The perceptual history of
consonance and dissonance
Counting vertical pitch-class sets in vocal polyphony
Richard Parncutt, Andreas Fuchs, Andreas Gaich, Fabio Kaiser
Centre for Systematic Musicology, University of Graz, Austria
Medieval and Renaissance Music Conference
University of Birmingham, 3-6 July 2014
SysMus Graz
Abstract
How were consonance and dissonance perceived in early polyphony? We
are complementing existing theory by counting vertical sets of three pitch
classes. Our sample includes works attributed to Perotin, Savio, Halle
(13th century); Machaut, Landini, Ciconia, Magister Andreas (14th);
Dufay, Dunstable, Ockeghem, Obrecht, Isaac, Le Rouge, de Insula (15th);
Lassus, Palestrina, Desprez, Byrd, Gabrieli (16th). We use electronic scores
available in the internet; we have not systematically addressed ficta. With
the Humdrum Toolkit we count unprepared sonorities (tones beginning
simultaneously) and prepared sonorities (one or more ties). As expected,
the most consonant pc-sets in the 14th-16th centuries correspond to
today’s major, minor, suspended and diminished triads in that order, plus
025/035 (e.g. CDF, DFG). With time, major and minor became relatively
more common. Suspended (057) and 025/035 were common in the 13th.
The data allow us to test psychological models of consonance and
dissonance based on smoothness (lack of beating), harmonicity (similarity
to harmonic series), diatonicness (scale belongingness) and evenness
(spacing around chroma cycle). All four predictions correlate with mean
results for the 13th-14th century, but only roughness and harmonicity
correlate with 15th and 16h separately, consistent with gradually
increasingly sensitivity to roughness and harmonicity.
Consonance and dissonance (C/D)
in early music
An interdisciplinary question!
Orientation
Humanities
Sciences
People
Music history
Music psychology
Information
Music theory
Music computing
Perotin: Mors
“Gm/A”: unprepared!
pc set:
GABbD
Perotin: Viderunt Omnes
“Cadd9/G”
pc set:
CDEG
Perotin: Viderunt Omnes
“C7/G” – unprepared!
pc set:
EGBbC
Alfonso el Sabio:
Santa Maria, strela do dia
“D7/C” – unprepared!
pc set:
F#ACD
Alfonso el Sabio:
Santa Maria, strela do dia
D/E ... ??? ... E7/sus4 ... D7sus/C ... ???
pc sets:
DEF#A
ABD ABDE
GACD
EGA
Dissonant sonorities in early music
How should we approach them?
•
•
•
•
Are pc-set labels appropriate?
Are dissonances accidental or deliberate?
Are they products of voice-leading rules?
Should we look at individual examples or
do statistics?
• Is frequency of occurrence (prevalence) a
useful measure of their consonance?
Why do statistical analysis?
Assumptions:
1. “Consonant” sonorities are more
prevalent
2. Composers are more likely to “prepare”
dissonant than consonant sonorities
 Two indirect measures of C/D
Preparation of dissonance
prepared
unprepared
less dissonant
more dissonant
Psychological explanations
Stream segregation reduces dissonance (Wright & Bregman, 1987)
Roughess depends on relative amplitude (Terhardt,1974)
Pitch-class sets – Tn-types
John Rahn (1980): Basic atonal theory
Intervallic inversion
037= minor; 047 = major
 same pc-set “3-11”
 different Tn-types “3-11A”, “3-11B”
There are…
19 Tn-sets of cardinality 3 (012, 013...)
43? Tn-sets of cardinality 4 (0123, 0124...)
Familiar examples
036 = dim, 048 = aug, 027 = sus (=702=057)
025/035: no name
All Tn-types of cardinality 3
What is the C/D of a Tn-set?
Three approximate measures
1. Consensus among music theorists (past and present)
2. Prevalence in musical scores
3. Psychological predictions
Here, we test 3 by comparing predictions with 2.
Psychological theories of C/D
of sonorities (vertical C/D)
Familiarity
Completely learned
(Cazden, Krumhansl…)
Roughness
Peripheral (ear); innate
(Helmholtz, Plomp…)
Diatonicity
Diatonic scale is “overlearned”
(Deutsch…)
Harmonicity (fusion)
Central (brain); partly learned
(Stumpf, Terhardt…)
The interval vector
Interval-based C/D models
Minor and minor triads have the same
interval vector: <001110>
i.e. both chords have:
0 m2s, 0 M2s, 1 m3s, 1M3s, 1 P4s, 0 TTs
(plus intervallic inversions)
Assumption
The C/D of a pc-set depends approximately
on its interval vector
(cf. interval-based Renaissance theory)
Roughness
A measure of C/D?
