# Quanta in sound, the sound of quanta:
a voice-informed quantum theoretical
perspective on sound

Maria Mannone1,2 and Davide<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1European Centre for
Living Technology (ECLT),
Ca’ Foscari University of Venice, Italy
2 Department of Mathematics and Computer Sciences,
University of Palermo, Italy

###### Abstract

Humans have a privileged, embodied way to explore the world of sounds, through
vocal imitation. The Quantum Vocal Theory of Sounds (QVTS) starts from the
assumption that any sound can be expressed and described as the evolution of a
superposition of vocal states, i.e., phonation, turbulence, and supraglottal
myoelastic vibrations. The postulates of quantum mechanics, with the notions
of observable, measurement, and time evolution of state, provide a model that
can be used for sound processing, in both directions of analysis and
synthesis. QVTS can give a quantum-theoretic explanation to some auditory
streaming phenomena, eventually leading to practical solutions of relevant
sound-processing problems, or it can be creatively exploited to manipulate
superpositions of sonic elements. Perhaps more importantly, QVTS may be a
fertile ground to host a dialogue between physicists, computer scientists,
musicians, and sound designers, possibly giving us unheard manifestations of
human creativity.

To appear in the book Quantum Computing in the Arts and Humanities: An
Introduction to Core Concepts, Theory and Applications. E. R. Miranda (Ed.).
Cham: Springer Nature, 2022 to appear. Pre-publication draft: 2021

##  1 Sound$\stackrel{{\scriptstyle voice}}{{\leftrightarrow}}$Quanta

Sometimes, when kids imitate sounds around them, they are blamed for producing
weird noises. However, they are unknowingly using their own voice as a probe
to investigate the world of sounds, and thus they are probably performing some
experiments. Some of the these kids will become sound designers, other ones
composers, other sound engineers and physicists; some other ones, will blame
future kids, and the cycle repeats.

What is a vocal imitation? It is the attempt to reproduce some essential
features of a sound, thought of or actually heard, with the human voice. The
imitation can refer to characteristics of the sound, or to its hypothetical
sources [22].

The human brain catches some salient sound features, and the voice attempts to
reproduce them. Sometimes, even poets (some of the kids above became poets as
well) coined new words as to include auditory dimensions in their poetry,
producing examples of onomatopoeia. This happened for example at the beginning
of 20th Century, with the poems by the futurist Filippo Tommaso Marinetti,
where the words Zang Tumb Tumb imitate motor and war noises [35], and with La
fontana malata (The sick fountain) by Aldo Palazzeschi, where the words Clof,
clop, cloch [41] mimic the intermittent flow of water and the noise of falling
drops.

Onomatopoeia gives poetical dignity to vocal imitations. Vocal imitations
raised the interest of science as well. In the framework of a recent European
project, the voice has been shown to be a powerful means to produce sound
sketches, which can be transformed into refined sound designs through
interactive voice-driven sound manipulations. In this sense, the machine
extracts sound from the embodied imagination of the sound designer [17]. Tools
of this kind, taking the form of an augmented microphone, have been prototyped
[49] and, with the purpose of turning the microphone into a music controller,
even developed into products [58].

A few scholars may argue that there are precise tools to investigate sounds,
such as the Fourier formalism, which uses decompositions based on sinusoidal
functions, and all formalisms inspired by Fourier’s approach [25, 28].
However, this formalism is not immediately understandable in everyday
communications, and it is less directly manipulable than vocal imitations.
While the Fourier formalism is powerful at the level of persons with some
education in sound and music, it is not the way laypersons communicate and
reason about sonic realities.

A powerful support to both qualitative and quantitative communication and
reasoning on sound is given by sound visualizations. Spectrograms display the
spectrum of frequencies through time of a sound. With spectrograms, we can
easily compare sounds and investigate how their properties change through
time. However, the vocal imitations of a natural or an artificial sound, which
appears as completely intuitive to humans (again, think about the kid giving
voice to a toy car), might be hard to find by comparison of the spectrogram of
the vocal imitation with the spectrogram of the original sound. It can be
possible to investigate some emerging properties of both sounds, but it can be
really hard while dealing, for example, with a vocal imitation of a motor, or
some other mechanical noise, that has a really different spectral profile than
human voice. Thus, Fourier-driven sound visualization has some limitations in
revealing the embodied perceptual features of sounds.

Which are the characteristics of human voice? The utterances of humans and
many mammals can be decomposed into overlapping chunks that fall within three
primitive classes: phonation, turbulence, and supraglottal myoelastic
vibrations [20]. In phonation, the source is in the vocal folds. In
turbulence, the source is in chaotic motion of inhaled or exhaled air.
Supraglottal myoelastic vibrations include several kinds of low-frequency
oscillations or pulse trains generated with different parts of the vocal
apparatus, such as lips or tongue. We can build up a new formalism to describe
the sound based on these components.

And what are the characteristics of sound as it is produced out of our body?
Sound is made of waves of rarefaction and compression, produced by vibrating
strings, air-filled pipes, vibrating membranes or plates, and so on. Consider
the simplest of these systems, which is probably the flexible string fastened
to a rigid support at both ends. This is one of the most important models in
physics, which has been used to demonstrate fundamental phenomena, in
acoustics as well as in other areas of physical sciences. In fact, whilst
vibrating strings have often been used as a paradigm for quantum mechanics,
the vice versa, that is, using quantum mechanics as a paradigm to understand
sound, was proposed in the nineteen-forties by Dennis Gabor [21], who imagined
how sound analysis and synthesis could be based on acoustical quanta, or
wavelets.222A wavelet is a wave-like oscillation under a finite temporal
envelope. His seminal work has been extensively carried on and expanded both
by scientists and musicians, and is certainly at the root of granular
approaches to sound and music [44].

A variety of ideas and methods of quantum mechanics have been applied to
describe forms and phenomena pertaining to that form of art whose medium is
sound, that is music. For example, tonal attractions have been modeled as
metaphorical forces [3], quantum parallelism has been proposed to model music
cognition [15], the quantum formalism has been proposed as a notational tool
for music-visual “atomic” elements [32], the non-Markovianity of open quantum
systems has been proposed as a measure of musical memory within a score [34].
Quantum computing, that is computation based on actual physical quantum
processes [39], starts being used to control sound synthesizers and computer-
generated music [36, 37]. The opposite practice, that is using sonification as
a means to make the actions of quantum algorithms perceivable as musical lines
and recognizable as patterns, has appeared with the flourishing of quantum
computing as an area of theoretical computer science [61].

This chapter is part of a book on quantum-theoretical and -computational
approaches to art and humanites, and its first chapter provides an excellent
introduction to quantum theory and quantum computing. Nevertheless, we give a
few basic notions, essentially the postulates of quantum mechanics, in Section
1.1. For now, we can say that quantum mechanics is a branch of physics, where,
in a nutshell:

  * •

Matter and energy, seen at the level of subatomic particles, are described as
discrete;

  * •

We describe particles as points or as probability waves to find them in some
places;

  * •

If we know the momentum of a particle, we don’t know its position, and vice
versa;

  * •

The measurement influences the state of what is measured: the observer
(subject) influences the observed (object).

In 1935, Albert Einstein tried to resist to quantum mechanics, postulating
hidden variables to justify such a bizarre behavior [18]. Some further studies
showed that, if Einstein was right, some inequalities should be satisfied [4],
but quantum-mechanical systems can be conceived and implemented that actually
violate such inequalities [2]. This means that a local realistic view of the
world does not apply to quantum phenomena. According to another Nobel prize,
Richard Feynman, nobody really understood quantum mechanics [19].

One might ask: if quantum mechanics is so difficult to be interpreted and
understood, why is it so often invoked to explain mundane affairs that have
nothing to do with particle physics? As a possible explanation, the formalism
is based on a few postulates, it assumes linearity and unitary (energy-
preserving) time evolution, and it gives a probabilistic framework capable to
explain concurrent and interfering phenomena.

Let us go back to sound and voice. If quantum mechanics can be joined with the
sound, and the sound with the voice, thus quantum mechanics can be joined with
the voice, and this is our idea: a Quantum Vocal Theory of Sound [48].

This approach is not opposed to the richness and complexity of Fourier
formalism, spectrograms, and so on. It presents a different paradigm, a
different starting point, using the primitive components of human voice. The
novelty is that these components appear as useful not only to investigate the
voice itself, but also to face the complexity of the world of general sound.
It is a strong statement, but it actually follows the intuition: each kid
knows well how to imitate the vroom vroom of a car, a long time before
learning how to read an equation and how to interpret a graph.

The QVTS approach can be exploited to investigate sound, decomposing it into
its essential features through the analysis step. But QVTS can also help do
the opposite, that is, create new sounds, in the synthesis step. Sound
synthesis can lead to creative applications; some possible applications are
described later on in this chapter.

The structure of the chapter is the following. In Section 1.1, we remind of
some basics of quantum mechanics. In Section 2, we present the fundamental
ideas of QVTS. In Section 3, examples of sound processing based on the QVTS,
with audible and interpretable outcomes, are given. In Section 4, we describe
our vision on the future of QVTS keeping an eye on interdisciplinary
collaborations and creative applications. As an example of possible creative
applications, we sketch the structure of a piece based on vocal states.

###  1.1 Some postulates to live by

An observable is a physical quantity, that can be described as a mathematical
(Hermitian, linear) operator. Each operator acts on a complex vector space,
the state space. The space where quantum observables live is the separable
Hilbert space. It’s separable, because we can distinguish the components along
different axes. In the case of QVTS, the space is related with the vocal
primitives, and it is separable as well, as we will see in the next section.

