Musical Tonality Existence of multiple tonalities raises a

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: C D E F G A B Major Chords: C E G F A C G B D E C G C Major A E B F C G D C Minor B F C G D G# D# A F C B G D 0 Major 9 4 11 5 0 7 2 0 Minor 11 5 0 7 2 8 3

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

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

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

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

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

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

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

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

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 11 4 5 10 9 8 7 (0 1 4 5 7 8 9 10 11)

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