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
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?
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?
Eberlein, R. (1994).
Die Entstehung der tonalen Klangsyntax.
Frankfurt: Peter Lang.
History of tonal syntax: Processes
Perceptual
universals
History
of ideas
Stylistic or compositional norms
(statistical regularities)
Music perception
(expectations)
Rules of
composition
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
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

History of triads
“pretonal”
“emergence” of MmT
12th
13th

2-part counterpoint, discant improvisation

3- and 4-part ctpt, 3rds & 6ths, imperfect consonances
14th

15th
Cent

16th
Cent

17th
Cent

Ars Nova (Vitry, Machaut)
 double-leading-tone cadence
Dunstable, Dufay, Ockeghem
 falling fifth cadence in 3 and 4 parts
 Fauxbourdon: parallel 6/3 triads
 Falsobordone: chains of root positions
Palestrina, Lassus
 most sonorities are major and minor triads
 final fifth replaced by triad; tierce de Picardie
all final sonorities become triads
 seventh chords, clear SDT progressions
Historical emergence of triads
an educated guess
proportion (%)
100
thirds
triads
final triads
0
1000
1200
1400
year
1600
1800
Causal relation
between the
three lines?
History of triadic theory
Century
Idea
14th lowest voice
governs sonority
15th triad as intervals
Theorists
Tewkesbury (mid 14th), other
contrapunctus tracts
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)
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…
1.
2.
3.
physics: measured frequencies, durations
experience: perceived pitches, durations
notation: symbolic pitches and durations
Emergence of Mm triads & tonalities
in “Popperian cosmology”
World 1
World 2
World 3
(physics)
(experience)
(knowledge)
Familiarity
Conceptualization
Represen- Performance
tation
(notation)
Period
14th-16th C.
(tonal cognition) (verbal cognition)
15th-17th C.
16th-18th C.
Causal chain: Each stage is a pre- or co-requisite for the next
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
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
 neural processing

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
023 034 045 056
035 046
047
inversion
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).
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
Missing fundamentals of a major triad
missing
fundamentals
some higher
harmonics
notes
harmonics
(up to C7)
C4
C5 G5 C6 E6
G6 Bb6 C7
A3
E6 G6 A6 B6
6 7 8 9
E4
E5 B5 E6 G#6 B6
F3
C6 F6 G6 A6
6 8 9 10
G3
G4 D5 G5 B5
D6 F6 G6 A6 B6
D3
C6 D6 E6 A6
7 8 9 12
Missing fundamentals of a minor triad
missing
fundamentals
higher
harmonics
notes
harmonics
(up to C7)
C4
C5 G5 C6 E6
G6 Bb6 C7
F3
C6 Eb6 F6 G6 A6
6 7 8 9 10
Eb4
Eb5 Bb5 Eb6
G6 Bb6
Ab3
C6 Eb6 Bb6
5 6 9
G3
G4 D5 G5 B5
D6 F6 G6 A6 B6
D3
C6 D6 E6 A6
7 8 9 12
Missing fundamentals of a major triad
octave generalized model – assuming octave equivalence
notes
harmonics
missing
fundamentals
C
C G E Bb D
A
E G (B)
E
E B G# D F#
F
C G (A)
G
GDBFA
D
C E (A F#)
harmonics
Missing fundamentals of a minor triad
octave generalized model – assuming octave equivalence
notes
harmonics
missing
fundamentals
C
C G E Bb D
F
C Eb G (A)
Eb
Eb Bb G Db F
Ab
C Eb (Bb)
G
GDBFA
D
C (D A E)
harmonics
Experiment on pitch salience
in musical chords
major triad 047
minor triad 037
3
goodness of fit 
3
2
1
0
-1 0
1 2
3 4 5
6 7 8
pc 
9 10 11 12
2
1
0
-1 0 1
2 3 4
5
6
7 8
9 10 11 12
pc 
Parncutt, R. (1993). Pitch properties of chords of octave-spaced tones.
Contemporary Music Review, 9, 35-50.
Krumhansl, C. L., & Kessler, E. J. (1982).
Tracing the dynamic changes in perceived tonal
organization in a spatial representation of musical keys.
Psychological Review
Krumhansl’s key profiles
pc-stability profiles
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?
Lerdahl’s “basic pitch space”
for the key of C major – after Deutsch & Feroe
level a
level b
C
C
level c
level d
C
C
level e
C Db D Eb E F F# G Ab A Bb B
hierarchical
depth
5 1 2 1 3 2 1 4 1 2 1 2
G
D
E
E F
G
G
A
B
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, 503-522.
Open triangles: pc stability profile of MmT1
Full squares: pc salience profile of tonic triad2
(a) C major
(b) C minor
7
K&K82
7
pc-weight/3 (Pmo88)
average rating (K&K82)
Pmo88
5
5
3
3
1
1
C
D
E F
G
A
B
-1
C
D
E F
G
A
B
12
chroma
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
Prevalence of pitches in
Gregorian chant
350
initial tone
prevalence
300
final tone
250
any tone /10
200
150
100
50
0
0
1
2
3
4
5
6
7
8
9
10 11
chroma (semitones above C)
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.).
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?
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”?)
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
Triads in Palestrina: Canticum
Canticorum (1583-84), Motet 1
root major
C
14+3+1=18
minor
4+7+1=12
sus
1+0+0=1
dim
0+0+0=0
total
19+10+2=31
D
6+4+0=10
18+5+3=26
8+1+0=9
0+0+0=0
32+10+3=45
Eb
2+3+0=5
0+0+0=0
0+0+0=0
0+0+0=0
2+3+0=5
E
0+0+0=0
1+0+0=1
0+0+0=0
2+0+0=2
3+0+0=3
F
30+5+0=35
0+0+0=0
1+0+0=1
0+0+0=0
31+5+0=36
G
17+1+0=18
28+8+1=37
2+0+0=2
0+0+0=0
47+9+1=57
A
5+2+0=7
4+1+0=5
3+0+0=3
0+0+0=0
12+3+0=15
Bb
29+6+0=35
0+0+0=0
1+0+0=1
0+0+0=0
30+6+0=36
tot
103+24+1=128
55+21+5=81
16+1+0=17 2+0+0=2
each cell: Root position + first inversion + second inversion = total
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)
Triads in Palestrina: Canticum
Canticorum (1583-84), Motet 1
number of occurrences
30
25
20
15
10
5
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
Interval between successive roots (semitones)
Eberlein, R. (1994). Die Entstehung der tonalen
Klangsyntax (pp. 422-423). Frankfurt: Peter Lang.
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
rising falling rising falling rising falling total
P4
P4
3rd
3rd
M2
M2
maj-maj
64
19
0
0
6
2
91
maj-min
60
1
2
9
5
0
77
min-maj
5
20
1
15
5
3
49
min-min
21
150
5
45
0
3
0
24
1
17
0
5
27
244
total
Wanted!
Experts in different
European countries
ESF Exploratory Workshop
“Evolution of Western tonal syntax”
•
•
•
•
historians
theorists
computer scientists
psychologists
parncutt@uni-graz.at