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Estimating historical changes in
consonance by counting prepared and
unprepared dissonances in musical scores
Richard Parncutt and Fabio Kaiser
Centre for Systematic Musicology, University of Graz, Austria
Craig Sapp
CCARH, Department of Music, Stanford University
Internat. Conference on Music Perception and Cognition
Thessaloniki, Greece
23-30 July 2012
SysMus Graz
How did people experience consonance
and dissonance (C/D) in the Middle Ages?
An interdisciplinary question
Orientation
Humanities
Sciences
People
Music history
Music psychology
Information Music theory
Music computing
This is a job for…
ystematic musicology!
Cognition of Early Polyphony
Shoppingverse.de
ESF Exploratory Workshop, Graz 2012
 Special issue of JIMS
ed. Barbara Tillmann & Frans Wiering
Preparation of dissonance
The dissonant m7
is prepared
The dissonant m7
is unprepared
 Sounds less
 Sounds more
dissonant
dissonant
Psychological explanations:
Stream segregation reduces dissonance (Wright & Bregman, 1987)
Roughess depends on relative amplitude (Terhardt,1974)
Hypothesis 1
We can learn about C/D from
statistical analysis of scores
Assumptions:
• Consonant sonorities are more prevalent
• Composers tend to prepare dissonant
sonorities (more so than consonant)
head-fi.org
 Indirect (implicit, unbiased?) measures of C/D
 Measures of experience of dead participants
Hypothesis 2
The C/D of a sonority has
three main psychological
components:
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• Roughness (peripheral)
• Harmonicity (central)
• Familiarity (central)
C/D part 1: Roughness
consonance
Origin: Inner ear (Helmholtz, Plomp…)
Interference between nearby partials (<1CBW)
1.5
1
0.5
dissonance
0
1
2
3
4
5
6
-0.5
Interval class (semitones) 
-1
-1.5
Huron, D. (1994). Interval-class content in equally tempered pitch-class sets:
Common scales exhibit optimum tonal consonance. Music Perception, 11, 289-305.
The harmonicity of an interval or chord
•
•
“Harmonic”= “Similar to harmonic series”
No standard quantitative formulation
Seems to explain why…
• Major triad more consonant/frequent than minor
• Triads/dyads in root position more cons/freq than inversion
• E.g. dissonance of P4 is due to harmonicity not roughness
www.unc.edu/~chapman3/westernscale.html
C/D part 2: Harmonicity
Harmonicity or periodicity?
Spectral versus temporal representations of complex sounds
• Mathematically equivalent
• Both present at different places on auditory pathways
Which correspond to conscious experience? (How?)
Mind-body problem is unsolved!
Careful scientific approach: Don’t jump to conclusions!
Focus on parameters whose relationship to conscious
experience has been clarified by previous research:
1. Physical attributes of signal
2. Measures of experienced pitch (standard empirical methods)
NOT Physiological measures
 C/D depends on harmonicity in the sense of
experienced patterns among experienced partials
C/D part 3: Familiarity
Scale-step stability is learned from musical experience (Krumhansl & Kessler, 1982)
but ultimately depends on pitch-salience profile of tonic triad (Parncutt, 1988, 2011)
Krumhansl, C. L., & Kessler, E. J. (1982). Tracing dynamic changes in perceived tonal
organization in a spatial representation of musical keys. Psychol Rev, 89, 334-368.
Parncutt, R. (2011). The tonic as triad: Key profiles as pitch salience profiles of tonic
triads. Music Perception, 28, 333-365.
