Music and Mathematics - Frequency and Harmony

Frequency and Harmony

A musical scale is a discrete set of pitches used in making or describing music. The most important scale in the Western tradition is the diatonic scale but many others have been used and proposed in various historical eras and parts of the world. Each pitch corresponds to a particular frequency, expressed in hertz (Hz), sometimes referred to as cycles per second (c.p.s.). A scale has an interval of repetition, normally the octave. The octave of any pitch refers to a frequency exactly twice that of the given pitch. Succeeding superoctaves are pitches found at frequencies four, eight, sixteen times, and so on, of the fundamental frequency. Pitches at frequencies of half, a quarter, an eighth and so on of the fundamental are called suboctaves. There is no case in musical harmony where, if a given pitch be considered accordant, that its octaves are considered otherwise. Therefore any note and its octaves will generally be found similarly named in musical systems (e.g. all will be called doh or A or Sa, as the case may be). When expressed as a frequency bandwidth an octave A2–A3 spans from 110 Hz to 220 Hz (span=110 Hz). The next octave will span from 220 Hz to 440 Hz (span=220 Hz). The third octave spans from 440 Hz to 880 Hz (span=440 Hz) and so on. Each successive octave spans twice the frequency range of the previous octave.

Because we are often interested in the relations or ratios between the pitches (known as intervals) rather than the precise pitches themselves in describing a scale, it is usual to refer to all the scale pitches in terms of their ratio from a particular pitch, which is given the value of one (often written 1/1), generally a note which functions as the tonic of the scale. For interval size comparison cents are often used.

Common name Example
name Hz 		Multiple
of fundamental		 Ratio
within octave		 Cents
within octave
Fundamental A2, 110 1x 1/1 = 1x 0
Octave A3 220 2x 2/1 = 2x 1200
2/2 = 1x 0
Perfect Fifth E4 330 3x 3/2 = 1.5x 702
Octave A4 440 4x 4/2 = 2x 1200
4/4 = 1x 0
Major Third C♯5 550 5x 5/4 = 1.25x 386
Perfect Fifth E5 660 6x 6/4 = 1.5x 702
Harmonic seventh G5 770 7x 7/4 = 1.75x 969
Octave A5 880 8x 8/4 = 2x 1200
8/8 = 1x 0

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