# A Geometric Framework for Pitch Estimation on Acoustic Musical Signals

Tom Goodman∗ ∗School of Computer Science, University of Birmingham Karoline
van Gemst† †School of Mathematics and Statistics, University of Sheffield and
Peter Tino∗

###### Abstract

This paper presents a geometric approach to pitch estimation (PE) - an
important problem in Music Information Retrieval (MIR), and a precursor to a
variety of other problems in the field. Though there exist a number of highly-
accurate methods, both mono-pitch estimation and multi-pitch estimation
(particularly with unspecified polyphonic timbre) prove computationally and
conceptually challenging. A number of current techniques, whilst incredibly
effective, are not targeted towards eliciting the underlying mathematical
structures that underpin the complex musical patterns exhibited by acoustic
musical signals. Tackling the approach from both a theoretical and
experimental perspective, we present a novel framework, a basis for further
work in the area, and results that (whilst not state of the art) demonstrate
relative efficacy. The framework presented in this paper opens up a completely
new way to tackle PE problems, and may have uses both in traditional
analytical approaches, as well as in the emerging machine learning (ML)
methods that currently dominate the literature.

###### keywords:

Pitch Estimation, Signal Processing, Geometry, Visualisation, Music
Information Retrieval

##  1 Introduction

Music Information Retrieval is increasingly gaining momentum as a cross-
disciplinary field of research (Muller et al., 2011), pulling together
techniques and researchers from computer science, cognitive science,
musicology, and electrical engineering, amongst others. Despite this, many
problems that on the surface appear to be trivially solvable from a human
perspective have proved intractable for computers, and thus, have remained
unsolved.

Pitch estimation is one such problem - the ability to take a musical signal as
an input, and at any given position in the signal, be able to ascertain what
notes (pitch chroma/pitch height pairs) are present. This is made difficult
because of the sheer volume of timbres that could be present, and notes that
could be played with various amplitudes, amongst other things. The remarkable
variety that exists within music, especially with increasing levels of
polyphony, render the problem of multi-pitch estimation (MPE) incredibly
challenging indeed.

Many approaches to pitch estimation are restricted to specific instruments
(for example, guitar tuning devices) (Böck and Schedl, 2012; Steinberger,
1996; Schramm et al., 2017). As a result, researchers are able to exploit
certain properties and assumptions (pertaining to the specific context in
which the methods will operate) to increase the accuracy of their approaches.
This helps to circumvent the greater difficulty of developing approaches that
function from a more generalised perspective.

Relatively recently, there has been a notable influx of approaches to problems
in Music Information Retrieval that utilise state-of-the-art machine learning
techniques (Kelz, Böck, and Widmer, 2019; Wu, Chen, and Su, 2018). Whilst the
accuracy of such approaches has proved good, the lack of ability to inspect
(and further, understand), the inner workings of them has resulted in a lack
of deep insight - especially into the underlying mathematical structures that
they are approximating. This provides a strong motivation to develop novel
algorithmic approaches (Goodman and Batten, 2018), in an attempt to discern
the intrinsic models - perhaps even those that we, as humans, exploit to
perform these tasks.

Additionally, given the growing use of processing power for these approaches
(Strubell, Ganesh, and McCallum, 2019), and the increasingly pressing nature
of climate change - with the IPCC special report on global warming of 1.5°C
concluding that our CO${2}$ emissions need to be halved by 2030 (Djalante,
2019) to curtail the increasing numbers of natural disasters - it is
imperative that we, as researchers, consider the efficiency of our algorithms
and approaches. With the era of Moore’s Law waning (Leiserson et al., 2020),
it is ever important to work towards algorithmic approaches that may prove
more computationally efficient. Indeed, although the cloud may appear to be
some ephemeral resource, somewhere there are real computers using real
electricity, and having real and profound impacts - both financially, and
environmentally. Looking again to model acoustic musical signals (and more
generally, AI problems) from mathematical points of view could give rise to
novel and efficient approaches that, whilst less efficacious in the short-term
than their machine-learning counterparts, may have the potential to open up
new branches of research in the long-term, with a particular focus on
understanding the problems being solved.

In this paper, a novel geometric perspective and methodology for both mono-
and multi-pitch estimation is presented. Section 2 gives a broad overview of
the related work, and a number of inspirations for the paper, following which
Section 3 introduces the model, which is then formalised in Section 4.
Building on this, Section 5 examines the ‘edge cases’ that arise in more
detail. Sections 6 and 7 present an approach to examine the prevalence of edge
cases, and apply it to data sampled randomly from the total space. Moving on
from the more theoretical perspective, Section 8 applies the model to ‘real-
world’ data, with Section 9 more closely analysing the performance of simple
algorithms working on the proposed model. Finally, Section 10 proposes future
direction for the work, and a number of potential applications of the
framework.

##  2 Related Work

Figure 1: Demonstration of the progression from Spectogram to Tentogram, and
then to Pitchogram (Elowsson, 2020).

Over the past few years, a plethora of approaches have been taken to tackle
the challenge of pitch estimation - with methods utilising the wavelet
transform (Kumar and Kumar, 2020), two-dimensional spectra (Zhang, Chen, and
Yin, 2020), and visual infomation (i.e. by viewing the physical instrument
itself) (Koepke, Wiles, and Zisserman, 2019), amongst others. Though varied,
much of the cutting-edge research is reliant on machine learning (ML)
techniques - not necessarily seeking to better-understand the underlying
structures present, and opting rather to maximise efficacy of the respective
approaches.

Elowsson (Elowsson, 2020) proposed a method for MPE that relies on ‘deep
layered learning’ (Elowsson and Friberg, 2014; Elowsson, 2018). It uses a
multi-stage system of neural networks and processing steps to elicit pitch
contours - i.e. pitch information coupled with an onset and offset for each
distinct note. From the MPE side, they opted to create a ‘Tentogram’ (i.e. a
tentative spectogram) through a spectral summation (their Section IV-F),
whitening, and logistic regression, which provided a much cleaner basis from
which to detect pitch peaks. A neural network was then used to convert the
Tentogram into a ‘Pitchogram’, using parabolic interpolation to achieve a 1
cent resolution. From the Pitchogram, ‘blobs’ are then identified (Miron,
Carabias-Orti, and Janer, 2014), with subsequent regions then merged (where
related), and finally a peak ridge (1D contour) was extracted. This method
exhibited state-of-the-art performance on the MAPS (Emiya et al., 2010),
Bach10 (Duan and Pardo, 2015), TRIOS, and MIREX Woodwind quintet datasets.

Kelz, Rainer and Böck looked to derive pitch contours from polyphonic audio
specifically from piano (Kelz, Böck, and Widmer, 2019). Previous ML approaches
tended to use many networks to extract the various features, but they instead
opted to use a shared representation (Ngiam et al., 2011) to simultaneously
predict the ADSR (Attack, Decay, Sustain, Release) aspects of each note. By
approaching the problem in this way, they were able to increase the potential
for generalisation to other instruments, as the representations that prove
useful to this task can be adapted to others. Further, rather than opting to
train another network to learn the ADSR envelopes of the input, they chose to
handcraft a Hidden Markov Model (HMM) with states corresponding to each
envelope, as well as an additional state to represent that a note is not
currently sounding. Similarly to this approach, it could be possible to use
the proposed framework (Section 4) in a kindred manner (albeit from a
different perspective).

The autocorrelation function (ACF) of a signal $\\{x(n)\\}_{n=0}^{N-1}$ at lag
$m$,

$r_{xx}(m)=\sum\limits_{n=m}^{N-1}x(n)x(n-m),$

has been widely used to elicit pitch information from time-domain signals
(Rabiner, 1977; Amado and Vieira Filho, 2008; Kraft and Zölzer, 2015).
Recently, de Obaldía, and Zölzer conducted a study looking at the efficacy of
the ACF on non-stationary sounds (to extract the fundamental period), and
presented a number of augmentations that allowed them to improve on the
current state-of-the-art approaches for monophonic pitch estimation (de
Obaldía and Zölzer, 2019). They achieved this by utilising musicological
knowledge to construct a heuristic that identifies non-related jumps in the
pitch contour, and subsequently modifying the signal to compensate for these.
They reported state-of-the-art results on both speech (including PTDB-TUG
(Pirker et al., 2011)) and musical signals (including Bach10).

Figure 2: A Tonnetz without identified edges - clearly showing its periodic
nature in both the vertical and horizontal directions (Image from (Bergstrom,
Karahalios, and Hart, 2007)).

First described by Euler in 1739, the Tonnetz (Figure 2) (Euler, 1739)
presents a way to spatially demonstrate the relationship between chords. Each
row corresponds to the circle of fifths, with each subsequent row
corresponding to the previous one with each element shifted from position $n$,
to position $(n+3)\bmod 12$, and aligned such that the closest two notes
diagonally ‘upwards’ correspond to a major third in each direction. By
identifying both vertically and horizontally, it is clear that the Tonnetz is
in fact toroidal in nature. It has proven particularly useful in describing
voice leadings in music - with distance between triangles (i.e. chords) on the
Tonnetz corresponding to musical ‘distance’ between chords.

In addition to his work on the generalised Tonnetz (Tymoczko, 2012), Tymockzo
posits a geometrical treatment of music theory in his book ‘A Geometry of
Music’ (Tymoczko, 2010) \- driven by underlying musicological knowledge. Where
Lewin tackled this kind of formalisation from a group-theoretical perspective
(Lewin, 1987), Tymockzo (whilst still basing his approach on symmetries) opted
to employ a more visual approach; considering pitch/chord spaces in such a way
that proves useful to composers and musicologists alike, without a necessarily
profound understanding of the underpinning mathematics.

Tymockzo defines musical objects, which are essentially ordered collections of
notes (Eg. (C4, E4, G4)), and five “OPTIC” operations -

  * •

Octave - changing the pitch height of a note;

  * •

Permutation - reordering the object (which has the effect of changing which
voice is playing which note);

  * •

Transposition - uniformly shifting all notes in an object by a given offset
(and direction);

  * •

Inversion - essentially reflection about a point in pitch space (i.e. pitches
ordered chromatically along a 1D line);

  * •

Cardinality change - introducing a new voice that duplicates a note that is
already present in the object.

