Musical Tonality Existence of multiple tonalities raises a variety of questions What is the relation of one tonality to another? Can we measure relatedness?

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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?

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

Interkey Distances The Circle of Fifths

Interkey Distances, con’t Diatonic set overlaps

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 5 th scale degree C Major: C D E F G A B C Minor: C D Eb F G Ab B

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

Interkey Distances, con’t Interkey correlations

Interkey Distances, con’t Interkey correlations, graphed

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

Interkey Distances, con’t The four-dimensional torus

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

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

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

Models of Key Finding Longuet-Higgens & Steedman (1971) C Major Diatonic Scale:CDEFGAB Major Chords:CEG FAC GBD EAB CGFCGD C Major0 Major AEB9411 FCGD5072 C Minor0 Minor B11 FCGD5072 G#D#83

Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Map of key space AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # C # AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # C # AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # C #

Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Notes in Fugue Subject: EF#BC#D#ED#EF#G#A C#D#EF#G#A#B

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

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

Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Map of key space AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # C # AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # C # AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # 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

Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Map of key space AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # C # AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # C # AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # 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

Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Map of key space AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # C # AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # C # AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # 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

Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Map of key space AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # C # AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # C # AEBF # C # G # D # A # FCGDA FCGDA EBF # C # G # D # A # F C # G # D # A # FCGDAEBF # 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

Models of Key Finding, con’t Longuet-Higgens & Steedman (1971) Notes in Fugue Subject: CDbCBEFBbAAbG ( )

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

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

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

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

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

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

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