1. Historical origins of major-minor tonality (MmT) A psychological approach
Richard Parncutt
Center for Systematic Musicology
University of Graz, Austria
Presented at Ren Med 2010, Royal Holloway, Egham GB, 5-8 July 2010
Refers to the following article in press in Music Perception:
The tonic as triad: Key profiles as pitch salience profiles of tonic triads

2. Explaining MmT’s hegemony
Like it or lump it...
most music heard today is based on
major & minor triads
major & minor keys
Why?
In the “West”
polyphony, ficta, triads?
Beyond the “West”
political? psychological?

3. Explaining musical structure the “Why is the sky blue?” approach
Explaining musical structure the “Why is the sky blue?” approach
MmT:
Why is it like it is? And not quite different? (Eberlein, 1994)
Early music:
Why did certain structures and patterns emerge in one century and disappear again in another?

4. History of tonal syntax: Processes
History of tonal syntax: Processes
History
of ideas
Perceptual
universals
Die Entstehung der tonalen Klangsyntax.
Eberlein, R. (1994).
Frankfurt: Peter Lang.
Stylistic or compositional norms
(statistical regularities)
Music perception
(expectations)
Rules of
composition

5. Music ficta and MmT’s “emergence” a theory focusing on notation
Music ficta and MmT’s “emergence” a theory focusing on notation
Mixolydian  major, Dorian  minor, usw.
Musica ficta can explain the scale steps in major/minor keys.
But it cannot explain their relative stability

6. Epistemology and approach
Epistemology and approach
Favor simpler theories (Ockam)
details are important (Dahlhaus)
but simpler theories are easier to falsify (Popper)
Favor generative theories (Lerdahl)
identify underlying principles or axioms
non-circular arguments, causeeffect
Favor interdisciplinarity (CIM, JIMS)
relevant knowledge should be considered
multidisciplinary theories are easier to falsify

7. History of triads 12th 13th 14th 15th Cent 16th 17th
History of triads
“pretonal”
12th
2-part counterpoint, discant improvisation
13th
3- and 4-part ctpt, 3rds & 6ths, imperfect consonances
14th
Ars Nova (Vitry, Machaut)
double-leading-tone cadence
“emergence” of MmT
15th
Cent
Dunstable, Dufay, Ockeghem
falling fifth cadence in 3 and 4 parts
Fauxbourdon: parallel 6/3 triads
Falsobordone: chains of root positions
16th
Palestrina, Lassus
most sonorities are major and minor triads
final fifth replaced by triad; tierce de Picardie
17th
all final sonorities become triads
seventh chords, clear SDT progressions

8. Historical emergence of triads an educated guess
Historical emergence of triads an educated guess
Causal relation between the three lines?

9. History of triadic theory
History of triadic theory
Century
Idea
Theorists
14th
lowest voice governs sonority
Tewkesbury (mid 14th), other contrapunctus tracts
15th
triad as intervals
Tinctoris (1477), Podio (1495), Gafori (1496)
16th
triad as sonority
Zarlino (1558), Sancta Maria (1565), Avianus (1581)
17th
root and inversion
Burmeister 1606), Harnisch (1608), Lippius (1612), Campion (1618), Crüger (1630)
18th
implied roots
Rameau (1721)

10. Karl Popper’s “three worlds” and Medieval music perception
Karl Popper’s “three worlds” and Medieval music perception
World 1: physical, material
World 2: experience, subjectivity
World 3: knowledge, information
We need to clearly separate…
physics: measured frequencies, durations
experience: perceived pitches, durations
notation: symbolic pitches and durations

11. Emergence of Mm triads & tonalities in “Popperian cosmology”
World 1 (physics)
World 2 (experience)
World 3 (knowledge)
Represen-tation
Performance (notation)
Familiarity
(tonal cognition)
Conceptualization (verbal cognition)
Period
14th-16th C.
15th-17th C.
16th-18th C.
Causal chain: Each stage is a pre- or co-requisite for the next

12. What is special about Mm triads?
What is special about Mm triads?
Frequency ratios?
major: 4:5:6 seems ok
minor: 10:12:15 is not so “simple”
Is tuning pure or Pythagorean?
Harmonic dualism?
overtones exist
undertones do not
root of C minor is C not G

13. Psychoacoustics of consonance 3 well established psychological factors
Roughness (Helmholtz)
nearby partials on basilar membrane
peripheral physiology
Fusion (Stumpf)
holistic perception of complex sounds
neural processing
Familiarity (Cazden, Tenney)
exposure promotes liking

14.
pc-set theory and consonance: 19 Tn-types of cardinality 3 after Rahn (1980)
prime form
012
013
014
015
016
024
025
026
027
036
037
048
inversion
023
034
045
056
035
046
047
012 = e.g. C-C#-D
013 = e.g. C-C#-D#
037 = minor triad
047 = major triad
The major and minor triads are by far the most consonant Tn-types of cardinality 3.
Only they have a P4 or P5 (fusion)
and no M2 or m2 (roughness).

