Psychoacoustics 3 - Perception of Pitch and Intervals

1. Alexis Baskind Psychoacoustics 3 Perception of Pitch and Intervals Alexis Baskind, https://alexisbaskind.net

2. Alexis Baskind Psychoacoustics 3 - Perception of Pitch and Intervals Course series Fundamentals of acoustics for sound engineers and music producers Level undergraduate (Bachelor) Language English Revision January 2020 To cite this course Alexis Baskind, Psychoacoustics 3 - Perception of Pitch and Intervals, course material, license: Creative Commons BY-NC-SA. Full interactive version of this course with sound and video material, as well as more courses and material on https://alexisbaskind.net/teaching. Except where otherwise noted, content of this course material is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International License. Psychoacoustics 3 - Perception of Pitch and intervals

3. Alexis Baskind Outline 1. What is pitch 2. Concert Pitch 3. Perception of intervals 4. Harmonicity and consonance 5. Temperaments 6. Complex sounds Psychoacoustics 3 - Perception of Pitch and intervals

4. Alexis Baskind Pitch and frequency • Frequency is the physical number of cycles per second of a periodic signal, expressed in Hertz • The Fundamental Frequency is the lowest frequency of a periodic signal • Pitch is a subjective psychoacoustical attribute “Pitch is the attribute of auditory sensation in terms of which sounds may be ordered on a musical scale” (American Standards Association – 1960) => so: Pitch and Frequency are not equal! Fundamental frequency is indeed often closely correlated with pitch perception, but not always ! Psychoacoustics 3 - Perception of Pitch and intervals

5. Alexis Baskind Let’s consider the case of a train of impulses (easy to generate with a DAW): • When the frequency of the impulses is too low (< approx. 20 Hz, depending on the sound), the related pitch is not perceived • If the original impulse itself contains its own pitch (for instance if it results from a strong resonance), two pitches may or may not be perceived, depending on: – The frequency of the train of impulses – The relation between the frequency of the impulse, and the frequency at which it’s repeated Lower limit of pitch perception Psychoacoustics 3 - Perception of Pitch and intervals

6. Alexis Baskind Pitch height and chroma • Pitch chroma is the position of the pitch within an octave (i.e. “A”, “B”, “F#”,…) • Pitch height is related to the octave number Psychoacoustics 3 - Perception of Pitch and intervals (from http://acousticslab.org/psychoacoustics/PMFiles/Module06.htm) Chroma Height

8. Alexis Baskind History of Musical Pitch Reference • The reference pitch (nowadays often A 440Hz) has changed during history • Contrary to a common conception, it hasn't just increased constantly over time: 1640 Vienna Franciscan Organ A457.6 1699 Paris Opera A404 1711 John Shore's tuning fork, frequency A423.5. He invented the tuning fork 1714 Strasbourg Cathedral organ A391 1780 Stein's tuning fork A422.6 1800 Broadwood's C fork, 505.7, which is about half a semitone lower than that of today 1812 Paris Conservatoire A440, as modern pitch 1813 George Smart adopted for the Philharmonic Society the pitch of A423.3. 1836 Pleyel's Pianos A446 1879 Covent Garden Opera A450 1885 At an international exhibition in London a pitch of A452 was adopted 1896 Philharmonic pitch A439 1939 At an international conference A440 was adopted. (from http://www.piano-tuners.org/history/pitch.html) Psychoacoustics 3 - Perception of Pitch and intervals

9. Alexis Baskind History of Musical Pitch Reference • On top of that, those historical references only concern western music • Thus the pitch reference is highly cultural • Nowadays, the “A440” standard is far from being always respected:  Some musical styles (for instance blues) will use a lower pitch reference  Soloist in concerts often play with a higher pitch reference (442 Hz, or sometimes even 444 Hz) • Why has pitch reference constantly changed ? Mostly since it has a strong influence on the tone color  For instance, if the tuning gets higher, most instruments will sound brighter Psychoacoustics 3 - Perception of Pitch and intervals

11. Alexis Baskind • In practice, the perception of intervals depends on many parameters • First, are we talking about melodic or harmonic intervals ? – The Melodic interval is the perceived interval between two successive notes – The Harmonic interval is the perceived interval between two simultaneous notes Perception of intervals Psychoacoustics 3 - Perception of Pitch and intervals

12. Alexis Baskind • An interval corresponds to a frequency ratio: therefore, adding intervals means multiplying the ratios • For example: – if an octave corresponds to a ratio of 2:1 (not always true, see later) – if a perfect fifth corresponds to a ratio of 3:2 Then: – an octave + a fifth corresponds to a ratio of 3:1 – a perfect fourth (the complement of a fifth to an octave) corresponds to a ratio of 4:3 Perception of intervals Psychoacoustics 3 - Perception of Pitch and intervals