These describe transformations between musical objects. Further, he goes on to
define a variety of musical constructs (such as chords and tone rows) in terms
of the set of OPTIC transformations under which each construct remains
invariant.

Building on this framework, he defines a two-note chord space, on which
progressions between two-note musical objects lie (Eg. (C4, E4)$\to$(C4,
E$\flat$4)). By enumerating the whole space, and identifying the edges with a
twist (which is necessary as, when enumerated fully, the vertical edges of the
two-note chord space are the reverse of one another), the two-note chord space
forms a Möbius strip. The use of this space in analysis is then demonstrated
practically by applying it to elicit musical insights on pieces (such as
Brahms’ Op. 116, No. 5), that would otherwise be obscure if viewed in
traditional notation. He goes on to provide a generalisation of n-note chord
spaces in higher dimensions - with a three-note chord space forming a twisted
triangular two-torus.

Inspired in particular by Tymockzo’s work, and historic algorithms such as
Noll’s Harmonic Pitch Spectrum (HPS) (Noll, 1970) and De Cheveigné and
Kawahara’s YIN algorithm (De Cheveigné and Kawahara, 2002), this paper sets
out to look at the problem of pitch estimation from a geometric point of view,
and construct algorithms from the building blocks laid out herein.

##  3 Reaching a Model

Figure 3: The build-up of a simplistic probabilistic representation.

Consider a frequency-sorted (low to high) set of notes, for example, {(C, 4),
(C, 5), (E, 5)}. One can imagine wanting to build up some representation of
where each note is likely to have originated. Figure 3 shows the iterative
construction of one such model - which is somewhat adjacent conceptually to
Markov chains. This could instead be viewed as a directed graph, with an edge
from each node to every other node that it could potentially be a harmonic of.
Weights are chosen from some node, B, to another node, A (of which B is
potentially a harmonic) to be i, such that B is the i${}^{\text{th}}$ harmonic
of A (Figure 4). Here, each weight corresponds to some measure of the
likelihood that a given note is generated by another. For example, given the
presence of C4, the probability that C5 is a fundamental (i.e. generated by
itself) may be 20%, whereas there may be an 80% probability that it is instead
a harmonic of C4. These are, of course, toy values, and it is more than likely
more realistic to readjust all weights following the addition of each note,
but the underlying concept remains the same.

Figure 4: Visualisation of the graphical structure overlaid onto the grid.
Figure 5: Visualisation of the final grid structure, where green cells
represent a Boolean value of true.

Further, one can imagine that trying to represent a large number of nodes (and
therefore a likely larger number of edges) renders this representation
visually messy, and hard to follow or decipher. By placing each node into a
grid (with the horizontal axis representing the pitch chromas ordered
according to the circle of fifths, and the vertical axis representing the
pitch height), and restricting the edges to only the first three harmonics,
this is alleviated. From here, it becomes clear that it is in fact possible to
dispose of the notion of this representation of a graph altogether: by
removing the edges (the information from which becomes implicit), and instead
assigning a Boolean value to each cell, representing whether the note is
audible or not (Figure 5). In order to apply any proposed algorithms, it is
necessary to convert the audio signal into the proposed format. Doing so
effectively will require reasonably extensive preprocessing (the full extent
of which is beyond the scope of this paper), which will need to include,

  * •

taking the signal from the time-domain to the frequency domain, and
discretising the frequency-domain signal (by binning the continuous signal
into the semitones of the western musical scale), likely through a constant-Q
transform (Brown and Puckette, 1992), or other Non-Stationary Gabor Transform
(NSGT);

  * •

eliminating the inharmonic part of the signal (in any domain). NB: the more
effectively this noise is reduced, the better the algorithms will perform (as
is true of most approaches); and

  * •

translating the resulting signal into the proposed grid format (Algorithm 1).
Computationally, this can be efficiently represented as a sparse matrix of 0s
and 1s,

in some order.

The problem of pitch detection is then reduced to the problem of finding the
decomposition of the grid (into shapes, or configurations) that corresponds to
the notes played in the input signal. As becomes apparent, this involves
discarding a number of false positive cases from the interpretation. From a
graphical perspective, this is somewhat equivalent to identifying the nodes
that correspond to fundamentals, removing them, and repeating the process
until no more are present.

Figure 6: (left) A directed graph depicting C3 and G4 (and each of their first
three harmonics sounding). The degrees of each node are shown in blue. The
following steps represent the steps of the simple algorithm described. The
bolded/underlined note at each step is the one selected as a fundamental.

For example, consider the graph in which (C, 3) and (G, 4) are sounding along
with their first three harmonics (Figure 6). By annotating each node, $\nu$,
with its indegree ($\text{deg}^{-}(\nu)$) and outdegree
($\text{deg}^{+}(\nu)$), some nodes present as sinks (i.e. with degree
$(n^{-},0)$ for some $n^{-}>0$), and some as sources (i.e. with degree
$(0,n^{+})$ for some $n^{+}>0$). For the purposes of pitch estimation, and
because of the chosen edge direction (from harmonic to fundamental), a sink
with indegree three is always a fundamental with its first three harmonics
present. Thus, a simple (yet somewhat effective algorithm) is to take each
node with indegree three (for each distinct part of the graph, as it may not
be connected), categorise them as fundamentals, and remove all categorised
nodes. This can then be iteratively applied to the graph until no sinks with
indegree three remain (as shown in Figure 6). Clearly this algorithm is a vast
oversimplification of the problem, but it nicely illustrates the benefits of
visual approaches.

The following sections will build upon this grid-based model, proving some
useful properties about it, and presenting some algorithms that utilise them.

Figure 7: Visualisation of a sequence of interpretations (representing
temporal slices), indexed by $\tau$.

##  4 The Proposed Model

As in Section 3, let $\chi^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$
be the set of pitch chromas, ordered by fifths - $\\{C,G,D,A,...,F\\}$. Let
the grid, $\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern
0.5pt}}}\coloneqq\chi^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern
0.5pt}}}\times\mathbb{Z}$, where any note in the western harmonic scale is a
point in $\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$,
i.e a pitch chroma/pitch height pair
$\nu_{i,j}\in\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$.
That is, $\nu_{i,j}$ represents pitch chroma indexed by
$i\in\chi^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$
(i.e. the circle of fifths), with pitch height $j\in\mathbb{Z}$.
$\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$
forms the backbone of the model. As the circle of fifths exhibits a periodic
nature, the left and right edges of the grid may be identified, or glued, to
realise $\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$
as a discretised infinite cylinder. Further, define the predicate,
$\mathcal{I}_{\tau}\colon\mathcal{N}^{\text{\makebox[6.25pt]{\hbox
to0.0pt{\vrule width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox
to0.0pt{\vrule width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern
0.5ptV\kern 0.5pt}}}\to\mathbb{B}$,

$\displaystyle\mathcal{I}_{\tau}(\nu_{i,j})=\left\\{\begin{array}[]{cc}\top&\hskip
14.22636pt\text{ if }\nu_{i,j}\text{ is observed at $\tau$}\\\ \bot&\hskip
14.22636pt\text{ if }\nu_{i,j}\text{ is not observed at $\tau$. }\\\
\end{array}\right.$ (3)

This is called an interpretation (i.e. of
$\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern
0.5pt}}}$), and can be seen as a single time slice of a signal, indexed by an
instantaneous point in time, $\tau$. By viewing a musical signal as a sequence
of temporal slices, an interpretation of the signal for any given $\tau$ by
the pair $(\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern
0.5pt}}},\mathcal{I}_{\tau})$ is obtained (Figure 7).

By viewing the ordered collection of pairs as a whole, one can uniformly
stretch each slice, and identify the appropriate
$\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$
faces to create a three-dimensional heatmap, with each note now represented by
a cube as opposed to a square. Thus, the interpretations are now indexed by an
interval, where previously they had been indexed by an instantaneous point in
time, that is,

$(\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern
0.5pt}}},\mathcal{I}_{\tau})\longmapsto(\mathcal{N}^{\text{\makebox[6.25pt]{\hbox
to0.0pt{\vrule width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox
to0.0pt{\vrule width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern
0.5ptV\kern 0.5pt}}},\mathcal{I}_{[\tau,\tau+\text{$1$})}).$ (4)

In general, when referring to any interpretation henceforth,
$\mathcal{I}_{\tau}$ may be replaced by $\mathcal{I}$ for simplicity.

By projecting onto hyperplanes parallel to the faces of the cuboid, one can
consider the signal from different perspectives - that is, with constant time,
constant pitch chroma, or with constant pitch height. For example, considering
the projection with constant pitch height, one can elicit a pitch contour
representation of the signal. Further, by viewing the heatmap as a translucent
construction, it is possible to consider all aspects simultaneously.

Let $f_{i}$ denote the $i^{\text{th}}$ harmonic, with $f_{0}$ being the
corresponding fundamental. Depending on the pitch chroma, the
2${}^{\text{nd}}$ harmonic, $f_{2}$, may or may not cross the octave boundary
(i.e. be in the next octave up from the 1${}^{\text{st}}$ harmonic). For
example, the first three harmonics of $(C,n)$ are $(C,n+1)$, $(G,n+1)$, and
$(C,n+2)$, whereas the first three harmonics of $(F,n)$ are $(F,n+1)$,
$(C,n+2)$, and $(F,n+2)$. By considering each chroma in
$\chi^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$,
it is clear that the presence of
$\\{f_{0},f_{1},f_{2},f_{3}\\}\subset\mathcal{N}^{\text{\makebox[6.25pt]{\hbox
to0.0pt{\vrule width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox
to0.0pt{\vrule width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern
0.5ptV\kern 0.5pt}}}$ make up one of two shapes; a turnstile shape, $\vdash$,
or a gamma shape, $\Gamma$, depending on the position of the fundamental.
Denote the set

$\chi_{\vdash}=\\{C,C\sharp,D,E\flat,E\\},\quad\text{with}\quad\chi_{\Gamma}=\chi^{\text{\makebox[6.25pt]{\hbox
to0.0pt{\vrule width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox
to0.0pt{\vrule width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern
0.5ptV\kern 0.5pt}}}\setminus\chi_{\vdash}\quad\text{as its complement, }$

and let $\pi_{\chi}$, $\pi_{y}$ be the projection of
$\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$
onto the horizontal and vertical axes respectively. Then, when
$\pi_{\chi}(f_{0})\in\chi_{\vdash}$ one observes the $\vdash$ shape, and
$\Gamma$ otherwise.

