Historical origins of major-minor tonality (MmT)
A psychological approach
Richard ParncuttCenter 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?
History of tonal syntax: Processes
Perceptualuniversals
Music perception(expectations)
Stylistic or compositional norms(statistical regularities)
Historyof ideas
Rules ofcomposition
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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, causeeffect
Favor interdisciplinarity (CIM, JIMS) relevant knowledge should be considered multidisciplinary theories are easier to falsify
History of triads
“pre
ton
al”
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
“em
erg
en
ce
” of M
mT
15th
Cent
Dunstable, Dufay, Ockeghem falling fifth cadence in 3 and 4 parts Fauxbourdon: parallel 6/3 triads Falsobordone: chains of root positions
16th
Cent
Palestrina, Lassus most sonorities are major and minor triads final fifth replaced by triad; tierce de Picardie
17th
Cent
all final sonorities become triads seventh chords, clear SDT progressions
0
100
1000 1200 1400 1600 1800
year
pro
po
rtio
n (
%)
thirds
triads
final triads
Historical emergence of triadsan educated guess
Causal relation between the three lines?
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)
Karl Popper’s “three worlds”and Medieval music perception
World 1: physical, materialWorld 2: experience, subjectivityWorld 3: knowledge, information
We need to clearly separate…1. physics: measured frequencies, durations 2. experience: perceived pitches, durations 3. notation: symbolic pitches and durations
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
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 consonance3 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)
012 = e.g. C-C#-D
013 = e.g. C-C#-D#
prime form 012 013 014 015 016 024 025 026 027 036 037 048
inversion 023 034 045 056 035 046 047
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
notesharmonics(up to C7)
missing funda-
mentals
some higherharmonics
C4C5 G5 C6 E6 G6 Bb6 C7
A3E6 G6 A6 B66 7 8 9
E4 E5 B5 E6 G#6 B6 F3C6 F6 G6 A66 8 9 10
G3G4 D5 G5 B5
D6 F6 G6 A6 B6D3
C6 D6 E6 A67 8 9 12
Missing fundamentals of a minor triad
notesharmonics(up to C7)
missing funda-
mentals
higherharmonics
C4C5 G5 C6 E6 G6 Bb6 C7
F3C6 Eb6 F6 G6 A66 7 8 9 10
Eb4Eb5 Bb5 Eb6
G6 Bb6Ab3
C6 Eb6 Bb65 6 9
G3G4 D5 G5 B5
D6 F6 G6 A6 B6D3
C6 D6 E6 A67 8 9 12
Missing fundamentals of a major triadoctave generalized model – assuming octave equivalence
notes harmonicsmissing funda-
mentalsharmonics
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#)
Missing fundamentals of a minor triadoctave generalized model – assuming octave equivalence
notes harmonicsmissing funda-
mentalsharmonics
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)
Experiment on pitch salience in musical chords
major triad 047
0
1
2
3
-1 0 1 2 3 4 5 6 7 8 9 10 11 12
minor triad 037
pc pc
goo
dn
ess
of f
it
Parncutt, R. (1993). Pitch properties of chords of octave-spaced tones. Contemporary Music Review, 9, 35-50.
0
1
2
3
-1 0 1 2 3 4 5 6 7 8 9 10 11 12
Krumhansl’s key profilespc-stability profiles
Kru
mh
ans
l, C
. L.,
& K
ess
ler,
E. J
. (1
982
).
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Prevalence model of key profiles
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?
major key minor key
Lerdahl’s “basic pitch space”for the key of C major – after Deutsch & Feroe
level a C
level b C G
level c C E G
level d C D E F G A B
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
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
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
1
3
5
7
chroma
(a) C major
Pmo88
K&K82
C D E F G A B
1
3
5
7
-1 1 2
(b) C minor
avera
ge r
ati
ng
(K
&K
82
)
pc-
weig
ht/
3 (
Pm
o8
8)
C D E F G A B
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.).
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 ismore commonmore 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 cadenceprevalence 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 minor sus dim totalC 14+3+1=18 4+7+1=12 1+0+0=1 0+0+0=0 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
num
ber
of o
ccur
renc
es
Prevalence of 2-chord progressions
rising P4
falling P4
rising 3rd
falling 3rd
rising M2
falling M2
total
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 5 0 0 1 0 27
total 150 45 3 24 17 5 244Eb
erle
in, R
. (1
994
). D
ie E
nts
tehu
ng
de
r to
nal
en
Kla
ng
syn
tax
(pp.
422
-42
3). F
rank
furt
: Pe
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La
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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
Wanted! Experts in different European countries
ESF Exploratory Workshop
“Evolution of Western tonal syntax”• historians• theorists• computer scientists• psychologists