A quantum state, that is a unit-length vector in the state space, can be seen
as a superposition of values with some probabilities. An eigeinstate is a
characteristic state of some operator. After the measurement process, the
probability wave collapses to a certain value: it is the eigenvalue of a
certain operator, and the system is in an eigenstate of that operator.

Probability is a key concept in quantum mechanics. According to the principle
of uncertainty, we cannot know, let’s say, the position and the momentum of a
particle with the same precision. The more is the information we have on
position, the less we know about momentum, and vice versa.

Let us say more on the idea of quantum measurement. Consider a Cartesian
framework with axes $x$, $y$, and $z$—a tridimensional space with three
mutually orthogonal axes. We can perform measurements along each of the axes.
Quantum measure implies a change in the measured entity. If the measurement
along the direction $x$ can give a positive result, in all subsequent
measurements along the same direction we will have a positive value. A measure
along x would zero out the y and z components, while leaving only the x
component with value 1. If, before measurement, there is a given probability
to get a specific outcome, after the effective measurement of that outcome,
the probability to get the same value in each subsequent measurement along the
same direction is $100\%$. In fact, in quantum mechanics, the measurement of a
state implies the destruction of part of the initial information, and thus the
process is called destructive measure. A quantum state is a superposition of
eigenstates, which are reduced to a single state after the measurement. Such
state collapse happens in the context including both the system and the
measuring entity, through the interaction of the two [50]. Intuitively, it’s
like observing and taking a picture of a person, and blocking him or her as
the represented image along that specific shooting direction. (Be careful the
next time you’ll take pictures).

Dennis Gabor first exploited the paradigm of quantum theory to investigate
sound [21], instead of doing the usual vice versa, with sound and strings used
as metaphors to understand quantum waves. Gabor proposed the concept of
quantum of sound, as a unit-area cell in the time-frequency plane, which could
be called phon, from the Greek φωνή. On the other hand, we start from a vocal
description of sound, to define the phon as the set of vocal primitive
operators.

##  2 The quantum vocal theory of sound

In a recent article [48], we have proposed the basics for a Quantum Vocal
Theory of Sound (QVTS). Here, we summarize its main ideas, and then we propose
some hints for future developments.

First of all, let us define the phon formalism, where the word phon indicates
the quantum of sound, expressed in the state space of vocal primitives. With
the phon formalism, we can define vocal states, and extend the quantum bit
(qubit) language to the human voice. Some quantum-mechanical concepts, such as
state preparation and measurement, can be extended to the domain of the voice
as well.

Consider a space with three independent directions: $x$, $y$, and $z$. In the
QVTS, the three axes of this “phonetic space” have a vocal meaning:

  * •

$z$ is the phonation, giving waveforms of different pitches;

  * •

$x$ is the turbulence, giving noises of different brightnesses;

  * •

$y$ is the myoelasticity, giving pulsations at different tempos (thought of as
slow pulse trains).

Such three-dimensional space is sketched in figure 1 where, at the
intersections between each of the axes and the unit (called Bloch) sphere, we
can find two mutually orthogonal vectors, each depicted as a tiny sketchy
spectrogram.

Figure 1: The Bloch sphere adapted to QVTS, to represent the phon space. Hand-
drawing by D. Rocchesso.

Given a sound recording of human voice, if we measure phonation using a
specific computational tool (such as SMS-tools [7]), it is possible to
separate such component from the rest, and all subsequent measurements of
phonation will be giving the same result. If we measure a primitive component
first, and then another one, the result is generally dependent on the order of
the two operations: A fact that is known as non-commutativity. Figure 5 shows
a couple of example spectrograms illustrating the difference.

A vocal state can be described as a superposition of phonation, turbulence,
and myoelasticity with certain probabilities. We can thus define a phon
operator $\overline{\sigma}$ as a 3-vector operator, providing information on
the x, y, and z components through its specific directions in the 3d phonetic
space. Each component of $\overline{\sigma}$ is represented by a linear
operator, so we have $\sigma_{x}$, $\sigma_{y}$, and $\sigma_{z}$.

###  2.1 Preparation and measurement along axes

According to the postulates of quantum mechanics, it is possible to perform
measurements along one arbitrary axis of the 3d phonetic space and, as a
result, we will have prepared the phon along that specific axis.

A quantum measurement is represented by an operator, called a projector,
acting on the state, and provoking its collapse onto one of its eigenvectors.
If the system is in a state $|\psi\rangle$ and then we make a measurement, the
probability to get the result j is given by:

$p_{j}\coloneqq
p_{m}(j|\psi)=\bra{\psi}M_{j}\ket{\psi}=\bra{\psi}M_{j}^{\dagger}M_{j}\ket{\psi},$
(1)

where the set {$M_{j}$} is a projector system in the Hilbert space of states.
{$M_{j}$} is a complete set of Hermitian and idempotent matrices. An Hermitian
matrix is a complex matrix, that is equal to its transposed conjugate
(indicated by the $\dagger$ symbol in equation 1). It has real eigenvalues.
Idempotent means that, if we apply multiple times an operator, the result is
the same as if we applied the operator just once. With an orthonormal basis of
measurement vectors $|a_{j}\rangle$, the elementary projectors are
$M_{j}=|a_{j}\rangle\langle a_{j}|$, and the system collapses into
$|a_{j}\rangle$.

####  2.1.1 Measurement along z

A measurement along the z axis is performed through the operator $\sigma_{z}$.
The eigenvectors (or eigenstates) of $\sigma_{z}$ are $\ket{u}$ and $\ket{d}$,
corresponding to pitch-up phonation and pitch-down phonation, with eigenvalues
$\lambda_{u}=+1$ and $\lambda_{d}=-1$, respectively:

$\sigma_{z}\ket{u}=\ket{u},\,\,\,\,\sigma_{z}\ket{d}=-\ket{d}.$

The eigenstates $\ket{u}$ and $\ket{d}$ are orthogonal, i.e.,
$\braket{u}{d}=0$, and they can be represented as column vectors

$\ket{u}=\begin{bmatrix}1\\\ 0\end{bmatrix},\,\ket{d}=\begin{bmatrix}0\\\
1\end{bmatrix}.$ (2)

The operator $\sigma_{z}$ can also be represented in matrix form as

$\sigma_{z}=\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}.$ (3)

Applying a measurement along the $z$ direction to a generic phon state
$\ket{\psi}$ corresponds to pre-multiply it by one of the measurement
operators (or projectors)

$M_{u}=\ket{u}\bra{u}=\begin{bmatrix}1&0\\\ 0&0\end{bmatrix}$

or

$M_{d}=\ket{d}\bra{d}=\begin{bmatrix}0&0\\\ 0&1\end{bmatrix},$

and to normalize the resulting vector to have length one. Such operators
satisfy the completeness relation $M_{u}+M_{d}=I$, summing up to the unit
operator.

A generic phon state $\ket{\psi}$ can be expressed as

$\ket{\psi}=\alpha_{u}\ket{u}+\alpha_{d}\ket{d},$ (4)

where the coefficients are complex numbers, $\alpha_{u}=\braket{u}{\psi}$, and
$\alpha_{d}=\braket{d}{\psi}$. Being the system in state $\ket{\psi}$, the
probability to measure pitch-up is

$p_{u}=\bra{\psi}M^{\dagger}_{u}M_{u}\ket{\psi}=\braket{\psi}{u}\braket{u}{u}\braket{u}{\psi}=\braket{\psi}{u}\braket{u}{\psi}=\alpha_{u}^{\ast}\alpha_{u}$
(5)

and, similarly, the probability to measure pitch-down is
$p_{d}=\braket{\psi}{d}\braket{d}{\psi}=\alpha_{d}^{\ast}\alpha_{d}$, where
$\ast$ denotes complex conjugation. The completeness relation ensures that
$p_{u}$ and $p_{d}$ sum up to one.

If we repeatedly prepare a state $\psi$ and measure it along the $z$
direction, we get the average value

$\braket{\sigma_{z}}\coloneqq\sum_{m=\\{u,d\\}}\lambda_{m}p_{m}=\braket{\psi}{\left(\sum_{m=\\{u,d\\}}\lambda_{m}M^{\dagger}_{m}M_{m}\right)}{\psi}=\braket{\psi}{\sigma_{z}}{\psi},$
(6)

where the sum within brackets is called the observable of the measurement.

In quantum computing terminology, the vectors333In quantum computing, the
vectors of the computational basis are normally called $\ket{0}$ and
$\ket{1}$. (2) give the computational basis of a qubit vector space. The
operator (3) corresponds to a Z gate, which acts as a phase flip on the second
state of the computational basis.

####  2.1.2 Measurement along x

The eigenstates of the operator $\sigma_{x}$ are $\ket{r}$ and $\ket{l}$,
corresponding to turbulent primitive sounds having different spectral
distributions, one with the rightmost (or highest-frequency) centroid and the
other with the lowest-frequency centroid. Their respective eigenvalues are
$\lambda_{r}=+1$ and $\lambda_{l}=-1$, such that

$\sigma_{x}\ket{r}=\ket{r},\,\,\,\,\sigma_{x}\ket{l}=-\ket{l}.$

If the phon is prepared $\ket{r}$ (turbulent) and then the measurement
apparatus is set to measure $\sigma_{z}$, there will be equal probabilities of
getting pitch-up or pitch-down phonation as an outcome. This measurement
property is satisfied if $\ket{r}$ is defined as

$\ket{r}=\frac{1}{\sqrt{2}}\ket{u}+\frac{1}{\sqrt{2}}\ket{d}.$ (7)

A similar definition is given for $\ket{l}$, such that the two eigenstates of
turbulence are orthogonal ($\braket{r}{l}=0$):

$\ket{l}=\frac{1}{\sqrt{2}}\ket{u}-\frac{1}{\sqrt{2}}\ket{d}.$ (8)

In matrix form, the turbulence operator is expressed as

$\sigma_{x}=\begin{bmatrix}0&1\\\ 1&0\end{bmatrix},$ (9)

and its eigenvectors are

$\ket{r}=\begin{bmatrix}\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{2}}\end{bmatrix},\,\,\ket{l}=\begin{bmatrix}\frac{1}{\sqrt{2}}\\\
-\frac{1}{\sqrt{2}}\end{bmatrix}.$ (10)

Applying a measurement along the $x$ direction to a generic phon state
$\ket{\psi}$ corresponds to pre-multiply it by one of the measurement
operators

$M_{r}=\ket{r}\bra{r}=\frac{1}{2}\begin{bmatrix}1&1\\\ 1&1\end{bmatrix}$

or

$M_{l}=\ket{l}\bra{l}=\frac{1}{2}\begin{bmatrix}1&-1\\\ -1&1\end{bmatrix},$

and to normalize the resulting vector to have length one. Such operators
satisfy the completeness relation $M_{r}+M_{l}=I$.