This is shown on Figure 8, where a fundamental is denoted by $\bullet$ and its
harmonics by $\times$.

Figure 8: Demonstration of the $\vdash$ and $\Gamma$ shapes in
$\mathcal{N}^{\text{\makebox[5.725pt]{\hbox to0.0pt{\vrule
width=5.725pt,height=0.16273pt,depth=0.16273pt\hss}\hbox to0.0pt{\vrule
width=5.725pt,height=4.38635pt,depth=-4.06088pt\hss}\kern 0.5ptV\kern
0.5pt}}}$.

This model (particularly the use of $\chi^{\text{\makebox[6.25pt]{\hbox
to0.0pt{\vrule width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox
to0.0pt{\vrule width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern
0.5ptV\kern 0.5pt}}}$ as opposed to a chromatically-ordered column set) is
chosen such that the pattern exhibited by a fundamental and its first three
harmonics (i.e. $\vdash/\Gamma$) appears spatially-compact. This serves to
make these patterns more easily discernible, and is also of use when looking
to decompose more complicated polyphonic signals into their constituent parts.

The different cells, or notes, on the cylinder can be related to each other by
considering a group action on $\mathcal{N}^{\text{\makebox[6.25pt]{\hbox
to0.0pt{\vrule width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox
to0.0pt{\vrule width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern
0.5ptV\kern 0.5pt}}}$. Let $\delta$ and $\omega$ denote the generators of
$\mathbb{Z}_{12}$ (the integers modulo 12) and $\mathbb{Z}$, respectively.
Then define a group action
$\mathbb{Z}_{12}\times\mathbb{Z}\circlearrowleft\mathcal{N}^{\text{\makebox[6.25pt]{\hbox
to0.0pt{\vrule width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox
to0.0pt{\vrule width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern
0.5ptV\kern 0.5pt}}}$ as follows. $\delta$ and $\omega$ induce maps on
$\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$
by

$\displaystyle(\delta,\mathbbm{1}_{\mathbb{Z}}):\ $
$\displaystyle\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern
0.5pt}}}\rightarrow\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern
0.5pt}}}\qquad\
{(\mathbbm{1}_{\mathbb{Z}_{12}},\omega):\mathcal{N}^{\text{\makebox[6.25pt]{\hbox
to0.0pt{\vrule width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox
to0.0pt{\vrule width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern
0.5ptV\kern 0.5pt}}}\rightarrow\mathcal{N}^{\text{\makebox[6.25pt]{\hbox
to0.0pt{\vrule width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox
to0.0pt{\vrule width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern
0.5ptV\kern 0.5pt}}}\ \ \ }$
$\displaystyle\nu_{i,j}\mapsto\nu_{i+1,j},\qquad\qquad\qquad\nu_{i,j}\mapsto\nu_{i,j+1},$

where $\mathbbm{1}_{\mathbb{Z}}$ and $\mathbbm{1}_{\mathbb{Z}_{12}}$ are the
identity elements in $\mathbb{Z}$ and $\mathbb{Z}_{12}$, respectively. In
other words, $(\delta,\mathbbm{1}_{\mathbb{Z}})$ acts on the cylinder by
rotating it clockwise by one cell, while applying
$(\mathbbm{1}_{\mathbb{Z}_{12}},\omega)$ corresponds to a vertical shift of
one cell downwards. Hence a map

$\mathbb{Z}_{12}\times\mathbb{Z}\times\mathcal{N}^{\text{\makebox[6.25pt]{\hbox
to0.0pt{\vrule width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox
to0.0pt{\vrule width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern
0.5ptV\kern 0.5pt}}}\rightarrow\mathcal{N}^{\text{\makebox[6.25pt]{\hbox
to0.0pt{\vrule width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox
to0.0pt{\vrule width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern
0.5ptV\kern 0.5pt}}}:\ (\delta^{k},\omega^{l},\nu_{i,j})\mapsto\nu_{i+k,j+l},$
(5)

is achieved, where $k,l\in\mathbb{Z}$, and $\delta^{12n}$ for any integer $n$
is the identity. For notational simplicity $(\delta,\mathbbm{1}_{\mathbb{Z}})$
is identified with $\delta$, and similarly for $\omega$. Note that this means
that, relative to a reference point, $\delta$ translates the note by a fifth,
and $\omega$ moves the note up an octave.

In terms of this action,

$\omega(f_{0})=f_{1},\qquad\omega\delta(f_{0})=f_{2},\qquad\text{and}\qquad\omega^{2}(f_{0})=f_{3},$

for $\pi_{\chi}(f_{0})\in\chi_{\vdash}$, where $\omega\circ\delta$ is
identified with $\omega\delta$. For $\pi_{\chi}(f_{0})\in\chi_{\Gamma}$ the
above holds with the exception of $f_{2}$ which in this case is given by

$\omega^{2}\delta(f_{0})=f_{2}.$

Furthermore, using this action, the $\vdash$ and $\Gamma$ shapes may be
written as,

$\vdash\,=\\{\mathbbm{1},\omega,\omega\delta,\omega^{2}\\},\qquad\Gamma=\\{\mathbbm{1},\omega,\omega^{2}\delta,\omega^{2}\\},$
(6)

where it is understood that by applying all elements of $\vdash$ to a note
traces out the turnstile shape, and similarly for $\Gamma$. In other words,
considering the $\vdash$ case, for each fundamental which is mapped to $\top$
by $\mathcal{I}$, there exist harmonics $\omega(f_{0})$,
$\omega\delta(f_{0})$, and $\omega^{2}(f_{0})$ such that each of these are
also mapped to $\top$ by $\mathcal{I}$,

$\forall_{\nu\in\mathcal{N}^{\text{\makebox[4.75pt]{\hbox to0.0pt{\vrule
width=4.75pt,height=0.12915pt,depth=0.12915pt\hss}\hbox to0.0pt{\vrule
width=4.75pt,height=3.48123pt,depth=-3.22292pt\hss}\kern 0.5ptV\kern
0.5pt}}}}\big{[}(\mathcal{F}(\nu))\to(\mathcal{I}(\omega(\nu))\land\mathcal{I}(\omega\delta(\nu))\land\mathcal{I}(\omega^{2}(\nu)))\big{]},$
(7)

given that $f_{1}$, $f_{2}$, and $f_{3}$ are observed (audible). Here
$\mathcal{F}(\nu)$ is a predicate that returns $\top$ iff $\nu$ is a
fundamental. Of course, such a construct does not exist in practice, but in
essence, the end result of a perfect pitch estimation algorithm is this
function, such that it best describes the ground truth of the signal. As
before, the $\Gamma$ case is equivalent under the replacement
$\omega\delta\mapsto\omega^{2}\delta$.

By observation of the corresponding $\vdash$ and $\Gamma$ shapes over the
circle of fifths, it is noted that three two-shape configurations exist -
namely $\Gamma\Gamma$, $\Gamma\vdash$, and $\vdash\Gamma$. These are used to
categorise a number of properties of the model.

######  Definition 4.0.1 (Configuration).

A configuration (denoted as $\Gamma\Gamma$, $\Gamma\vdash$, or $\vdash\Gamma$)
represents the shapes generated by fundamentals residing in two adjacent
columns in $\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$.
When referring to a false fundamental, $\otimes$, the second column of the
configuration always corresponds to that in which $\otimes$ lies. Thus, the
first column corresponds to the preceding one - i.e.
$\pi_{\chi}(\delta^{-1}(\otimes))$.

Whilst every fundamental, together with its first three harmonics, exhibit one
of the two aforementioned shapes (i.e $\Gamma$ or $\vdash$), the inverse
statement is not true. Namely, the presence of a $\vdash$ or $\Gamma$ shape
does not imply that the note concerned is a fundamental, as shown in Figure 9.
Here, $\otimes$ denotes a harmonic which presents as a fundamental. Such
harmonics are called false fundamentals.

Figure 9: Counterexample showing that not all $\vdash$ shape-exhibiting notes
are fundamentals.

Similarly to how a fundamental and its first three harmonics corresponds to
either a $\vdash$ or a $\Gamma$, a given harmonic could only have arisen from
a fundamental related to it by the inverse shape, i.e either  $\vdash$ or  L .

The inverse shapes may be written as,

$\text{\reflectbox{$\vdash$}}=\\{\sigma^{-1}|\ \sigma\in\
\vdash\\}=\\{\mathbbm{1},\omega^{-1},(\omega\delta)^{-1},\omega^{-2}\\},$ (8a)
$\text{\reflectbox{L}}=\\{\sigma^{-1}|\
\sigma\in\Gamma\\}=\\{\mathbbm{1},\omega^{-1},(\omega^{2}\delta)^{-1},\omega^{-2}\\},$
(8b)

where $\sigma^{-1}$ is the inverse of $\sigma$ with respect to the group
structure.

Additionally, define the function $\Psi(\chi_{i})$ for any
$\chi_{i}\in\chi^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$
as,

$\displaystyle\Psi(\chi_{i})=\left\\{\begin{array}[]{cc}\vdash&\hskip
14.22636pt\text{ if }\chi_{i}\in\chi_{\vdash}\\\ \Gamma&\hskip
14.22636pt\text{ otherwise}.\\\ \end{array}\right.$ (11)

Intuitively this function takes a given chroma (i.e.
$\pi_{\chi}(\nu)\in\chi^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern
0.5pt}}}$), and returns the set of group elements that trace out the
corresponding shape when applied to a note with this chroma.

The generator of a note is defined as the fundamental that deposited the
corresponding frequency. Note that the generators of a note may sit both in
the same, or proceeding column to itself. This means then when enumerating the
possible generators in most cases (i.e. not $\Gamma\Gamma$), it is necessary
to consider notes in  $\vdash$ $\bigcup$ L . In many cases there may be
multiple generators for a single note.