Pc-set
012 013 014 015 016 024 025 026 027 036 037 048
Inversion
023 034 045 056
035 046
047
#semitones 2
1
1
1
1
0
0
0
0
0
0
0
#tritones
0
0
0
0
1
0
0
1
0
1
0
0
sum
2
1
1
1
2
0
0
1
0
1
0
0
C/D of interval classes
convergent evidence from different sources
Harmonicity
A measure of C/D?
Pc-set
012 013 014 015 016 024 025 026 027 036 037 048
Inversion
# Fourths
023 034 045 056
0
0
0
1
0
035 046
0
1
0
047
1
0
1
0
Diatonicity
A measure of C/D?
Pc-set
012 013 014 015 016 024 025 026 027 036 037 048
Inversion
Diatonicity
023 034 045 056
0
2
0
2
1
035 046
3
4
1
047
5
1
3
0
Possible justifications:
• Notation: Practical limitations of diatonic notation system
• Psychology: Deep familiarity of diatonic scale since antiquity
Music database
• Vocal polyphony
• Mainly sacred, some secular
• Mainly 4 parts; sometimes 3, 5, 6, or 8
Sources
•
•
•
•
•
•
Kern scores
CPDL (Choral Public Domain Library)
Elvis (Electronic Locator of Vertical Interval Successions)
PMFC (Polyphonic Music of the Fourteenth Century)
musicalion.com
IMSLP.org
“Composers” in database
13th century
Perotin (1150/65-1200/25, French)
Alfonso el Sabio (1221-1284, Spanish)
Adam de la Halle (1250-1310, French)
Montpellier Codex (1250-1300, French)
14th century
Guillaume de Machaut (1300-1377, French)
Landini, Francesco (1325-1397, Italian)
Johannes Ciconia (c.1335 or c.1370, French)
Philippe de Vitry (1291-1361, French)
Jacopo da Bologna (1340-1386, Italian)
Egardus (fl. c. 1370 – after 1400, Flemish)
Composers in database
15th century
Guillaume Dufay (1397-1474, Franco-Flemish)
John Dunstaple (1390-1453, English)
Johannes Ockeghem (1410/30-1497, Franco-Flemish)
Jacob Obrecht (1450-1505, Flemish)
Heinrich Isaac (1450-1517, Franco-Flemish)
Guillaume le Rouge (fl. 1450-1465, Netherlands)
Simon de Insula (fl. c.1450-60, English or French)
16th century
Orlando de Lassus (1532-1594)
Giovanni Pierluigi da Palestrina (1514/15-1594)
Josquin Desprez (1450/55-1521)
William Byrd (1540-1623)
Giovanni Gabrieli (1555-1612)
Andrea Gabrieli (1532-1585)
Composers in database
17th century
Claudio Monteverdi (1567-1643)
Heinrich Schütz (1585-1672)
Adriano Banchieri (1568-1634)
Girolamo Frescobaldi (1583-1643)
Ruggero Giovannelli (1560-1625)
18th century
Johann Joseph Fux (1660-1741)
Georg Philipp Telemann (1681-1767)
Johann Sebastian Bach (1685-1750)
Georg Friedrich Händel (1685-1759)
Giovanni Battista Pergolesi (1710-1736)
Niccolo Jommelli (1714-1736)
Christof Willibald Gluck (1714-1787)
Carl Philipp Emanuel Bach (1714-1788)
Johann Friedrich Doles (1715-1797)
Joseph Haydn (1732-1809)
Dmitri Stepanowitch Bortniansky (1751-1825)
Wolfgang Amadeus Mozart (1756-1791)
Composers in database
19th century
Ludwig van Beethoven (1770-1827)
Franz Schubert (1797-1828)
Felix Mendelssohn Bartholdy (1809-1847)
Robert Schumann (1810-1856)
Charles Gounod (1818-1893)
Anton Bruckner (1824-1896)
Robert Lowry (1826-1899)
Johannes Brahms (1833-1897)
Josef Gabriel Rheinberger (1839-1901)
Peter Iljitsch Tschaikowsky (1840-1893)
Antonin Dvorak (1841-1904)
Nikolai Rimski-Korsakow (1844-1908)
13th-C sample in database
Perotin
•
•
•
•
Magnum Liber Organi
Viderunt omnes
Sederunt
Mors
Montpellier Codex
•
•
•
•
#66 Mater Dei – Mater Virgo – Eius
#78 Dieus Mout me fet sovent fremir
#158 Mal d'amors presnes m'amie
#319 On parole – A Paris – Frese
nouvele
• #339 Alle psallite cum luya
Adam de la Halle
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Fi Maris de vostre Amour
Je muir je muir d'amourete
Li dous regars de ma dame
Hareu li maus d'amer M'ochist
A dieu commant amouretes
Dame or sui trais
Amours et ma dame aussi
Or est Baiars en la pasture Hure
A jointes mains vous proi
He Diex quant verrai
Diex comment porroie
Trop desire aveoir
Bonne amourete
Tant con je vivrai
14th-C sample in database
Guillaume de Machaut
•
•
•
•
Messe de nostre dame (Kyrie, Gloria)
Hoquetus David
Comment puet on mieux dire
De toutes Flours
Francesco Landini
