Download The demise of number ratios in music theory

Intervals as distances, not ratios:
Evidence from tuning and
intonation
Richard Parncutt
Centre for Systematic Musicology
University of Graz, Austria
Graham Hair
Department of Contemporary Arts,
Manchester Metropolitan University, UK
SysMus Graz
ISPS, 28-31 August 2013, Vienna
International Symposium on Performance Science
Abstract
Many music theorists and psychologists assume a direct link between musical
intervals and number ratios. But Pythagorean ratios (M3=61:84) involve
implausibly large numbers, and just-tuned music (M3=4:5) only works if scale
steps shift from one sonority to the next. We know of no empirical evidence
that the brain perceives musical intervals as frequency ratios. Modern
empirical studies show that performance intonation depends on octave
stretch, the solo-accompaniment relationship, emotion, temporal context,
tempo, and vibrato. Just intonation is occasionally approached in the special
case of slow tempo and no vibrato, but the reason is to minimize roughness
and beating - not to approach ratios. Theoretically, intonation is related to
consonance and dissonance, which depends on roughness, harmonicity,
familiarity, and local/global context. By composing and performing music in
19-tone equal temperament (19ET), the second author is investigating how
long it takes singers to learn to divide a P4 (505 cents) into eight roughly
equal steps of 63 cents, or a M2 (189 cents) into three; and whether the
resultant intonation is closer to 19ET or 12ET. Given that the average size of
an interval depends on both acoustics (nature) and culture (nurture), it may
be possible to establish a sustainable 19ET performance community.
Boethius
Italian philosopher, early 6th century
“But since the nete synemmenon to the
mese (3,456 to 4,608) holds a sesquitertian
ratio -- that is, a diatessaron -- whereas the
trite synemmenon to the nete synemmenon
(4,374 to 3,456) holds the ratio of two
tones....”
The major third interval (M3)
 perceptual
category
“Pythagorean tuning”
reflects motion tendencies (leading tone rises)
emphasizes difference between major and minor
“Just tuning”
minimizes beats between almost-coincident harmonics
- only if spectra are harmonic and steady (slow, non-vibrato)
The difference
81/80 = 22 cents = “syntonic comma”
Much smaller than category width of M3 = 100 cents
The major scale in 3 tuning systems
ratios and cents
Scale step ^2
^3
^4
^5
^6
^7
^8
12ET*
200
400
500
700
900
1100
1200
Pythagorean
8:9
204
64:81 3:4
408
498
2:3
702
16:27 128:243 1:2
906
1110
1200
Just**
8:9
4:5
3:4
2:3
3:5
8:15
1:2
204
386
498
702
884
1088
1200
*12ET = 12-tone equally-temperament
Most intervals have 2 ratios
Would the real ratio please stand up?
interval
P1
m2
M2
m3
M3
P4
TT
P5
m6
M6
m7
M7
P8
note
C
C#
D
D#
E
F
F#
G
G#
A
A#
B
C
chr.
0
1
2
3
4
5
6
7
8
9
10
11
12
pure/just Pythagorean
1:1
1:1
16:15
256:243
9:8 or 9:10
9:8
6:5
32:27
5:4
81:64
4:3
4:3
45:32
729:512
3:2
3:2
8:5
128:81
5:3
27:16
9:5 or 7:4
16:9
15:8
243:128
2:1
2:1
Strange ideas of ratio theorists
Pythagoreans
since 6th Century BC
The universe is number and music reflects it
Monochord mathematics
• first four numbers (tetraktys) are special (1+2+3+4=10)
• all intervals by multiplying and dividing these numbers
Music of the spheres
Planets and stars move to these ratios
 a cosmic symphony!
Pythagoras could hear it! Did he have tinnitus? ;-)
Saint Bonaventure
Italian medieval theologian and philosopher, 1221 – 1274
God is number
“Since all things are beautiful and to some
measure pleasing; and there is no beauty and
pleasure without proportion, and proportion is
found primarily in numbers; all things must have
numerical proportion. Consequently, number is
the principal exemplar in the mind of the Creator
and as such it is the principal trace that, in things,
leads to wisdom. Since this trace is extremely
clear to all and is closest to God, it … causes us to
know Him in all corporeal and sensible things”
Itinerarium mentis in Deum, II, 7
Giovanni Battista Benedetti
Italian mathematician, 1530 –1590
Consonance is all about waves
• sound consists of air waves or vibrations
• in the more consonant intervals the shorter, more
frequent waves concurred with the longer, more
frequent waves at regular intervals
(letter to Cipriano de Rore dated around 1563)
Johannes Kepler
German mathematician, astronomer (1571-1630)
Music helps you understand the solar system
Third law of planetary motion:
• The square of the orbital period of a planet is
directly proportional to the cube of the semimajor axis of its orbit.