Figure 10: Figure showing where a false fundamental and each of its apparent
harmonics could have been generated from, with colours tracing out  $\vdash$
and  L for each note (overlaying both $\Gamma$ and $\vdash$ situations).

Suppose an interpretation is is given such that a false fundamental is
present. By investigating the possible positions for this note in
$\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$,
there is a finite region containing the fundamentals that could have created
the false fundamental and its first three apparent harmonics. In any of the
possible positions for the false fundamental, this region is contained in the
region given by $\varhexagon(\otimes)$, with
$\varhexagon\coloneqq\\{\sigma^{\prime}\sigma|\sigma^{\prime}\in\text{\reflectbox{$\vdash$}}\bigcup\text{\reflectbox{L}},\sigma\in\vdash\bigcup\Gamma\\}$,
as shown in Figure 10. This holds by construction.

Consequently, in order to check whether a note could be generated by some
other note, it is sufficient to search a $3\times 5$ area centred on the note.
Though this proves sufficient from the perspective of generators, there are a
number of edge cases (i.e. false fundamentals) that will naively result in
false positives.

Hence, the problem of multi-pitch estimation is reduced to that of
distinguishing between fundamentals ($\bullet$) and harmonics masquerading as
fundamentals ($\otimes$).

##  5 Fantastic Edge Cases (and Where to Find Them)

This section considers only cases in which a single false fundamental
($\otimes$) is present. In reality, this is expected to be enough of a
generalisation as long as algorithms consider false fundamentals sequentially,
such that $\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$
is traversed along

$(\delta^{i}(\omega\delta^{12})^{j-1})(1,1),\
\forall_{i\in\\{1,\cdots,12\\}}\forall_{j\in\\{1,\cdots,10\\}},$ (12)

that is, left-to-right, bottom-to-top.

######  Definition 5.0.1 (Edge Case).

An edge case is a set of notes (i.e. fundamentals and their first three
harmonics), in which a note that presents as a fundamental, is in fact not one
- as seen in Figure 9. That is,

$\otimes(\nu)\ =\
\big{(}\forall_{\sigma\in\Psi(\nu)}\big{[}\mathcal{I}(\sigma\nu)\big{]}\big{)}\land\neg\mathcal{F}(\nu).$

By considering  $\vdash$ $\bigcup$ L for each constituent note (similar to
Figure 10), it is possible to construct logical expressions for the generators
of any edge case. Through knowledge of the specific configuration (which is
always known for a given note), it is possible to use the appropriate subset
of  $\vdash$ $\bigcup$ L . The logical expressions are as follows,

$f_{0}=\omega^{-1}\lor\,\omega^{-2}\lor\left\\{\begin{array}[]{ll}(\omega\delta)^{-1}&\text{
if }\Psi(\pi_{\chi}(f_{0})-1)\ =\ \vdash\\\ \omega^{-2}\delta^{-1}&\text{
otherwise},\\\ \end{array}\right.$ (13)
$f_{1}=\omega\lor\omega^{-1}\lor\left\\{\begin{array}[]{ll}\delta^{-1}&\text{
if }\Psi(\pi_{\chi}(f_{0})-1)\ =\ \vdash\\\ (\omega\delta)^{-1}&\text{
otherwise},\\\ \end{array}\right.$
$f_{2}=\omega\delta\lor\,\delta\lor\left\\{\begin{array}[]{ll}\omega^{-1}\delta&\text{
if }\Psi(\pi_{\chi}(f_{2}))\ =\ \vdash\\\
\omega^{2}\delta&\text{otherwise},\\\ \end{array}\right.$
$f_{3}=\omega\lor\,\omega^{2}\lor\left\\{\begin{array}[]{ll}\omega\delta^{-1}&\text{
if }\Psi(\pi_{\chi}(f_{0})-1)\ =\ \vdash\\\ \delta^{-1}&\text{ otherwise}.\\\
\end{array}\right.$

Note that these actions are all relative to the fundamental, and the
interpretation functions are omitted for brevity - that is,
$\mathcal{I}_{\tau}(\sigma)$ $\mapsto$ $\sigma$ for all
$\sigma=\omega^{i}\delta^{j}$, $(i,j)\in\mathbb{Z}\times\mathbb{Z}$.

Further, it is possible to define basic edge cases;

######  Definition 5.0.2 (Basic Edge Case).

A basic edge case is an edge case such that each constituent note has at most
one generator.

More specifically, this means that each apparent harmonic (including the false
fundamental itself) has precisely one generator (as if any part had zero
generators it wouldn’t be a false fundamental). This can be trivially
represented by replacing the logical ors ($\lor$) with xors ($\oplus$) - for
example for $f_{0}$,

$f_{0}=\omega^{-1}\oplus\,\omega^{-2}\oplus\left\\{\begin{array}[]{ll}(\omega\delta)^{-1}&\text{
if }\Psi(\pi_{\chi}(f_{0})-1)\ =\ \vdash\\\ \omega^{-2}\delta^{-1}&\text{
otherwise}.\\\ \end{array}\right.$

The same replacement holds for $f_{1}$, $f_{2}$, and $f_{3}$.

Following from this, it is possible to enumerate every basic edge case for a
specific configuration, and ascertain the total number. Initially the answer
for four choices, each with three options would simply be $3^{4}=81$. Due to
overlap in which generators satisfy the constituent, however, the actual
result is significantly lower, and can be enumerated with a simple counting
method (Table 1). Note that $f_{2}$ has been omitted as it has no overlap (and
therefore, multiplying the end result by 3 is sufficient).

$f_{0}$ | $f_{1}$ | $f_{3}$
---|---|---
$\omega^{-1}$ | - | $\omega^{2}$
|  | $\omega\delta^{-1}$
$\omega^{-2}$ | $\omega$ | -
| $\delta^{-1}$ | $\omega^{2}$
|  | $\omega\delta^{-1}$
$(\omega\delta)^{-1}$ | $\omega$ | -
| $\delta^{-1}$ | $\omega^{2}$
|  | $\omega\delta^{-1}$
| Total | 8 $\times$ 3 = 24
Table 1: Table showing the enumeration of possible basic edge cases for the
$\vdash\Gamma$ configuration.

The same holds true for the $\Gamma\Gamma$ and $\Gamma\vdash$ configurations,
as although the composed actions are different, there are still the same
overlaps ($f_{0}/f_{1}:\omega^{-1},\ \text{and}\ f_{1}/f_{3}:\omega$), and the
same number of overall choices.

One might be tempted to claim, therefore, that there are $24\times 3=72$ basic
edge cases. Though technically this may be true, the choice was made to
instead define a number of basic edge types - similar to Lent Davis and
Maclagan’s definition of cap types regarding the card game SET (Davis and
Maclagan, 2003). This allows for comparisons to be made irrespective of
configuration. Further, there are a number of invariants that hold across all
configurations, for all basic edge types.

Let

$g:\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern
0.5pt}}}\to\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$
(14)

be the map sending a note to its generating fundamental.

Further consider the triple $g(f_{0},\ f_{1},\ f_{3})=(g(f_{0}),\ g(f_{1}),\
g(f_{3}))$ constructed from letting $g$ act on a fundamental and its first and
third harmonics. It is not necessary to consider $f_{2}$ as it has no overlap
with the other constituent parts (as shown below, with the overlaps made bold
for clarity),

$\displaystyle f_{0}:\qquad$ $\displaystyle\bm{\omega^{-1}}$
$\displaystyle\oplus\omega^{-2}$ $\displaystyle\oplus(\omega\delta)^{-1}$
(15a) $\displaystyle f_{1}:\qquad$ $\displaystyle\bm{\omega}$
$\displaystyle\oplus\bm{\omega^{-1}}$ $\displaystyle\oplus\delta^{-1}$ (15b)
$\displaystyle f_{2}:\qquad$ $\displaystyle\omega^{2}\delta$
$\displaystyle\oplus\omega\delta$ $\displaystyle\oplus\delta$ (15c)
$\displaystyle f_{3}:\qquad$ $\displaystyle\bm{\omega}$
$\displaystyle\oplus\omega^{2}$ $\displaystyle\oplus\omega\delta^{-1}.$ (15d)

Any basic edge case can be associated with such a triple, which corresponds to
the generating set of the false fundamental and its first and third apparent
harmonics.

######  Definition 5.0.3.

(Basic Edge Type)

Two basic edge cases are of the same type iff their two corresponding triples
are related by

$\delta\mapsto\omega^{k}\delta,\ k\in\\{-1,0,1\\}.$ (16)

######  Remark.

Note that if a basic edge case is obtained from another through
$\delta\mapsto\omega^{k}\delta$, then it is possible to move in the opposite
direction using the inverse map $\delta\mapsto\omega^{-k}\delta$.

For any given type, there will be precisely three members - with precisely one
being associated to each of the three configurations. The members of each type
(by configuration) are related as according to Figure 11.

Figure 11: Diagram showing the relationships between members of the same type
between different configurations.

This becomes clearer following inspection of Table 2. Intuitively, the only
difference between the $\vdash$ and $\Gamma$ shapes is the shifting of
$f_{2}$. The $f_{2}$ in a $\Gamma$ shape is obtained from the corresponding
$\vdash$ shape by acting $\omega$ on its $f_{2}$, and its inverse on the
contrary.

######  Lemma 5.0.1.

Being of the same type is an equivalence relation

###### Proof.

For equivalence, the relation must be reflexive, symmetric, and transitive.
The reflexive and symmetric properties may be shown by choosing $k=0$, and
letting $k\mapsto-k$ in Definition 16, respectively. Further, transitivity
holds by virtue of the diagram in Figure 11. ∎

######  Remark.

Note that the types of basic edge case are independent of note configuration.

Figure 12 visually represents the eight basic edge types shown in Table 2.

There are a number of invariants that hold within each type. These are
properties which are the same across each edge type, and provide information
about their geometric structure. The invariants considered are; back-$\delta$
count ($|\delta^{-1}|$), edge characteristic ($\epsilon$), and generating
structure ($G.S$) which are to be defined in the following.

