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Musical Systems

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

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Musical Systems

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  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 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

  2. Musical Scales The Chromatic Scale Difference: 1 Semitone ┌─┐┌─┐ └───┘└───┘ Difference: 2 Semitones

  3. 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

  4. Intervals and Frequency Ratios Interval Note Frequency Ratio Name Name Equal Unison C 1.000 Minor Second C#1.059 Db1.059  Major Second D 1.122 Minor Third D#1.189 Eb1.189 Major Third E 1.260  Perfect Fourth F 1.335  Tritone F#1.414 Gb1.414  Perfect Fifth G 1.498  Minor Sixth G#1.587 Ab1.587  Major Sixth A 1.682  Minor Seventh A#1.782 Bb1.782  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?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

  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? • 2:1 – Octave • 3:2 – Musical fifth

  7. Intervals and Frequency Ratios Interval Note Frequency Ratio Name Name Equal Just Unison C 1.000 1.000 Minor Second C#1.059 1.067 Db1.059  1.067 Major Second D 1.122 1.111 (10:9) 1.125 (9:8) Minor Third D#1.189 1.200 Eb1.189 1.200 Major Third E 1.260  1.250 Perfect Fourth F 1.335  1.333 Tritone F#1.414 1.406 (45:32) Gb1.414  1.422 (64:45) Perfect Fifth G 1.498  1.500 Minor Sixth G#1.587 1.600 Ab1.587  1.600 Major Sixth A 1.682  1.667 Minor Seventh A#1.782 1.777 Bb1.782  1.800 Major Seventh B 1.888  1.875 Octave C 2.000 2.000

  8. Intervals and Frequency Ratios Interval Note Frequency Ratio Name Name Equal Just Pythagorean Unison C 1.000 1.000 1.000 Minor Second C#1.059 1.067 1.053 (28:35) Db1.059  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) Eb1.189 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) Gb1.414  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) Ab1.587  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) Bb1.782  1.800 1.802 (310:215) 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, con’t Major tonality Tonality of C Major

  11. Musical Tonality, con’t Minor tonality Tonality of C Minor (Harmonic) C Minor (Natural) C Minor (Melodic)

  12. 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”

  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, con’t 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, con’t Major and Minor Key Profiles (Krumhansl & Kessler, 1982)

  20. The Tonal Hierarchy, con’t C and F# Major Key Profiles

  21. Perceiving Bitonality The Petroushka Chord (Krumhansl & Schmuckler, 1986)

  22. Perceiving Bitonality, con’t The Petroushka Chord (Krumhansl & Schmuckler, 1986) C Major Ratings F# Major Ratings

  23. Perceiving Bitonality, con’t The Petroushka Chord (Krumhansl & Schmuckler, 1986) Bitonal Ratings

  24. Perceiving Atonality Serial Music (Krumhansl, Sandell, & Sargent,1987) Tone Rows for Schoenberg’s Wind Quintet (1924) and String Quartet no. 4 (1936).

  25. Perceiving Atonality, con’t 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, con’t Classical Indian Music (Castellano, Bharucha, & Krumhansl,1984)

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