Musical Systems Facts about musical systems Musical cultures

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

Musical Scales The Chromatic Scale 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” Difference: 1 Semitone ┌─┐┌─┐ C C# D D# E F F# G G# A A# B C └───┘ Difference: 2 Semitones

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

Intervals and Frequency Ratios Interval Name Note Name Frequency Ratio Equal Unison C 1. 000 Minor Second C# Db 1. 059 Major Second D 1. 122 Minor Third D# Eb 1. 189 Major Third E 1. 260 Perfect Fourth F 1. 335 Tritone F# Gb 1. 414 Perfect Fifth G 1. 498 Minor Sixth G# Ab 1. 587 Major Sixth A 1. 682 Minor Seventh A# Bb 1. 782 Major Seventh B 1. 888 Octave C 2. 000

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 21/24 • http: //www. youtube. com/watch? v= Nxrfoar 3 Hf. Q • 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

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

Intervals and Frequency Ratios Interval Name Note Name Frequency Ratio Equal Just Unison C 1. 000 Minor Second C# Db 1. 059 1. 067 Major Second D 1. 122 Minor Third D# Eb 1. 189 1. 111 (10: 9) 1. 125 (9: 8) 1. 200 Major Third E 1. 260 1. 250 Perfect Fourth F 1. 335 1. 333 Tritone F# Gb 1. 414 1. 406 (45: 32) 1. 422 (64: 45) Perfect Fifth G 1. 498 1. 500 Minor Sixth G# Ab 1. 587 1. 600 Major Sixth A 1. 682 1. 667 Minor Seventh A# Bb 1. 782 1. 777 1. 800 Major Seventh B 1. 888 1. 875 Octave C 2. 000

Intervals and Frequency Ratios Interval Name Note Name Frequency Ratio Equal Just Pythagorean Unison C 1. 000 Minor Second C# Db 1. 059 1. 067 1. 053 (28: 35) 1. 068 (37: 211) Major Second D 1. 122 1. 125 Minor Third D# Eb 1. 189 1. 111 1. 125 1. 200 Major Third E 1. 260 1. 250 1. 265 Perfect Fourth F 1. 335 1. 333 Tritone F# Gb 1. 414 1. 406 1. 422 1. 407 (210: 36) 1. 424 (36: 29) Perfect Fifth G 1. 498 1. 500 Minor Sixth G# Ab 1. 587 1. 600 1. 580 (27: 34) 1. 602 (38: 212) Major Sixth A 1. 682 1. 667 1. 688 Minor Seventh A# Bb 1. 782 1. 777 1. 800 1. 788 (24: 32) 1. 802 (310: 215) Major Seventh B 1. 888 1. 875 1. 900 Octave C 2. 000 1. 186 (25: 33) 1. 201 (39: 214)

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”

Musical Tonality, con’t Major tonality Tonality of C Major Level 1: C Level 2: E G 3 rd and 5 th scale degrees Level 3: D F A B Diatonic scale degrees Level 4: Tonic, 1 st scale degree 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

Musical Tonality, con’t Minor tonality Tonality of C Minor (Harmonic) Level 1: C Tonic, 1 st scale degree Level 2: Eb G 3 rd and 5 th scale degrees Level 3: D F Ab B Diatonic 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 C Minor (Natural) C Minor (Melodic)

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”

Diatonic Sets Scale # 0 Major C major C G major G D major D Natural minor C A minor A E minor E Harmonic minor C 1 2 3 D A E 4 5 6 7 8 E F B C F# G G D A D Eb B C F# G F D A G E B Ab F C D F G Ab Eb 9 10 11 A E B B F# C# Bb G D B

Diatonic Set Overlaps C C# D C Major G major F major A major F# major C C C D# E F F# G G# A A# B Overlap D E F D D C# D# G A B E F# G A B E F G A Bb E F# G# A B F F# G# A# B 6 6 4 2 Natural minor C A minor C G minor C D D D Eb F G Ab Bb G A Bb 4 7 5 Harmonic minor C D Eb G Ab 5 F B

Diatonic Set Overlaps, con’t The Circle of Fifths

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

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.

The Tonal Hierarchy Krumhansl & Shepard (1979)

The Tonal Hierarchy, con’t Major and Minor Key Profiles (Krumhansl & Kessler, 1982)

The Tonal Hierarchy, con’t C and F# Major Key Profiles

Perceiving Bitonality The Petroushka Chord (Krumhansl & Schmuckler, 1986)

Perceiving Bitonality, con’t The Petroushka Chord (Krumhansl & Schmuckler, 1986) C Major Ratings F# Major Ratings

Perceiving Bitonality, con’t The Petroushka Chord (Krumhansl & Schmuckler, 1986) Bitonal Ratings

Perceiving Atonality Serial Music (Krumhansl, Sandell, & Sargent, 1987) Tone Rows for Schoenberg’s Wind Quintet (1924) and String Quartet no. 4 (1936).

Perceiving Atonality, con’t Serial Music (Krumhansl, Sandell, & Sargent, 1987) Probe Tone Ratings Group 1 Group 2

Perceiving Non-Western Tonality Classical Indian Music (Castellano, Bharucha, & Krumhansl, 1984)

Perceiving Non-Western Tonality, con’t Classical Indian Music (Castellano, Bharucha, & Krumhansl, 1984)