15. Why is ear training so difficult?
We do not hear
frequencies (World 1), notes (World 3)
We hear pitches (World 2)
and extrapolate to notes by
musical experience
theoretic knowledge
What about missing fundamentals?
e.g. voice on telephone
Mm triads have missing fundamentals at 2nd, 4th and 6th above root

16. Missing fundamentals of a major triad
notes
harmonics
(up to C7)
missing funda-mentals
some higher C4
C5 G5 C6 E6
G6 Bb6 C7 A3
E6 G6 A6 B6 E4
E5 B5 E6 G#6 B6 F3
C6 F6 G6 A6 G3
G4 D5 G5 B5
D6 F6 G6 A6 B6 D3
C6 D6 E6 A6

17. Missing fundamentals of a minor triad
notes
harmonics
(up to C7)
missing funda-mentals
higher C4
C5 G5 C6 E6
G6 Bb6 C7 F3
C6 Eb6 F6 G6 A6
Eb4
Eb5 Bb5 Eb6
G6 Bb6
Ab3
C6 Eb6 Bb6 G3
G4 D5 G5 B5
D6 F6 G6 A6 B6 D3
C6 D6 E6 A6

18. missing funda-mentals
Missing fundamentals of a major triad octave generalized model – assuming octave equivalence
notes
harmonics
missing funda-mentals C
C G E Bb D A
E G (B) E
E B G# D F# F
C G (A) G
G D B F A D
C E (A F#)

19. missing funda-mentals
Missing fundamentals of a minor triad octave generalized model – assuming octave equivalence
notes
harmonics
missing funda-mentals C
C G E Bb D F
C Eb G (A) Eb
Eb Bb G Db F Ab
C Eb (Bb) G
G D B F A D
C (D A E)

20. Experiment on pitch salience in musical chords
Experiment on pitch salience in musical chords
major triad 047
minor triad 037
goodness of fit 
pc 
pc 
Parncutt, R. (1993). Pitch properties of chords of octave-spaced tones.
Contemporary Music Review, 9,

21. Krumhansl’s key profiles pc-stability profiles
Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys. Psychological Review
Krumhansl, C. L., & Kessler, E. J. (1982).

22. Prevalence model of key profiles
major key
minor key
Aarden, B. (2003). Dynamic melodic expectancy.
PhD dissertation, Ohio State University.
Why is G more prevalent that C in C major - but C is more stable?

23. Lerdahl’s “basic pitch space” for the key of C major – after Deutsch & Feroe
level a C
level b G
level c E
level d D F A B
level e Db Eb F# Ab Bb
hierarchical depth 5 1 2 3 4
Lerdahl, E. (2001). Tonal pitch space (p. 47). New York: Oxford.
Deutsch, D., & Feroe, J. (1981) The internal representation of pitch sequences in tonal music. Psychological Review, 88,

24. Open triangles: pc stability profile of MmT1 Full squares: pc salience profile of tonic triad2
1Krumhansl, C. L., & Kessler, E. J. (1982). Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys. Psychological Review
2Parncutt, R. (1988). Revision of Terhardt's psychoacoustical model of the root(s) of a musical chord. Music Perception

25. Prevalence of pitches in Gregorian chant
B (11) is the least frequent tone at any position.
Source of data: Bryden, J. R., & Hughes, D. G. (1969).
An index of Gregorian chant. Cambridge, MA: Harvard University Press.).

26. Chant: Why are some pitches more common than others?
Theory:
Tones whose harmonics correspond to diatonic scale steps are more consonant
 preferred  more prevalent
Implication for mi-fa:
fa is
more common
more stable
 origin of leading tone?

27. What is a music psychologist doing at MedRen?
Long-term project:
history of tonal syntax and perception
humanities: music history, music theory
sciences: psychology, computing
Planned first step:
ESF strategic workshop
15-30 speakers, many European countries
1-3 days, plenty of discussion
follow-up research project
*ESF = European Science Foundation (“science” = “Wissenschaft”?)

28.
Double leading-tone cadence prevalence of cadence and contexts in different periods?
Origin: two-part cadences (12th Century)
major sixth  octave; major third  fifth; etc.
 double-leading-tone cadence (14th)
two intervallic resolutions simultaneously
 falling-fifth cadence (16th)
transition from 3 to 4 voices
voicing GDGB-CCGC avoids parallels

29. Triads in Palestrina: Canticum Canticorum (1583-84), Motet 1
root
major
minor
sus
dim
total C
14+3+1=18
4+7+1=12
1+0+0=1
0+0+0=0
=31 D
6+4+0=10
18+5+3=26
8+1+0=9
=45 Eb
2+3+0=5 E
2+0+0=2
3+0+0=3 F
30+5+0=35
31+5+0=36 G
17+1+0=18
28+8+1=37
47+9+1=57 A
5+2+0=7
4+1+0=5
12+3+0=15 Bb
29+6+0=35
30+6+0=36
tot
=128
=81
16+1+0=17
each cell: Root position + first inversion + second inversion = total

30. Sonorities in Renaissance polyphony
Hierarchy of chord types:
major triad
minor triad
suspended triad
diminished triad
Hierarchy of chord positions:
root position
first inversion
second inversion
Psychological theory
guiding principle is consonance
hierarchy of psychoacoustic components:
fusion (brain; perception of complex tones)
smoothness (inner ear; frequency analysis)

31. Triads in Palestrina: Canticum Canticorum (1583-84), Motet 1
number of occurrences

32. Prevalence of 2-chord progressions
Prevalence of 2-chord progressions
Eberlein’s sample
J. S. Bach
7 chorales; kleine harmonische Labyrinth
Händel
Trio sonata Op. 5 No. 5
Mozart
Missa brevis KV 65 (Kyrie, Gloria, Agnus Dei)
Beethoven
Mass in C (Kyrie, Gloria)
Mendelssohn
Motets Op. 78, Nos. 1 & 2
Eberlein, R. (1994). Die Entstehung der tonalen Klangsyntax (pp ). Frankfurt: Peter Lang.
rising P4
falling P4
rising 3rd
falling 3rd
rising M2
falling M2
total
maj-maj 64 19 6 2 91
maj-min 60 1 9 5 77
min-maj 20 15 3 49
min-min 21 27
150 45 24 17
244

33. Wanted! Experts in different European countries
Wanted! Experts in different European countries
ESF Exploratory Workshop
“Evolution of Western tonal syntax”
historians
theorists
computer scientists
psychologists