In quantum computing, the operator (9) corresponds to a X gate, which is the
equivalent of the NOT gate in classical logic circuits, as it flips the states
of the computational basis. The vectors (7) and (8) form the Hadamard basis,
often denoted with the symbols $\\{\ket{+},\ket{-}\\}$.

Preparation in one of the states of the Hadamard basis
$\\{\ket{r},\ket{l}\\}$, followed by measurement along the $z$ axis, results
in an operation that is equivalent to coin flipping, $+1$ or $-1$ being
obtained with equal probability.

####  2.1.3 Measurement along y

The eigenstates of the operator $\sigma_{y}$ are $\ket{f}$ and $\ket{s}$,
corresponding to slow myoelastic pulsations, one faster and one slower,444In
describing the spin eigenstates, the symbols $\ket{i}$ and $\ket{o}$ are often
used, to denote the in–out direction. with eigenvalues $\lambda_{u}=+1$ and
$\lambda_{d}=-1$, such that

$\sigma_{y}\ket{f}=\ket{f}\,\,\,\,\sigma_{y}\ket{s}=-\ket{s}.$

If the phon is prepared $\ket{f}$ (pulsating) and then the measurement
apparatus is set to measure $\sigma_{z}$, there will be equal probabilities
for $\ket{u}$ or $\ket{d}$ phonation as an outcome. This measurement property
is satisfied if

$\ket{f}=\frac{1}{\sqrt{2}}\ket{u}+\frac{i}{\sqrt{2}}\ket{d},$ (11)

where $i$ is the imaginary unit.

Likewise, the $\ket{s}$ state can be defined in such a way that the two
eigenstates of pulsation are orthogonal ($\braket{f}{s}=0$):

$\ket{s}=\frac{1}{\sqrt{2}}\ket{u}-\frac{i}{\sqrt{2}}\ket{d}.$ (12)

In matrix form, the pulsation operator is expressed as

$\sigma_{y}=\begin{bmatrix}0&-i\\\ i&0\end{bmatrix},$ (13)

and its eigenvectors are

$\ket{f}=\begin{bmatrix}\frac{1}{\sqrt{2}}\\\
\frac{i}{\sqrt{2}}\end{bmatrix},\,\,\ket{s}=\begin{bmatrix}\frac{1}{\sqrt{2}}\\\
-\frac{i}{\sqrt{2}}\end{bmatrix}.$

Applying a measurement along the $y$ direction to a generic phon state
$\ket{\psi}$ corresponds to pre-multiply it by one of the measurement
operators

$M_{f}=\ket{f}\bra{f}=\frac{1}{2}\begin{bmatrix}1&-i\\\ i&1\end{bmatrix}$

or

$M_{s}=\ket{s}\bra{s}=\frac{1}{2}\begin{bmatrix}1&i\\\ -i&1\end{bmatrix},$

and to normalize the resulting vector to have length one. Such operators
satisfy the completeness relation $M_{f}+M_{s}=I$.

The matrices (3), (9), and (13) are called the Pauli matrices. In quantum
computing, these are all useful one-qubit gates.

###  2.2 Measurement along an arbitrary direction

Orienting the measurement apparatus in the phonetic space along an arbitrary
direction $\overline{n}=\left[n_{x},n_{y},n_{z}\right]^{\prime}$ means taking
a weighted mixture of Pauli matrices:

$\sigma_{n}=\overline{\sigma}\cdot\overline{n}=\sigma_{x}n_{x}+\sigma_{y}n_{y}+\sigma_{z}n_{z}=\begin{bmatrix}n_{z}&n_{x}-in_{y}\\\
n_{x}+in_{y}&-n_{z}\end{bmatrix}.$ (14)

####  2.2.1 Sines+ models and the phon space

The Harmonic plus Noise model [7] is well suited to describe measurement and
preparation in the phonation-turbulance planar section of the 3d phonetic
space. An arbitrary direction in such plane is described by the operator

$\sigma_{n}=\begin{bmatrix}\cos\theta&\sin\theta\\\
\sin\theta&-\cos\theta\end{bmatrix},$ (15)

where $\theta$ is the angular direction, pointing to a superposition of
phonation and turbulence (see figure 1). The eigenstate for eigenvalue $+1$ is

$\displaystyle\ket{\lambda_{1}}=\left[\cos\theta/2,\sin\theta/2\right]^{\prime},$
(16)

the eigenstate for eigenvalue $-1$ is

$\displaystyle\ket{\lambda_{-1}}=\left[-\sin\theta/2,\cos\theta/2\right]^{\prime},$
(17)

and the two are orthogonal. Suppose we prepare the phon to pitch-up $\ket{u}$.
If we rotate the measurement system along $\overline{n}$, the probability to
measure $+1$ is

$p(+1)=\braket{u}{\lambda_{1}}\braket{\lambda_{1}}{u}=\left|\braket{u}{\lambda_{1}}\right|^{2}=\cos^{2}\theta/2,$
(18)

and the probability to measure $-1$ is

$p(-1)=\left|\braket{u}{\lambda_{-1}}\right|^{2}=\sin^{2}\theta/2.$ (19)

The expectation (or average) value of measurement is therefore

$\braket{\sigma_{n}}=\sum_{j}\lambda_{j}p(\lambda_{j})=(+1)\cos^{2}\theta/2+(-1)\sin^{2}\theta/2=\cos\theta.$
(20)

Symmetrically, if we prepare the phon in state $\ket{\lambda_{1}}$ and we
measure along the $z$ axis, we get a pitch-up with probability
$\cos^{2}\theta/2$ and a pitch-down with probability $\sin^{2}\theta/2$.

More generally, the Sines plus Noise plus Transients model [57] may be
suitable to describe measurement and preparation in the whole 3d phonetic
space, where supraglottal myoelastic vibrations are made to correspond to
transient pulse trains. For example, consider the vocal fragment555It is one
of the example vocal sounds considered in [45], and taken from [38]. whose
spectrogram is represented in figure 2. An extractor of pitch salience and an
extractor of onsets666The feature extractors are found in the Essentia library
[6]. have been applied to highlight respectively the phonation (horizontal
dotted line) and myoelastic (vertical dotted lines) components in the
spectrogram. In the $z-y$ plane, there would be a measurement orientation and
a measurement operator that admit such sound as an eigenstate.

Figure 2: Spectrogram of a vocal sound which is a superposition of phonation
and supgraglottal myoelastic vibration. A salient pitch (horizontal dotted
line) as well as quasi-regular train of pulses (vertical dotted lines) are
automatically extracted.

###  2.3 Purity and Mixing

In quantum mechanics, the density operator is a mathematical object that
describes the statistical (pure or mixed) state of a quantum system, and it is
usually represented as a matrix. A pure state is not referred to a moral
condition, but to a separability of states. A mixed state indicates an
inseparability of states from the viewpoint of the observer, who has some
degree of epistemic uncertainty. Thus, the concept of density matrix
generalizes the concept of state superposition. The density operator is
defined as

$\rho=\sum_{j}p_{j}\ket{\psi_{j}}\bra{\psi_{j}},$ (21)

where $p_{j}$ indicates the probability for the $j$-state. The density
operator for a pure state is $\rho=|\psi\rangle\langle\psi|$, and the trace of
the associated density matrix is $\rm{tr}[\rho^{2}]=1$. For a mixed state,
$\rm{tr}[\rho^{2}]<1$. It can be shown that the density matrix (21) can be
expressed as a composition of Pauli matrices as in (14), with the addition of
the identity matrix. From such representation, pure states can be shown to lay
on the surface of the Bloch sphere, while mixed states stay inside the sphere,
with the completely chaotic state being found at the origin [12]. A pure state
can contain a superposition, but such a composition is defined with certainty.
A mixed state is a probabilistic mixing. The mixed state is inseparable. The
generalization introduced by mixed states can represent the audio concept of
mixing, thus coming useful in composition of auditory scenes.

###  2.4 Not too sure? Uncertainty can be measured

In the wonderland of quantum mechanics, it can happen that, the better we know
something, the lesser we know something else. In QVTS, the more precise our
knowledge of phonation, the less precise our measurement of turbulence. In
quantum mechanics, if we measure two observables ${\bf L}$ and ${\bf M}$
simultaneously in a single experiment, the system is left in a simultaneous
eigenvector of the observables only if ${\bf L}$ and ${\bf M}$ commute, i.e.,
if their commutator $\left[{\bf L,M}\right]={\bf LM-ML}$ vanishes. When the
commutator is different from zero, we say that the two operators do not
commute. This happens with measurement operators along different axes. It is
the case of $\left[\sigma_{x},\sigma_{y}\right]=2i\sigma_{z}$. As a
consequence for QVTS, phonation and turbulence cannot be simultaneously
measured with certainty.

