Is there a ‘mathematical’ reason that the music notes in chords ‘work’ together?

Question

I recently learned that the wavelength of a note is half that of the same note one octave lower. Do the wavelengths of the notes in a chord have some sort of similarities? Is there another reason that the notes sound good together?

In: 21

Answer 1

The Western system of music breaks tones down into groups of 12 notes (counting the white and black keys on a piano).

The gaps between the notes of a musical scale are called “intervals”, and music theory is largely concerned with these intervals. The simplest interval to understand is the “octave” – where we go from one note – say a middle C – to the same note but higher up the scale – say the next C (one “octave” higher). That interval is defined by a doubling of the frequency of the note’s pitch and going up a full octave takes the full set of 12 (white and black) keys on a piano.

Other intervals between the notes in a scale can be described using a system of “just intonation”, where the ratios between the frequencies of notes on the scale are described in simple ratios such as 3:2 or 4:3. Certain intervals – with simple relationships like this – strike our ears as pleasant when the two notes are played together. This system proved impractical for tuning different instruments together and for transposing music between different keys, so since the 18th century we’ve generally used the “12-tone equal temperament” tuning system.

In this system, as you go up the 12 intervals in the scale, each note has a main frequency that is a fixed proportional amount higher than the frequency of the note before. Since it takes 12 intervals to double the frequency, each interval has a proportional increase in frequency of the 12th-root of 2.

The 12th-root of 2 is approximately 1.05946, so if we take the frequency (in hertz) of any note on the piano, the frequency of the next note (including white and black keys) will be 1.05946 times higher. Repeat this 12 times and we get a doubling of the frequency because we’ve gone up an octave.

This 12-tone equal temperament (“12-TET”) system loses the simpler, more natural, relationships between notes, and some people with very well-trained ears have a sense that some notes, played on well-tuned instruments, don’t feel quite right – they seem slightly sharp or flat.

There’s a video about this stuff here: https://youtu.be/bCYcS57eCqs