Musical%20Systems

1
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
  dont actually use the entire set
• Octave equivalence repeat notes with 21
  frequency ratio
• Collapse across octaves, have 12 distinct tones
  called chromatic set

2
Musical Scales
The Chromatic Scale
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
Difference 1 Semitone --
C C D D E F F G G A A B C
------ Difference 2 Semitones
3
Musical Systems

• Chromatic Set
• Octave equivalence
• Tones with 21 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

4
Intervals and Frequency Ratios
Interval Note Frequency Ratio Name 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
5
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?vNxrfoar3HfQ
• 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

6
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?
• 21 Octave
• 32 Musical fifth

7
Intervals and Frequency Ratios
Interval Note Frequency Ratio Name Name Equal Jus
t Unison C 1.000 1.000 Minor
Second C 1.059 1.067 Db 1.059  1.067 Major
Second D 1.122 1.111 (109) 1.125
(98) Minor Third D 1.189 1.200 Eb 1.189
1.200 Major Third E 1.260  1.250 Perfect
Fourth F 1.335  1.333 Tritone F 1.414 1.406
(4532) Gb 1.414  1.422 (6445) Perfect
Fifth G 1.498  1.500 Minor Sixth G 1.587
1.600 Ab 1.587  1.600 Major Sixth A 1.682 
1.667 Minor Seventh A 1.782 1.777 Bb 1.782 
1.800 Major Seventh B 1.888  1.875 Octave C 2.0
00 2.000
8
Intervals and Frequency Ratios
Interval Note Frequency Ratio Name Name Equal Jus
t Pythagorean Unison C 1.000 1.000 1.000 Minor
Second C 1.059 1.067 1.053 (2835) Db 1.059 
1.067 1.068 (37211) Major Second D 1.122
1.111 1.125 1.125 Minor Third D 1.189
1.200 1.186 (2533) Eb 1.189 1.200 1.201
(39214) 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
(21036) Gb 1.414  1.422 1.424 (3629) Perfect
Fifth G 1.498  1.500 1.500 Minor
Sixth G 1.587 1.600 1.580 (2734) Ab 1.587 
1.600 1.602 (38212) Major Sixth A 1.682 
1.667 1.688 Minor Seventh A 1.782 1.777 1.788
(2432) Bb 1.782  1.800 1.802 (310215) Major
Seventh B 1.888  1.875 1.900 Octave C 2.000
2.000 2.000
9
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

10
Musical Tonality, cont
Major tonality Tonality of C Major
Level 1 C C Tonic, 1st scale degree
Level 2 E G E G 3rd and 5th scale degrees
Level 3 D F A B D F A B Diatonic scale degrees
Level 4 C D F G A C D F G A Non-diatonic scale tones
Diatonic Scale Diatonic Scale C D E F G A B C C D E F G A B C
Semitones Semitones 2 2 1 2 2 2 1 2 2 1 2 2 2 1
11
Musical Tonality, cont
Minor tonality Tonality of C Minor
(Harmonic) C Minor (Natural) C
Minor (Melodic)
Level 1 C C Tonic, 1st scale degree
Level 2 Eb G Eb G 3rd and 5th scale degrees
Level 3 D F Ab B D F Ab B Diatonic scale degrees
Level 4 C E F A A C E F A A Non-diatonic scale tones
Diatonic Scale Diatonic Scale C D Eb F G Ab B C C D Eb F G Ab B C
Semitones Semitones 2 1 2 2 1 3 1 2 1 2 2 1 3 1
12
Musical Tonality, cont

• 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

13
Diatonic Sets
Scale 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
14
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
15
Diatonic Set Overlaps, cont
The Circle of Fifths
16
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

17
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.
18
The Tonal Hierarchy
Krumhansl Shepard (1979)
19
The Tonal Hierarchy, cont
Major and Minor Key Profiles (Krumhansl
Kessler, 1982)
20
The Tonal Hierarchy, cont
C and F Major Key Profiles
21
Perceiving Bitonality
The Petroushka Chord (Krumhansl Schmuckler,
1986)
22
Perceiving Bitonality, cont
The Petroushka Chord (Krumhansl Schmuckler,
1986)
C Major Ratings
F Major Ratings
23
Perceiving Bitonality, cont
The Petroushka Chord (Krumhansl Schmuckler,
1986)
Bitonal Ratings
24
Perceiving Atonality
Serial Music (Krumhansl, Sandell, Sargent,1987)
Tone Rows for Schoenbergs Wind Quintet (1924)
and String Quartet no. 4 (1936).
25
Perceiving Atonality, cont
Serial Music (Krumhansl, Sandell, Sargent,1987)
Probe Tone Ratings
Group 1
Group 2
26
Perceiving Non-Western Tonality
Classical Indian Music (Castellano, Bharucha,
Krumhansl,1984)
27
Perceiving Non-Western Tonality, cont
Classical Indian Music (Castellano, Bharucha,
Krumhansl,1984)