Sample of music for this study
•
•
•
•
Representative scores from four centuries
Vocal polyphony
Mainly sacred, some secular
Mainly 4 parts, sometimes 3, 5, 6, or 8
Internet sources
•
•
•
•
•
Kern scores
Choral Wiki
PMFC (Polyphonic Music of the Fourteenth century)
Elvis (Electronic Locator of Vertical Interval Successions)
CPDL.org (free choral music)
13th-Century sources
Perotin (1150/65 – 1200/25, French)
Magnum Liber Organi, Viderunt omnes, Sederunt, Mors
Alfonso el Sabio (1221-1284, Spanish)
Virgen Santa Maria, Cantigas de Santa Maria
Adam de la Halle (1250-1310, French)
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 amourete, 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
Montpellier Codex (1250-1300, Anonymus, French)
#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
14th-Century sources
Guillaume de Machaut (1300-1377, French)
Messe de nostre dame (Kyrie and Gloria), Hoquetus David, Comment
puet on mieus dire, De toutes Flours
Landini, Francesco (1325-1397, Italian)
madrigal from Squarcialupi Codex, Deh! dimmi tu, A le sandra lo
spirto, Cara mi donna, Quanto piu caro fay, Si dolce non sono
Johannes Ciconia (c.1335 or c.1370, French)
O felix templum jubila, Petrum Marcellum venetum - O Petre antistes
inclite, O Padua, sidus praeclarum, Venetie mundi splendor, Gloria
Philippe de Vitry (1291 – 1361, French)
Lugentium siccentur, Rex quem metrorum, Virtutibus laudabilis, Vos
Qui Admiramini Gratissima virginis species
Jacopo da Bologna (1340 - 1386, Italian)
Aquila Altera, I Senti Za Como Larcho Damore, In Verde Prato
Magister Andreas: Sanctus
Egardus: Gloria
15th-Century sources
Guillaume Dufay (1397 – 1474, Franco-Flemish)
O tres piteulx/Omnes amici, Missa Ave Regina Coelorum,
Missa Se la face ay pale, Adieu ces bons Vins
John Dunstaple (1390 - 1453)
Veni Creator, Descendi in ortum meum, Veni Sancte Spiritus
Johannes Ockeghem (1410/30 - 1497)
Missa l’homme armé (Kyrie)
Jacob Obrecht (1450 - 1505)
Parce Domine
Heinrich Isaac (1450-1517)
A la battaglia, Innsbruck ich muss dich lassen
Guillaume le Rouge (1450-1465, Netherlands)
Missa Soyez amprantiz, Stella celi (So ys emprentid)
Simon de Insula (fl. c.1450–60, English or French)
Missa O admirabile
16th-Century sources
Orlando de Lassus (1532 - 1594)
Matona mia cara, Salve regina mater
Giovanni Pierluigi da Palestrina (1514/15 - 1594)
Dies sanctificatus, Adoramus Te, Hodie Christus Natus Est, Missa Brevis
(Credo), Missa Aeterna Christi munera
Josquin Desprez (1450/55 - 1521)
Ave Maria, Missa l’homme armé, Salve Regina, Missa Pange lingua, El grillo,
Nymphes des Bois a 5, Tulerunt Dominum meum
William Byrd (1540 – 1623)
Ave verum corpus, Ne irascaris domine
Giovanni Gabrieli (1555-1612)
O Magnum Mysterium, Plaudite
Andrea Gabrieli (1532-1585): Musica tolta da i madrigali di Claudio
Monteverdi, e d'altri autori, e fatta spirituale; Bonum est et suaue; Ne confide
Counting pc sets: Procedure
No. of voices varies from 1 to 8
At every onset in any voice: count
number of pcs in vertical sonority
If 2: count a “dyad”
If 3: count a “triad”
Pitch-class set terminology
6 pc-sets with cardinality 2:
• 2-1 = m2, M7, m9 etc.
• 2-2 = M2, m7, M9 etc.
Intervallic inversion (within octave): capital letters
• 2-1A = m2, m9 etc.
• 2-1B = M7, M14 etc.
Chordal inversion (diff. tone in bass): small letters
• 3-11A = minor triad
• 3-11Aa = root-position minor triad
Dyads
13th-Century sample
Plus all transpositions, inversions and compounds
unprep
prep
Centuries:
13 14
15 16
Dyads
unprep
prep
Centuries:
13 14
15 16
2-1 (m2, M7)
Both are always dissonant; m2 more so (roughness)
unprep
prep
Centuries:
13 14
15 16
2-2 (M2, m7)
both are always dissonant
unprep
prep
Centuries:
13 14
15 16
2-3 (m3, M6)
…historical change from dissonant to intermediate
…smaller interval (≈1CBWrough) became less dissonant
unprep
prep
Centuries:
13 14
15 16
2-4 (M3, m6)
…changed from dissonant to intermediate
……M3 was always more consonant than m6
unprep
prep
Centuries:
13 14
15 16
2-5 (P4, P5)
…P5 always more consonant than P4
…clearly cons; later, unprep. P5 avoided (reduce fusion?)