The uncertainty principle is one of the key ideas of quantum mechanics. It is
based on Cauchy-Schwarz inequality in complex vector spaces. According to the
uncertainty principle, the product of the two uncertainties is at least as
large as half the magnitude of the commutator:

$\Delta{\bf L}\Delta{\bf M}\geq\frac{1}{2}\left|\braket{\psi}{\left[{\bf
L,M}\right]}{\psi}\right|$ (22)

Let ${\bf L}=\mathscr{T}=t$ be the time operator, and ${\bf
M}=\mathscr{W}=-i\frac{d}{dt}$ be the frequency operator. Applying them to the
complex oscillator $Ae^{i\omega t}$, we get a time-frequency uncertainty,
where the uncertainty is minimized by the Gabor function (a sinusoid windowed
by a Gaussian) [24].

####  2.4.1 The order matters

Kids learn that multiplying $a$ times $b$, with $a,\,b$ natural numbers, is
the same as multiplying $b$ times $a$—and, since early age, they think that
commutativity is always verified. Reading a book and then going for a walk
might be the same stuff as going for a walk and then reading a book (maybe).
Quantum mechanics does not work that way, and the same for QVTS. If we record
a singer, we take away vowels, and then we take again away vowels, the result
is the same—the recording is in an autostate of no-vowels. If we take away
vowels, and then we take away the noise, the result is different from what we
could hear if we do the opposite, that is, taking away the noise and then the
vowels. More precisely, the measurement operators oriented along different
axes do not commute. For example, let $A$ be an audio segment. The measurement
(by extraction) of turbulence by the measurement operator turbulence-right
$M_{r}=|r\rangle\langle r|$ leads to $M_{r}(A)=A^{\prime}$. A successive
measurement of phonation by the measurement operator pitch-up
$M_{u}=|u\rangle\langle u|$ gives $M_{u}(A^{\prime})=A^{\prime\prime}$, thus
$M_{u}(A^{\prime})=M_{u}M_{r}(A)=A^{\prime\prime}$. If we perform the
measurements in the opposite order, with phonation first and turbulence later,
we obtain $M_{r}M_{u}(A)=M_{r}(A^{\ast})=A^{\ast\ast}$. We expect that
$[M_{r},M_{u}]\neq 0$, and thus, that $A^{\ast\ast}\neq A^{\prime\prime}$. The
diagram in figure 3 shows non-commutativity in the style of category theory.

In bra-ket notation, this fact can be expressed as

$\begin{split}&{M_{r}}{M_{u}}\ket{A}=\ket{r}\braket{r}{u}\braket{u}{A}=\braket{r}{u}\ket{r}\braket{u}{A}\neq\\\
&M_{u}M_{r}\ket{A}=\ket{u}\braket{u}{r}\braket{r}{A}=\braket{u}{r}\ket{u}\braket{r}{A}.\end{split}$
(23)

Given that $\braket{r}{u}$ is a scalar and $\braket{u}{r}$ is its complex
conjugate, and that $\ket{u}\bra{r}$ is generally non-Hermitian, we get

$\begin{split}[M_{r},M_{u}]=\ket{r}\braket{r}{u}\bra{u}-\ket{u}\braket{u}{r}\bra{r}=\\\
=\braket{r}{u}\ket{r}\bra{u}-\braket{u}{r}\ket{u}\bra{r}\neq 0,\end{split}$
(24)

or, in terms of matrices

$\begin{split}[{M_{r}},{M_{u}}]=\frac{1}{2}\left(\begin{matrix}1&1\\\
1&1\end{matrix}\right)\left(\begin{matrix}1&0\\\
0&0\end{matrix}\right)-\frac{1}{2}\left(\begin{matrix}1&0\\\
0&0\end{matrix}\right)\left(\begin{matrix}1&1\\\ 1&1\end{matrix}\right)=\\\
=\frac{1}{2}\left(\begin{matrix}1&0\\\
1&0\end{matrix}\right)-\frac{1}{2}\left(\begin{matrix}1&1\\\
0&0\end{matrix}\right)=\frac{1}{2}\left(\begin{matrix}0&-1\\\
1&0\end{matrix}\right)=\frac{i}{2}\sigma_{y}\neq 0.\end{split}$ (25)

On audio signals, measurements of phonation and turbulence can be performed
using the sines + noise model [7], as described in figure 4. The measurement
of phonation is performed through the extraction of the sinusoidal component,
while the measurement of turbulence is performed through the extraction of the
noise component with the same model. The spectrograms for $A^{\prime\prime}$
and $A^{\ast\ast}$ in figure 5 show the results of such two sequences of
analyses on a segment of female speech, confirming that the commutator
$\left[M_{r},M_{u}\right]$ is non-zero.

Figure 3: A non-commutative diagram representing the non-commutativity of
measurements of phonation ($M_{u}$) and turbulence ($M_{r}$) on audio $A$.

Figure 4: On the left, an audio segment is analyzed via the sines+noise model.
Then, the noise part is submitted to a new analysis. In this way, a
measurement of phonation follows a measurement of turbulence. On the right,
the measurement of turbulence follows a measurement of phonation. This can be
described via projectors through equation (23), and diagrammatically in figure
3.

Consider again figure 1, which shows a representation of the phon space using
the Bloch sphere. There are small spectrograms at the extremities, in
correspondence of $\ket{s}$, $\ket{f}$, $\ket{u}$, $\ket{d}$, $\ket{r}$, and
$\ket{l}$. Applying $\sigma_{z}\sigma_{x}$ to a state, we get the flipped
state we would obtain if we had applied $\sigma_{x}\sigma_{z}$. If we apply a
pitch operator and then a turbulence operator (or vice versa) to a slow
impulse train ($\ket{s}$ or $\ket{f}$), we get another impulse train.

Figure 5: Top: spectrogram corresponding to a measurement of phonation $M_{u}$
following a measurement of turbulence $M_{r}$, leading to
$M_{u}M_{r}(A)=A^{\prime\prime}$; bottom: spectrogram corresponding to a
measurement of turbulence $M_{r}$ following a measurement of phonation
$M_{u}$, leading to $M_{r}M_{u}(A)=A^{\ast\ast}$.

###  2.5 Time flies

The variation of quantum states in time can be obtained through the
application of time evolution operators on them. Similarly, suitable time
operators can make the density matrix vary in time as well. Given a density
operator $\rho(t_{0})$ at time $t_{0}$, its time variation is obtained
applying a unitary operator $U(t_{0},t)$:

$\rho(t)={\bf U}^{{\dagger}}(t_{0},t)\rho(t_{0}){\bf U}(t_{0},t).$ (26)

This is the most general definition: There are no assumptions on states (mixed
or pure), and the only assumptions on the operator ${\bf U}$ are that it is
unitary, i.e., ${\bf U}^{{\dagger}}{\bf U}={\bf I}$, with I the identity
matrix, and that it depends only on $t$ and $t_{0}$.

But actually there is more. The unitary operator U, evaluated at a tiny time
increment $\epsilon$, is related to the Hamiltonian ${\bf H}$, describing the
energy of the system:

${\bf U}(\epsilon)={\bf I}-i\epsilon{\bf H}.$ (27)

For a closed and isolated system, H is time-independent, and the unitary
operator becomes ${\bf U}(t)=e^{i{\bf H}(t-t_{0})}$. However, nature is more
complex, things are not isolated, and usually H is time-dependent, and the
time evolution is given by an integral. To complicate things even more, with a
non-commutative Hamiltonian, an explicit solution cannot be found. The problem
can be circumvented by considering local time segments where the Hamiltonian
is locally commutative.

An evolving state can, at a certain time, be subject to measurement. The
quantum measurement operator (or projector) acts on the state and make it
collapse onto one of its eigenvectors. If we have a mixed state, the system
collapses into an ensemble of states.

In the QVTS, the phon state evolution is subject to restoring forces, and the
Hamiltonian depends on the state orientation in the phon space. Such evolution
is alike that of a spin in a magnetic field. The Hamiltonian can be expressed
as

${\bf
H}=\frac{\omega}{2}\overline{\sigma}\cdot\overline{n}=\frac{\omega}{2}\begin{bmatrix}n_{z}&n_{x}-in_{y}\\\
n_{x}+in_{y}&-n_{z}\end{bmatrix},$ (28)

whose energy eigenvalues are $E_{j}=\pm\frac{\omega}{2}$, with energy
eigenvectors $\ket{E_{j}}$.  An initial phon $\ket{\psi(0)}$ can be expanded
in the energy eigenvectors as

$\ket{\psi(0)}=\sum_{j}\alpha_{j}(0)\ket{E_{j}},$ (29)

where $\alpha_{j}(0)=\braket{E_{j}}{\psi(0)}$, and the time evolution of the
state is

$\ket{\psi(t)}=\sum_{j}\alpha_{j}(t)\ket{E_{j}}=\sum_{j}\alpha_{j}(0)e^{-iE_{j}t}\ket{E_{j}}.$
(30)

Where do the restoring forces come from, in the sound domain? Broadly
speaking, they come from the local temporal sound production context.
Similarly to the concept of coarticulation in phonetics, the locally-defined
Hamiltonian is determined by neighboring sounds, extending their effects in
the short-term future or past [16]. In practice, we can rely on an audio
analysis system, such as the Short-Time Fourier Transform (STFT), to extract
and manipulate slowly-varying features such as pitch salience or spectral
energy to determine the components of the Hamiltonian (28).  Considered a
slice of time and an audio signal, the initial phon state can be made to
evolve subject to a time-dependent yet commutative Hamiltonian expressed as

$H(t)=e^{-kt}{\bf S},$ (31)

where ${\bf S}$ is a time-independent Hermitian matrix and $k$ governs the
spreading of coarticulating features. Such Hamiltonian evolution has been
inspired by a quantum approach to image segmentation [63], or figure-ground
segregation. For evolution in the phon space, the matrix ${\bf S}$ can be set
to assume the structure (28), where the components of potential energy can be
extracted as audio features through time-frequency analysis. For example, the
$n_{z}$ component can be made to correspond to the extracted pitch salience,
and the $n_{x}$ component can be made to correspond to the extracted
noisiness. In the time slice under examination, an initial $\ket{u}$ state
will evolve to a final state

$\ket{\psi(t)}=e^{-i\int_{0}^{t}{\bf H}(\tau)d\tau}\ket{u}={\bf
U}(0,t)\ket{u},$ (32)

which in general will be a superposition (4) in the phon space. A measurement
in the computational (phonation) basis will make it collapse to $\ket{u}$ or
$\ket{d}$ according to the probabilities $\alpha_{u}^{*}\alpha_{u}$ or
$\alpha_{d}^{*}\alpha_{d}$, respectively. If there are two competing and
concurrent pitch lines, the Hamiltonian evolution followed by measurement may
thus make a pitch following process stay on one line or jump to the other one.
In this way, auditory streaming processes [8] and figure-ground segregation
can be mimicked.