13. Alexis Baskind • Another common approximation: “a perceived octave corresponds to a frequency ratio of 2:1” Actually this is true only in the case of harmonic intervals, and not necessarily for melodic intervals • In practice, a melodic octave corresponds to a frequency ratio which is always greater than 2 => Octave Stretching phenomenon Melodic intervals Psychoacoustics 3 - Perception of Pitch and intervals

14. Alexis Baskind • The perception of two simultaneous tones depends on the frequency distance between them: 1. If the frequency distance between both tones is sufficient (more than about a third octave), both tones are perceived distinctly (except in some cases if they are in harmonic relation, see next part) 2. If both frequencies are very close to each other, only one amplitude-modulated pitch (=beat) is heard 3. In the transition region, roughness occurs. The bigger the frequency distance, the more obvious both pitches are heard distinctly Harmonic intervals Psychoacoustics 3 - Perception of Pitch and intervals

15. Alexis Baskind • When two tones are superimposed, the resulting waveform may be considered as unique sinusoid with an amplitude modulation Harmonic intervals - Beats Psychoacoustics 3 - Perception of Pitch and intervals Tone1 with frequency f1=100 Hz Tone2 with frequency f1=120 Hz Tone1 + Tone2 = tone with frequency 110 Hz, modulated at 20 Hz + =

16. Alexis Baskind • When two tones are superimposed, the resulting waveform may be considered as unique sinusoid with an amplitude modulation => The superposition is equivalent to a unique tone at the mean frequency, modulated in amplitude by a second sinusoid, which frequency is the difference between both original frequencies Harmonic intervals - Beats Psychoacoustics 3 - Perception of Pitch and intervals

17. Alexis Baskind Hearing cannot clearly distinguish two tones when the interval is smaller than its frequency resolution (called critical bandwidth) • This phenomenon is closely related to simultaneous masking (see lesson on perception of loudness): both tones interfere and mask each other in the hearing system Harmonic intervals – critical bands Psychoacoustics 3 - Perception of Pitch and intervals

19. Alexis Baskind Harmonicity • In acoustics, Harmonicity is the objective degree of periodicity of a sound • This notion has to be distinguished from the musical notion of harmony (i.e. the organisation of pitches in a given musical context) • Harmonicity is highly correlated with the organisation of the frequency components (i.e. fundamental + overtones) of an harmonic sound Psychoacoustics 3 - Perception of Pitch and intervals

20. Alexis Baskind The harmonic series • The harmonic series in music is based on the mathematical harmonic series (all frequencies are multiple of the fundamental frequency) Psychoacoustics 3 - Perception of Pitch and intervals (From J.Meyer, “Acoustics and the Performance of Music”) Example with C2 as fundamental:

21. Alexis Baskind • If two pure tones are in a harmonic relation, they may be fused or not, depending on their relative level and position on the harmonic series • This leads to the concept of consonance • Consonance is the ability of sounds to be fused as a single sound when they are mixed together (consonance means “to sound with”) • The basic idea of consonance is: two harmonic sounds in harmonic relation share several common frequencies through their overtones. The more common frequencies there is, the more both sounds fuse with each other The harmonic series Psychoacoustics 3 - Perception of Pitch and intervals

22. Alexis Baskind • According to an unverified legend (the “Pythagorean hammers”), the first historical definition of consonance was given by Pythagoras, and is based on the ratio between the frequencies of two tones with respect to their position in the harmonic series – If the ratio between frequencies is a fraction of two small integers, the interval is said to be consonant – As soon as the ratio becomes more and more complex, the sound is more and more dissonant – If the ratio cannot be expressed as a fraction of integers, the sound is dissonant => For example, an octave (ratio 2:1) is more consonant than a fifth (3:2) or a fourth (4:3), which is itself more consonant than a major third (5:4) or a minor sixth (8:5)… Consonance – pythagorean definition Psychoacoustics 3 - Perception of Pitch and intervals

23. Alexis Baskind Interval Frequency ratio Unison 1:1 Octave 2:1 Perfect fifth 3:2 Perfect fourth 4:3 Major third 5:4 Minor sixth 8:5 Minor third 6:5 Major sixth 5:3 Minor seventh 9:5 Major second 9:8 Major seventh 15:8 … … Consonance – Just intonation Psychoacoustics 3 - Perception of Pitch and intervals consonant dissonant This theory of consonance lead to the Just temperament (see later)

24. Alexis Baskind • Initially, a consonant interval was an interval that produced a sound which is “pleasant” to our ears. A dissonant interval was judged “unpleasant” • But since more and more complex intervals and chords were introduced in western music over ages, the distinction “pleasant/unpleasant”, and thus the concept of consonance itself, evolved a lot • Nowadays consonance is not anymore considered as an absolute criterion, but as a subjective and highly cultural concept Consonance – modern definition Psychoacoustics 3 - Perception of Pitch and intervals