unprep
prep
Centuries:
13 14
15 16
2-6 (A4, d5)
…both clearly diss, but A4 more frequent - can resolve
in 2 ways (to M3 or P5); d5 only to M3 (not P4)
unprep
prep
Triads
13th-Century sample
Plus all transpositions, inversions and compounds
unprep
prep
Centuries:
13 14
15 16
Triads: Samples from 4 centuries
unprep
prep
3-1 (012, e.g. CC#D)
That‘s a semitone cluster never happened until 20th C.
Centuries:
13 14
15 16
3-2 (013, 023, e.g. CDbEb, CDEb)
clearly diss; later, unprep. consistently avoided
023 preferably in “root position”
unprep
prep
Centuries:
13 14
15 16
3-3 (014, 034, e.g. CDbE, CD#E)
not enough data
unprep
prep
Centuries:
13 14
15 16
3-4 (015, 045. e.g. CDbF, CEF)
unprep
prep
clearly diss; not enough data in 13th
later, prep. 3-4Ab & 3-4Ba more common (part of M7 chord)
Centuries:
13 14
15 16
3-5 (016, 056, e.g. CDbGb, CFGb)
clearly diss; most completely avoided in 16th
3-5Ac & 3-5Bb more common (outer voices span P5)
unprep
prep
Centuries:
13 14
15 16
3-6 (024, e.g. CDE)
clearly diss; root position most common
(13th: not enough data)
unprep
prep
Centuries:
13 14
15 16
3-7 (025, 035, e.g. CDF, CEbF)
clearly diss, but not as diss as other chords
earlier, p5 above bass important (3-7Ac, 3-7Bc)
later, m7 above bass also important (3-7Ab)  m7 chord
unprep
prep
Centuries:
13 14
15 16
3-8 (026, 046, e.g. CDF#, CEF#)
clearly diss
3-8Ab: early Mm7 chord
unprep
prep
Centuries:
13 14
15 16
3-9 (027, e.g. CDG sus)
clearly diss
“sus4” (3-9c) is more common in 13th & 16th (why?)
unprep
prep
Centuries:
13 14
15 16
3-10 (036, e.g. CEbGb dim)
clearly diss, but not very diss; 1st inv most cons, as
expected; 13th C: too few data
unprep
prep
Centuries:
13 14
15 16
3-11 (037, 047 = min, maj)
more major than minor, root position than inversion
exception: minor 1st inversion in 14th
unprep
prep
Centuries:
13 14
15 16
3-12 (048, aug)
not enough data
unprep
prep
Evidence for 3 components of C/D
Roughness
• Explains most prevalence data in 13th-16th centuries
• Gradually less important  irrelevant for modern listeners?
(McDermott et al., 2010: McLaughlan et al., sub.)
Harmonicity
• Explains dissonance of P4, consonance of triads incl. P5,
preference for root position triads
Familiarity
• Moderate dissonances  more consonant
• Extreme dissonances  more avoided (expertise effect?)
Acknowledgment
Andreas Gaich
Student assistant
Centre for Systematic Musicology
Uni Graz, Austria
Literature
Fazio, R. H., & Olson, M. A. (2003). Implicit measures in
social cognition research: Their meaning and use.
Annual Review of Psychology, 54, 297-327.
McDermott, J. H., Lehr, A. J., & Oxenham, A. J. (2010).
Individual differences reveal the basis of consonance.
Current Biology, 20, 1035–1041.
McLachlan, N. M., Marco, D. J. T. & Wilson, S. J.
Consonance and pitch. Under review.
Parncutt, R., & Hair, G. (2011). Consonance and
dissonance in theory and psychology: Disentangling
dissonant dichotomies. Journal of Interdisciplinary
Music Studies, 5 (2), 119-166.

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