Psychoacoustics 3 - Perception of Pitch and Intervals

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

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,
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Psychoacoustics 3 - Perception of Pitch and intervals

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

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 !

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

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
(from http://acousticslab.org/psychoacoustics/PMFiles/Module06.htm)
Chroma
Height

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)

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

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

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

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

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

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

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

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

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

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)
(From J.Meyer, “Acoustics and the Performance of Music”)
Example with C2 as
fundamental:

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

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

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
consonant
dissonant
This theory of
consonance lead to the
Just temperament (see
later)

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

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
Just temperaments

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
rsemitone = 212
»1.05946
rcent = rsemitone
100 = 21200
»1,000578

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
212
26
24
23
212
( )
5
2
212
( )
7
212
( )
8
212
( )
9
212
( )
10
212
( )
11

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

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

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
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

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

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

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

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

Alexis Baskind
Stretched tuning
Source: Brian Tung, Wikimedia

Alexis Baskind
• If the overtones are even more inharmonic, two or several
simultaneous pitches can be perceived (bells, gongs…)…
Inharmonic sounds

Alexis Baskind
• … or sometimes even no pitch at all
Example: cymbal roll with wool mallets
Inharmonic sounds

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

Alexis Baskind
Appendix: non-equal
Temperaments

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
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

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
3
2
æ
è
ç
ö
ø
÷
12
2( )
7

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
1 9:8 5:4 4:3 3:2 5:3 15:8 2:1

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

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

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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