Aims:
• understand the music of the spheres
• express planetary motion in music notation
(Did he have tinnitus too?)
Gottfried Wilhelm Leibniz
German mathematician and philosopher (1646-1716)
Consonance is about subconscious counting
“Die Freude, die uns die Musik macht, beruht auf
unbewusstem Zählen.”
“Musik ist die versteckte mathematische Tätigkeit
der Seele, die sich nicht dessen bewusst ist, dass sie
rechnet.”
(Letters)
Leonhard Euler
Swiss mathematician and physicist (1707-1783)
Consonance is based on numbers
“…the degree of softness of ratio 1:pq,
if p and q are prime numbers … is p+q-1."
Tentamen novae theoriae musicae ex certissimis
harmoniae principiis dilucide expositae (1731)
(A attempt at a new theory of music, exposed in
all clearness according to the most well-founded
principles of harmony)
Ross W. Duffin
Dept of Music, Case Western Reserve U, Cleveland OH
You can hear number ratios directly
“12ET major thirds are … the invisible elephant in our musical
system today. Nobody notices how awful the major thirds are. (…)
Asked about it, some people even claim to prefer the elephant.
(…) But I’m here to shake those people out of their cozy state of
denial. It’s the acoustics, baby: Ya gotta feel the vibrations.“
How equal temperament ruined harmony (and why you should
care). London: Norton, 2007 (pp. 28-29)
Kurt Haider
Institut für Musiktheorie und harmonikale
Grundlagenforschung, Wien
Ratios can explain almost everything
• harmonikale Grundlagenforschung: eine mathematische
Strukturwissenschaft (Pythagoreer, Platon, Neuplatoniker)
• seit Kepler: auch eine empirische Wissenschaft
• führt die Struktur der Naturgesetze auf ganzzahlige
Proportionen zurück
• durch die Intervallempfindung der ganzzahligen Proportionen
werden nun qualitative Parameter wie Form, Gestalt oder
Harmonie wieder Gegenstand der Wissenschaften
kurthaider.megalo.at/node/49
Clarence Barlow
composer of electroacoustic music
Ratios help you compose
“Harmonicity” of an interval depends on “digestibility”
of the numbers in its ratio (prime factors)
 Systematic enumeration of the most harmonic ratios
within an octave
1:1, 15:16, 9:10, 8:9, 7:8, 6:7, 27:32, 5:6, 4:5, 64:81, 7:9,
3:4, 20:27, 2:3, 9:14, 5:8, 3:5, 16:27, 7:12, 4:7, 9:16, 5:9,
8:15, 1:2.
Two essays on theory. Computer Music Journal, 11, 44-59 (1987)
Laurel Trainor
(Music) Psychologist, McMaster University
Infants process frequency ratios
“Effects of frequency ratio simplicity on infants'
and adults' processing of simultaneous pitch
intervals with component sine wave tones”
(abstract)
Effects of frequency ratio on infants' and adults'
discrimination of simultaneous intervals.
Journal of Experimental Psychology: Human
Perception and Performance, 23 (5), 1427-1438
(1997)
Opposition to ratio theory
Aristoxenus “Harmonics”
(4th Century BC; pupil of Aristotle)
There is more to music than number
“Mere knowledge of magnitudes does not enlighten one as to the
functions of the tetrachords, or of the notes, or of the differences
of the genera, or, briefly, the differences of simple and compound
intervals, or the distinction between modulating and nonmodulating scales, or the modes of melodic construction, or
indeed anything else of the kind.”
“we must not follow the harmonic theorists in their dense
diagrams which show as consecutive notes those which are
separated by the smallest intervals [but] try to find what intervals
the voice is by nature able to place in succession in a melody”
Macran, H. S. (1902). The harmonics of Aristoxenus. London: Oxford UP.
Jean-Philippe Rameau
French composer and theorist (1683 -1764)
First tried to explain triads using ratios:
• major triad
20:25:30 (4:5:6)
• Mm7
20:25:30:36
• minor triad
20:24:30 (10:12:15)
• m7
25:30:36:45
Later referred to the corps sonore:
Foundation of harmony is the intervals between the harmonic
partials of complex tones in the human environment
Hermann von Helmholtz
German physiologist and physicist, 1821-1894
“Even Keppler (sic.), a man of the deepest
scientific spirit, could not keep himself free from
imaginations of this kind … Nay, even in the most
recent times theorizing friends of music may be
found who will rather feast on arithmetical
mysticism than endeavor to hear out partial
tones” (p. 229).
