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Joseph An

The Divergent Infinite Series: The Harmonic Series

The Harmonic Series is a concept that was first publicly introduced into the scientific world

by Nicole Oresme. Oresme, however, is accredited with the main glory of revealing the existence of

the harmonic series, but rather were Johann Bernoulli and Jacob Bernouilli, who revamped the topic

into the scientific community three centuries later in the 1600s.

The harmonic series was historically used for architectural purposes primarily in the Baroque

period to establish harmonic relationships between the exterior and interior features of floor plans

and of elevations (Hersey). In music, the first term, or the fundamental wavelength (Pierce), is

considered to produce the pitch the human ear perceives when a note is played by an instrument.

However, when a harmonic-capable instrument produces a note, a harmonic series is produced,

with smaller partial waves being emitted as well (Pierce). Although the note itself is perceived as

the first term of the harmonic series, the whole harmonic series produced by the instrument creates a

timbre, or quality of sound unique to an instrument (professional musicians can distinguish between

different instruments when the same note is being played by multiple instruments. This is a result of

their attuning to the harmonic effects produced by the individual instruments and ultimately, the

timbre of each sound). Timbre is also referred to as the tone of the sound (Thompson).

The first term, the fundamental wavelength

when a string of an instrument is played.

(The fundamental wavelength, and the

overtones/harmonics produced when a string is

played)

Examples of instances when harmonic sounds are producedbow pulling on a string,

strumming a guitar.

Examples of instances when inharmonic sounds are producedplucking a string, piano

hammer striking a piano string.

(Note that the term inharmonicity refers to the occurrence of when a tone is produced, the


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partials produced do not strictly adhere to the rules of the harmonic series which are explained

below).

A principle example of a diverging infinite series denoted by:

, this summation represents the sum of fractions of constant integer multiples of one another. As the

terms increase, the numerical value of each term is less than the terms preceding it ( a3 is less than a2

ora1). Graphically, the series is represented by a logarithmic curve that is exponentially decreasing.

(Graph by Jim Belk, an assistant

professor of mathematics at Bard

College)

Note that many students misinterpret the harmonic series as a sequence of terms, not as a

summation of terms. For example, students misinterpret as the limit of 1/n as it

approaches infinityessentially a sequence. Although the infinite sequence of 1/n converges to 0,

the infinite series of 1/n diverges to infinity. Although we cannot measure if the harmonic series

reaches infinity (infinity being an abstract concept), we can accurately hypothesize it does so. One

relatively simple method to evaluate the harmonic series is by use of one of Oresmes postulates.

Oresmes Postulate (pre-grouping)

= 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 +1/7 + 1/8 +. . . + . . . =

Oresme postulated that since the summation of 1/1 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + . . . + . . .

equals infinity., and that grouping the harmonic series by the 2, 4, 8, 16 , terms (excluding the first

two terms), results in each group being greater than 1/2 the harmonic series must diverge toinfinity. The grouping is illustrated below.


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Oresmes Postulate (post-grouping)

= 1/1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 +1/7 + 1/8) +. . . + . . . = (line 1)

= 1/1 + 1/2 + (7/12) + (533/840) + . . . + . . . = (line 2)

If line 2 was extended, it would be evident that each grouping that is deviated from line 1 is

greater than 1/2. Therefore, since 1/1 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + . . . + . . . = , line 2 must

also be true, and the harmonic series diverges to infinity.


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Works Cited

Hersey, George L.,Architecture and Geometry in the Age of the Baroque, p 11-12 and

p37-51.

Pierce, John R. (2001). "Consonance and Scales". In Perry R. Cook.Music, Cognition, and Computerized Sound. MIT Press.ISBN978-0-262-53190-0.http://books.google.com/books?id=L04W8ADtpQ4C&pg=PA169&dq=musical+tone+harmonic+partial+fundamental+integer.

Thompson,William Forde (2008).Music, Thought, and Feeling: Understanding the

Psychology of Music. p. 46. ISBN978-0-19-537707-1.

http://www.oup.com/us/catalog/general/subject/Psychology/CognitivePsychology/?view=usa&ci=97

80195377071.
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