Twelve-tone scales

This page attempts to explain how musical notes relate to one another. It turns out to be one of those things that is much more complicated than it first seems.

First things first. You will be able to play the scales on the page, but just must make sure that your device is not muted, and the volume is turned up (but not too loud).

Once that's done, click the button to start the synth!

If at any point, the sound stops being made, simply click the button again and it should work! If not, a page reload, then button click should do the trick!

Some terms

Before we get started, let's introduce some of the terms we'll be using.

The interval abbreviations used on the page below are:

U

Unison

m2

Minor second

M2

Major second

m3

Minor third

M3

Major third

P4

Perfect fourth

TT

Tritone!

P5

Perfect fifth

m6

Minor sixth

M6

Major sixth

m7

Minor seventh

M7

Major seventh

O

Octave

Octave

The fundamental musical relationship is the octave. This is a doubling or halving of frequency.

If we start with the note A at 440 Hz, the octave above is two times this: 880 Hz. Similarly, we can drop an octave by halving the frequency to 220 Hz.

You can see this relationship on the spiral scale below. Each rotation inwards represents an increase of an octave.

Hover over or click the dark circles to play the root note (or unison) and the octave above it.

Dividing the octave

Octaves are all very well, but the harmonic possibilites are severely limited. How do we divide an octave up?

The next most fundamental ratio is 1.5 times the root frequency. This is known as a perfect fifth. A tonic of 440 Hz gives a perfect fifth (P5) of 660 Hz. Expressed as a ratio, this is 3/2 in relation to the tonic.

If you drop down below the tonic, dividing by 1.5, you end up at a frequency of ~293 Hz. We can move this up an octave by multiplying by 2, to end up at a freqency of ~587 Hz. Expressed as a ratio, the lower frequency 2/3 of the root, and the higher frequency is 4/3 above. This is the perfect fourth.

We're starting to build the scale of A:

Pythagorean tuning

If we take the fifth and the fourth, and apply the same process again, we will eventually generate all the notes in the key of A.

This is known as Pythagoreaon tuning or three-limit tuning. We are using ratios derived from the number 3 to generate our whole palette of notes. This is known as a "just intonation", as it is able to generate pure harmonies. We'll come back to this.

There are other schemes that enable generation of just intonation, allowing other numbers such as five and seven to be introduced into the ratios. These are known as five and seven limit tunings.

The full table of tones is listed below, in the order the are generated from the smallest ratio (in column two) to the largest. The notes are scaled into the 'home' octave, either dividing by or multiplying by two until the scaled ratio is between 1 and 2.

The eagled-eyed amongst you with musical knowledge will notice that the augmented fourth and diminished fifth are slightly different frequencies. In western music, these twins are generally considered equivalent. I refer to this as the tritone in the page. In an Pythagorean tuning for the scale of A, D♯ and E♭ are not quite the same. The gap between them is known as a Pythagorean comma.

Here's the Pythagorean scale in all its glory, including the two tritones!

12 Tone Equal Temperament

The problem with scales derived from just intonation is that they work really well in the home key, generating lovely pure harmonic intervals. So an A scale sounds right when playing in the key of A. Shift this to a different key (say the key of F), and what we would call a C when derived from the A tonic would be different. This results in disharmonious, clashy sounds known as wolf notes.

This is a massive problem for tuned instruments like pianos. It's not really any good to have a piano that you can only play in a small number of keys related to A, or whatever is the home key.

For this reason, a tuning scheme known as twelve-tone equal temperament (12-TET) was invented. This divides the octave into twelve equal ratios. There are 1200 cents in an octave, so a semitone (minor second) in 12-TET is 100 cents. This generates a good approximation for tuned instruments that plays well in all the scales.

Giles-tone™

Now comes the fun part. I recently had a go at coming up with my own scale, using a scheme that I dreamed up.

My plan was, multiply the previous note in the scale (starting with the tonic) by an increasing number (2, 3, 4, etc), then scale that down to the home octave by dividing by powers of 2. This would naturally generate duplicates, which I would discard. For every new note, I'd assign that to the nearest interval on the scale. I'm referring to this as Giles-tone™, or "2 Unlimited" tuning (no no, no no no no, no no no no, no no there's no limit). Here's how that panned out.

So how did I do? Well, it's good in parts, although generally pretty shocking. A few interesting points to note:

• The first three intervals that the scale generates are the perfect fifth, the major third and the minor seventh.
• The ratios of these are pretty harmonious. We're off to a good start!
• The minor seventh is technically the harmonic seventh, which is meant to be "sweeter in quality" than the just minor seventh. Fancy!
• The lower part of the octave is pretty OK.
• Things start to fall apart as we get to the fourth. This is far from perfect!
• From then on in it's a pack of wolves, particularly around the sixths!