On the Sensations of Tone as a Physiological Basis for the
Theory of Music, 1863; 4th ed. transl. A. J. Ellis
(but Helmholtz theorized with ratios too…)
Ratios in Western music theory
1. Pythagoras (6th C. BC)  Boethius (6th C. AD)
• Musical intervals are ratios
• Based on prime numbers 2 & 3
• Spiritual, cosmic, religious
Ratios in Western music theory
2. Renaissance theorists
Ratios can include factors of 5  “just”
• Ramos de Pareja (1482)
• Gioseffo Zarlino (1558)
• Giovanni Battista Benedetti (1585)
Can that explain the sonority of triads?
Ratios in Western music theory
3. Scientific revolution (18th-19th C.)
New concept of musical intervals
audible relationships between partials in
harmonic complex tones
Consonance based on
• harmonicity (Rameau, Stumpf)
• roughness (Helmholtz)
Shift of emphasis
from maths to physics, physiology, psychology
Ratios in Western music theory
4. 20th-C. experiments on intonation in music
• ≈12ET generally preferred
• Pythagorean preferred over just
(e.g. rising leading tones)
• Just intonation: only for slow, steady tones
with no vibrato
Many studies!
Ambrazevicius, Devaney, Duke, Fyk, Green, Hagerman
& Sundberg, O’Keefe, Loosen, Karrick, Kopiez,
Nickerson, Rakowski, Roberts & Matthews...
Just tuning: Impossible in practice
The fifth between ^2 (8:9) and ^6 (3:5) is not 2:3!
 Must constantly shift scale steps to stay in tune
 If you don’t like it when your choir gradually
goes flat or sharp, “just tuning” is not for you!
Renaissance choral polyphony
“Renaissance performers would have preferred solutions that
favor just intonation wherever and whenever possible …
deviations from it would have been momentary adjustments to
individual intervals, rather than wholesale adoption of
temperament schemes”
Ross W. Duffin (2006). Just Intonation in Renaissance Theory and
Practice. Music Theory Online
Johanna Devaney
with Ichiro Fujinaga, Jon Wild, Peter Schubert, Michael Mandel
Participants: professional singers
Task: sing an exercise by Benedetti
(1585) to illustrate pitch drift in just
Main results:
• Intonation close to 12ET
• Standard deviation of pitch is
typically 10 cents (!)
• Small drift in direction of
Benedetti’s prediction
Limited precision of “Ideal tuning”
Just noticeable difference in middle register
for simultaneous or successive pitches
under ideal conditions: 2 cents
Uncertainty in f0 of singing voice
vocal jitter of best non-vibrato voices: 3 cents
Intervals in the audible harmonic series
all are stretched - physics & perception! 10 cents
M2 = 8:9 (204 cents) or 9:10 (182 cents): 20 cents
Structure
Must tune all intervals between all scale steps!
Expression
Expressive intonation: 50 cents
So why do people sing in 12ET?
1. Familiarity with piano
2. Compromise between Pythagorean and Just
We don’t know which!
Point 1: since 18th Century
Point 2: for millennia!
 Gregorian chant: Pythagorean? Or 12ET?
 Renaissance polyphony: just? Or 12ET?
Thomas Kuhn’s “paradigm shift”
or scientific revolution
Paradigm
•
•
Entire landscape of knowledge and implications in a discipline
Universally accepted
Long process of change
•
•
Gradual increase in number of anomalies  crisis
Experimentation with new ideas  intellectual battles
Features of change
•
•
Old and new are incommensurable
Shifts are more dramatic in previously stable disciplines
Examples
Physics: Classical mechanics  relativity and quantum mechanics
Psychology: Behaviorism  cognitivism
Music theory: Math & notation  performance & perception
Carl Dahlhaus
German musicologist, 1928-1989
“Whereas in the ancient-medieval tradition number ratios were
considered to be the foundation or formal cause of consonance,
in modern acoustics and music theory they paled to an external
measure that says nothing about the essence of the matter. … In
the music theory of the 18th and 19th Centuries, the overtone
series is the natural archetype of the interval hierarchy upon
which rules of composition are founded. … The surrender of the
Platonic idea of number meant nothing less than the collapse of
the principle that had carried ancient and medieval music theory.”
C. Dahlhaus (Ed.), Einführung in die Systematische Musikwissenschaft (1988)
Interval perception is not about ratios - it is about
Categorical perception
Color
e.g. range of wavelengths of the color red
– “nature”:
• physiology of rods and cones
– “nurture”:
• mapping between color words and light spectra
Speech sounds
e.g. range of formant frequencies of vowel /a/
– “nature”:
• vocal tract resonances near 500 and1500 Hz
– “nurture”:
• learned formant frequencies of each vowel
Categorical perception of musical intervals
Burns & Campbell (1994)
Stimuli:
Melodic intervals
of complex tones;
all ¼ tones up to
one octave
Participants:
Musicians
Task:
name the interval
using regular
interval names
(semitones)
The ear acquires relative pitch categories
…from the distribution of pitches in performed music
F
F#/Db
G
In music, pitch varies on a continuous scale.