##  3 Quantum vocal sound processing

In this section, we present some examples that show how the quantum formalism,
as assimilated by the QVTS, can be used together with classical signal
processing, for creative yet controllable analysis/synthesis tasks.  Given the
time-frequency representation of an audio signal, as provided by the STFT, the
elements of the ${\bf S}$ matrix of the Hamiltonian (31) can be computed from
decimated audio features. For example, pitch salience can be extracted from
time-frequency analysis [51], and used as the $n_{z}$ component. The
exponential factor can be set to $g(m)=e^{-km}$, where $m$ is the frame number
within a segment of $M$ frames. The time evolution (32) can be computed by
approximating the integral with a cumulative sum. Starting from an initial
state (e.g., $\ket{u}$), the phon goes through repeated cycles of Hamiltonian
evolution, measurement, and collapse. The decision to measure phonation or
turbulence can be based on the degree of pitchiness that the evolution within
a certain audio segment has reached. Since the observable $\sigma_{z}$ has
eigenvalues $\pm 1$ for eigenvectors $\ket{u}$ and $\ket{d}$, a measure of the
degree of pitchiness can be given by the distance of
$\norm{\sigma_{z}\ket{\psi}}$ from $\norm{\ket{\psi}}$. The degrees of
noisiness and transientness can be similarly determined using the observables
$\sigma_{x}$ and $\sigma_{y}$, respectively.

When doing actual audio signal processing based on the QVTS, several degrees
of freedom are available to experiment with: The decimation factor or number
$M$ of frames in a segment; The exponential damping factor $k$; The thresholds
for switching to a certain measurement direction in phon space; The decision
to collapse or not – this is a freedom we have if we are using a classical
computer!

###  3.1 Playing with pure states

####  3.1.1 Fugue following

Consider the beginning of the Fugue from the Toccata and Fugue in D Minor, BWV
565, by Johann Sebastian Bach (figure 6). In this fragment, there is only one
voice, played by the left hand. However, the design of this sequence actually
creates the illusion of two voices, with an upper line with an ostinato A, and
a lower line with notes G, F, E, D, C$\sharp$, D, E, F, and so on.

Figure 6: Toccata and Fugue in D minor BWV 565 by J. S. Bach: beginning of the
Fugue.

The score fragment of figure 6 was automatically rendered with piano samples
at 100bpm and analyzed via the STFT,777Sample rate $44100\rm{Hz}$, window size
$2048$, transform size $4096$, hop size $1024$. with pitch salience and noise
energy extracted via the SMS-tools [7]. Setting the parameters frame
decimation $M=10$, exponential damping $k=0.1$, threshold of pitchiness $0.9$,
collapse decimation $5$, we obtain a phon evolution from pitch-up phonation
represented by the green dots of figure 7. The red and yellow lines represent
the two most salient pitches, as features extracted from the STFT. In the
first part of the evolution, the green dots mainly correspond to the ostinato
of note A, while, in the second part, they mainly follow the ascending scale
fragment (A B C$\sharp$ D…). Even without any noise added, transient and noise
components are inherent in piano samples (e.g., key noise) and, therefore, the
phon is subject to non-negligible forces and actually moves from the initial
$\ket{u}$ state. Different runs will give different evolutions, as the
collapse is governed by probability amplitudes, but in general we observe that
the Hamiltonian listener follows the upper or the lower melodic line for some
fractions of time.  Interestingly, the melodic line following is stable, or
even more stable, if we add a strong white noise to the signal, with an
amplitude that is about one tenth of the signal. An example evolution is
depicted in figure 8, where the effect of the added noise is only apparent
after second $5$, when effectively there is no signal.  Figure 9 shows the
phon evolution when the fugue is drowned into noise. In this case the melodic
contour following is more easily disrupted. The zero-pitch green dots
represent points where measurement and collapse have been oriented to the $x$
direction, for the effect of thresholding in pitchiness. In a resynthesis,
these dots correspond to noise bursts, while the other dots come from
$z$-oriented measurements and produce pitched notes.

Figure 7: The synthetic recording of the excerpt in figure 6 through
Hamiltonian evolution. The upper line with the repeated A is evident in the
first part of the graph, while the second part contains a fragment of the
melody of the lower line.

Figure 8: The synthetic recording of the excerpt in figure 6 through
Hamiltonian evolution, with the noise added (amplitude 0.1). The upper line
with the repeated A is evident in the first part of the graph, while the
second part contains a fragment of the melody of the lower line.

In repeated runs of Hamiltonian fugue following, we can see multiple melodic
lines emerging as the time evolution of some initial sound/vocal state. The
collapse of the phon to a state or another can be interpreted as the attention
shift from figure to ground, or vice versa [8, 5].

The proposed example can be the starting point for a wider investigation in
the field of auditory bistability. Bistability is an intriguing topic in
cognition. As a reference for quantum effects in cognition, especially
regarding superposition and non-classical probability, we may refer to [62]
for a theoretical quantum-based approach to explain cognitive acts. The idea
of bistability is also faced in an article on mathematics and image/music
[32]. It exploits the Dirac formalism used in quantum mechanics to represent
images as superpositions of essential visual forms. There is a minimum number
of forms which allows the recognizability of the form. With a little abuse of
terminology, we can consider this limiting, minimum value of simple forms as
the limit of a Gestalt neighborhood, as the discrete version of a topological
neighborhood, having in the center the initial, complete, not-approximated
visual form. When an image is bistable, we can imagine to have two
neighborhoods as the two faces of a thin cylinder. We can see one face or the
other one; but we cannot see the two faces together. This is the core idea of
bistability. While classic examples of bistability are visual, also auditory
illusions can be constructed, e.g., with different auditory streamings [10].
These cases might be analyzed with the help of QVTS, as we did with the
beginning of the Fugue from the Toccata and Fugue BWV 565.

Figure 9: The same fragment of figure 6 completely drowned into noise
(amplitude 1.0).

####  3.1.2 Glides tunneling

Continuity effects have been very important to derive a perceptual
organization of sound for auditory scene analysis [8]. Gestalt principles such
as proximity or good continuation are often used to describe how listeners
follow concurrent pitch lines and extract temporal patterns from a scene. A
simple yet significant case is that of two gliding and crossing tones
interrupted, at the crossing point, by a short burst of noise [13].

Figure 10: Tracing the two most salient pitches and noise energy for two
crossing glides interrupted by noise

Figure 10 (top) shows the spectrogram of two gliding and crossing tones,
interrupted by a $200\rm{ms}$-band of white noise, intervening at time
$1.5\rm{s}$. The red and yellow lines are the traces of the two most salient
pitches, as extracted using the Essentia library [6]. With stimuli such as
this, listeners most often report perceiving a single frequency-varying
auditory object tunneling [60] the interruption. Depending on the temporal
extension and intensity of the noise burst, a perceived V-shaped trajectory
may be predominant over a rectilinear continuation, thus making proximity
prevail over good continuation.

It is interesting to use the case of crossing glides interrupted by noise as a
test for Hamiltonian evolution, with the matrix ${\bf S}$ of the Hamiltonian
(31) computed from decimated audio features such as pitch salience and noise
energy. As a result of such feature extraction from a time-frequency
representation, we obtain two potentials, for phonation and turbulence, which
drive the Hamiltonian evolution. Figure 10 also displays (middle) the computed
salience for the two most salient pitches and (bottom) the energy traces for
two bands of noise ($1\rm{kHz}$ – $2\rm{kHz}$ and $2\rm{kHz}$ – $6\rm{kHz}$).
It is clear how the pitch extractor becomes uncertain when the two tones get
close in frequency and start beating, and it wiggles around during the noisy
interruption.

Figure 11: Tracking the phon state under Hamiltonian evolution from pitch-up.

Figure 11 shows an example evolution of the phon state, starting from
$\ket{u}$. In this specific run of the evolution, the phon sticks to phonation
(one of the two pitches) until well inside the noise band, even though pitch
has very little salience within the interruption (see figure 10, middle), with
only occasional switches to turbulence (the zero-pitch green dots in figure
11). Right after the noise interruption, the phon evolution is still
uncertain, until it steadily takes a $\ket{u}$ state, thus giving an overall
V-shaped bouncing trajectory. In this instance, proximity is shown to prevail
over good continuation. Due to the statistical nature of quantum measurement,
another run of the evolution may well produce a downward-crossing trajectory,
and turbulence bursts may be found at different times. Such uncertainty on
noise location is consistent with the known perceptual fact that bursts of
noise overlapped to a noise transition are not precisely located, with errors
that can be up a few hundred milliseconds [59].

