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Musical Tonality

Musical Tonality. Existence of multiple tonalities raises a variety of questions What is the relation of one tonality to another? Can we measure relatedness? How do we represent relatedness? Questions regarding perception of tonality How do we find our sense of key?

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Musical Tonality

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  1. Musical Tonality • Existence of multiple tonalities raises a variety of questions • What is the relation of one tonality to another? • Can we measure relatedness? • How do we represent relatedness? • Questions regarding perception of tonality • How do we find our sense of key? • Are we accurate in sense of key? • What models do we have for determining key?

  2. Theoretical Maps of Interkey Distances • Are there ways of deriving interkey distances? • Put differently, can we determine key maps? • Key distance an important issue in music-theoretic descriptions of tonal music because of key modulation • Musical composition written in home key • Compositions move to other keys, or modulate • Usually return to home key • Practice codified in interkey distances – key distances considered close if modulations are frequent

  3. Interkey Distances The Circle of Fifths

  4. Interkey Distances, con’t Diatonic set overlaps

  5. Interkey Distances, con’t Diatonic set overlaps • The problem of minor keys • Different forms of the minor • Relation between the major and minor keys • The Relative Minor (C major and A minor) • Identical scale degrees • C Major: C D E F G A B • A Minor: A B C D E F G • The Parallel Minor (C major and C minor) • Common tonic and 5th scale degree • C Major: C D E F G A B • C Minor: C D Eb F G Ab B

  6. Interkey Distances, con’t Krumhansl & Kessler (1982)

  7. Interkey Distances, con’t Interkey correlations

  8. Interkey Distances, con’t Interkey correlations, graphed

  9. Interkey Distances, con’t Krumhansl & Kessler (1982) Multidimensional scaling solution

  10. Interkey Distances, con’t The four-dimensional torus

  11. Interkey Distances, con’t Krumhansl & Kessler (1982) Map of key space

  12. Interkey Distances, con’t Schoenberg (1954/1969) Chart of key distance

  13. Interkey Distances, con’t Werts (1983) • Analyzed key progressions between compositions • Expressed as movement from one key to neighboring keys • Produced a multidimensional scaling solution based on number of times keys moved from one to another • Similar to MDS solution just seen

  14. Models of Key Finding Longuet-Higgens & Steedman (1971) C Major Diatonic Scale:C D E F G A B Major Chords:C E G F A C G B D E A B C G F C G D C Major 0 Major A E B9 4 11 F C G D5 0 7 2 C Minor 0 Minor B 11 F C G D 5 0 7 2 G# D# 8 3

  15. Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Map of key space A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C# A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C# A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C # 9 4 11 6 1 8 3 10 5 0 7 2 9 5 0 7 2 9 4 11 6 1 8 3 10 5 1 8 3 10 5 0 7 2 9 4 11 6 1 9 4 11 6 1 8 3 10 5 0 7 2 9 5 0 7 2 9 4 11 6 1 8 3 10 5 1 8 3 10 5 0 7 2 9 4 11 6 1 9 4 11 6 1 8 3 10 5 0 7 2 9 5 0 7 2 9 4 11 6 1 8 3 10 5 1 8 3 10 5 0 7 2 9 4 11 6 1

  16. Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Notes in Fugue Subject: E F# B C# D# E D# E F# G# A 4 6 11 1 3 4 3 4 6 8 9 C# D# E F# G# A# B 1 3 4 6 8 9 11

  17. Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Map of key space A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C# A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C# A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C # Possible Keys: Note 1 (E) : B, E, A, D, G, C, F#

  18. Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Map of key space A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C# A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C# A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C # Possible Keys: Note 1 (E) : B, E, A, D, G, C, F# Note 2 (E, F#) : B, E, A, D, G

  19. Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Map of key space A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C# A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C# A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C # Possible Keys: Note 1 (E) : B, E, A, D, G, C, F# Note 2 (E, F#) : B, E, A, D, G Note 3 (E, F#, B) : B, E, A, D, G

  20. Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Map of key space A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C# A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C# A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C # Possible Keys: Note 1 (E) : B, E, A, D, G, C, F# Note 2 (E, F#) : B, E, A, D, G Note 3 (E, F#, B) : B, E, A, G Note 4 (E, F#, B, C#) : B, E, A

  21. Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Map of key space A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C# A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C# A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C # Possible Keys: Note 1 (E) : B, E, A, D, G, C, F# Note 2 (E, F#) : B, E, A, D, G Note 3 (E, F#, B) : B, E, A, G Note 4 (E, F#, B, C#) : B, E, A Note 5 (E, F#, B, C#, D#) : B, E

  22. Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Map of key space A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C# A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C# A E B F# C# G# D# A# F C G D A F C G D A E B F# C# G# D# A# F C# G# D# A# F C G D A E B F# C # Possible Keys: Note 1 (E) : B, E, A, D, G, C, F# Note 2 (E, F#) : B, E, A, D, G Note 3 (E, F#, B) : B, E, A, G Note 4 (E, F#, B, C#) : B, E, A Note 5 (E, F#, B, C#, D#) : B, E . . . Note 10 (E, F#, B, C#, D#, G#) : B, E Note 11 (E, F#, B, C#, D#, G#, A) : E

  23. Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Notes in Fugue Subject: C Db C B E F Bb A Ab G 0 1 0 11 4 5 10 9 8 7 (0 1 4 5 7 8 9 10 11)

  24. Models of Key Finding, con’t Krumhansl-Schmuckler Key-Finding Algorithm (Krumhansl & Schmuckler, 1986, Schmuckler & Tomovski, 1995) Schubert, Op. 94, no. 1 – Tone durations

  25. Models of Key Finding, con’t Krumhansl-Schmuckler Key-Finding Algorithm (Krumhansl & Schmuckler, 1986) Bach, C Minor Prelude – Tone durations

  26. Models of Key Finding, con’t Krumhansl-Schmuckler Key-Finding Algorithm (Schmuckler & Tomovski, 2005)

  27. Models of Key Finding, con’t Krumhansl-Schmuckler Key-Finding Algorithm (Schmuckler & Tomovski, 2005)

  28. Models of Key Finding, con’t Brown & Butler (1981) Intervals of the diatonic set Major Scale

  29. Models of Key Finding, con’t Brown & Butler (1981) Trichords and the rare interval

  30. Models of Key Finding, con’t Brown & Butler (1981) Intervals of the diatonic set Minor Scale

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