When some pitches are more common, categories crystalize.
These categories are the REAL ORIGINAL “musical intervals”.
In real performance, Just and Pythagorean
have no physical existence at all!
Just
Pythagorean
Bimodal distribution
Normal distribution
with tendency toward
• pure (M3 = 386 cents) or
• Pythagorean (M3 = 408 cents)
sd ≈ 20 cents
+ 1 sd = acceptable tuning
+ 2 sd = pitch category
We never find this!
We generally find this!
Does the brain have a
ratio-detection device?
If it did, we might expect:
1. bimodal interval performance and preference distributions
2. low tolerance to mistuning of harmonics in complex tones
3. an evolutionary basis for ratio detection
In fact:
1. distributions are unimodal
2. harmonics mistuned by a quartertone or semitone (!) are still
perceived as part of the complex tone (Moore et al., 1985)
3. environmental interaction depends on identification of
sound sources via synchrony, harmonicity… (Bregman, 1990)
For the psychoacousticians:
This is not a spectral approach!
Pitch is an experience of the listener – not a
physical or physiological measure
Pitch generally depends on both temporal and
spectral processing, which are inextricably
mixed and hidden in neural networks.
What influences intonation?
Real-time adjustment of frequency in performance
Perceptual effects (individual tones)
– octave stretch (small intervals compressed)
– beating of coinciding partials
Cognitive effects (musical structure)
– less stable tones are more variable in pitch
– rising implication of leading tone; major-minor distinction
Effects of performance
– solo versus accompaniment (soloists tend to play sharp)
– technical problems or limitations of instruments
Effects of interpretation
– intended emotion (e.g. tension-release)
– intended timbre (e.g. deep = low)
“Authentic” Renaissance polyphony?
Some choristers practise just tuning with
real-time computer feedback
Pros:
• improve intonation skills
Cons:
• suppress expression
• construct fake authenticity
• just tuning produces pitch drift
• we cannot separate timbre & tuning
So what about quartertones?
• Quartertones simply lie
between half-tone steps
• Like half-tones, they are
pitch categories - not ratios.
Non-western music theories
• Ratio theories exist in many
music traditions
• All are problematic for the
same reasons
Microtonal composition
• Intervals are ALWAYS learned
• ANY microtonal scale can be learned, but:
A new scale is easier to learn if
• similar to existing scales
• roughly equal small intervals (JND)
• unequal larger intervals (asymmetry)
Relevance of ratios
• Approximate: yes
(familiar harmonic series; minimize roughness)
• Exact: no
Microtonal composition
ETs that most closely approximate simple ratios
have 5, 7, 12, 19, 31, 53 tones per octave
Logical next step is 19ET:
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19
C
C#
Db
D
D#
Eb
E
E#
Fb
F
F#
Gb
G
G#
Ab
A
A#
Bb
B
B#
Cb
C
Cf. 12ET:
0
1
2
3
4
5
6
7
8
9
10
11
12
C
C#
Db
D
D#
Eb
E
F
F#
Gb
G
G#
Ab
A
A#
Bb
B
B
In this music, 19ET is like 12ET!
• 12 pitch categories – not 19
exact pitches
• based on a 7-tone diatonic
subset
• Tuning is more important for
anchor tones which may be
grouping, metrical, melodic,
harmonic, durational accents
Conclusions
Musical intervals are:
• cultural and psychological (not mathematical)
• approximate (categorical)
• learned from music (an aural tradition)
Exact musical interval size depends on:
•
•
•
•
•
musical familiarity
consonance: harmonicity, roughness
physical and perceptual stretch
structure and voice leading
emotion and expression
Origin of Western intervals
• Familiarity of harmonic complex tones in
speech (audible harmonic series)
• Prehistoric emergence of scales (= sets of
psychological pitch categories)
• Consonance of tone combinations in music
We don’t need ratios to explain…
Major and minor triads; harmonic cadences
• harmonicity, fusion, smoothness
Tuning of violin versus piano accompaniment
• octave stretch, leading tones, expression
Character of Renaissance choral music
• pitch structure, rhythm, timbre, expression
Ratio-based microtonality (e.g. Partch )
• Form, development, timbre
Music’s meaning, beauty, magic
• chains of associations
Imagine: A music theory without ratios
We can explain the
structure, beauty, power
of music without ratios
But there is a paradox:
You have to understand ratios…
• to understand intervals
• to realise that intervals are not ratios