Transcript C/D
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 SysMus Graz Medieval and Renaissance Music Conference University of Birmingham, 3-6 July 2014
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 People Information
Humanities
Music history Music theory
Sciences
Music psychology 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
less dissonant
unprepared
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…)
Diatonicity
Diatonic scale is “overlearned” (Deutsch…)
Roughness
Peripheral (ear); innate (Helmholtz, Plomp…)
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 Inversion
#semitones 2
012 013 014 015 016 024 025 026 027 036 037 048 023 034 045 056 035 046 047
1 1 1 1 0 0 0 0 0 0 0 #tritones sum 0 2 0 1 0 1 0 1 1 2 0 0 0 0 1 1 0 0 1 1 0 0 0 0
C/D of interval classes
convergent evidence from different sources
Harmonicity
A measure of C/D?
Pc-set Inversion
# Fourths
012 013 014 015 016 024 025 026 027 036 037 048
0
023 034 045 056
0 0 1 0 0
035 046
1 0 1 0
047
1 0
Diatonicity
A measure of C/D?
Pc-set Inversion Diatonicity 012 013 014 015 016 024 025 026 027 036 037 048
0
023 034 045 056
2 0 2 1 3
035 046
4 1 5 1
047
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 17 th century
Claudio Monteverdi (1567-1643) Heinrich Schütz (1585-1672) Adriano Banchieri (1568-1634) Girolamo Frescobaldi (1583-1643) Ruggero Giovannelli (1560-1625)
18 th 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 19 th 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 • • • Jacopo da Bologna Aquila Altera I Senti Za Como Larcho Damore In Verde Prato • • • • • 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 Sanctus • Egardus Gloria • 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) Runner-up: ic5 (P4/P5)
Roughness 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”: “Entropy”: defined in Parncutt (1988) 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 13 th 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
13 th C prepared: 027, 037, 047, 035, 025, … 0247, 0257, 0358, 0357 14 th -16 th : more 047 and 037, less diversity 13 th …16 th : 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