This blog is part of a series wth all my music notes. In the Series page you can find all the posts of the series.

Music representations

There are a large number of music representations. Most of them fall into two categories:1

  • Literal notation: a perfect representation of the music sound usually given as a time series.
  • Symbolic notation: high-level representation of the structure of the music sound.

Literal notation is used for the most part for recording or signal processing. For example, digital recording uses files of sampled sounds. A mono sound file is simply a large array containing the sound amplitude for each sampled point in time.

Alternatively, most musicians and music researchers use different type of symbolic notations. These notations hide the details of the sound, concentrating instead more in the salient feature of the music. This approach to music notation is designed to assist musicians when performing, as is the case for example of music sheets.

Musical note

The most straightforward symbolic representation is the musical note. In music, a note is a symbol denoting sound.2 At a minimum, notes express the duration and pitch of a sound. The duration is just how long a sound persists. The pitch of a sound represents a perceptual property of the sound that makes it possible to judge sounds as “higher” and “lower” in a sense associated with musical melodies.3

I want to emphasize that a note’s pitch is an abstract musical object. It cannot be extracted directly from a sound because a pitch represents a property of a sound and the way a listener perceives it. However, it is possible to account for the listener’s perception using reference sounds like a simple-tone (sinusoidal) sound produce by a generator. For example, the Acoustical Society of America (ASA) defines the pitch of a sound may be described by the frequency of that simple tone having a specified sound pressure level that listeners judge to produce the same pitch.4

Simple pitch space

Based on the above discussion, I will advance the following definition.

Definition (simple pitch space): Pitch is an abtract musical object related to a musical note. A collection of single pitches forms a simple pitch space that is denoted by $\mathcal{S}$. Associated to the pitch space there is a function $p: \mathbb{R}_{>0} \rightarrow \mathcal{S}$ such as for each frequency $f$ we can assign the pitch $p(f)$.

I like to emphasize that each element of the simple pitch space is a single or simple pitch. I will discuss musical objects based on two or more pitches later in the series. Also, it is always possible to assign a pitch for a given frequency value. However, the other way around is often not true, as one pitch might be related to a collection of frequencies.5

I will say that any set that satisfies the definition of simple-pitch space is a particular instance or representation of a simple-pitch space. Below, I show two examples of simple-pitch space instances.

Example (Positive reals as single pitch space): The set $\mathbb{R}_{>0}$ and with the function $p: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}; f \mapsto cf$ for any constant $c \in \mathbb{R}_{>0}$ is a single pitch space.
Example (Reals as single pitch space): The set $\mathbb{R}$ and with the function $p: \mathbb{R} \rightarrow \mathbb{R}_{>0}; f \mapsto b \log_a(cf)$ for any constants $a, b, c \in \mathbb{R}_{>0}$ is a single pitch space.

The simple pitch space as a concept is not very practical due to its lack of structure or regularity. The definition, in essence, says that we can map it to an abstract object called a pitch for a frequency value. In the next blog, I will improve the above description by identifying a more exciting collection of pitches within a simple pitch space.

Disclaimer

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Expect conceptual errors and constant updates to the whole blog series. Be skeptical and critical of this series content. Ultimately, you are responsible for the information you consume.

References

  1. This clasification was inspired by Rhythm and transforms by William Sethares, chap2, page 23 ↩

  2. Wikipedia entry: Musical note. ↩

  3. Wikipedia entry: Pitch (music). ↩

  4. ASA definition of pitch Annotation 2. ↩

  5. In chromatic scale, the standard pitch $A_4$ (or $A$ of fourth octave) is defined as having a frequency of $440 \text{khz}$. However, if octave is ignored (equivalence of the octaves) then $A$ is associated to the following set of frequencies $\big\lbrace 2^{(n-4)} \cdot 440 \text{khz} \vert n \in \mathbb{Z} \big\rbrace$. The index $n$ enumerates the octaves. ↩