With this example, we have given a demonstration of how quantum evolution of
the phon state can be set to reproduce relevant phenomena in auditory
perception, with possible applications in computational auditory scene
analysis.

###  3.2 Playing with mixed states

The quantum concept of mixing, briefly described in section 2.3, can be
related to the familiar audio concept of mixing. At the start of a Hamiltonian
evolution, the initial state may be mixed, i.e., known only as a probabilistic
mixture. For example, at time zero we may start from a mixture having
$\frac{1}{3}$ probability of $\ket{u}$ and $\frac{2}{3}$ probability of
$\ket{d}$. The density matrix would evolve in time according to equation (26).

When a pitch measurement is taken, the outcome is up or down according to

$P[m=i|\rho]=Tr[\rho M_{i}],$ (33)

and the density matrix that results from collapsing upon measurement is given
by

$\rho^{(i)}=\frac{M_{i}\rho M_{i}}{Tr[\rho M_{i}]}.$ (34)

The density matrix can be made audible in various ways, thus sonifying the
Hamiltonian evolution. For example, the completely chaotic mixed state,
corresponding to the half-identity matrix $\rho=\frac{1}{2}{\bf I}$, can be
made to sound as noise, and the pure states can be made to represent distinct
components of an audio mix.

Figure 12: Amplitudes of components $\ket{u}$, $\ket{d}$, and noise resulting
from a Hamiltonian evolution from a mixed state.

Figure 13: Sound synthesis obtained from the density matrix evolution from a
mixed state, using the component amplitudes depicted in figure 12.

####  3.2.1 Glides tunneling

Given the same audio scene of the two crossing glides interrupted by noise
(figure 10), we may follow the Hamiltonian evolution from an initial mixed
state. We can choose to make the pure states to sound like the upper or the
lower of the most salient pitches, and the completely mixed state to sound
like noise. These three components can be mixed for states with intermediate
degrees of purity. If $p_{u}$ and $p_{d}$ are the respective probabilities of
$\ket{u}$ and $\ket{d}$ as encoded in the mixed state, the resulting mixed
sound can be composed by a noise having amplitude $\min{(p_{u},p_{d})}$, by
the upper pitch weighted by $p_{u}-\min{(p_{u},p_{d})}$, and by the lower
pitch weighted by $p_{d}-\min{(p_{u},p_{d})}$. Figure 12 shows an example of
evolution from the mixed state having probabilities $\frac{1}{3}$ and
$\frac{2}{3}$, with periodic measurements and collapses ruled by equation
(34).

The analyzed audio scene and the model parameters, including the computed
Hamiltonian, are the same as used in the evolution of pure states described in
section 3.1.2. The amplitudes of the three components can be used as automated
knobs to control two oscillators and a noise generator, producing the sound of
spectrogram 13, characterized by a prevailing upward tone with a downward
bifurcation and a noisy tail.

####  3.2.2 Vocal superposition

As another example of mixed state evolution, we consider again the vocal sound
whose spectrogram is depicted in figure 2. It was chosen as an example of
actual superposition of phonation and slow myoelastic vibration. Despite the
presence of only one definite pitch, we can prepare the phon in an initial
mixed state, having $\frac{1}{3}$ probability of $\ket{u}$ and $\frac{2}{3}$
probability of $\ket{d}$, and compute a Hamiltonian evolution based on
potentials deduced from the time-frequency analysis, namely pitch salience,
noise component, and detected onsets. As in the example of section 3.2.1, we
chose to assign phonation amplitudes equal to $p_{u}-\min{(p_{u},p_{d})}$ and
$p_{d}-\min{(p_{u},p_{d})}$ to the components $\ket{u}$ and $\ket{d}$,
respectively, and turbulence amplitude $\min{(p_{u},p_{d})}$ to the noise
component. In addition, here we extract a pulsating component as well,
corresponding to slow myoelastic vibration, whose amplitude is derived from
the probabilities $p_{f}$ and $p_{s}$ of fast or slow pulsation. For example,
$p_{f}$ is derived from $Tr[\rho\ket{f}\bra{f}]$, which is similar to equation
(33).

Figure 14 shows the amplitude profiles that are extracted from the Hamiltonian
evolution, where we chose to measure phonation when $\min{(p_{u},p_{d})}>0.5$,
otherwise measuring along the slow myoelastic vibration axis. With non-
physical freedom, we collapsed the mixed state, along $\ket{u}$, $\ket{d}$,
$\ket{f}$, or $\ket{s}$, using an equation similar to (34), once every five
measurements. The resulting sound, which can be considered as a quantum-
inspired audio effect, has the spectrogram depicted in figure 15, where the
most salient pitch and the onsets have been extracted again and superimposed.

Figure 14: Amplitudes of components $\ket{u}$, $\ket{d}$, turbulence, and slow
myoelastic vibration, resulting from a Hamiltonian evolution from a mixed
state, run on the vocalization of figure 2.

Figure 15: Sound synthesis obtained from the density matrix evolution from a
mixed state, using the component amplitudes depicted in figure 14, and all
three components of phonation, turbulence, and slow myoelastic vibration. The
most salient pitch and onsets, as extracted from the synthetic sound, are
displayed as red dashed lines.

###  3.3 From signal processing to quantum computing, and back

In digital audio, talking about quantization does not mean referring to
quantum theory. Instead, quantization is meant to be the reduction of a
continuous range of signal amplitude values to a finite set of discrete
values, with a cardinality that depends on the number of bits dedicated to
represent each discrete-time sample. Signal quantization introduces a kind of
noise, which tends to have a spectrotemporal structure that somehow follows
the signal, thus becoming audible as a distortion for low-amplitude signals. A
cure for quantization noise is dithering, i.e., adding some tiny broadband
noise to the audio signal itself, before quantization, thus making
quantization noise more spectrally uniform and perceptually tolerable [42].
That injecting dither noise to signals and systems can make human and machine
processing more robust is a fact that has been known for a long time, and
widely applied in a variety of fields, including audio and image processing.
In quantum-inspired sound processing, as illustrated in the example of section
3.1, dithering can be used to control how erratic leading-pitch attribution
can be, in auditory scenes of competing sources.

As opposed to low-amplitude noise, that may actually make the pitch evolution
of a phon more stable, when a high-amplitude noise burst is encountered, it
actually acts as a bounce on the phon state, making it rotate by an angle
$\theta$. A sequence of bursts, such as that of the example in section 3.2, is
much like a sequence of bounces between billiard balls of highly different
weights.  Recently, the classically mechanic behavior of balls, whose weights
are in ratios of powers of 100, has been shown to be perfectly analogous to
the kernel of the Grover algorithm for quantum search [9], which is based on
unitary reflections in the state space.

In the examples of sections 3.1 and 3.2, the quantum evolution is driven by
potentials that are derived from the same audio that is being processed. To
turn these evolutions into quantum algorithms we should freeze a reference
audio segment, extract the feature-based potentials from the time-frequency
representation, and convert the elementary unitary transformations into
quantum gates, arranged along left-to-right wires. Each stage of the quantum
algorithm would represent a bounce or a measurement in the phon space, as long
as the operators are consistent with the postulates of quantum mechanics. It
should be noted that we have only been considering single-qubit (or single-
phon) operators. The universe of multiple and entangled phons remains to be
explored.

In both quantum mechanics and sound signal processing, unitary operators and
unitary transformations have a central role. In fact,  in physically-inspired
sound synthesis and digital audio effects, unitary matrix transformations are
often found, as scattering elements in feedback delay networks for artificial
reverberation [43]. In these structures, if the feedback matrix ${\bf A}$ is
chosen to be unitary, an initial pulse bounces indefinitely, at each bounce
scattering into a multiplicity of other pulses. In the ball-within-the-box
(BaBo) model [46], the matrix ${\bf A}$ can be interpreted as a scattering
ball that redistributes the energy from incoming wavefronts into different
directions, each corresponding to a planar wave loop, which produces a
harmonic series. Indeed, the matrix ${\bf A}$ does not have to be unitary for
the feedback structure to be lossless [52]. However, even staying within the
class of unitary matrices, ${\bf A}$ can be chosen for its scattering
properties, ranging from the identity matrix (no scattering at all) to
maximally-diffusive structures [47, 53]. A promising perspective for future
quantum sound processing, is to find realizable quantum operators for such
matrices. In particular, the Hadamard operator and the Householder reflection
are extensively used in quantum algorithms, and these were proposed as
reference matrices for feedback delay networks with maximally-diffusive
properties [26]. In the context of QVTS, a Hadamard gate ${\bf H}$ converts a
phon from $\ket{r}$ to $\ket{u}$, and from $\ket{l}$ to $\ket{d}$. If followed
by a measurement in the computational basis, it can be used to discriminate
between the two turbulent states. If inserted in a phon manipulation sequence,
it determines a switch in the vocal state of sound. Loops are not allowed in
quantum computing [39], but by spatially unfolding feedback, a reverberator
based on feedback delay networks may be converted to a quantum algorithm with
several stages of unitary operators, acting as scattering elements on a
multiplicity of phons. As a non-negligible detail, banks of delay lines of
different lengths, in the order of tens of milliseconds, should be interposed
between consecutive scattering sections.

