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Topic: Harmonic series mathematics

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	Series (mathematics) - Wikipedia, the free encyclopedia
		In mathematics, a series is the sum of a sequence of terms.
		Series may be finite, or infinite; in the first case they may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way.
		Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète.
	www.wikipedia.org /wiki/Infinite_series (1682 words)

	Harmonic series (music): Definition and Links by Encyclopedian.com - All about Harmonic series (music) (Site not responding. Last check: 2007-11-07)
		Pitched musical instruments are usually based on some sort of harmonic oscillator, for example a string or a column of air, which can oscillate at a number of frequencies.
		The second harmonic (or first overtone) is twice the frequency of the fundamental, which makes it an octave higher.
		On most wind instruments, for example the saxophone, oboe, or bassoon, there is an octave key which opens a small hole in the tube, prompting the instrument to oscillate at the second harmonic and giving the second octave of the instrument.
	www.encyclopedian.com /ha/Harmonic-series-(music).html (429 words)

	Series (mathematics) (Site not responding. Last check: 2007-11-07)
		In mathematics, a series is a sum of a sequence of termss.
		Fourier series were being investigated as the result of physical considerations at the same time that Gauss, Abel, and Cauchy were working out the theory of infinite series.
		Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann (1701) and still earlier by Viète.
	www.sciencedaily.com /encyclopedia/series__mathematics_ (1548 words)

	Harmonic series (music)
		Put another way: since the harmonic series is an arithmetic series (1f, 2f, 3f, 4f...), and the octave, or octave series, is a geometric series (f, 2×f, 4×f, 8×f...), this causes the overtone series to divide the octave into increasingly smaller parts as it ascends.
		Similarly, the fourth harmonic partial is four times the frequency of the fundamental; it is a perfect fourth above the third partial (two octaves above the fundamental).
		Some harmonics correspond very nearly to named pitches; others, for example the 7th harmonic, are signifigantly off from the equal tempered tones.
	www.brainyencyclopedia.com /encyclopedia/h/ha/harmonic_series__music_.html (709 words)

	Harmonic series (music) -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		Some harmonics correspond very nearly to named pitches of the (Click link for more info and facts about equal tempered) equal tempered scale; others, for example the 7th harmonic, are significantly off from the equal tempered tones.
		The different harmonics are accessed by increasing the vibration of the lips against the mouthpiece, essentially by tightening the embouchure and blowing the air faster.
		This is not so useful as the same note could be sounded by pushing the string all the way to the (A narrow strip of wood on the neck of some stringed instruments (violin or cello or guitar etc) where the strings are held against the wood with the fingers) fingerboard at this point.
	www.absoluteastronomy.com /encyclopedia/h/ha/harmonic_series_(music).htm (1270 words)

	Encyclopedia: Harmonic series (music) (Site not responding. Last check: 2007-11-07)
		If the first 15 harmonics are transposed into the span of one octave, they approximate what the West has adopted as the major scale based on the fundamental tone.
		Rather than perceiving the individual harmonics of a musical tone, we perceive them together as a tone color or timbre, and we hear the overall pitch as the fundamental of the harmonic series being experienced.
		Notes that are not in the harmonic series are played by changing the effective length of the resonator, usually by opening a venting hole in the side of the instrument.
	www.nationmaster.com /encyclopedia/Harmonic-series-(music) (3688 words)

	Harmonic series (mathematics): Definition and Links by Encyclopedian.com - All about Harmonic series (mathematics) (Site not responding. Last check: 2007-11-07)
		In mathematics, the harmonic series is the infinite series
		This can be proved by noting that the harmonic series is term-by-term larger than or equal to the series
		This is a consequence of the Taylor series of the natural logarithm.
	www.encyclopedian.com /ha/Harmonic-series-(mathematics).html (231 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		In the case of a convergent series this could enable someone to determine the sum of an infinite series to a higher degree of accuracy than would be possible by simply adding a large number of terms.
		The $\omega$-series is a series of the form $\omega(j;k\sb 1,\cdots,k\sb n)=\sum\sb {i=0}\sp \infty(-1)\sp i/(j+s\sb i)$, where $j$ and $k\sb i (i=1,2,\cdots,n)$ are positive integers, $s\sb 0=0$, $s\sb n=S$, $s\sb i=[i/n]S+\sum\sb {t=1}\sp {\text i,\text{mod}\,n}k\sb t$.
		The harmonic series, both with and without alternating signs, and their relationship with the series obtained by continuing the progression backwards to $-\infty$ are discussed.
	www.mathematik.uni-bielefeld.de /~sillke/PUZZLES/harmonic-series (3362 words)

	harmonic
		In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integral multiple of the fundamental frequency.
		It is the amplitude and placement of harmonics and partials which give different instruments different timbre (despite not usually being detected separately by the untrained human ear), and the separate trajectories of the overtones of two instruments playing in unison is what allows one to perceive them as separate.
		Harmonics may be used to check at a unison the tuning of strings which are not tuned to the unison.
	www.fact-library.com /harmonic.html (544 words)

