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 C C# D D# E F F# G G# A A# B C Db Eb Gb Ab Bb Note Names: “C” “D” “E” “F” “G” “A” “B” “C Sharp” “D Sharp” “F Sharp” “G Sharp” “A Sharp” “D Flat” “E Flat” “G Flat” “A Flat” “B Flat” The Chromatic Scale C C# D D# E F F# G G# A A# B C 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 2 1/12, or 1.059

4. Intervals and Frequency Ratios Interval NoteFrequency Ratio NameNameEqual UnisonC1.000 Minor SecondC#1.059 Db1.059 Major SecondD1.122 Minor ThirdD#1.189 Eb1.189 Major ThirdE1.260 Perfect FourthF1.335 TritoneF#1.414 Gb1.414 Perfect FifthG1.498 Minor SixthG#1.587 Ab1.587 Major SixthA1.682 Minor SeventhA#1.782 Bb1.782 Major SeventhB1.888 OctaveC2.000

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 19 th century, uses freq ratio of 2 1/24 http://www.youtube.com/watch?v= Nxrfoar3HfQ 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 NoteFrequency Ratio NameNameEqualJust UnisonC1.0001.000 Minor SecondC#1.059 1.067 Db1.059 1.067 Major SecondD1.122 1.111 (10:9) 1.125 (9:8) Minor ThirdD#1.189 1.200 Eb1.189 1.200 Major ThirdE1.260 1.250 Perfect FourthF1.335 1.333 TritoneF#1.414 1.406 (45:32) Gb1.414 1.422 (64:45) Perfect FifthG1.498 1.500 Minor SixthG#1.587 1.600 Ab1.587 1.600 Major SixthA1.682 1.667 Minor SeventhA#1.782 1.777 Bb1.782 1.800 Major SeventhB1.888 1.875 OctaveC2.000 2.000

8. Intervals and Frequency Ratios Interval NoteFrequency Ratio NameNameEqualJustPythagorean UnisonC1.0001.0001.000 Minor SecondC#1.059 1.0671.053 (2 8 :3 5 ) Db1.059 1.0671.068 (3 7 :2 11 ) Major SecondD1.122 1.1111.125 1.125 Minor ThirdD#1.189 1.2001.186 (2 5 :3 3 ) Eb1.189 1.2001.201 (3 9 :2 14 ) Major ThirdE1.260 1.2501.265 Perfect FourthF1.335 1.3331.333 TritoneF#1.414 1.4061.407 (2 10 :3 6 ) Gb1.414 1.4221.424 (3 6 :2 9 ) Perfect FifthG1.498 1.5001.500 Minor SixthG#1.587 1.6001.580 (2 7 :3 4 ) Ab1.587 1.6001.602 (3 8 :2 12 ) Major SixthA1.682 1.6671.688 Minor SeventhA#1.782 1.7771.788 (2 4 :3 2 ) Bb1.782 1.8001.802 (3 10 :2 15 ) Major SeventhB1.888 1.8751.900 OctaveC2.000 2.0002.000

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 Level 1:CTonic, 1 st scale degree Level 2:E G3 rd and 5 th scale degrees Level 3:D F A BDiatonic scale degrees Level 4:C# D# F# G# A#Non-diatonic scale tones Diatonic Scale: C D E F G A B C Semitones: 2 2 1 2 2 2 1

11. Musical Tonality, con’t Minor tonality Tonality of C Minor (Harmonic) C Minor (Natural) C Minor (Melodic) Level 1:CTonic, 1 st scale degree Level 2:Eb G3 rd and 5 th scale degrees Level 3:D F Ab BDiatonic scale degrees Level 4:C# E F# A A#Non-diatonic scale tones Diatonic Scale: C D Eb F G Ab B C Semitones: 2 1 2 2 1 3 1

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 #01234567891011 Major C majorCDEFGAB G majorGABCDEF# D majorDEF#GABC# Natural minor C minorCDEbFGAbBb A minorABCDEFG E minorEF#GABCD Harmonic minor C minorCDEbFGAbB

14. Diatonic Set Overlaps CC#DD#EFF#GG#AA#B Overlap C MajorCDEFGAB Major G majorCDEF#GAB6 F majorCDEFGABb6 A majorC#DE F#G#AB4 F# majorC#D#F F# G# A#B2 Natural minor C minorCDEbFGAbBb4 A minorCDEFGAB7 G minorCDEbFGABb5 Harmonic minor C minorCDEbFGAbB5

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)