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

1. 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 • ISPS, 28-31 August 2013, Vienna • International Symposium on Performance Science SysMus Graz

2. 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.

3. BoethiusItalian philosopher, early 6th century “But since the netesynemmenon to the mese (3,456 to 4,608) holds a sesquitertian ratio -- that is, a diatessaron -- whereas the trite synemmenon to the netesynemmenon (4,374 to 3,456) holds the ratio of two tones....”

4. The major third interval (M3)  perceptualcategory • “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

5. The major scale in 3 tuning systemsratios and cents *12ET = 12-tone equally-temperament

6. 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

7. Strange ideas of ratio theorists

8. Pythagoreanssince 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? ;-)

9. 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” Itinerariummentis in Deum, II, 7

10. Giovanni Battista BenedettiItalian 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)

11. Johannes KeplerGerman 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?)

12. Gottfried Wilhelm Leibniz German mathematicianandphilosopher (1646-1716) Consonance is about subconscious counting “Die Freude, die uns die Musikmacht, beruht auf unbewusstemZählen.” “Musikist die verstecktemathematischeTätigkeit der Seele, die sichnichtdessenbewusstist, dasssierechnet.” (Letters)

13. 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 theoriaemusicae ex certissimisharmoniaeprincipiisdilucideexpositae  (1731) (A attempt at a new theory of music, exposed in all clearness according to the most well-founded principles of harmony)

14. 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: Yagotta feel the vibrations.“ How equal temperament ruined harmony (and why you should care). London: Norton, 2007 (pp. 28-29)

15. Kurt HaiderInstitut für Musiktheorie und harmonikale Grundlagenforschung, Wien Ratioscanexplainalmosteverything • harmonikaleGrundlagenforschung: 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

16. 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)

17. 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)

18. Opposition to ratio theory

19. 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.

20. 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

21. 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 theorizedwithratiostoo…)

22. Ratios in Western music theory1. Pythagoras (6th C. BC)  Boethius (6th C. AD) • Musical intervals areratios • Based on prime numbers 2 & 3 • Spiritual, cosmic, religious

23. Ratios in Western music theory2. 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?

24. Ratios in Western music theory3. 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

25. Ratios in Western music theory4. 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...

26. 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!

27. 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

28. Johanna Devaneywith Ichiro Fujinaga, Jon Wild, Peter Schubert, Michael Mandel • Participants: professional singers • Task: singan 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

29. 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

30. 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?

31. 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

32. 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 SystematischeMusikwissenschaft(1988)

33. Interval perception is not about ratios - itis 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

34. Categorical perception of musical intervalsBurns & Campbell (1994) Stimuli: Melodicintervals of complextones; all ¼ tones up to one octave Participants: Musicians Task: name the intervalusingregularintervalnames (semitones)

35. The earacquires relative pitchcategories…fromthedistributionofpitches in performedmusic 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”.

36. In real performance, Just and Pythagorean have no physical existence at all! Just Pythagorean • Bimodaldistribution • withtendencytoward • pure (M3 = 386 cents) or • Pythagorean (M3 = 408 cents) • Wenever find this! Normal distribution sd ≈ 20 cents + 1 sd = acceptabletuning + 2 sd = pitchcategory Wegenerally find this!

37. Doesthebrainhave a ratio-detectiondevice? Ifitdid, wemightexpect: • bimodalintervalperformanceandpreferencedistributions • lowtolerancetomistuningofharmonics in complextones • an evolutionarybasisforratiodetection In fact: • distributionsareunimodal • harmonicsmistunedby a quartertone orsemitone (!) are still perceivedaspartofthecomplex tone (Moore et al., 1985) • environmental interactiondepends on identificationofsoundsources via synchrony, harmonicity… (Bregman, 1990)

38. Forthepsychoacousticians: This is not a spectralapproach! Pitch is an experienceofthelistener– not a physicalorphysiologicalmeasure Pitch generallydepends on both temporal andspectralprocessing, whichareinextricablymixedandhidden in neuralnetworks.

39. 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)

40. “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

41. So whataboutquartertones? • Quartertones simplyliebetween half-tone steps • Like half-tones, theyarepitchcategories- not ratios. Non-western musictheories • Ratio theoriesexist in manymusictraditions • All areproblematicforthe same reasons

42. Microtonalcomposition • Intervalsare ALWAYS learned • ANY microtonalscalecanbelearned, but: A newscaleiseasiertolearnif • similartoexistingscales • roughlyequalsmallintervals (JND) • unequal larger intervals (asymmetry) Relevanceofratios • Approximate: yes (familiarharmonicseries; minimizeroughness) • Exact: no

43. Microtonalcomposition ETs thatmostcloselyapproximate simple ratioshave 5, 7, 12, 19, 31, 53 tones per octave Logical nextstepis 19ET: Cf. 12ET:

45. In thismusic, 19ET islike 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

46. 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

47. 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

48. 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

49. Imagine: A music theory without ratios We can explain the structure, beauty, power of music without ratios • But thereis a paradox: • Youhavetounderstandratios… • tounderstandintervals • torealisethatintervalsare not ratios