	Rearranging The Alternating Harmonic Series (Intro)
		Conditionally convergent series are those series that converge as written, but do not converge when each of their terms is replaced by the corresponding absolute value.
		It is clear, however, that even with such a simple example as the alternating harmonic series one cannot hope for a closed form solution to the problem of rearranging it to sum to an arbitrary real number.
		The latter two questions are completely answered by Riemann's theorem for rearrangements of arbitrary conditionally convergent series, but our goal is to provide a more concrete setting of Riemann's results within the context of the alternating harmonic series with the hope that the reader will then have a better understanding of the general theory.
	ecademy.agnesscott.edu /~lriddle/series/rear.htm (552 words)

	BAIN: The Harmonic Series (Overtone Series)
		For example, the series of frequencies 1000, 2000, 3000, 4000, 5000, 6000, etc., given in Hertz (Hz.), is a harmonic series; so is the series 500, 1000, 1500, 2000, 2500, 3000, etc. Notice that the difference between adjacent members of both series is constant, that is to say, the harmonics are equally-spaced.
		So that rather than perceiving the many individual harmonics of a musical tone, we ordinarily perceive an identifiable tone color, or timbre, whose pitch is associated with the fundamental of the harmonic series being experienced.
		During the Middle Ages (c476-1453) a Pythagorean tuning, a tuning derived from a series of six 3:2 fifths, was an accepted "standard" for tuning the seven tones of a diatonic scale.
	www.music.sc.edu /fs/bain/atmi02/hs/index-audio.html (3444 words)

	Encyclopedia: Harmonic series (mathematics) (Site not responding. Last check: 2007-11-07)
		Even the sum of the reciprocals of the prime numbers diverges to infinity (although that is much harder to prove; see proof that the sum of the reciprocals of the primes diverges).
		If p > 1 then the sum of the series is ζ(p), i.e., the Riemann zeta function evaluated at p.
		This can be used in the testing of convergence of series.
	www.nationmaster.com /encyclopedia/Harmonic-series-%28mathematics%29 (298 words)

	Harmonic series (music) - InfoSearchPoint.com (Site not responding. Last check: 2007-11-07)
		The difference in terms of frequency (measured in Hertz (Hz)) is the same between all partials, but the ear responds in a logrithmic fashion, so the partials sound 'closer' together.
		On most wind instruments, for example the saxophone, oboe, or bassoon, there is an octave key which opens a small hole in the tube, prompting the instrument to oscillate at the second harmonic partial and giving the second octave of the instrument.
		The amplitude and placement of different partials determine the timbre of different instruments, and among a number of psychoacoustic factors, the separate envelopes of the partials two instruments playing in unison is what allows one to perceive them as separate.
	www.infosearchpoint.com /display/Harmonic_series_(music) (606 words)

	Harmonic series (mathematics) (Site not responding. Last check: 2007-11-07)
		Even the sum of the reciprocals of the prime numbers diverges to infinity (although that is much harder to prove; see here).
		The series is convergent if p>1 and divergent otherwise.
		When p=1, the series is the harmonic series.
	www.ukpedia.com /h/harmonic-series-mathematics-.html (202 words)

	Rane Professional Audio Reference (H)
		harmonicity The degree to which a sound's timbre conforms to a harmonic series (Thanks to Scott Wilkinson for this succinct definition).
		A series whose terms are in harmonic progression, such as 1 + 1/2 + 1/3 + 1/4 + 1/5 +...
		A series of tones consisting of a fundamental tone and the overtones produced by it, and whose frequencies are consecutive integral multiples of the frequency of the fundamental.
	www.rane.com /par-h.html (3531 words)

	Proof that the Harmonic Series Diverges
		Many series have a boundary that marks the limit of their growth.
		The series of powers of ½ is an example of this type of series with a limit and is called the "Harmonic Series."
		This topic, the limits of infinite series, is one of the most fascinating topics in all of mathematics.
	users.rcn.com /mwhitney.massed/Harmonic_Series/Harmonic_Series.html (645 words)

	Harmonic Analysis of Spherical Functions on Real Reductive Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. ... (Site not responding. Last check: 2007-11-07)
		The harmonic analysis of spherical functions treated here contains the essentials of large parts of harmonic analysis of more general functions on semisimple Lie groups.
		Since the latter involves many additional technical complications, it will be very illuminating for any potential student of general harmonic analysis to see how the basic ideas emerge in the context of spherical functions.
		Mathematicians and graduate students as well as mathematical physicists interested in semisimple Lie groups, homogeneous spaces, representations and harmonic analysis will find this book stimulating.
	www.uni-protokolle.de /buecher/isbn/3540183027 (272 words)