•
•
•
•
•
•
Squarcialupi Codex: madrigal
Deh! dimmi tu
A le sandra lo spirto
Cara mi donna
Quanto piu caro fay
Si dolce non sono
Johannes Ciconia
•
•
•
•
•
O felix templum jubila
Petrum Marcellum venetum
O Padua, sidus praeclarum
Venetie mundi splendor
Gloria
Philippe de Vitry
•
•
•
•
Lugentium siccentur
Rex quem metrorum
Virtutibus laudabilis
Vos Qui Admiramini Gratissima virginis
Magister Andreas
Jacopo da Bologna
• Sanctus
• Aquila Altera
• I Senti Za Como Larcho Damore
• In Verde Prato
• Gloria
Egardus
Solage
• Fumeux fume par fumee
Size of database in each century
Counting “chords”: Method
Database in Kern format
Analyse using Humdrum Toolkit (Huron)
Count Tn-types of cardinality 3 and 4
Distinguish prepared from unprepared sonorities
Correlate counts with model predictions
All trichords 1200-1900
“Cases“: all trichords and tetrachords
Next slides: only the top ten trichords
The main 10 trichords
in 13th-C vocal polyphony
ignoring register, “inversion”, spacing, doubling...
Tn-type
All tetrachords 1300-1900
“Cases”: all trichords and tetrachords
Next slides: only the top ten
The main tetrachords, 13th century
Trichords before 1600
Trichords after 1600
Tetrachords before 1600
Tetrachords after 1600
Nb 17th-C
sample is
missing
1672-1700
Which is the best C/D model?
• Predict C/D using different models
• Compare with prevalence data
• Which model performs better?
• Implications for history of C/D?
Which interval class determines C/D?
Correlation coefficient between predictions and prevalence
UNprepared trichords and tetrachords
 Winner:
ic 1 (m2/M7)  Roughness
 Runner-up: ic5 (P4/P5)  Harmonicity
Comparison of roughness models
Correlation between predictions and prevalence
UNprepared trichords
 Roughness is generally a good predictor
 Winner: Huron model (the added complexity helps)
Comparison of roughness models
Correlation between predictions and prevalence
Prepared trichords
Conclusion: as before
Comparison of roughness models
Correlation between predictions and prevalence: tetrachords
Comparison of harmonicity models
Correlation between predictions and prevalence
UNprepared trichords
Again, the more complex models are better
Comparison of harmonicity models
Correlation between predictions and prevalence
Prepared trichords
Again, the more complex models are better
Comparison of pitch clarity models
Correlation between predictions and prevalence: trichords
All 3 measure the “peakedness” of the pc-salience profile (Parncutt, 1988).
“Salience”:
salience of most salient pc
“Root ambiguity”:
defined in Parncutt (1988)
“Entropy”:
equation from statistical mechanics.
Comparison of pitch clarity models
Correlation between predictions and prevalence: tetrachords
UNprepared
Prepared
Comparison of best models
Correlation between predictions and prevalence. trichords
In 13th C, all 3 are important. Later, roughness and harmonicity
are equally important; diatonicity becomes irrelevant
Comparison of best models
Correlation between predictions and prevalence: tetrachords
More complex, here diatonicity is the best predictor.
These are new calculations, we need to check them.
Specific conclusions
Main sonorities
• 13th C prepared: 027, 037, 047, 035, 025, …
0247, 0257, 0358, 0357
• 14th-16th: more 047 and 037, less diversity
• 13th…16th: More unprepared dissonances
Roughness and harmonicity
• Explain prevalence of triads in 13th-16th C
 prefer 5ths & avoid 2nds  major/minor
Familiarity
• Moderate dissonances  more common
• Extreme dissonances  less common
General conclusions
Vertical or horizontal?
Prevalence of sonorities in multi-voiced western
music is determined mainly by vertical C/D!
 not by voice-leading rules?
Psychological C/D-concepts
Vertical C/D is determined mainly by 3 factors:
roughness, harmonicity, familiarity
Future research
To explain C/D, we need humanities & sciences:
• history of music theory
• psychological and statistical studies