######  Definition 5.0.4 (back-$\delta$ count).

The back-$\delta$ count, $|\delta^{-1}|$, is a positive integer representing
the number of $\delta^{-1}$ occurring in a triple $(g(f_{0}),\ g(f_{1}),\
g(f_{3}))$ associated to to a given edge case.

######  Example.

The triple $(g(f_{0}),\ g(f_{1}),\ g(f_{2}))=(\omega^{-1},\ \omega^{-1},\
\omega\delta^{-1})$ has back-$\delta$ count $|\delta^{-1}|=1$.

######  Definition 5.0.5 (Edge Characteristic).

The edge characteristic, $\epsilon$, is given by the number of distinct
fundamentals that are generators for the given note. That is, the number of
distinct elements in $(g(f_{0}),\ g(f_{1}),\ g(f_{3}))$.

######  Example.

The triple $(g(f_{0}),\ g(f_{1}),\ g(f_{3}))=(\omega^{-1},\ \omega^{-1},\
\omega\delta^{-1})$ has edge characteristic $\epsilon=2$.

In addition to $|\delta^{-1}|$ and $\epsilon$, another invariant is the
“Generating Structure” (G.S.), which not only considers the number of
generating fundamentals, but also which pairs satisfy overlap (i.e. pairs
generated by the same note). Naively, there are 5 possible generating
structures,

  1. I.

$f_{0}=f_{1}=f_{3}$

  2. II.

$f_{0}=f_{1}\neq f_{3}$

  3. III.

$f_{0}=f_{3}\neq f_{1}$

  4. IV.

$f_{0}\neq f_{1}=f_{3}$

  5. V.

$f_{0}\neq f_{1}\neq f_{3}$

By construction, cases 1 and 3 can never occur. Hence, all basic edge cases
exhibit a generating structure of either 2, 4, or 5.

These invariants help to distinguish between different basic edge cases, as
well as potentially aiding in decomposing non-basic cases into basic ones. The
invariants for each class are enumerated in Table 2.

Type | $|\delta^{-1}|$ | $\epsilon$ | G.S. | Cases
---|---|---|---|---
$\Gamma\Gamma/\Gamma\vdash$ | $\vdash\Gamma$
1 | 0 | 2 | $f_{0}=f_{1}\neq f_{3}$ | $\\{\omega^{-1},\omega^{2}\\}$
2 |  |  | $f_{0}\neq f_{1}=f_{3}$ | $\\{\omega^{-2},\omega\\}$
3 | 1 | 2 | $f_{0}=f_{1}\neq f_{3}$ | $\\{\omega^{-1},\delta^{-1}\\}$ | $\\{\omega^{-1},\omega\delta^{-1}\\}$
4 |  |  | $f_{0}\neq f_{1}=f_{3}$ | $\\{\omega^{-2}\delta^{-1},\omega\\}$ | $\\{(\omega\delta)^{-1},\omega\\}$
5 |  | 3 | $f_{0}\neq f_{1}\neq f_{3}$ | $\\{\omega^{-2},(\omega\delta)^{-1},\omega^{2}\\}$ | $\\{\omega^{-2},\delta^{-1},\omega^{2}\\}$
6 | 2 | 3 | $f_{0}\neq f_{1}\neq f_{3}$ | $\\{\omega^{-2},(\omega\delta)^{-1},\delta^{-1}\\}$ | $\\{\omega^{-2},\delta^{-1},\omega\delta^{-1}\\}$
7 |  |  |  | $\\{\omega^{-2}\delta^{-1},(\omega\delta)^{-1},\omega^{2}\\}$ | $\\{(\omega\delta)^{-1},\delta^{-1},\omega^{2}\\}$
8 | 3 | 3 | $f_{0}\neq f_{1}\neq f_{3}$ | $\\{\omega^{-2}\delta^{-1},(\omega\delta)^{-1},\delta^{-1}\\}$ | $\\{(\omega\delta)^{-1},\delta^{-1},\omega\delta^{-1}\\}$
| Total number of cases | $8\cdot 3=24$
Table 2: Table showing the different types of basic edge case, together with
their invariants, and elements (excluding $f_{2}$). Figure 12: The eight basic
edge types represented visually. Each $\bullet$ represents a generator, with
$\otimes$ representing the false fundamental, and the unfilled generators
containing $\vdash$ or $\Gamma$ denoting the shape drawn out in the
$\delta^{-1}$ column (i.e. corresponding to
$\Psi(\pi_{\chi}(\delta^{-1}(\otimes)))$.

As can be seen in Table 2, there is only a single case in which the type is
not uniquely determined by $|\delta^{-1}|$, $\epsilon$, and G.S. - namely
types 6 and 7. These can be distinguished by considering the position of the
generator that sits in the same column as the false fundamental - for type 6,
the generator sits below the false fundamental ($\omega^{-2}$), whereas for
type 7, the generator sits above the false fundamental ($\omega^{2}$).

######  Remark.

Note that, as before, the multiplication by 3 in Table 2 is due to there being
no restrictions on the choice of generator for the second harmonic.

######  Lemma 5.0.2.

The second “harmonic” ($f_{2}$) of a false fundamental ($\otimes$) must be
generated from the column directly to the right of the false fundamental (i.e.
$\Psi(\pi_{\chi}(\delta(\otimes)))$).

###### Proof.

Assume that there exists some generator
$g_{i}\in\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$
for $f_{2}$ that lies in the same column as the false fundamental111Note that
the only other choice would be the column directly to the right of $\otimes$..
There are two possible cases;

######  Case 1:

$\pi_{\chi}(g_{i})\in\chi_{\vdash}$

For $g_{i}$ to generate the $f_{2}$ at $\omega\delta(\otimes)$, it would have
to lie at $\mathbbm{1}(\otimes)$. Hence, $\otimes$ would no longer be a false
fundamental - resulting in a contradiction. $\bot$.

######  Case 2:

$\pi_{\chi}(g_{i})\in\chi_{\Gamma}$

The same argument as Case 1, applying the map
$\omega\delta\mapsto\omega^{2}\delta$. Hence, $\bot$.

Thus, as $\bot$ is reached in both possible cases, there can be no such
generator for $f_{2}$. Further, as

$\lnot\exists_{x}.P(x)\equiv\forall_{x}.\lnot P(x),$ (17)

all generators for any $f_{2}$ must lie in the column directly to the right of
the false fundamental. ∎

######  Proposition 5.1.

There are 24 basic edge types.

###### Proof.

This follows from Table 2, and Lemmas 6.1.1 and 6.1.2. ∎

######  Corollary 5.1.1.

There are no 5-fundamental basic edge cases

###### Proof.

Following from Table 2, the largest $\epsilon$ is 3. Including the generator
for $f_{2}$, this leads to a 4-fundamental case. By pigeonhole principle,
adding another generator would doubly-satisfy at least one note, meaning that
it would no longer be a basic edge case. Hence, there are no 5-fundamental
basic edge cases. ∎

It also proves possible to define restrictions on the presence of generators
for a false fundamental (with respect to the chroma configuration in which it
sits). In order to do this, the minimum number of generators that must fall in
certain proximity (such as the von Neumann and Moore neighbourhoods) to a
false fundamental can be considered.

In terms of the operators $\omega,\delta$, the von Neumann- and Moore-
neighbourhoods of some note $\nu$, are the notes generated by acting the
elements of the sets

$\\{\delta^{\pm 1},\omega^{\pm 1}\\}\quad\text{ and }\quad\\{\delta^{\pm
1},\omega^{\pm 1},\omega^{\pm 1}\delta^{\pm 1}\\},$ (18)

on $\nu$, respectively.

Figure 13: The potential generators (a ($f_{0}$), b ($f_{1}$), c ($f_{2}$), d
($f_{3}$)) for each configuration, and the von Neumann (red) and Moore (blue)
neighbourhoods.

The problem of choosing basic edge cases with the least generators in these
neighbourhoods can be reduced to the problem of choosing some a, b, c, and d
(corresponding to generators for $f_{0},...,f_{3}$) from Figure 13. This can
be achieved by choosing generators according to the following order,

  1. I.

Outside both neighbourhoods;

  2. II.

Inside Moore neighbourhood, outside of von Neumann neighbourhood;

  3. III.

Inside von Neumann neighbourhood (and $\therefore$ inside Moore
neighbourhood).

If multiple choices are available, it is sufficient to choose any one, without
loss of generality, as they have the same effect on the final count as one
another. Table 3 shows the resulting (minimum) counts of generators in the
neighbourhoods for false fundamentals of certain configurations.

| $\Gamma\Gamma$ | $\vdash\Gamma$ | $\Gamma\vdash$
---|---|---|---
a |  $\omega^{-2}$ (I) |  $\omega^{-2}$ (I) |  $\omega^{-2}$ (I)
b |  $(\omega\delta)^{-1}$ (II) |  $\delta^{-1}$ (III) |  $(\omega\delta)^{-1}$ (II)
c |  $\omega^{2}\delta$ (I) |  $\omega^{2}\delta$ (I) |  $\omega\delta$ (II)
d |  $\omega^{2}$ (I) |  $\omega^{2}$ (I) |  $\omega^{2}$ (I)
M | 1 | 1 | 2
v.N. | 0 | 1 | 0
Table 3: Table showing the minimum generators in the von Neumann (v.N.) and
Moore neighbourhoods of a false fundamental given its chroma configuration.

Though it may seem that choosing a generator that satisfies multiple parts
(i.e. $\omega^{-1}$ or $\omega$) may reduce the overall counts, it is always
possible in these cases to instead make choices that don’t reside in the von
Neumann neighbourhood.

Better understanding the occurence of edge cases is an important step towards
identifying them in practice, and gives a deeper understanding of the proposed
model itself. Sections 6 and 7 go on to look at reduction of edge cases to
potential basic cases, and the experimental prevalence of basic edge types,
and Sections 8 and 9 investigate the theoretical basis of the model from a
more experimental standpoint.

##  6 Reduction and Reducibility of Edge Cases

In order to gain a better understanding of the occurrence of edge cases (and
therefore the problem of pitch estimation), it proves useful to be able to
classify edge cases by which basic edge types they are related to. In order to
achieve this, it is necessary to reduce edge cases (i.e. remove redundancy) by
removing potential generators such that the false fundamental in question is
still preserved.

