Where Discovery Begins

# Music and Math, Mathematical Concepts in Music, Scales Harmony and Ratios, Math and Music

Music and Math, Mathematical Concepts in Music, Scales Harmony and Ratios, Math and Music – The Connection Between Math and Music – https://youtu.be/HLjWuxBsKvw

Music is based upon scales.

Enjoy the mathematical development of the most often used, octave based, 12 tone interval scale. Two versions developed here are 1) the 12 tone equal tempered scale (what we see and hear on typical keyboards) and 2) the 12 tone Just scale. These two scales differ by the different frequency ratios of the 12 tones in the scale in going from the tonic (ratio 1) up to the octave (ratio 2).

Examine possibilities of alternate numbered interval scales (other than 12) and see why the most popular octave based scale is the 12 tone equal tempered scale (equal frequency ratios between notes in the scale). This scale lends to the proficiency of keyboards and guitars. Most Western orchestras tune to this scale.

Guitars, keyboards, and wind instruments produce complex harmonic notes. Contrary to simple tones, these notes contain the fundamental frequency of the note in addition to an associated set of upper harmonic tones (partials) that are integer multiples of the frequency of the fundamental of the note being sounded. Learn how these associated upper harmonics play a key role in distinguishing the characteristic sound of different instruments (wind and string) playing the same fundamental note.

Especially for guitar players, learn how the position of the pluck point on a string on a fret board has a dramatic influence on the characteristics of the upper harmonics that sound with the fundamental note being played.

When two or more notes are sounded simultaneously, see how the characteristic harmony (consonant or dissonant), is mathematically based upon the frequency ratios of the two notes (intervals) or the three notes (triads) that are sounded together. This concept is mathematically and graphically demonstrated.

See how the perfect fifth interval with interval ratio 3/2 (3 to 2) explains the coincidence of every third harmonic of the root note with every second harmonic of the fifth. This high level of coincidence explains why the perfect fifth interval (example C-G) has the most satisfying (consonant) harmony between any two notes within an octave based, 12 tone scale.

See how the minor second interval with interval ratio 15/16 (15 to 16) explains the coincidence of every sixteenth harmonic of the root note with every fifteenth harmonic of the minor second note. This very low level of coincidence explains why the minor second interval (example C-C#) is the most unsatisfying (dissonant) harmony between any two notes within an octave based, 12 tone scale.