 1.200Eb 1.189 1.200

Major Third E 1.260  1.250

Perfect Fourth F 1.335  1.333

Tritone F# 1.414 1.406 (45:32)Gb 1.414  1.422 (64:45)

Perfect Fifth G 1.498  1.500

Minor Sixth G# 1.587 1.600Ab 1.587  1.600

Major Sixth A 1.682  1.667

Minor Seventh A# 1.782 1.777Bb 1.782  1.800

Major Seventh B 1.888  1.875

Octave C 2.000 2.000

Intervals and Frequency RatiosInterval Note Frequency RatioName Name Equal Just PythagoreanUnison C 1.000 1.000 1.000

Minor Second C# 1.059 1.067 1.053 (28:35)Db 1.059  1.067 1.068 (37:211)

Major Second D 1.122 1.111 1.125 1.125

Minor Third D# 1.189 1.200 1.186 (25:33)Eb 1.189 1.200 1.201 (39:214)

Major Third E 1.260  1.250 1.265

Perfect Fourth F 1.335  1.333 1.333

Tritone F# 1.414 1.406 1.407 (210:36)Gb 1.414  1.422 1.424 (36:29)

Perfect Fifth G 1.498  1.500 1.500

Minor Sixth G# 1.587 1.600 1.580 (27:34)Ab 1.587  1.600 1.602 (38:212)

Major Sixth A 1.682  1.667 1.688

Minor Seventh A# 1.782 1.777 1.788 (24:32)Bb 1.782  1.800 1.802 (310:215)

Major Seventh B 1.888  1.875 1.900

Octave C 2.000 2.000 2.000

Musical Tonality

• Tonality:

• One note functions as a reference point for all of the tones

• Called the “tonic” or “tonal center”

• Other pitches have well-defined relation to tonal center – called “tonal function”

Musical Tonality, con’t

Major tonality

Tonality of C Major

Level 1: C Tonic, 1st scale degree

Level 2: E G 3rd and 5th scale degrees

Level 3: D F A B Diatonic scale degrees

Level 4: C# D# F# G# A# Non-diatonic scale tones

Diatonic Scale: C D E F G A B C

Semitones: 2 2 1 2 2 2 1

Musical Tonality, con’t

Minor tonality

Tonality of C Minor (Harmonic)

C Minor (Natural)

C Minor (Melodic)

Level 1: C Tonic, 1st scale degree

Level 2: Eb G 3rd and 5th scale degrees

Level 3: D F Ab B Diatonic scale degrees

Level 4: C# E F# A A# Non-diatonic scale tones

Diatonic Scale: C D Eb F G Ab B C

Semitones: 2 1 2 2 1 3 1

Musical Tonality, con’t

• Additional points about tonality

• Can be transposed to begin on ANY of the 12 chromatic pitches

• Thus, there are 12 major and 12 minor tonalities

• 24 tonalities in all

• Tonalities vary in terms of how related they are to one another

• Relation between tonalities can be assessed in terms of overlap between notes of “diatonic set”

Diatonic SetsScale # 0 1 2 3 4 5 6 7 8 9 10 11

Major

C major C D E F G A B

G major G A B C D E F#

D major D E F# G A B C#

 

Natural minor

C minor C D Eb F G Ab Bb

A minor A B C D E F G

E minor E F# G A B C D

Harmonic minor

C minor C D Eb F G Ab B

Diatonic Set Overlaps

C C# D D# E F F# G G# A A# B Overlap

C Major C D E F G A B

Major

G major C D E F# G A B 6

F major C D E F G A Bb 6

A major C# D E F# G# A B 4 

F# major C# D# F F# G# A# B 2

Natural minor

C minor C D Eb F G Ab Bb 4

A minor C D E F G A B 7

G minor C D Eb F G A Bb 5

Harmonic minor

C minor C D Eb F G Ab B 5

Diatonic Set Overlaps, con’t

The Circle of Fifths

Significance of Tonal Structure

• What is the psychological significant of tonal structure?

• Psychological principle that certain perceptual and conceptual objects have special psychological status

• Classic work by Rosch (1975)

• Certain members in a group are normative, best example of category

• Cognitive reference points for judging members of category

• Exs, vertical and horizontal lines, numbers that are multiples of 10, focal colors

• Evidence for this structure?

• Ratings of goodness or typicality

• Memory for exemplars

• Description of hierarchical ordering seems applicable to tonality

The Probe Tone Method

Krumhansl & Shepard (1979)

Context:

Probe Tone(s):

Task: Rate how well the probe tone fit with the previous passage in a musical sense.

The Tonal Hierarchy

Krumhansl & Shepard (1979)

The Tonal Hierarchy, con’t

Major and Minor Key Profiles

(Krumhansl & Kessler, 1982)

The Tonal Hierarchy, con’t

C and F# Major Key Profiles

Perceiving Bitonality

The Petroushka Chord

(Krumhansl & Schmuckler, 1986)

Perceiving Bitonality, con’t

The Petroushka Chord

(Krumhansl & Schmuckler, 1986)

C Major

Ratings

F# Major

Ratings

Perceiving Bitonality, con’t

The Petroushka Chord

(Krumhansl & Schmuckler, 1986)

Bitonal

Ratings

Perceiving Atonality

Serial Music

(Krumhansl, Sandell, & Sargent,1987)

Tone Rows for Schoenberg’s Wind Quintet (1924) and String Quartet no. 4 (1936).

Perceiving Atonality, con’t

Serial Music

(Krumhansl, Sandell, & Sargent,1987)

Probe Tone Ratings

Group 1

Group 2

Perceiving Non-Western Tonality

Classical Indian Music

(Castellano, Bharucha, & Krumhansl,1984)

Perceiving Non-Western Tonality, con’t

Classical Indian Music

(Castellano, Bharucha, & Krumhansl,1984)