	Series (mathematics)
		Examples of simple series include arithmetic series which is a sum of a arithmetic progression which can be written as:
		and geometric series which is a sum of a geometric progression which can be written as:
		This argument doesn't prove that the sum is equal to 2, but it does prove that it is at most 2 -- in other words, the series has an upper bound.
	news-server.org /s/se/series__mathematics_.html (628 words)

	Annals of Mathematics, II. Series, Vol. 149, No. 3, pp. 785-829, 1999 (Site not responding. Last check: 2007-11-07)
		The author uses a general analysis on the defect measures and energy concentration sets associated with a weakly convergent sequence of stationary harmonic maps between Riemannian manifolds $M$ and $N$.
		Secondly, he studies the asymptotic behavior at infinity of stationary harmonic maps from $\bbfR^n$ into a compact Riemannian manifold $N$ with bounded normalized energies.
		He also shows that if analytic target manifolds do not carry any harmonic $S^2$, then the singular sets of stationary maps are $m\le n-4$ rectifiable.
	www.emis.ams.org /journals/Annals/149_3/3.html (214 words)

	Real and Complex Analysis (Higher Mathematics Series)
		Furthermore, its problems are not mere extensions of the proofs given in the text or trivial applications of the results- many of the results are alternate proofs to major theorems or different theorems not proved.
		This, in turn, is used to prove Plancherels theorem and the uniqueness of Fourier transforms as a character homomorphism.lt;pgt;The tenth chapter, on basic complex analysis, essentially covers an entire undergraduate course on the subject, with added results based on a solid knowledge of topology on the plane.
		lt;pgt;Most of the basic results from the power series perspective are covered in the text, but while the geometric view is examined, it is still done in a very analytic, formula-based way that does not allow the reader to gain too much intuition.
	www.wkonline.com /a/Real_and_Complex_Analysis_0070542341.htm (1370 words)

	Mathematical Series Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-07)
		Looking For mathematical series - Find mathematical series and more at Lycos Search.
		Find mathematical series - Your relevant result is a click away!
		Look for mathematical series - Find mathematical series at one of the best sites the Internet has to offer!
	www.karr.net /search/encyclopedia/Category:Mathematical_series (215 words)

	Mathematics 216: Assignment 1
		Consider next the question of whether the partial sums approach a limit L. This happens if the n-th partial sum s(n) gets closer and closer to L as n gets larger and larger.
		One way a series can diverge is by getting larger and larger, with out bound.
		This fact was already known in the fourteenth century to Nicolas Oresme, a thirteenth century Scholastic philosopher.
	www.math.utah.edu /classes/216/assignment-01.html (401 words)

	Mudd Math Fun Facts: Thinned-Out Harmonic Series
		You're probably already aware that the harmonic series, which is the sum of the reciprocals of all natural numbers, diverges.
		For instance, it may appear that this series is divergent, especially when contrasting it with sums of reciprocals of numbers with one or more 9's.
		That series diverges (easy to show), and this series seems to have "more" terms in it...
	www.math.hmc.edu /funfacts/ffiles/20005.3.shtml (303 words)

	CBMS Regional Conference Series in Mathematics, Vol. 94: Saltman, J.:
		mathematical gems still used in university curriculae today.
		mathematics in a conceptual, entertaining, and accessible way.
		This work is at the crossroads of a number of mathematical areas,
	www.yurinsha.com /101/p2.html (758 words)

	Harmonic series (Site not responding. Last check: 2007-11-07)
		SeeHarmonic series (music)Harmonic series (mathematics)These two concepts are related.
		This is a disambiguation page; that is, one that just points to other pages that might otherwise have the same name.
		It is cruel that we must be torn apart who unconquerable power and authority rose in me. The fighting must be some way, and John Carter, who has fought his way conscious mind a series of nine long forgotten sounds.
	www.termsdefined.net /ha/harmonic-series.html (243 words)

	Harmonic Analysis and Integral Geometry (Research Notes in Mathematics Series): 紀伊國屋書店BookWeb
		Harmonic Analysis and Integral Geometry (Research Notes in Mathematics Series): 紀伊國屋書店BookWeb
		Comprising a selection of expository and research papers, Harmonic Analysis and Integral Geometry grew from presentations offered at the July 1998 Summer University of Safi, Morocco-an annual, advanced research school and congress.
		Several articles are devoted to the new theory of Radon transforms on trees.With its related presentations addressing recent developments in various aspects of these intriguing areas of study, Harmonic Analysis and Integral Geometry becomes an important addition not only to the Research Notes in Mathematics series, but to the general mathematics literature.
	bookweb.kinokuniya.co.jp /htmy/1584881836.html (169 words)

	Musical Mathematics: Manuscript Pages
		a practice in the mathematics of tuning instruments and analyzing scales
		Forster's 1300-page manuscript, please visit our Musical Mathematics page, which shows the Table of Contents of this work.
		If you would like to see Musical Mathematics in print, please write to us so that we may include your emails and letters in our grant applications.
	www.chrysalis-foundation.org /Manuscript.htm (309 words)

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