musical systems
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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 - PowerPoint PPT PresentationTRANSCRIPT
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)