HARMONIC SERIES

.

Crosswords: HARMONIC SERIES

English words defined with "HARMONIC SERIES": first harmonic, fundamental, fundamental frequency. (references)

Specialty definitions using "HARMONIC SERIES": Craig effect ♦ harmonic series of sounds, harmonics of the earth's gravitational fields. (references)

See

• Harmonic 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. If you followed a link here, you might want to go back and fix that link to point to the appropriate specific page.

---

Harmonic series (mathematics)

(From Wikipedia, the free Encyclopedia)

See harmonic series (music) for the (related) musical concept.

In mathematics, the harmonic series is the infinite series

It diverges, albeit slowly, to infinity. This can be proved by noting that the harmonic series is term-by-term larger than or equal to the series

which clearly diverges. Even the sum of the reciprocals of the prime numbers diverges to infinity (although that is much harder to prove; see here). The alternating harmonic series converges however:

This is a consequence of the Taylor series of the natural logarithm.

If we define the n-th harmonic number as

then H_n grows about as fast as the natural logarithm of . The reason is that the sum is approximated by the integral

whose value is ln(n).

More precisely, we have the limit:

where γ is the Euler-Mascheroni constant.

Lagarias proved in 2001 that the Riemann hypothesis is equivalent to the statement

where σ(n) stands for the sum of positive divisors of n.

The generalised harmonic series, or p-series, is (any of) the series

for p a positive real number. The series is convergent if p>1 and divergent otherwise. When p=1, the series is the harmonic series. 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.

See also

• The harmonic mean.
  
  

  ---

  
  
  

  Harmonic series (music)

  (From Wikipedia, the free Encyclopedia)

  This article is about the harmonic series in music theory. See harmonic series (mathematics) for the related mathematical concept.

  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 integer multiples of the lowest frequency make up the harmonic series.

  The lowest of these frequencies is called the fundamental or first partial. This is the note you get when with normal bowing of a stringed instrument or from the lowest octave of a woodwind instrument. All of the other frequencies in the harmonic series are integer multiples of the fundamental. 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.

  The second partial 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 partial and giving the second octave of the instrument. On brass instruments, the second harmonic is the lowest playable note. The fundamental is called a pedal note or pedal tone and can be faked.

  The third harmonic partial, at three times the frequency of the fundamental, is a perfect fifth above the second harmonic. 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). Note that double the partial number means double the frequency, which in turn means the 'pitch' is an octave higher. For example, the 6th partial G is an octave higher than the 3rd partial G.

  After that the harmonics come thick and fast, getting closer and closer together. Some harmonics correspond exactly to named pitches; others, for example the 7th harmonic, lie between the semitones.

  For example, given a fundamental of C', the first 16 harmonics are:

  • 1st C'
  • 2nd C
  • 3rd G
  • 4th c
  • 5th e (this, and the following odd-numbered partials are "out of tune" in terms of equal temperament)
  • 6th g
  • 7th b-sesquiflat
  • 8th c'
  • 9th d'- (the so-called 'major tone')
  • 10th e' - (the so-called 'minor tone')
  • 11th f'-semisharp
  • 12th g'
  • 13th a'-semiflat (but out of tune)
  • 14th b'-sesquiflat
  • 15th b' natural
  • 16th c''
  • 17th c-sharp'' but out of tune

  
  An illustration of the harmonic series above, as musical notation. Not all the "wrong" notes are marked as such - see text for more details.

  If you have a player capable of reading Vorbis files (for example Winamp 3), you can listen to A'' (110 Hz) and 15 partials by clicking here.

  In just intonation all notes are exact in regards to the harmonic series, and all intervals are based on ratios of the lower integers. In modern equal temperament, notes are approximate, so that music can be played in any key without retuning. See musical tuning.

  Harmony in Western music, especially the major chord, is based on the lower pitches of the overtone series. Since it uses equal temperament, however, only the octave is exactly in tune.

  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. Not all musical instruments have partials that exactly match the harmonic partials as described here. The partials of Piano, and other, strings are increasingly sharper than partials because the strings are stiff, leading to nonlinear, inharmonic effects. See Piano acoustics.

  See also:

  • harmony
  • dissonance
  • FM synthesis
  • missing fundamental
  • overtone