Theory of Music #4

We already know single tones. The natural order of things would be to consider pairs of pitches now.

Why? Make an experiment and try to play any random song from YouTube on any instrument. You will start by locating the first sound by the method of trial and error. There's nothing wrong with that – only a person gifted with absolute pitch is able to recognize the frequency without a prior “calibration of the ear”.

Later it will get more difficult. You will be trying short sequences and checking if “it works” every couple of minutes. In the end it will “somewhat” work, but some phrases may sound suspicious. Does that mean that you have a tin ear and it's time to give up on music? No. Many people are capable of playing by ear and it's pretty improbable that all of them are phenomenally talented.

You just don't know how to look for pitches yet. It turns out that it is the distances between the subsequent pairs of sounds, and not the absolute values of frequency, what should be our guide to melodies. Specific distances have a very distinct sound, so soon you will be able to recognize, name, play, and sing them.

First, a little terminology. The distance between two pitches is called an interval. The number of scale degrees between two pitches determines the name of the interval. The number of semitones decides whether the interval will be “minor” or “major” (we will talk more about this soon).

A pair of sounds is melodic if two pitches happen directly one after another, or harmonic when they happen simultaneously (the latter option rather cannot be sung if you don't know the techniques of polyphonic singing).

Above we introduced the definitions of the so-called simple intervals, that is intervals no larger than an octave. If you ever come across a compound interval (meaning larger than an octave), for example the nona, you will easily handle it.

Simple examples (first we give the lower and then the higher pitch)

Why don't the fourth, fifth, and octave come in “minor” and “major” versions? First, notice that a “major fourth” would be the same as “minor fifth”. Second, on the pitches of the major and minor scales we would be able to build only one “major fourth” and only one “minor fifth” (check it!). Additionally, the prima, fourth, fifth, and octave are the nicest resonating intervals. That's why they are called “pure”.

Definitions of intervals are just words. Where are the promised “distinct sounds”? Well, you can try and remember the sounds of intervals through persistent playing, listening, and guessing. However we suggest a different method: try associating intervals with famous songs:

Unfortunately it's not enough to just read the theory in order to become proficient at recognizing intervals. You also have to practice. From now on, whenever you hear a radio jingle or the announcement at a train station, ask yourself: what intervals are heard in it? In your spare time try to play two sounds on the piano and without looking at the keyboard guess the distance between them. Don't worry if the beginnings are hard – it's especially easy to mix up the wide intervals. Meanwhile, we suggest to go back to theory for a little bit.

There are two reasons for this: first, we omitted one interval (which one?). Second, in the previous lessons we promised to familiarize you with enharmonics. For our examples to be stronger, we will try to maximally complicate them, and in order to do so we will introduce the concept of derivative intervals.

Derivative intervals are defined as our known intervals, but expanded or narrowed by a semitone. We are then speaking of, respectively, diminished and augmented intervals.

Examples of derivative intervals:

As you are already aware, there is no “major fourth” and “minor fifth. We can however, consider the augmented fourth and diminished fifth. Both intervals are equal to 6 semitones and they have received a common name: tritone. The tritone is a quite disturbing, “drilling”, and less common interval.

So far we have managed to pass in silence the existence of such chromatic signs as the double sharp (𝄪) and double flat (𝄫). Now we will need them, but there is nothing to fear: the double sharp raises the pitch by two semitones, and the double flat lowers it by two semitones. The resulting pitches have names that are intuitive enough (C𝄪, D𝄪, E𝄪...; C𝄫, D𝄫, E𝄫...).

Why so redundant? Why define, e.g. augmented third, which has the same span as pure fourth? Why define augmented sixth if it has exactly same span as minor seventh? Well, as you know, we can achieve a given pitch by approaching it from the top with a flat (or double flat) or by approaching it from the bottom with a sharp (or double sharp). Meanwhile, the name of the interval is determined by the number of scale degrees separating the unmodified pitches (“non-sharp” and “non-flat”). In some cases it can therefore turn out that the correct naming requires the usage of augmented and diminished intervals...

Without further ado, advanced examples (we are first indicating the lower and then the higher pitch):

Please, promise that you will thoroughly study each one of the above examples – thanks to them you will understand why sometimes we write “G♯” and other times “A♭”, despite having the same pitch in mind. We're warning you that most of these examples are purely theoretical and it's hard to imagine a situation where they would come to existence. However that doesn't excuse us from understanding the correct notation of enharmonic pitches!

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