Maggie's Farm

The News Junkie: Sadly I can not understand this

As the work of Pythagoras implied, every musical tone entails the notes that make up modern harmony and melody. Bach showed how it all fits together, exploring the far reaches of this insight. Many of Bach's discoveries were not mined until the Romantics and then Jazz.

Pythagoras, experimenting with the length of strings, found that the most pleasing musical relationships were formed from simple ratios, such as 3:2 (perfect fifth) and 4:3 (perfect fourth).

When a string vibrates, it doesn't typically vibrate in a simple manner, all at once as it were, but many different vibrations are superimposed, multiples of the primary vibration.
https://flypaper.soundfly.com/wp-content/uploads/2016/01/string-harmonics-750.png
All of the pictured vibrations can occur on the same string, but are in a rational number relationship. The different amounts of each vibration creates the timbre of the instrument. For instance, a scratchy violin has more amplitude on the higher harmonics than does a sonorous flute.
https://www.musiccrashcourses.com/images/other/fourier_flute_violin.png

On the first diagram, look at the musical notion below the string. When you sound a C-note on the string, the string will also sound the C an octave higher (2x frequency), and the G above that (3x frequency), the C above that (4x) above that and so on. From the fourth harmonic, you can see the basic C-E-G triad chord, which, it turns out, sounds as harmonics from a simple pluck of a single string.

The following diagram shows how three harmonics add up to a non-sinusoidal wave form. The ear hears the first harmonic, the fundamental, as the pitch or note, and the sum of the harmonics as the timbre.
https://electrical-engineering-portal.com/wp-content/uploads/2016/01/harmonics-graphic.png

If were to have two strings tuned, one to C and one to a harmonic of C, and plucked the C, then the harmonic string will also ring in sympathy. The closer they are harmonically, the more it will ring. If you press the damper pedal on a piano, lifting the dampers on all the strings, and strike a note, then not only will that string ring, but so will all the strings harmonic to the struck note. This creates a richer timbre than if you strike the same note without pressing the damper pedal which lifts the damper on only the struck string. If you divide a string exactly in the middle (such as on a guitar), it will sound the C one octave higher. If you divide it to a third of its length, it will sound the G above that. And so on.

This is the basis of chords, harmonic progression, and even melody (the upper harmonics of the fundamental note).