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Musical Systems

Explore different musical systems, including the chromatic scale, octave equivalence, intervals, and tonality. Learn about pitch variation, tuning systems, and the significance of tonal structure in music.

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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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