In music, the most harmonious frequency interval is the octave. Two notes are separated by an octave if one is twice the frequency of the other. In western music, the octave is divided into \(12\) notes, with the frequency ratio between adjacent notes being \(2^{1/12}\). A frequency interval with this ratio is called a semitone. The \(12\) notes that divide the octave are called the chromatic scale. The following table shows the frequency ratio of each note to the first note. The notes are numbered from \(0\) to \(12\). Note \(12\) is one octave up in frequency from note \(0\).
| Note | Frequency Ratio | Rational approx |
|---|---|---|
| 0 | 2^0 = 1.0 | 1/1 |
| 1 | 2^(1/12) = 1.059463094359295 | 18/17 = 1.0588 |
| 2 | 2^(1/6) = 1.122462048309373 | 9/8 = 1.125 |
| 3 | 2^(1/4) = 1.189207115002721 | 6/5 = 1.2 |
| 4 | 2^(1/3) = 1.259921049894873 | 5/4 = 1.25 |
| 5 | 2^(5/12) = 1.334839854170034 | 4/3 = 1.3333 |
| 6 | 2^(1/2) = 1.414213562373095 | 7/5 = 1.4 |
| 7 | 2^(7/12) = 1.498307076876682 | 3/2 = 1.5 |
| 8 | 2^(2/3) = 1.587401051968199 | 8/5 = 1.6 |
| 9 | 2^(3/4) = 1.681792830507429 | 5/3 = 1.67 |
| 10 | 2^(5/6) = 1.781797436280679 | 9/5 = 1.8 |
| 11 | 2^(11/12) = 1.887748625363387 | 15/8 = 1.875 |
| 12 | 2^(1) = 2.0 | 2/1 |
Two notes with frequency ratios that can be approximated as a ratio of small integers will be the most harmonious. So note \(0\) and note \(7\) will be the most harmonious, since their frequency ratio is approximately \(3/2\). The next most harmonious combination is note \(0\) and \(5\) which has a ratio of \(4/3\). Two notes next to each other are the most dissonant.
Let's take a \(12\) bead necklace with beads of two colors (also called binary) to represent chords (simultaneously played notes) of the chromatic scale. Using colors black and white, there are \(30\) binary necklaces with no two adjacent beads being black, ignoring the one with all white beads. These are shown below.
If we take the rightmost bead to be the \(C\) note, and label beads counterclockwise with the \(12\) notes (C,C#,D,D#,E,F,F#,G,G#,A,A#,B) in ascending order of frequency, taking a black bead to mean the corresponding note is played, we can listen to these \(30\) necklaces as chords below.
30 chords with no adjacent notes
© 2010-2017 Stefan Hollos and Richard Hollos
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