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.

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Mathematical Concepts in Music, Scales Harmony and Ratios.

Often, you hear the phrase ” Music and Mathematics are the same”. This book addresses some of the basic underlying mathematical concepts and connections with the music we listen to. The focus here is quite rigorous and pragmatic and very much unlike most typical books in music. The Pythagorean, the just intoned, and the equal tempered scales are the most prominent octave-based musical scales that simply differ in the distribution of successive note ratios in going from the tonic up to the octave. The mathematical basis for the note ratios that define these three scales is developed here. The 12 tone scale is the most prominent of all equal tempered scales. A procedure for generating alternate numbered equal tempered intervals is shown for targeting preferred ratios such as perfect fifths and major thirds within scales. Vibrating strings are ubiquitous in the concert piano, string instruments, and guitars. The Fourier analysis and harmonic synthesis of vibrating waves on strings demonstrates the unique dependence of harmonic structure and sound quality upon the initial setup conditions of the string. Beat phenomena within intervals of simple tones leads to the analysis of intervals of complex harmonic notes and the interactions of the upper partials of these notes. Based on work done by Helmholtz and, more recently, by Plomp and Levelt, we present the development of a mathematical model that provides a quantitative measure of the sensory perception of consonance and dissonance in tonal harmony. The concept of tonality is explored in terms of the mathematical structure of triads and seventh chords. This mathematical approach lends itself very nicely to the numeric representation and quantitative analysis of musical objects. The author, George Articolo is an emeritus professor of mathematics at Rutgers University and is an avid jazz musician.