On some previous blogs i’ve written on, I had a series of articles that I called “The God Chord” where I talked about tuning, the overtone series, and all the magical connections between numbers, math, frequencies, geometry and music. I think that the simplicity and beauty of the numbers behind music have been obscured behind arcane music theory systems that confuse the intuitiveness of the origin of the fundamental materials of sound and music.
I’ve talked to a lot of people and musicians who are intimidated by music theory. I think there is a reason for this–western music theory doesn’t make any sense! I consider all the words and ideas in old-school music theory to be the product of a centuries long cultural evolution that never gave a second thought to being clear or logical.
Sine waves (odd number harmonics) combing to create a square-wave shaped waveform.
Anyway, enough ranting. Let’s get down to the numbers. After familiarizing yourself with how musical frequencies are related, thinking about music from a mathematical standpoint is a lot more intuitive and direct. There are two basic materials that everything in music is based on: octaves, and the overtone series.
Musical notes are described numerically as frequencies, or in “cycles per second,” denoted by the term “hertz” or “hz.” The wonderful thing about frequencies is that they are also used to describe the electromagnetic spectrum, including visible light, radio, micro, gamma, and x-rays, or, if you want, the rotation and orbits of planets, anything that repeats periodically. The incredible thing about all of this, is that musical frequencies can be related to all kinds of phenomena–the light emitted by the sun, the resonances in minerals and atoms, with simple ratios. This begins to explain why I’ve titled this series “the god chord.” Because, I think, if God really exists somewhere, I think God is a frequency. Or a melody. Or, most likely, a CHORD.
If you need to catch up on your frequency theory, check WIKIPEDIA!
So, as I mentioned before, I wanted to describe octaves, and overtones. Let’s start with frequencies, and octaves.
Musical frequencies are Logarithmic. For instance, if we are in the key of A (440 hertz), the next octave up would be at 880 hz, and below, 220 hz, and so on. All of these frequencies, that are powers of 2 of the original frequency, are considered the same note, but just at different locations. This phenomenon is highly useful and fundamental to nearly all music in all cultures. If you were to describe this in ratios, with 440 hz being 1/1, then 880hz would be 2/1, 1760hz (the next octave up) would be 4/1, and so on. 220 hz would be 1/2, 110hz would be 1/4, etc. incidentally, 2/1 and 4/1 are both Overtones. Any frequency you can imagine has duplicates of itself in octaves going toward infinity both above and below, beyond the range of human hearing, and the scale of human perception.
OK. Now, let’s talk about the overtone series. The overtone series is also described by mathematicians as the Harmonic Series. Like the octaves I was just describing, the overtone series is infinite, only limited by the range of human hearing and the physical properties of whatever object is projecting the frequencies. When analyzing a musical tone, you can clearly see the overtone series in its frequency components:
Harmonic partials in a piano tone.
Let’s start with a frequency of 440 hertz.
The overtone series is simply the original frequency, multiplied by every integer, starting with the number one. so,
440*1 = 440
440*2 = 880
440*3 = 1320
440*4 = 1760
440*5 = 2200
and so on, until infinity.
In musical notation, the overtone series looks approximately like this, though most notes in the overtone series can’t be accurately represented in this kind of notation:
approximate representation of the overtone series on a musical staff
This very simple mathematical series is the root of all musical scales and harmonies.
It sounds like this:
It might sound vaguely familiar to you. If you’ve ever played the harmonics up and down a guitar string, those are the same frequencies. Periodic (repeating) musical tones are all built from these frequencies. In fact, here is a simple recording of me playing the harmonics up a guitar string, until the 8th harmonic in the harmonic series.
You can count along with the notes, and each number will be the corresponding harmonic: Just count:
1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1.
The first harmonic is the “fundamental” and that is just the sound of me playing an open string. If you listen close, you can hear certain intervals. For instance, the 3rd harmonic is a perfect fifth, the 5th harmonic is a major third (doesnt’ make sense, right?). I personally love the sound of the seventh harmonic, which is a lot like a “minor seventh,” but is actually about 30% of a semitone flat from the interval you would play on a piano.
As opposed to an octave, this series is an Arithmetic Progression, not a logarithmic one. the octaves of the fundamental frequency, are the fundamental multiplied by 2, 4, 8, 16, 32, 64…. So, you can see that, between each octave further up the overtone series, the overtones become more closely spaced. At some point, the frequencies become so close together that it is difficult to for the human ear to distinguish them.
Allright. I hope you’re still with me! After these fundamentals, comes the interesting stuff. Now, we can combine the concept of the octave and harmonics to make all kinds of scales and harmonies. The trick is to take the frequencies in the overtone series and move them downwards (or upwards) with octave displacement to make the tones fit into a normal musical scale. For instance:
As described above, the fifth overtone of A 440 is 2200 hz. 2200 divided by 4 (this moves the note down two octaves) is 550 hertz. That is the exact frequency of a harmonically pure major third (the note C# in the key of A). On a well-tuned piano, the note would be about 554 hertz. A small but noticeable difference. And when you put more of these notes together in harmonies, you can really hear the beauty of the tones interlocking with their pure mathematical relationships!
Here is a demo I made just to show some very simple, overtone-based harmonies. Hopefully you can hear the quality of the overtone series, that I showed in the previous audio examples here, but applied in a looser context. Just a very simple improvisation, but I can’t get enough of those pure harmonies.
Hope you made it all the way through, and I hope I grabbed your interest! Check back, I will be writing more about this subject, with more illustrations and audio examples.