Shor’s algorithm for factorization of a large integer $N$ relies on an
efficient way of finding the periodicity (modulo $N$) of the function $a^{x}$,
constructed from a randomly chosen integer $a$ that is smaller than $N$ and
coprime with it. To compute the periodicity of a function, the Quantum Fourier
Transform (QFT) operator is used, which transforms a superposition of
$N=2^{n}$ computational basis states on $n$ qubits, with coefficients ${\bf
x}=\left[x_{0},x_{1},\dots,x_{N-1}\right]$, into another superposition with
coefficients ${\bf X}=\left[X_{0},X_{1},\dots,X_{N-1}\right]=DFT({\bf x}$),
that is the Discrete Fourier Transform of ${\bf x}$. Using quantum
parallelism, such DFT is implemented with $O(n^{2})$ quantum gates, while
classically that would take $O(n2^{n})$ steps. Recently, a direct
transposition of the Fast Fourier Transform (FFT) into the form of a quantum
circuit has been proposed and called the Quantum FFT (QFFT) [1]. Instead of
the amplitude encoding used for the QFT, a basis encoding is used, where a
data sequence ${\bf x}$ is expressed as a tensor product of vector spaces
$\bigotimes_{j=0}^{N-1}\ket{x_{j}}$. A potential impact on audio signal
processing would be that quantum parallelism would allow to perform all frames
of a $N$-bins STFT simultaneously, with $O(N\log_{2}{N})$ gates.  The
aggregate features of an audio segment would then be encoded in the resulting
vector of qubits.

##  4 Quantum evolution of the state (of the art)

A respectable scientific theory helps find new results, confirms expectations,
extends the validity of known laws bringing them toward the realm of the
unknown and (formerly) inexplicable, and so on.

An exciting scientific theory leaves room for imagination and artistic
creativity. New ideas can arise from the interdisciplinary dialogue between
people of different fields. QVTS is intrinsically interdisciplinary, and we
think it can enhance the dialogue between worlds.

Interchanges between music and quantum mechanics constitute a relatively new
and flourishing research area. Our contribution to this field is the addition
of the human voice, and the use of vocal primitives as a probe to more
generally investigate the world of sounds. In section 3 we have proposed some
examples of a creative use of QVTS, where the Hamiltonian evolution is the
starting point for sound synthesis. In this section, we suggest some further
creative applications.

The density matrix can be exploited to improve source separation techniques.
In fact, the operation of partial trace on density matrix allows us to
separate a system from the environment, the reservoir. Choosing on which part
of the whole system we are making the operation of partial trace, we can
interchangeably choose which part we are neglecting. For example, given a
polyphonic vocal recording, we can establish that singer 1 is the system and
singers 2, 3, and 4 are the environment (thus, we can perform the partial
trace on singers 2-3-4), or that singer 2 is the system and singers 1, 3, and
4 are the environment, and so on. In fact, as a practical interest in the
domain of QVTS, we can think of a general recording, with multiple voices, and
interpret it as a statistic mixture of states. Voices might be organized as a
solo singing voice against a a background of several other voices of a choir—a
quantum choir. Therefore, QVTS may help analyze choral music. In addition, it
can give us hints also on how to create music. Creativity can precisely take
off from mixtures of states and vocal polyphony.

Because QVTS constitutes a bridge between sounds and quantum formalism, we can
play with the symmetries of particle processes and transform them to musical
symmetries, thus giving voice to quantum processes. We can create
correspondences between certain quantum properties of particles and the
sounds, their transformations and musical transformations. For example, an
inversion of the spin could be musically rendered with an inversion of the
pitch interval; a quantum superposition can be rendered with the simultaneous
playing of different orchestral sections to create a “cloud of sound.” A
quantum measurement, with the subsequent collapse of the sound wave, could be
rendered with the sudden silence of other orchestral sections, and with the
remaining sound of a section, or even only one instrument sound. Musical
structures can be thought of as transformations over time of “states” (short
musical sequences or essential musical ideas for example). According to this
metaphor, we might describe the time evolution of quantum states, including
density matrices describing inseparable state superpositions through generated
musical structures. These hints should be confronted with perceptual criteria,
to create an idea of the processes in the mind of the listener.

Finally, we may imagine an interface where the user can modify states on the
Bloch sphere, modifying the synthesis in real time. Such an interface might
allow a “Quantum Synthesis,” maybe the Gabor’s dream. A quantum synthesizer
with potential for development has indeed been recently proposed [14], where
the quantum circuits such as the one for Grover’s search can be run on a
simulator or a quantum computer, and probability distributions and computation
steps can be heard, with auditory exploitation of quantum noise.

We end this section with a fun, original musical fragment, or, better, a set
of instructions to make music directly out of the QVTS-Bloch’s sphere. A
suitable synthesizer as the one hypothesized above could make this attempt a
concrete tool for creative purposes. Let us imagine a short musical
composition with two vocal (not instrumental) lines, created out of moving
states on the Bloch’s sphere. As another homage to Bach, we can be inspired by
the structure of the Two-Part Inventions, with the parts imitating each other,
as in a simple counterpoint. Thus, we can provocatively call our attempt Two-
Part Quantum Invention No. 1. Figure 16 shows a tentative notation, with a
schematic sphere derived from figure 1, and the sequence of state variations.
Generalization of the proposed idea to more voices and intricate counterpoints
is up the reader. This structure could be used as a set of instructions for
vocal improvisation, similarly to the “quantum improvisations” Pauline
Oliveros used to conduct [40]. Conceptually, the two voices can be instances
of the same evolving phon, from which we can, in principle, extract infinite
counterpoint lines. If the parallel motion of parts causes troubles in
classical counterpoint because of the feeling of sameness, intrinsic
parallelism is the real advantage of quantum computation, eventually leading
to quantum supremacy for some computational problems. Music counterpoint may
actually give voice to quantum parallel computations.

Figure 16:  Sketch for a music composition based on Bloch’s sphere for QVTS
(by M.M.).

###  4.1 Concluding remarks

Starting from kids’ playing and moving toward futuristic scenarios of quantum
choirs and Quantum Inventions, in this chapter we presented the fundamental
ideas of the Quantum Vocal Theory of Sound (QVTS), along with some proposal of
future developments.

We aimed to discuss a supplemental formalism to describe sound (with the vocal
probe), rather than proving any “wrongness” or obsolescence of the classical
formalism, such as Fourier analysis, for sound. Our supplemental formalism is
an alternative one, it gives a new perspective, and it has the advantage of
providing more information, especially regarding Gestalt-related phenomena, as
in the case of bistable states.

QVTS is interdisciplinary in nature, as it provides a bridge between sound
sources, sound production, human perception, and the intuitive identification
of sounds and sound sources. In addition, the uncertainty (or fuzziness) that
is proper to quantum thinking might be compared with the approximation of
human intuitive assessments about sounds, sound sources, and sound
identification.

Beyond the theoretical foundation, our aim is to foster the creation of
machines that measure sounds in terms of vocal primitives and express sonic
processes as evolving superpositions of vocal primitives. We hope that our
presentation of QVTS may lead to further questions, research, developments, as
well as to artistic contributions.

May it be a simple metaphor, or a quantitative tool, the core of quantum
mechanics is more and more inspiring for musicians, performers, and
scientists, shedding light on new and unexplored collaborations and insights.

All of that, can lead to the sound of quanta.

