Intervals, It Is All The Story

Barbican Center, London. Monday, March 3rd, 2025.

I spent the last few days in diving more into Music Composition world. What intriguing me is the Logical structure in it. A perfect application for mathematical thinking.

I started with two books, the first "Harmony, Melody & Composition" by Paul Sturman, which looks like a simplified composition course textbook. The second, "Fundamentals of Musical Composition" by Arnold Schoenberg, an "advanced" composition book, with analytical & critical approach, rather than just a textbook.

All Music books, agreed on the importance of "Intervals" in Music, and for me, I think, it's all the story. Let's try to anatomize it. First, what we mean by Intervals in Music? simply, and loosely saying, it's the distance. However, it more about differentiation and extraction rather than a measurement. So, distance of what? Distance of the frequency of sounds. How far is "Pitch" D from "Pitch" C, without getting into physical matters (real distance like we do in the "Dozan" process).

As known, music built over "Octave", the same Octal Numbering in Computer Science, except the notation is different. "They" divided the distance into:

Unison. U1.
Major/minor Second. (M/m)2.
Major/minor Third. (M/m)3.
Perfect Four. P4.
Perfect Fifth. P5.
Major/minor Sixth. (M/m)6.
Major/minor Seventh. (M/m)7.
Perfect Eighth. P8.

As you may notice, (Major/minor, Perfect), when we say Major/minor, we mean (Whole Tone, Half/Semi Tone) e.g. (C to D is Major Second (M2), while C to C# is minor second). The perfect, as they call it, "It perfect! so, it can't be semi!". The naming, as I read is back to Church Music, and their believes.

Each interval, has it personality. And, based on it we can build the Scale, which leads to The Harmony. However, and again, I emphisaize, it important to understand & look at these intervals as a way to differentiate more than measuring. So, instead of saying from "Pitch" middle C, go to Pitch "C3", you can say from "Middle C, play the P5(~~C3)". A massive simplication, and reduction.

I can't go more deep, unless I explain the Scales, and how it built & works. I can, based on my understanding so far, to say that scales is nothing more than applying specific intervals patterns to a "pitch", to derive a progression of "notes". The famous two Scale, is the Major & Minor scales. Their Patterns:

Major:

Whole Tone – Whole Tone – Half Tone – Whole Tone – Whole Tone – Whole Tone – Half Tone. Simply: (W W H W W W H).
Minor:

Whole Tone – Half Tone – Whole Tone – Whole Tone – Half Tone – Whole Tone – Whole Tone. Simply: (W H W W H W W).

So, to build a scale, let's say C Major Scale, pick the C pitch, and apply the intervals to. The result will be a set of tones, represent the language you should follow in your music. The scales is huge subject, just do a simple mathematical combinatorics process, to figure out how how many scales you can built. Needless to say, not all scales are great. And as they are an application of the intervals, so each scale have it personality & character!

Now, we built the scale, to get into composition, we need to understand it notes, and how it interact & affect the piece we want to write. "They" classify each tone as follow:

The First, is the Tonic Tone, which is the strongest & heavily affect, and the scale name is based on it (C is the Tonic in Major C).
The Second, is the Super Tonic.
The Third, is the Mediant.
The Fourth, is the Sub Domint.
The Fifth, is the Dominant.
The Sixth, is the Sub Mediant.
The Seventh, is the Leading.

Each one is a topic by it self! .. I have alot more, but no time I have.