Figure 14: An irreducible non-basic edge case.

Given a set of generators, $\mathcal{G}$, that lie in $\varhexagon(\otimes)$
for some false fundamental, they are reducible iff any part, $f_{n}\ \in\
\\{f_{0},\ f_{1},\ f_{2},\ f_{3}\\}$ is satisfied more than once (i.e. non-
basic) - barring the exception outlined below. Reduction (denoted as
$\rightarrow_{g}$) takes $\mathcal{G}$, and removes some given generator $g\
\in\ \mathcal{G}$ such that the false fundamental is still satisfied by
$\mathcal{G}\setminus g$,

$\mathcal{G}\ \rightarrow_{g}\ \mathcal{G}\setminus g.$ (19)

Such a removal of a generator seeks only to remove its ‘fundamentalness’ - it
is entirely possible that it could still be generated elsewhere. Indeed, this
must be the case for any reduction via a generator in $\vdash\bigcup\ \Gamma$,
such as in Figure 15. Note that reduction is not unique; there may be multiple
valid reductions that can be applied to a given set of generators.

Figure 15: A reduction removing a generator in $\vdash\bigcup\ \Gamma$. Note
that $g(f_{2})$ is omitted for brevity.

It would be reasonable to assume that any non-basic edge case can be reduced,
therefore, to one of the eight basic edge cases (Table 2). On the contrary,
however, there exists a case that is both non-basic (i.e. each of its parts
are not satisfied exactly once) and irreducible - that is, that no potential
generator could be removed whilst preserving the false fundamental (Figure
14). Importantly however, this is the only irreducible non-basic edge case.

######  Proposition 6.1.

The only irreducible non-basic edge cases contain both $\omega^{-1}$ and
$\omega$.

###### Proof.

First, it is important to identify the root cause of the irreducibility of
$\\{\omega^{-1},\ \omega,\ g(f_{2})\\}$. Examining the generators of each part
of the false fundamental yields,

$\displaystyle f_{0}$ $\displaystyle:\ \omega^{-1}$ $\displaystyle f_{1}$
$\displaystyle:\ \omega^{-1}\ \lor\ \omega$ $\displaystyle f_{2}$
$\displaystyle:\ g(f_{2})$ $\displaystyle f_{3}$ $\displaystyle:\ \omega.$

Note that as before, the specific choice of generator for $f_{2}$ does not
matter, as it has no impact on the other generators.

As $f_{1}$ is satisfied by multiple generators, it is not a basic edge case.
In order to reduce it to a basic edge case, it would be necessary to remove
one of the generators of $f_{1}$ \- either $\omega^{-1}$ or $\omega$. Removing
either results in $f_{0}$ or $f_{3}$ respectively no longer being satisfied,
however, and thus neither can be removed whilst still preserving the false
fundamental. Thus, there are clearly multiple forms of irreducibility - beyond
the basic case.

  * •

Basic irreducibility - each $f_{n}\ \in\ \\{f_{0},\ f_{1},\ f_{2},\ f_{3}\\}$
is satisfied exactly once. Hence, removing any number of generators will
result in the false fundamental no longer being satisfied.

  * •

Non-basic irreducibility - each $f_{n}\ \in\ \\{f_{0},\ f_{1},\ f_{2},\
f_{3}\\}$ is satisfied, but in such a way that at least one is saturated (i.e.
has more than one unique generator), and removing any of the additional
generators results in the false fundamental no longer being satisfied. By
observation of Figure 13, this overlap only occurs between $\omega^{-1}$ and
$\omega$.

It is clear that there exist irreducibile cases containing
$\omega^{-1}\oplus\omega$ \- i.e. classes I and II (Table 2). Hence, the only
non-basic irreducible edge cases must contain both $\omega^{-1}$ and $\omega$.
Further, any case constructed to contain $\omega^{-1}$, $\omega$, and some
number of other fundamentals is trivially reducible back to $\omega^{-1}$ and
$\omega$.

Thus, the only non-basically irreducible edge case is the one containing
$\omega^{-1}$ and $\omega$. ∎

It is worth noting that there are really three such cases, based on arbitrary
choice of the generator for $f_{2}$.

Figure 16: An example of a reduction graph, with each step (arrow) showing a
reduction in the set of generators. Note that the special case of $\\{\omega,\
\omega^{-1},\ g(f_{2})\\}$ is denoted as ‘Type $\varnothing$’, and the
configuration is $\Gamma\Gamma$ or $\Gamma\vdash$.

Through repeated application of all possible reductions to the nodes (to which
reduction is yet to be applied), one may obtain a reduction graph for a given
set of generators, $\mathcal{G}$ (Figure 16). Given that no reduction could
ever produce a set of generators larger than the input, this graph will
additionally be acyclic. Further, the terminating nodes (i.e.
$\text{deg}^{-}(\nu)=0$) of such graphs correspond to irreducible cases. Such
a graph can, therefore, be used in order to understand the potential basic
edge types that correspond to a given set of generators.

##  7 Prevalence of Basic Edge Types

As previously mentioned, it is important to understand the occurrence of false
fundamentals, in order to better differentiate them from genuine ones. One
such way is to consider the prevalence of each type in practice. This can be
achieved by constructing a reduction graph for sample interpretations (from
all combinatorial possibilities) with varying numbers of fundamentals (and
their first three harmonics).

In order to sample interpretations from the total sample space, it proves
sufficient to construct them by selecting n unique notes,
$\nu_{0},\nu_{1},...,\nu_{n}$ from $\mathcal{N}^{\text{\makebox[6.25pt]{\hbox
to0.0pt{\vrule width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox
to0.0pt{\vrule width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern
0.5ptV\kern 0.5pt}}}_{\alpha}$, treating all such notes as fundamentals, and
thus adding them (and their harmonics) to the interpretation. For the charts
in Figure 17, a sample size of 1000 interpretations was taken for each number
of simultaneous fundamentals (0, 120] For each of these interpretations, a
naive algorithm (simply classifying all $\vdash$ and $\Gamma$-exhibiting notes
as fundamentals) was applied, and a reduction graph derived from each
$\otimes$ \- where the difference between the input set and the result of the
naive algorithm is the set of false fundamentals. From each of these reduction
graphs, all terminal nodes were classified either as one of the basic edge
types, or the special case, $\varnothing$. In cases where multiple terminal
nodes were present, a value was added to each tally such that the sum of all
added values was one.

Though 1000 interpretations may at first appear to be a relatively small
sample size, it should be noted that this corresponds to 120,000
interpretations sampled on the whole, with an average of 15.75 (16) edge cases
per interpretation. These are, as expected, concentrated around the centre of
the distribution (of total simultaneous fundamentals), as the number of total
possible edge cases peaks around the centre. Thus, on average, each set of
1000 interpretations leads to 1575 edge cases to classify, but with relatively
sparse distribution to the tail-ends (i.e. $<10$ simultaneous fundamentals),
which resulted in $<100$ edge cases being classified per 1000 interpretations.
In order to ascertain a more reliable picture of the makeup of edge cases -
particularly with low numbers of simultaneous fundamentals, a significantly
larger sample size of 20000 interpretations was used (Figure 18).

Looking at Figure 17, it is clear that not all basic edge types are equally
common. Subfigure 17(a) gives the overall occurrence of each type, with type 3
being the most common (along with the other three-fundamental cases), and type
5 being the least common (along with the other four-fundamental cases). The
special case, $\varnothing$ appears to sit between the two, which intuitively
coheres with other observations, as it too is a three-fundamental case -
albeit not basic. In general, it is hard to draw meaningful insight from this,
which incentivises the use of Subfigure 17(c) \- looking at the trends of the
prevalences as the number of fundamentals changes.

As Subfigure 17(c) shows, at low numbers of simultaneous fundamentals, the
three-fundamental cases are significantly more dominant than the four-
fundamental cases - constituting almost 100% of the cases until around 16
simultaneous fundamentals. Beyond this point, the incidence of four-
fundamental cases increases significantly - particularly types 6 and 8 - with
the special case $\varnothing$ notably occurring increasingly less often. As
before, it is hard to directly relate these results to real-world data (i.e.
recorded music), which is much more structured than the random samples that
were used, but a number of conclusions can still be drawn,

  * •

With low numbers of simultaneous fundamentals (eg. string quartet), cases 5-8
are incredibly unlikely to occur.

  * •

From Subfigure 17(d), it is clear that even at large numbers of fundamentals,
the accuracy of even the naive algorithm on polyphonic music - with perfect
noise removal, recording, playing, etc. - is above around 75%.

Regarding Subfigure 17(b), graphs appear to have between three and five (of a
possible nine) terminal nodes, with the average broadly decreasing as the
number of fundamentals grows. The trend appears more turbulent towards the
left tail, which is likely due to the low number of samples for these numbers
of fundamentals.

By combining this knowledge with a heuristic for the number of fundamentals at
a given $\mathcal{I}_{\tau}$, it may be possible to more easily distinguish
between fundamentals and false fundamentals by comparing specific examples to
the profile laid out above.

Figure 18 considers specifically the cases for which there are a low ($<10$)
number of simultaneous fundamentals. In these cases, the total occurrence of
four-fundamental basic cases is, on average, 3.6%, with the majority of these
weighted towards interpretations with $>6$ simultaneous fundamentals
(Subfigures 18(a) & 18(c)). Particularly interestingly, the most common case
in this subset of interpretations is the special case, $\varnothing$, with
20.4% of the total. Overall, the trend of three-fundamental cases being more
common remains, but the ordering within these groupings change - most notably
(beyond $\varnothing$’s jump) with type 5 cases being significantly more
prevalent than their counterparts compared to the data in Figure 17.

(a) A pie chart showing the average (proportional) prevalence of
each edge type, and $\varnothing$.

(b) A graph showing the estimated average number of terminal nodes for varying
numbers of simultaneous unique fundamentals.

(c) A stacked bar chart showing the change in proportional
prevalence of basic edge types and $\varnothing$ as the number of
simultaneous unique fundamentals changes.

(d) Another stacked bar chart, mirroring subfigure 17(c), but including the
proportion of non-edge cases.