## References

  * [1] Asaka, R., Sakai, K., and Yahagi, R.: Quantum circuit for the fast Fourier transform. Quantum Information Processing, 19, 277 (2020)
  * [2] Aspect, A., and Dalibard, J., and Roger, G.: Experimental test of Bell’s inequalities using time-varying analyzers. Physical review letters 49 (25), 1804 (1982)
  * [3] Blutner, R., beim Graben, P.: Gauge models of musical forces. J. of Math. and Music, (2020) doi: 10.1080/17459737.2020.1716404
  * [4] Bell, J. S.: On the Einstein Podolsky Rosen Paradox. Physics Physique Fizika. _1_ (3): 195–200 (1964) doi:10.1103/PhysicsPhysiqueFizika.1.195
  * [5] Bigand, E., McAdams, S., and Forêt, S.: Divided attention in music. International Journal of Psychology 35 6, 270–278 (2000)
  * [6] Bogdanov, D., Wack, N., Gómez Gutiérrez, E., Gulati, S., Herrera Boyer, P., Mayor, O., Roma Trepat, G., Salamon, J., Zapata González, J. R., and Serra, X.: Essentia: An audio analysis library for music information retrieval. In: Proc. of the 14th Conference of the International Society for Music Information Retrieval (ISMIR), 493–498, Curitiba, Brazil, 2013
  * [7] Bonada, J., Serra, X., Amatriain, X., and Loscos, A.: Spectral processing, in Zölzer, U. (Editor): DAFX: Digital Audio Effects, 393–445, John Wiley & Sons, Ltd (2011)
  * [8] Bregman, A. S.: Auditory scene analysis: The perceptual organization of sound. MIT Press: Cambridge, MA (1994)
  * [9] Brown, A. R.: Playing Pool with $|\psi\rangle$: from Bouncing Billiards to Quantum Search. Quantum _4_ , 357, Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften (2020)
  * [10] Byrne, Á., Rinzel, J., Rankin, J.: Auditory streaming and bistability paradigm extended to a dynamic environment. Hearing Research _383_ , 107807 (2019)
  * [11] Cádiz, R., and Ramos, J.: Sound Synthesis of a Gaussian Quantum Particle in an Infinite Square Well. Computer Music Journal _38_ (4), 53–67 (2014)
  * [12] Cariolaro, G.: Quantum communications. Cham: Springer (2015)
  * [13] Ciocca, V., and Bregman, A. S.: Perceived continuity of gliding and steady-state tones through interrupting noise. Perception & Psychophysics 42 5, 476–484, Springer (1987)
  * [14] Costa Hamido, O., and Cirillo, G. A., and Giusto, E.: Quantum synth: a quantum-computing-based synthesizer. In: Proceedings of the 15th International Conference on Audio Mostly, 265–268 (2020)
  * [15] Dalla Chiara, M. L., Giuntini, R., Leporini, R., Negri, E., Sergioli, G.: Quantum information, cognition, and music. Frontiers in Psych. 6, 1583 (2015)
  * [16] Daniloff, R.G., and Hammarberg, R.E.: On defining coarticulation. Journal of Phonetics 1 (3), 239–248 (1973) http://www.sciencedirect.com/science/article/pii/S0095447019313889
  * [17] Delle Monache, S., Rocchesso, R., Bevilacqua, F., Lemaitre, G., Baldan, S., Cera, A.: Embodied sound design. International Journal of Human-Computer Studies 118, 47–59 (2018)
  * [18] Einstein, A., Podolsky, B., and Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Physical Review _41_ , 777 (1935) Retrieved online from http://www.drchinese.com/David/EPR.pdf
  * [19] Feynman, R.: The Character of Physical Law. Cambridge (MA), MIT Press (1995)
  * [20] Friberg, A., Lindeberg, T., Hellwagner, M., Helgason, P., Salom$\tilde{\mbox{a}}$o, G. L., Elowsson, A., Lemaitre,G., and Ternström, S.: Prediction of three articulatory categories in vocal sound imitations using models for auditory receptive fields. The Journal of the Acoustical Society of America 144 (3), 1467–1483 (2018)
  * [21] Gabor, D.: Acoustical quanta and the theory of hearing. Nature _159_ , 4044, 591 (1947)
  * [22] Gaver, W. W.: What in the world do we hear? An ecological approach o auditory event perception. Ecological Psychology. 5 (4), 285 (1993)
  * [23] Ghirardi, G.: Quantum superpositions and definite perceptions: envisaging new feasible experimental tests. Physics Letters A 262 (1), 1–14 (1999)
  * [24] Irino, T., and Patterson, R. D.: A time-domain, level-dependent auditory filter: The gammachirp. The Journal of the Acoustical Society of America, 101 (1), 412–419 (1997)
  * [25] von Helmholtz, H.: Die Lehre von den Tonempfindungen als physiologische Grundlagefürdie Theorie der Musik. Braunschweig, F. Vieweg und sohn (1870)
  * [26] Jot, J.-M.: Efficient models for reverberation and distance rendering in computer music and virtual audio reality. In: Proceedings: International Computer Music Conference 1997, Thessaloniki, Hellas, 25-30 September 1997, 236–243, The International Computer Music Association (1997)
  * [27] Juett, J. C.: Pauline Oliveros and Quantum Sound. Liminalities: A Journal of Performance Studies 6 (2), 1–10 (2010) http://liminalities.net/6-2/oliveros.pdf
  * [28] Koenig, D. M., and Delwin, D. F.: Spectral Analysis of Musical Sounds with Emphasis on the Piano. (New York, Oxford University Press, 2015)
  * [29] Kulpa, J., Campion, E., and Cella, C.: QuBits, a System for Interactive Sonic Virtual Reality. Proceedings of ICMC 2020,Santiago, Chile (2020). Retrieved fromhttp://www.carminecella.com/research/2020_ICMC_QuBits.pdf
  * [30] Kulpa, J.: QuBits, an Interactive Virtual Reality Project and Compositional Space for Sound and Image. Advisor: E. Campion. PhD dissertation, Berkeley, California (2019). Retrieved online from https://escholarship.org/uc/item/5tq9x2hb
  * [31] Lindeberg, T., and Friberg, A.: Idealized Computational Models for Auditory Receptive Fields. PLoS ONE 10 (3), e0119032. https://doi.org/10.1371/journal.pone.0119032
  * [32] Mannone, M., Favali, F., Di Donato, B., Turchet, L. Quantum GestART: identifying and applying correlations between mathematics, art, and perceptual organization. J. Math. Music (2020) doi: 10.1080/17459737.2020.1726691
  * [33] Mannone, M.: Introduction to gestural Similarity. An application of category theory to the orchestra. J. Math Music 12 (2), 63–87 (2018)
  * [34] Mannone, M., and Compagno, G.: Characterization of the degree of Musical non-Markovianity. ArXiV (2014) Retrieved online from https://arxiv.org/abs/1306.0229
  * [35] Marinetti, F. T.: Zang tumb tumb. Milano, Edizioni futuriste di poesia (1914) Retrieved online from http://parliamoitaliano.altervista.org/zang-tumb-tumb/
  * [36] Miranda, E. R.: Quantum Computer: Hello, Music! In: E. Miranda (Ed.), Handbook of Artificial Intelligence for Music: Foundations, Advanced Approaches, and Developments for Creativity. Springer Nature (2021)
  * [37] Miranda, E. R.: Creative Quantum Computing: Inverse FFT Sound Synthesis, Adaptive Sequencing and Musical Composition. In: A. Adamatzky (Ed.), Alternative Computing, World Scientific (2021)
  * [38] Newman, F.: MouthSounds: How to Whistle, Pop, Boing, and Honk… for All Occasions and Then Some. Workman Publishing (2004)
  * [39] Nielsen, M. A., and Chuang, I. L.: Quantum Computation and Quantum Information. Cambridge University Press (2010)
  * [40] Oliveros, P.: Quantum listening: From practice to theory (to practise practice). In: K. Siu Tong and C. Sin-wai, Culture and Humanity in the New Millennium: The Future of Human Values. The Chinese University Press, 27–44 (2002)
  * [41] Palazzeschi, A.: La fontana malata, from Poemi. Firenze, Cesare Blanc (1909) Retrieved online from https://www.pensieriparole.it/poesie/poesie-d-autore/poesia-38384
  * [42] Pohlmann, K. C.: Principles of Digital Audio, USA, McGraw-Hill, Inc., 3rd edition (1995)
  * [43] Pulkki, V., Lokki, T., and Rocchesso, D.: Spatial effects. DAFX: Digital Audio Effects, 139–183, Wiley (2011)
  * [44] Roads, C.: Microsound. Cambridge, MA: MIT Press (2001)
  * [45] Rocchesso, D., Mauro, D., and Drioli, C.: Organizing a sonic space through vocal imitations. J. Audio Eng. Soc. 64 7/8, 474–483 (2016) http://www.aes.org/e-lib/browse.cfm?elib=18333
  * [46] Rocchesso, D.: The Ball within the Box: A Sound-Processing Metaphor. Computer Music journal 4 (19), 47–57, The MIT Press (1995) http://www.jstor.org/stable/3680990
  * [47] Rocchesso, D.: Maximally diffusive yet efficient feedback delay networks for artificial reverberation. IEEE Signal Processing Letters, 4 (9), 252–255 (1997) 10.1109/97.623041
  * [48] Rocchesso, D., Mannone, M.: A quantum vocal theory of sound. Quantum Inf. Proc. 19, 292 (2020) doi: 10.1007/s11128-020-02772-9
  * [49] Rocchesso, D., Mauro, D. A., and Delle Monache, S.: MiMic: The Microphone as a Pencil. Proceedings of the TEI ’16: Tenth International Conference on Tangible, Embedded, and Embodied Interaction, in Eindhoven, Netherlands. Association for Computing Machinery, 357–364, TEI ’16, New York, NY, USA (2016) https://doi.org/10.1145/2839462.2839467
  * [50] Rovelli, C.: Helgoland. Penguin Books(2021)
  * [51] Salamon, J., and Gomez, E.: Melody Extraction From Polyphonic Music Signals Using Pitch Contour Characteristics. IEEE Transactions on Audio, Speech, and Language Processing 20 6, 1759–1770 (2012)
  * [52] Schlecht, S. J., and Habets, E. A. P.: On Lossless Feedback Delay Networks. IEEE Transactions on Signal Processing, 65 (6), 1554–1564 (2017) 10.1109/TSP.2016.2637323
  * [53] Schlecht, S. J.: FDNTB: The feedback delay network toolbox. In: Proceedings of the 23rd International Conference on Digital Audio Effects DAFx-20 (2020)
  * [54] Shamma S.: Spectro-Temporal Receptive Fields. In: Jaeger D., Jung R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY (2013) https://doi.org/10.1007/978-1-4614-7320-6_437-1
  * [55] Spence, C., Senkowski, D., and Röder, B.: Crossmodal processing. Exp Brain Res 198 (107), 107–111 (2009). Retrieved online from https://link.springer.com/article/10.1007/s00221-009-1973-4
  * [56] Spence, C.: Crossmodal correspondences: A tutorial review. Attention, Perception, & Psychophysics 73, 971–995 (2011) Retrieved online from https://doi.org/10.3758/s13414-010-0073-7
  * [57] Verma, T. S., Levine, S. N., and Meng, T. H.: Transient Modeling Synthesis: a flexible analysis/synthesis tool for transient signals. Proceedings of the International Computer Music Conference, 48–51 (1997)
  * [58] Vochlea microphone. Retrieved online from https://vochlea.com/
  * [59] Vicario, G. B.: La “dislocazione temporale” nella percezione di successioni di stimoli discreti (The “time displacement” in the perception of sequences of discrete stimuli). Rivista di Psicologia, 57 1, 17–87 (1963)
  * [60] Vicario, G: L’effetto tunnel acustico (The acoustic tunnel effect). Rivista di Psicologia 54, 41–52 (1960)
  * [61] Weimer, H.: Listen to Quantum Computer Music. http://www.quantenblog.net/physics/quantum-computer-music, Accessed: 2021-01-02 (2010)
  * [62] Yearsley, J.M., and Busemeyer, J.R.: Foundations of Probability Theory in Psychology and Beyond (special issue). Journal of Mathematical Psychology 74, 99 (2016)
  * [63] Youssry, A., El-Rafei, A., and Elramly, S.: A quantum mechanics-based framework for image processing and its application to image segmentation. Quantum Information Processing 14 (10), 3613–3638 (2015)