Intervals as distances, not ratios:Evidence from tuning and

intonation

Richard ParncuttCentre for Systematic Musicology

University of Graz, Austria

Graham HairDepartment of Contemporary Arts,

Manchester Metropolitan University, UK

ISPS, 28-31 August 2013, ViennaInternational Symposium on Performance ScienceSysMus Graz

AbstractMany 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.

BoethiusItalian 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)

“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 difference81/80 = 22 cents = “syntonic comma”

Much smaller than category width of M3 = 100 cents

perceptual category

The major scale in 3 tuning systemsratios and cents

Scale step ^2 ^3 ^4 ^5 ^6 ^7 ^8

12ET* 200 400 500 700 900 1100 1200

Pythag-orean

8:9 64:81 3:4 2:3 16:27 128:243 1:2204 408 498 702 906 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 note chr. pure/just Pythagorean

P1 C 0 1:1 1:1

m2 C# 1 16:15 256:243

M2 D 2 9:8 or 9:10 9:8

m3 D# 3 6:5 32:27

M3 E 4 5:4 81:64

P4 F 5 4:3 4:3

TT F# 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# 10 9:5 or 7:4 16:9

M7 B 11 15:8 243:128

P8 C 12 2:1 2:1

Strange ideas of ratio theorists

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 spheresPlanets and stars move to these ratios

a cosmic symphony!

Pythagoras could hear it! Did he have tinnitus? ;-)

Pythagoreanssince 6th Century BC

Saint BonaventureItalian 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 KeplerGerman mathematician, astronomer (1571-1630)

Music helps you understand the solar systemThird 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?)

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)

Gottfried Wilhelm Leibniz German mathematician and philosopher (1646-1716)

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. DuffinDept 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 HaiderInstitut 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 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 HelmholtzGerman 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…)

• Musical intervals are ratios• Based on prime numbers 2 & 3• Spiritual, cosmic, religious

Ratios in Western music theory1. Pythagoras (6th C. BC) Boethius (6th C. AD)

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 theory2. Renaissance theorists

New concept of musical intervals audible relationships between partials in harmonic complex tones

Consonance based on• harmonicity (Rameau, Stumpf)• roughness (Helmholtz)

Shift of emphasisfrom maths to physics, physiology, psychology

Ratios in Western music theory3. Scientific revolution (18th-19th C.)

• ≈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...

Ratios in Western music theory4. 20th-C. experiments on intonation in music

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 Devaneywith 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 registerfor 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 seriesall are stretched - physics & perception! 10 centsM2 = 8:9 (204 cents) or 9:10 (182 cents): 20 cents

Structure Must tune all intervals between all scale steps!

ExpressionExpressive 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?

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

ExamplesPhysics: Classical mechanics relativity and quantum mechanicsPsychology: Behaviorism cognitivismMusic theory: Math & notation performance & perception

Thomas Kuhn’s “paradigm shift”or scientific revolution

Carl DahlhausGerman 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 perceptionColor

e.g. range of wavelengths of the color red– “nature”:

• physiology of rods and cones– “nurture”:

• mapping between color words and light spectra

Speech soundse.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 intervalsBurns & 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 GIn 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!

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

We generally find this!

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

We never find this!

Just Pythagorean

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 tones3. an evolutionary basis for ratio detection

In fact:4. distributions are unimodal5. harmonics mistuned by a quartertone or semitone (!) are still

perceived as part of the complex tone (Moore et al., 1985)6. environmental interaction depends on identification of sound

sources via synchrony, harmonicity… (Bregman, 1990)

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.

For the psychoacousticians:

This is not a spectral approach!

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 skillsCons:• 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

• 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:

Cf. 12ET:

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

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

Microtonal composition

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

ConclusionsMusical 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