26. Alexis Baskind • A musical Temperament is a specification of all pitches in a given musical system • Many different temperaments (see Appendix) were used over ages, including: – Pythagorean temperament, with perfect fifths – Zarlino scale (late 16th century) – Meantone temperament (renaissance) – Well temperaments (baroque=>end of the 19th) – Equal temperament (late 18th century), nowadays the mostly used standard Temperament Psychoacoustics 3 - Perception of Pitch and intervals Just temperaments

27. Alexis Baskind • The principle of equal temperament is to divide the octave in 12 equal intervals (the semitones) • Therefore, as the octave corresponds to a frequency ratio of 2:1, each semitone corresponds to a frequency ratio of: • Thus no intervals are just except the octave. However, the error differs depending on the interval • To allow a more precise definition of tuning, a subdivision of the semitone is used, the cent, which is a hundredth of a semitone: Equal Temperament Psychoacoustics 3 - Perception of Pitch and intervals rsemitone = 212 »1.05946 rcent = rsemitone 100 = 21200 »1,000578

28. Alexis Baskind Name Frequency ratio Cents Just intonation interval Cents in just intonation Error (cents) Unison 1 0 1 0.00 0 Minor second = 1.059463 100 16/15 = 1.066667 111.73 −11.73 Major second = 1.122462 200 9/8 = 1.125 203.91 −3.91 Minor third = 1.189207 300 6/5 = 1.2 315.64 −15.64 Major third = 1.259921 400 5/4 = 1.25 386.31 +13.69 Perfect fourth = 1.334840 500 4/3 = 1.333333 498.04 +1.96 Augm. fourth = 1.414214 600 7/5 = 1.4 582.51 +17.49 Perfect fifth = 1.498307 700 3/2 = 1.5 701.96 −1.96 Minor sixth = 1.587401 800 8/5 = 1.6 813.69 −13.69 Major sixth = 1.681793 900 5/3 = 1.666667 884.36 +15.64 Minor seventh = 1.781797 1000 7/4 = 1.75 968.83 +31.17 Major seventh = 1.887749 1100 15/8 = 1.875 1088.27 +11.73 Octave 2 1200 2 1200.00 0 Equal Temperament – comparison with just intonation Psychoacoustics 3 - Perception of Pitch and intervals 212 26 24 23 212 ( ) 5 2 212 ( ) 7 212 ( ) 8 212 ( ) 9 212 ( ) 10 212 ( ) 11

29. Alexis Baskind • The error for fifths and fourths is low but audible (±2 cents) • Thirds are more out of tune • The minor seventh is too high from almost a sixth-tone The equal temperament is a compromise, and an average reference • In practice many instruments do not really use a perfect equal temperament: – Violin, Cello and Viola are tuned using perfect fifths – The tuning of Piano is stretched for notes of the low and high-register (stretched tuning, see later) Equal Temperament Psychoacoustics 3 - Perception of Pitch and intervals

31. Alexis Baskind • What happens for complex sounds with overtones ? • For harmonic sounds, overtones in the harmonic series are melted with the fundamental frequency, thus changing the timbre but not the pitch • When two complex sounds are played simultaneously, their overtones interact with each other • This may lead to roughness between overtones => This is one of the reasons why tritones were judged dissonant (“diabolus in musica”) Complex sounds Psychoacoustics 3 - Perception of Pitch and intervals

32. Alexis Baskind • The ear is able to perceive the pitch of a harmonic sound even if it does not have energy at the fundamental frequency • As a matter-of-fact, the perception of pitch does not only depend on the energy of the fundamental, but on the periodicity in the time domain as well Missing Fundamental – Virtual Pitch Psychoacoustics 3 - Perception of Pitch and intervals Source: skyhead/Wikipedia Example: mixing two tones at 200 Hz and 300 Hz create a time oscillation with the frequency of the missing fundamental at 100 Hz

33. Alexis Baskind • This is in practice why we are able to hear very low pitches (C0-C1) in music played by loudspeakers which cannot reproduce frequencies below 100 Hz • This is also why we can hear the actual pitch of the voice through telephone (which has a lower frequency cutoff of 300 Hz) Missing Fundamental – Virtual Pitch Psychoacoustics 3 - Perception of Pitch and intervals

34. Alexis Baskind • If some overtones of a harmonic sound are much louder than the others, they may sometimes not be perceptually fused with the rest of the sound, and are thus perceived as a separate pitch • This phenomenon is called spectral pitch • A very good example of this phenomenon is given by overtone singing (see previous course “The Overtone Spectrum”) Spectral Pitch Psychoacoustics 3 - Perception of Pitch and intervals

