Intervals as distances, not ratios: Evidence from tuning and intonation

SysMus Graz

Richard Parncutt

Centre for Systematic Musicology University of Graz, Austria

Graham Hair

Department of Contemporary Arts, Manchester Metropolitan University, UK 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

12ET* Pythag orean

200 8:9 204

Just**

8:9 204

^3 ^4

400 500 64:81 3:4

408

498 4:5

386

3:4 498

^5

700 2:3 702 2:3 702

^6 ^7 ^8

900 1100 1200 16:27 128:243 1:2

906 1110

1200 3:5

884

8:15

1088

1:2 1200 *

12ET

= 12-tone equally-temperament

Most intervals have 2 ratios

Would the real ratio please stand up?

interval note chr. pure/just Pythagorean P1 C 0 1:1 1:1 m2 C# M2 D 1 16:15 256:243 2 9:8 or 9:10 9:8 m3 D# M3 E 3 6:5 32:27 4 5:4 81:64 P4 F TT F# 5 4:3 4:3 6 45:32 729:512 P5 G 7 3:2 3:2 m6 G# 8 8:5 128:81 M6 A 9 5:3 27:16 m7 A# M7 10 9:5 or 7:4 16:9 B 11 15:8 243:128 P8 C 12 2:1 2:1

Strange ideas of ratio theorists

Pythagoreans

since 6 th 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 semi major 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”

(4 th 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 non modulating 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; 4 th ed. transl. A. J. Ellis (but Helmholtz theorized with ratios too…)

Ratios in Western music theory

1. Pythagoras (6 th 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 (18 th -19 th 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. 20 th -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 f 0 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 18 th 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  Experimentation with new ideas  crisis intellectual battles • •

Features of change

Old and new are incommensurable Shifts are more dramatic in previously stable disciplines

Examples

Physics: Classical mechanics  Psychology: Behaviorism  relativity and quantum mechanics 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 18 th and 19 th 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

with tendency toward • pure (M3 = 386 cents) or • Pythagorean (M3 = 408 cents)

We never find this!

Normal distribution

sd ≈ 20 cents + 1 sd = acceptable tuning + 2 sd = pitch category

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

0

C

Logical next step is 19ET:

1

C#

2

Db

3

D

4

D#

5

Eb

6

E

7

E# Fb

8

F

9

F#

10 11 12 13 14 15 16 17 18 19

Gb G G# Ab A A# Bb B B# Cb C

0

C

Cf. 12ET:

1

C# Db

2

D

3

D# Eb

4

E

5

F

6

F# Gb

7

G

8

G# Ab

9

A

10

A# Bb

11

B

12

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