Terminology for sounds and notes
I'm struggling with the terminology for all the concepts related to sounds and notes. Most of the texts I read are not very precise in their usage of different terms, which makes it hard for me as a beginner to fully grasp the different concepts and get a clear understanding of their differences.
From a theoretical point of view, I'd like to distinguish the following concepts:
Anything we can hear (i.e. an osciallation of pressure in a compressible media)
The same as #1 , but consisting of only a single frequency (e.g. a 440Hz wave, this is a purely theoretical construct)
The same as #2 , but including overtones (this one depends on the instrument used and is the "real" counterpart to #2 )
One element of a scale (e.g. a C, independent of frequency)
Any concept of the above + information about length/duration
The terms which I've so far encountered in relationship with the concepts above are:
Sound (corresponds to concept 1)
Tone (although I'm not sure whether this is used in standalone form in English)
Note (can correspond to concepts 2, 3, and 4, according to Wikipedia)
Noise (sometimes used for concept 1, as in "I heard a noise")
Pitch (according to Wikipedia, "Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale" and "may be quantified as a frequency")
Timbre (according to Wikipedia, this refers to the collection of properties which distinguish two entities of concept #3 for the same fundamental frequency, i.e. harmonics and envelope)
Amplitude and Loudness
There already exists a similar question and answer, but it distinguishes only pitch, note, timbre, and tone.
My background is mathematics, and hence I'm used to very precise definitions. I had expected that all these terms are defined more or less precisely, especially since music has strong ties to physics and mathematics. If this is not the case, how do musicians and music theorists usually resolve the resulting imprecision?
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Anything we can hear (i.e. an osciallation of pressure in a compressible media) - This is a "sound". Keep in mind that the media does not have to be compressible; we can hear sounds underwater.
The same as #1 , but consisting of only a single frequency (e.g. a 440Hz wave, this is a purely theoretical construct) - This is generally the definition of the countable noun "tone". The noun "tone" is both countable ("three distinct tones") and uncountable ("a bassoon's tone"); the uncountable noun is less concrete.
The same as #2 , but including overtones (this one depends on the instrument used and is the "real" counterpart to #2 ) - This starts to get into the definition of the "uncountable" noun "tone". It can also be described more precisely as "timbre"; the unique collection of fundamentals and overtones produced the "timbre" of an instrument's "tone" which distinguishes that instrument from any other when heard by a listener.
One element of a scale (e.g. a C, independent of frequency) - This may be termed a "note" but is more accurately a "pitch". The fact that the pitch is part of a scale is largely irrelevant; pitches are the calibrated, named semitonal steps in the Western music system.
Any concept of the above + information about length/duration - This is most properly a "note". "A C# quarter-note", given an overarching tempo and meter for the music in which the note is played, defines both the pitch and the space of time it should be played.
The mathematically-thinking musician is going to use these terms significantly differently from what might be common practice. That doesn't necessarily make this less complicated, but we can nail down quite a lot of definitions when given context and verbosity. (That is, a higher incidence of usage of specific terms is going to confine the definitions of those terms that would otherwise be less specific.)
"I heard a sound" and "I heard a noise" are functionally indistinguishable, but usually refer to "more pleasant" and "less pleasant" things, respectively, due to social constructs.
Note is one of the more liberally used terms, but when put in the context of verbose usage of other more specific terms, usually refers to a pitched sound of definable length. On paper, this would refer to a symbolic representation of that. Sometimes used the same way (although less in the paper context, hence the term notation), is the word "tone."
Tone is also used, perhaps moreso, in the context of "quality of sound." In this context, it is synonymous with timbre, although usually used at a more granular level. i.e. Timbre would be used to distinguish between one instrument, synthesizer, percussion instrument or another, whereas tone would be used to compare different types of a single instrument with the same playing technique, or the quality of sound production on any one instrument.
Pitch is a more specific way to refer to frequency (and only frequency) than "note". The wikipedia definition you cite is functionally identical to "it's the frequency of oscillation." A strict relationship of "note name" to frequency is dependent on the historical and practical contexts of tuning system. In music, we typically refer to pitch by a note name and then interpret specific frequencies only when necessary to match others not using an equal-tempered A440 basis.
Rhythm is important to define, since it refers to specifically quantized points in time marking the beginnings and ends of sounds.
To answer the last part of your question; where the terminology is imprecise, we usually rely on context to a very high degree to understand what is being communicated.
Look for a book on "Acoustics" which is a discipline of physics, sometimes also called "the physics of sound".
Some comments (IMHO):
One element of a scale is called a "pitch". A "pitch" is any sound which has a regular waveform oscillating at a stable fundamental frequency that can be clearly discerned. A pitch is quantified in Hz, or Hertz, or cycles per second.
"Noise" is sound which does not have a discernable oscillating, repeating waveform and therefore no dominant, discernable pitches.
However, notes created by musical instruments can contain noise components, such as the "thunk" of the hammer when a piano key is pressed, and many other examples.
A note is a pitch which is produced by a musical instrument and functions in a harmonic or musical context.
"Timbre" refers to the distinctive sound created by individual musical instruments, and involves the fundamental, all the overtones, the noise components, and how they change over time (the envelopes). However, timbre is such a complex phenomenon that it must be appreciated as an inherently subjective quality. How does the timbre of a Stradivarius violin compare to a Guarnieri violin, for example? How does the bouquet of one vintage of Cabernet Sauvignion compare to one of another vintage?
All musical instrument sounds, if examined in granular detail, change drastically over time. The first milliseconds of the strike of a note on the guitar sound very different than the sound being made several milliseconds later, and throughout the duration of the note. This is what we refer to as the envelope of the sound. Noise components, change in fundamental pitch, the change in amplitude of the various harmonic overtones, the overall change in volume or loudness, and the introduction or modulation of vibrato and tremolo are some factors. Yet to the untrained ear a clear single pitch is perceived. Find a reference work that displays three-dimensional graphs of fast Fourier transforms (FFTs) of various pitches played by different kinds of musical instruments to see what I'm referring to.
Unfortunately, while music, sound and acoustics are all forms of physics, musicians themselves usually approach music from a naïve perspective in that they know a lot about musical expression but they don't know anything about mathematics or physics as a formal discipline. Therefore musicians and music educators don't always use consistent, precise nomenclature.
If you look for books about the physics of sound and about audio engineering and acoustics, you will find more information that appeals to your mathematical background.
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