35. Alexis Baskind • Most of time, music instruments do not produce pure harmonic sounds • Hearing has a tolerance that allows a single pitch to be deduced, if the sound is not too inharmonic • But inharmonic overtones may modify the final pitch ! • This is one of the reasons a piano cannot be tuned according to the equal temperament Inharmonic sounds Psychoacoustics 3 - Perception of Pitch and intervals

36. Alexis Baskind • The strings in the low and high register are inharmonic because of their stiffness and density • This phenomenon gets stronger for upright piano: since the length of the strings is smaller, they must be thicker, thus more inharmonic Therefore the upper notes are tuned higher and the lower notes lower than what the pure maths suggests Stretched tuning of a Piano Psychoacoustics 3 - Perception of Pitch and intervals

37. Alexis Baskind Stretched tuning Psychoacoustics 3 - Perception of Pitch and intervals Source: Brian Tung, Wikimedia

38. Alexis Baskind • If the overtones are even more inharmonic, two or several simultaneous pitches can be perceived (bells, gongs…)… Inharmonic sounds Psychoacoustics 3 - Perception of Pitch and intervals

39. Alexis Baskind • … or sometimes even no pitch at all Example: cymbal roll with wool mallets Inharmonic sounds Psychoacoustics 3 - Perception of Pitch and intervals

40. Alexis Baskind • Pitch and pitch discrimination differ as a function of the context (melodic/harmonic intervals) • In a melodic context, octaves correspond to a frequency ratio which is greater than 2, pitch discrimination is good • In a harmonic context, octaves and consonant intervals correspond to the harmonic series, pitch discrimination is limited to critical bandwidths • There is no perfect temperament, it’s a question of compromises Conclusion Psychoacoustics 3 - Perception of Pitch and intervals

41. Alexis Baskind Appendix: non-equal Temperaments Psychoacoustics 3 - Perception of Pitch and intervals

42. Alexis Baskind • The Pythagorean temperament is based on the conception that all octaves, fifths and fourths have to be perfect • It is based on a stacking of perfect fifths, all pitched down to the same octave Pythagorean Temperament Psychoacoustics 3 - Perception of Pitch and intervals C G D A E B F# C# G# D# A# F Circle of Fifths Note that the major pentatonic scale uses the first 5 notes of the circle

43. Alexis Baskind • The Pythagorean temperament is makes all fifths and fourths just, except the last one ! • As a matter of fact : twelve stacked fifths (ratio ) are a little bit higher than seven octaves (ratio ) • Thus the last fifth (from F to C if the tonic is C) is too small => This is a so-called Wolf interval • The difference between a perfect fifth and this smaller fifth is called the Pythagorean comma • Also, all thirds are not just • This temperament is compatible with transpositions, but then the position of the wolf interval changes. Pythagorean Temperament Psychoacoustics 3 - Perception of Pitch and intervals 3 2 æ è ç ö ø ÷ 12 2( ) 7

44. Alexis Baskind • In the 16th century, the third played a more and more important role in music • Thus the Pythagorean temperament is not suitable, since all thirds are really out of tune • The scale invented by Joseph Zarlino (1517-1590), which is a diatonic scale, uses only just intervals: Zarlino Scale Psychoacoustics 3 - Perception of Pitch and intervals 1 9:8 5:4 4:3 3:2 5:3 15:8 2:1

45. Alexis Baskind • This scale, that may be extended to a chromatic scale, has the main advantage of providing only just intervals with respect to the tonic • But it can hardly be transposed: – Several fifths are not just – All minor seconds differ from each other – All major seconds differ from each other Zarlino Scale Psychoacoustics 3 - Perception of Pitch and intervals

46. Alexis Baskind • Meantone temperament, that was used a lot in the renaissance, was an attempt to make all thirds just • The principle is to reduce the size of all fifths from a very small value (a syntonic comma), so that all thirds can be just • Then 11 stacked fifths are a little bit out of tune, but this is acceptable • On the other hand, the “wolf fifth” is even worse than in the Pythagorean temperament • Thus in practice, some tonalities were not allowed, in order to avoid this fifth • This may be one of the reasons to the relation that is sometimes made between each tonality and a specific mood Meantone Temperament Psychoacoustics 3 - Perception of Pitch and intervals

47. Alexis Baskind • As modulations were more and more used in music, it became important to play all tonalities to be as just as possible, which is not possible with meantone temperament or well temperament • Equal Temperament is a solution to make all tonalities equal in terms of intervals • Equal temperament was first used in Europe in the early 17th century, but generalized as a standard in the late 19th century, as it possible to tune instruments to a cent-precision Equal Temperament Psychoacoustics 3 - Perception of Pitch and intervals

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