Figure 17: Charts depicting various properties relating to the prevalence of
basic edge types with respect to the number of simultaneous unique
fundamentals.

(a) A pie chart showing the average (proportional) prevalence of
each edge type, and $\varnothing$ for total simultaneous fundamentals (0, 10].

(b) A graph showing the estimated average number of terminal nodes for varying
numbers of simultaneous unique fundamentals (0, 10] - corresponding to the
left hand side of Subfigure 17(b).

(c) A stacked bar chart showing the change in proportional
prevalence of basic edge types and $\varnothing$ for (0, 10] simultaneous
unique fundamentals.

Figure 18: Charts mirroring those in Figure 17, but considering only
interpretations with (0, 10] simultaneous unique fundamentals.

##  8 Experimental Application

Though this model is useful theoretically, in practice, real-world
applications are rarely so clear-cut or clean - and will remain so unless
there exists some perfect approach to noise removal, amongst other
preprocessing. Hence, it is prudent to look not at the discrete, but at the
continuous in intepretations, $\mathcal{I}$ \- i.e.
$\mathcal{I}:\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern
0.5pt}}}\to\mathbb{B}$ becomes
$\mathcal{I}:\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern
0.5pt}}}\to\mathbb{R}$.

Figure 19: The note G4 being played on a variety of instruments, all
exhibiting the $\Gamma$ shape described in Section 4.

Doing so effectively creates a heatmap, in which this additional dimension
(perpendicular to $\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$)
represents an ‘intensity’ of each note - for example, their respective
amplitudes in the frequency domain 222 This construction can be viewed as a
real rank 1 trivial vector bundle, with
$\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern 0.5pt}}}$
as a base manifold with trivial topology. In this interpretation, a heatmap is
a slice through the topological bundle.. Even in this kind of construction,
however, the $\vdash$/$\Gamma$ shapes are very much still prominent - as
demonstrated when applied to some monophonic signals from the University of
Iowa (Electronic Music Studios) (Fritts, 2012) (Figure 19). Here, the
intensity is visualised through brightness, with brighter notes representing
more prominent frequencies. Though timbrally very different, all of the
instruments shown clearly exhibit the $\Gamma$ shape as anticipated.

Despite this, there are clear differences in the prominence of these shapes
between the various instruments. Though flute and trumpet exhibit
exceptionally clean examples, the clarity in the piano and violin heatmaps is
- whilst still interpretable - somewhat diminished. This is likely a result of
multiple media (in the case of piano and violin, strings) vibrating in
sympathy to the true fundamental - particularly given that the strings are
housed in a shared body. Further, the resonance of this body may also have
contributed to the noise.

To build these models, sliding windows were taken from the signal, with a
length of 4096 samples, and a hop size of 1024. A constant-Q transform (Brown,
1991) with a Hanning window (Podder et al., 2014) (using the Librosa
implementation (McFee et al., 2015)) was then applied to achieve a frequency
domain representation binned by the 120 semitones of the western musical scale
between C0 and B9 inclusive. These values were then normalised across the
signal (not just per window), and plotted as a heatmap using matplotlib
(Hunter, 2007). Further, each window used here corresponds to a unique
interpretation, $\mathcal{I}_{\tau}$, where $\tau$ is the start time index of
the window.

It is worth noting that the shapes that appear to mirror the fundamentals and
their harmonics in chromatically adjacent columns (i.e. F$\sharp$ and
G$\sharp$ in the case of Figure 19) are a result of spectral leakage, which
has not been entirely nullified by the Hanning window. In practice, this could
likely be removed, or otherwise accounted for in specific algorithms and
approaches.

Figure 20: Left: Side-on view of a 3D heatmap of the melody of Bach’s “Ach
Gott und Herr”, from the Bach10 dataset (Duan and Pardo, 2015), with darker
colours corresponding to greater amplitudes. Right: Projection of the heatmap
onto the $\mathbb{Z}_{12}\times\tau$ plane, eliciting piano roll notation of
the piece (albeit ordered by the circle of fifths, and not chromatically).

Looking further, at the three-dimensional heatmap described in Section 4 (with
each $\mathcal{I}$ indexed as $\mathcal{I}_{[\tau,\tau+1)}$), Figure 20 is
achieved. With a projection onto the $\mathbb{Z}_{12}\times\tau$ plane, piano-
roll notation is achieved as expected. Algorithms working in this space may be
able to smooth the estimation in the temporal domain by better-exploiting the
temporal aspects of music; it is certainly a great oversimplification to treat
each window (and therefore each interpretation) as independent of one another.

This was achieved using Python’s vpython (Scherer, Dubois, and Sherwood, 2000)
module, representing each note as a black (#ffffff) cube, each with opacity
proportional to the loudest note within each specific interpretation333as
opposed to in Figure 19, where amplitudes are normalised across the whole
signal,

$Opacity_{\nu}=\frac{|\nu|}{\mathcal{I}_{MAX}}+0.05,$ (20)

with slight linear scaling to make each individual note stand out better. Note
that $|\nu|$ here refers to the amplitude of a given note, $\nu$, and
$\mathcal{I}_{MAX}$ refers to the largest amplitudes in a given
interpretation. It is worth noting that an approach using varying shades of
grey as opposed to solid black yielded less optimal results.

Figure 21: A closer look at how heatmaps differ as the note progresses from
onset/attack, to transient, to offset/decay (left-to-right).

Figure 21 shows the notable differences between heatmaps constructed from
windows from the onset, transient, and decay of a single note. As expected,
the shapes are clearest during the transient, but in general this raises the
more profound issue of choosing an appropriate window, or windows, when given
chunks of a signal - such as following onset detection. A simple yet effective
heuristic is to consider both the total number of bins filled above some
threshold $\alpha$ (eg. 3.25$\mu$ (Goodman and Batten, 2018)), and the total
magnitude of all bins above this threshold in a given window. That is,

$\frac{\sum\limits_{\nu\in N}\nu}{|\\{\nu\ |\ \nu\geq\alpha,\ \nu\in N\\}|},\
N=\\{\nu\ |\ \mathcal{I}(\nu),\nu\in\mathcal{N}^{\text{\makebox[6.25pt]{\hbox
to0.0pt{\vrule width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox
to0.0pt{\vrule width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern
0.5ptV\kern 0.5pt}}}_{\alpha}\\};$ (21)

effectively the average amplitude of an audible note in the window described
by $\mathcal{I}$. Doubtless there are more sophisticated approaches, but this
serves its purpose if nothing else but as a benchmark. Should time efficiency
not be of particular concern, of course, it may be optimal to consider all
windows in a chunk (taking their average result) - only discarding a handful
of particularly noisy or otherwise useless ones.

Figure 22: An edge case (specifically $\varnothing$) being exhibited on real
data (trumpet) - with D4, A5, and D6 (blue dots) as the fundamentals, and D5
being the false fundamental, $\otimes$.

Figure 22 depicts an edge case built up of the notes D4, A5, and D6, all
played on trumpet - exhibiting the special case, $\varnothing$. Though masked
somewhat by the spectral leakage on the right-hand side of the image, it is
clear to see how such edge cases fool the naive algorithm, even when a
threshold is utilised to cut out noise. Of course, the use of Algorithm 1
alleviates this by attempting to remove inharmonic noise, but in doing so, may
cause false negatives to arise. Note that when using Algorithm 1 on real-world
data, the same switch from $\mathbb{B}$ to $\mathbb{R}$ applies, insofar as
the lines $\mathcal{M}[i,j]=1$ be replaced by $\mathcal{M}[i,j]=|\nu_{i,j}|$.
That is, the amplitude of the bin as opposed to a boolean representing whether
or not the note corresponding to the bin was audible.

##  9 Evaluation

This section presents a brief evaluation of both a naive algorithm on
monophonic music, and a more sophisticated (albeit still simplified) algorithm
on theoretical polyphonic samples - similar to those utilised in Section 7. As
noted beforehand, the intention of this paper (and investigation on the whole)
is not to achieve state of the art results on MPE problems, but rather to lay
the foundations for more geometrical approaches to them. Thus, the evaluation
is brief, but nonetheless provides insight - particularly surrounding future
work.

For monophonic signals, a naive algorithm that treats the relation between
fundamentals and $\vdash$/$\Gamma$-exhibiting notes as an equivalence was
used. As shown in Section 4, this is untrue due to the presence of edge cases,
but nonetheless when only one fundamental is present, such cases can never
occur. In response to the spectral leakage, the algorithm was modified
slightly to take not only the shape of the potential fundamental and its
harmonics into account, but also the corresponding shapes at $\delta^{\pm 5}$.

This was applied to a total of 1395 monophonic samples from the University of
Iowa dataset, spanning 17 instruments in total (some of which were categorised
into vibrato and non-vibrato playing), resulting in a mean accuracy of 73.74%.
Removing the outliers (violin/viola/cello/double bass (pizz.), and tuba), this
average becomes 88.27%. When considering just whether the pitch chroma is
correct (i.e. diregarding octave errors), this increases to 95.08%. Table 4
benchmarks this against an implementation of Noll’s Harmonic Product Spectrum
(HPS) algorithm, also using a Hanning window.

A table containing a full breakdown of results, broken down by instrument (and
vibrato/non-vibrato playing), can be found in Appendix B.

| HPS | Naive
---|---|---
Overall | 58.36% | 73.74%
No Outliers | 67.73% | 88.27%
Chroma Accuracy | 77.61% | 95.08%
Table 4: Table showing the average accuracy of both the naive algorithm and
the HPS algorithm as a benchmark when applied to the University of Iowa
samples. Figure 23: Confusion matrices for each set of samples. From top-left
to bottom-right: 1) Alto Flute (vib.); 2) Alto Sax (non-vib.); 3) Alto Sax
(vib.); 4) Bass (pizz.); 5) Bass (arco); 6) Bass Clarinet (non-vib.); 7) Bass
Trombone (non-vib.); 8) Bassoon (non-vib.); 9) B$\flat$ Clarinet (non-vib.);
10) Cello (pizz.); 11) Cello (arco); 12) E$\flat$ Clarinet (non-vib.); 13)
Flute (non-vib.); 14) Flute (vib.); 15) Oboe (non-vib.); 16) Soprano Sax (non-
vib.); 17) Soprano Sax (vib.); 18) Tenor Trombone (non-vib.); 19) Trumpet
(non-vib.); 20) Trumpet (vib.); 21) Tuba (non-vib.); 22) Viola (pizz.); 23)
Viola (arco); 24) Violin (pizz.); 25) Violin (arco).

