Theory of Music #1

What is music? It is the art of creating air vibrations. Air is to musician what canvas is to a painter or wood to a sculptor. Air carries vibrations which are registered by our hearing organs and interpreted by the brain. In the vacuum (despite what we see in “Star Wars”) there is no sound nor music.

To trivialize further, we cannot forget about the theory of memes (memetics), according to which music is a virus preying on human brains. The virus evolves when we consciously, or less consciously, introduce variations to melodies that are “stuck in our head”. It even rubs off on us – when someone else picks up the tune we've been humming. It was probably in the course of such evolution that the tribal percussion music has evolved to its current state.

Surely a less controversial perspective would be deeming music as one of the fine arts. However we will try to show that, despite its “fineness”, it is a very strongly formalized field which is governed by strict rules. In this and in the following lessons we will be gradually stripping music off its magic and breaking it into pseudomathematical rules.

If you think - “I am too old for this, because you're supposed to learn music in your childhood”, you are, of course, wrong – it is worth to bring up the example of Leonard Cohen here, who started his career after the age of thirty. We can provocatively declare: there won't ever be a better moment to learn music!

Let's start by identifying the sounds which music can operate on. In the spectrum of sound accessible to the human ear, which is approximately 20 – 20000 Hz, there are innumerably many frequencies. Frequency can be any real number within this range. Creating music would be incredibly complicated if we had to look for the right sound in such a huge collection, that is why humanity decided to simplify things. How? Well, we decided to use only a tiny subset, that is, several dozen “acceptable” frequencies. Every tone with an “unacceptable” frequency will be considered as “false” - for example, such sounds are emitted by an out of tune instrument.

Frequency (number expressed in hertz, being the number of air vibrations per unit time), describes what is commonly called the pitch of sound. Let's not confuse this quantity with volume (usually measured in decibels) or timbre (rather being a descriptive parameter).

What frequencies are acceptable? It is precisely defined by an international standard. There is a simple rule which will allow us to calculate them with arbitrary precision. The only information we need is:
1) one of the acceptable tones has a frequency of 440 Hz (musicians would call it “A from the one-line octave”, but we will discuss it later)
2) The frequencies of two closest acceptable tones differ from each other by the factor of $\sqrt[12]{2}$. Yes, by a factor, not a constant value. This implies that the “musical” frequencies make up a geometric, rather than arithmetic sequence.

If you place a spectrum analyzer (for example a guitar tuner) to a piano, you will notice that pressing any key will create a tone of a frequency equal to: $$f = 440 \cdot (\sqrt[12]{2})^n \;\textrm{[Hz]}$$ where integer parameter $n$ varies depending on which key we press. Excellent! We don't have to worry about the continuity of frequency, because modern music operates solely on “quantized” pitches. Why so? It is a difficult question, and the best answer is probably: because it has been adopted this way.

In history, all kinds of systems have been considered. For example, some Asian cultures have assumed a 22nd root as the distance between the closest frequencies. “Our” recipe for sounds (called the equal temperament system) is simple and very general, which will become evident when we get to the concept of key and the circle of fifths.

What exact property does the number $\sqrt[12]{2}$ hold? Among others $(\sqrt[12]{2})^{12}=2$, meaning every twelve keys on the piano there is a tone with double the frequency. Such “distance” between tones is called an octave. Synchronous playing of tones differing in an octave has a very nice effect – both sounds resonate without grumbling, because both waves have “matching” lengths!

The number $\sqrt[12]{2}$ is so important that it has earned itself a special name. Musicians call it a semitone, although they would have probably introduced this term differently. What is the exact value of the semitone? It is an irrational number equal approximately 1.0594. Now we can see why it is so difficult to sing or play the violin – a 1% mistake in frequency is audible as false, and a mistake of 5.94% is already the next tone! Pianists and guitarists don't know this problem!

If there is any solace to this, it is that singing can be learned, and it is something that the author of this text has been testing himself for years. Also, the beta-testers of Temptonik have made terrific progress in a couple of months. Really, it's worth trying!

We already know that “compatible” pitches appear every 12 semitones. This is what we will do now: we'll give names to eleven tones, and then we will say that those 11 names repeat in other octaves, just like subsequent steps are repeated on various floors of a building. Wait, wait, what do you mean “they repeat in other octaves”? The octave was supposed to be the distance between frequencies (equal to 12 semitones) and not some collection of pitches. Unfortunately the term octave has been used ambiguously. And so, not only distances between “compatible” pitches are called octaves, but also specific, eleven-pitch sets. A drawing is worth a thousand words.

To distinguish the octaves from lowest to highest the following names have been coined: sub-contra, contra, small, great, one-line, two-line, three-line... It is important to note here that in Temptonik it doesn't matter which octave you are singing in (that is, you can sing correct pitches, but from a different octave, with no influence on the result). Unfortunately the traditional names given to subsequent tones are not on point – e.g. the first tone in an octave (meaning: in the eleven-pitch set) is C. We need to remember that in Continental Europe the note B is called H (which looks even worse).

In our childhood we used to call the pitches differently: do, re, mi fa, sol, la si. They are the exact equivalents of C, D, E, F, G, A, B, respectively, but we're not going to use that convention here.

Why haven't we named the black keys yet? There is a reason for it: the black keys have ambiguous names. The names of the black keys come from the names of the neighboring white keys:

We can see that the following names can be used interchangeably: C♯=D♭, D♯=E♭, F♯=G♭, G♯=A♭, A♯=B♭. But... are they interchangeable for sure? For now we're not going to worry about that, but we guarantee that some small problems will emerge from it. If you already want to find out what they are you can read about the phenomenon of enharmonics, but please be warned that it might be a difficult journey.

In conclusion: we already know which frequencies we can use when creating music and the names that they bear. We also know that the situation is simple: there are 11 different pitches, and the rest are repetitions in the subsequent octaves. Unfortunately, we can still compose an ugly melody even if bound to the acceptable tones. It is impossible to extract an “unacceptable” pitch on a correctly tuned piano, but that does not guarantee that each sequence of pressed keys will sound beautiful. Composition of pitches is governed by separate rules, which we will touch upon in the next lessons.

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