Though, as expected, this approach does not reach state of the art results, it
still outperforms HPS by a significant margin. Figure 23 consists of confusion
matrices for each of the instruments (and playing types), showing the
algorithm’s input (x-axis) against its output (y-axis). Thus, the line $y=x$
is indicative of perfect accuracy, and deviations from this line correspond to
errors in the classification. Note that the axes are truncated to match the
range of notes tested on each instrument, with the y-axis running
chromatically upwards from bottom to top, and the x-axis running chromatically
upwards from left to right. Even at first glance, the outliers are relatively
clear, and this kind of visualisation has the potential to elicit more
profound understanding of how and where an algorithm is failing, and perhaps
even (by extrapolation) particular properties of certain instruments that make
them more troublesome for pitch detection approaches.

Figure 24: Simulated accuracy for a naive approach (blue), and simple
algorithm (red) when applied to sample polyphonic data.

Further, for polyphonic input, a simple extension to the naive monophonic
approach, whereby $\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern
0.5pt}}}_{\alpha}$ is traversed left-to-right, bottom-to-top, was utilised.
This exploited the assumption that - at least with acoustic music - there will
be no undertones. Thus, the bottom-leftmost note with amplitude above some
threshold will always be a fundamental (Goodman and Batten, 2018). The naive
algorithm is then applied to subsequent notes, and for each potential
fundamental, the possible generators for each of its harmonics are enumerated
and checked against the current list of perceived fundamentals (i.e. those
that have already been classified as such by the algorithm). If two or more
(of a maximum four) of the fundamental and its generators have one or more
harmonics that have already been classified as a fundamental, the note is
considered to be a false fundamental, and is discarded. The choice of
threshold here may seem somewhat arbitrary, but was chosen as there are a
significant number of generators that lie above or to the right of the note
being classified - most notably the harmonics themselves. Thus, whilst they
may themselves be fundamentals, it is unclear at this stage of the algorithm.
This choice was then tested empirically, with a value of two (from choices [1,
4]) resulting in the best performance.

As in Section 7, 1000 sample interpretations were taken for each number of
simultaneous unique fundamentals. The accuracy of this simple approach is
benchmarked against the accuracy of the naive approach in Figure 24. Though
there is a clear increase in accuracy, it is anticipated that it will be
possible to build on this simplistic approach using the analysis and
techniques outlined in Sections 5, 6, and 7, but the implementation of this is
beyond the scope of this paper.

##  10 Summary and Future Work

Moving forwards, there are a number of improvements and implementations that
can be created off of the back of this framework.

Firstly, the simple polyphonic approach can be refined and extended using the
characterisations of edge cases outlined in Section 5. This could further be
augmented by combining the approaches in Sections 6 and 7 to perform a
‘backward pass’ over $\mathcal{N}^{\text{\makebox[6.25pt]{\hbox to0.0pt{\vrule
width=6.25pt,height=0.18082pt,depth=0.18082pt\hss}\hbox to0.0pt{\vrule
width=6.25pt,height=4.87373pt,depth=-4.5121pt\hss}\kern 0.5ptV\kern
0.5pt}}}_{\alpha}$, working right to left, top to bottom, and utilising
reduction to reconsider the likelihood of notes presenting as false
fundamentals. This can further use the working count of currently perceived
generators as a heuristic for the number of distinct simultaneous
fundamentals. This kind of forward-backward approach is a fundamental concept
in latent variable models - a concept which itself aligns well with the
problem of pitch estimation as set out in this paper.

In addition, it may be possible to reformulate the problem as a decomposition
of the total heatmap into its constituent (albeit constructively overlapping)
$\vdash$ and $\Gamma$ shapes, potentially using spectrogram subtraction to
represent this decomposition. This pivots the work towards combinatorics
rather than a necessarily more algorithmic perspective.

Further, one could build on the three-dimensional model to create approaches
that effectively utilise the temporal aspects of music in their predictions.
One such example may be the extraction of $\vdash$ and $\Gamma$-shaped prisms
from the extended heatmap - representing a note, or notes, sounding for some
period of time.

In conclusion, this paper presents a different perspective on approaching
pitch estimation problems - doing so from a more geometric perspective. To
this end, an idealised model of fundamentals and their harmonics is introduced
(and later adapted to real-world scenarios). Importantly, from a geometrical
perspective in particular, this model results in spatially-close shapes,
namely $\vdash$ and $\Gamma$. Further, it provides a framework on which to
approach pitch estimation problems in this way, along with a thorough
investigation into the edge cases that occur in this model.

Though the simple algorithms outlined in Section 9 do not provide state of the
art results, the intention is instead to provide a solid foundation on which
to construct more sophisticated algorithms - particularly utilising the
characterisation of, and insight into, edge cases. Furthermore, this is
certainly a step towards more intuitive and innovative geometrical solutions
in the field of pitch estimation, and Music Information Retrieval on the
whole.

## Acknowledgements

This work was supported by the EPSRC under Grant: EP/R513167/1.

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##  Appendix A Creation of an Interpretation, $\mathcal{I}$

Algorithm 1 Creation of an Interpretation, $\mathcal{I}$, from a Sorted Set of
Notes

0: $\Phi$, a chromatically sorted set of notes

0: $\mathcal{M}$, a (matrix) interpretation of the notes in $\Phi$

$\mathcal{M}\longleftarrow zeroes(10,\ 12)$

for $\nu_{i,j}\in\Phi$ do

// Is $\nu_{i,j}$ a harmonic of another note?

$f_{1}\longleftarrow\mathcal{M}[i,\ j-1]$

$f_{2}\longleftarrow 0$

if $\Psi(\chi_{i-1})\ =\ \vdash$ then

$f_{2}\longleftarrow\mathcal{M}[i-1,\ j-1]$

else

$f_{2}\longleftarrow\mathcal{M}[i-1,\ j-2]$

end if

$f_{3}\longleftarrow\mathcal{M}[i,\ j-2]$

if $f_{1}\lor f_{2}\lor f_{3}$ then

$\mathcal{M}[i,\ j]\ =\ 1$

continue

end if

// Is $\nu_{i,j}$ a potential fundamental?

$\phi_{1}\longleftarrow\nu_{i,j}\in\Phi$

$\phi_{2}\longleftarrow\bot$

if $\Psi(\chi_{i-1})\ =\ \vdash$ then

$\phi_{2}\longleftarrow\nu_{i+1,\ j+1}$

else

$\phi_{2}\longleftarrow\nu_{i+1,\ j+2}$

end if

$\phi_{3}\longleftarrow\nu_{i,\ j+2}\in\Phi$

if $\phi_{1}\land\phi_{2}\land\phi_{3}$ then

$\mathcal{M}[i,\ j]\ =\ 1$

continue

end if

$\Phi\longleftarrow\Phi\setminus\nu_{i,j}$

end for

return $\mathcal{M}$

##  Appendix B Full Results - Naive Algorithm (Monophonic)

Instrument | Type | HPS | Naive
---|---|---|---
1 | 2 | 3 | 1 | 2 | 3
Alto Flute | Vib | 88.89% | 88.89% | 88.89% | 97.22% | 97.22% | 97.22%
Alto Sax | Nonvib | 75.00% | 75.00% | 81.25% | 100.00% | 100.00% | 100.00%
Vib | 68.75% | 68.75% | 75.00% | 100.00% | 100.00% | 100.00%
Bass | Pizz | 20.19% | - | - | 53.85% | - | -
Arco | 36.54% | 36.54% | 39.42% | 71.15% | 53.85% | 72.12%
Bass Clarinet | Nonvib | 63.04% | 63.04% | 65.22% | 100.00% | 100.00% | 100.00%
Bass Trombone | Nonvib | 0.00% | 0.00% | 29.63% | 44.44% | 44.44% | 62.96%
Bassoon | Nonvib | 45.00% | 45.00% | 62.50% | 75.00% | 75.00% | 95.00%
Bb Clarinet | Nonvib | 84.78% | 84.78% | 84.78% | 97.83% | 97.83% | 97.83%
Cello | Pizz | 18.00% | - | - | 46.00% | - | -
Arco | 65.26% | 65.26% | 68.42% | 88.42% | 88.42% | 88.42%
Eb Clarinet | Nonvib | 82.05% | 82.05% | 82.05% | 94.87% | 94.87% | 94.87%
Flute | Nonvib | 94.59% | 94.59% | 94.59% | 100.00% | 100.00% | 100.00%
Vib | 94.59% | 94.59% | 94.59% | 100.00% | 100.00% | 100.00%
Oboe | Nonvib | 77.14% | 77.14% | 97.14% | 100.00% | 100.00% | 100.00%
Soprano Sax | Nonvib | 84.38% | 84.38% | 87.50% | 90.63% | 90.63% | 90.63%
Vib | 78.13% | 78.13% | 81.25% | 90.63% | 90.63% | 90.63%
Tenor Trombone | Nonvib | 33.33% | 33.33% | 66.67% | 78.79% | 78.79% | 100.00%
Trumpet | Nonvib | 51.43% | 51.43% | 82.86% | 74.29% | 74.29% | 97.14%
Vib | 51.43% | 51.43% | 82.86% | 74.29% | 74.29% | 100.00%
Tuba | Nonvib | 18.92% | - | - | 48.65% | - | -
Viola | Pizz | 22.00% | - | - | 28.00% | - | -
Arco | 88.00% | 88.00% | 91.00% | 100.00% | 100.00% | 100.00%
Violin | Pizz | 25.27% | - | - | 38.46% | - | -
Arco | 92.22% | 92.22% | 96.67% | 97.78% | 97.78% | 100.00%
Table 5: Table showing the performance of the naive algorithm on monophonic
samples from the University of Iowa Electronic Music Studios dataset,
benchmarked against Noll’s HPS algorithm. 1, 2, and 3 correspond to the whole
dataset, sans outliers, and chroma accuracy respectively.