The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.

The geometric mean is also one of the three classical Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean.

For all positive data sets containing at least one pair of unequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between ( see Inequality of arithmetic and geometric means.

Replacing the arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result.

In this scenario, using the arithmetic or harmonic mean would change the ranking of the results depending on what is used as a reference.

In mathematics, a generalized mean, also known as power mean or Hölder mean ( named after Otto Hölder ), is an abstraction of the Pythagorean means including arithmetic, geometric, and harmonic means.

In mathematics, the harmonic mean ( sometimes called the subcontrary mean ) is one of several kinds of average.

The harmonic mean H of the positive real numbers x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub > is defined to be the reciprocal of the arithmetic mean of the reciprocals of x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub >:

The harmonic mean of 1, 2, and 4 is

If a set of weights, ..., is associated to the dataset, ...,, the weighted harmonic mean is defined by

The harmonic mean is a special case where all of the weights are equal to 1.

It is also equivalent to any weighted harmonic mean where all weights are equal.

It is possible to recursively calculate the harmonic mean ( H ) of n variates.

For the special case of just two numbers and, the harmonic mean can be written

In this special case, the harmonic mean is related to the arithmetic mean

Similar to the view of Leopold Kronecker that " God made the integers ; all else is the work of man ," musicians drawn to the alphorn and other instruments that sound the natural harmonics, such as the natural horn, consider the notes of the natural harmonic series — particularly the 7th and 11th harmonics — to be God's Notes, the remainder of the chromatic scale enabled by keys, valves, slides and other methods of changing the qualities of the simple open pipe being an artifact of mere mortals.

It is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal which has been buried under noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies.

If the speed of the fluttering is close to a harmonic of the control's movement, the resonance could break the control off completely.

His next important breakthrough was in the Opus 33 string quartets ( 1781 ), where the melodic and the harmonic roles segue among the instruments: it is often momentarily unclear what is melody and what is harmony.

Evaluation of the related second derivatives allows the prediction of vibrational frequencies if harmonic motion is estimated.

The vibrational energy is approximately that of a quantum harmonic oscillator:

Therefore x ( t ) = cos t. This is an example of simple harmonic motion.

Both natural harmonics and artificial harmonics, where the thumb stops the note and the octave or other harmonic is activated by lightly touching the string at the relative node point, extend the instrument's range considerably.

Of course, if two stations transmit on the same frequency, it is practically impossible for the receiver to separate them ; so instead of all stations transmitting at the same frequency, each chain was allocated a nominal frequency, 1f, and each station in the chain transmitted at a harmonic of this base frequency, as follows:

The euphonium ( like the baritone ; see below for differences ) is pitched in concert B, meaning that when no valves are in use the instrument will produce partials of the B harmonic series.

A harmonic is a frequency that is a whole number multiple of a lower register, or " fundamental " note of the flute.

In other contexts, it is more common to abbreviate it as f < sub > 1 </ sub >, the first harmonic.

( The second harmonic is then f < sub > 2 </ sub > = 2 ⋅ f < sub > 1 </ sub >, etc.

According to Manolo Sanlúcar E is here the tonic, F has the harmonic function of dominant while Am and G assume the functions of subdominant and mediant respectively.

Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis.

which is Laplace's equation, the solutions to which are called harmonic functions by mathematicians.

Schmidt is generally, if erroneously, regarded as a conservative composer ( such labels rest upon yet-to-be-resolved aesthetic / stylistic arguments ), but the rhythmic subtlety and harmonic complexity of much of his music belie this.

This last summation is the harmonic series, which diverges.

The resulting theory is a central part of harmonic analysis.

In certain situations, especially many situations involving rates and ratios, the harmonic mean provides the truest average.

For instance, if a vehicle travels a certain distance at a speed x ( e. g. 60 kilometres per hour ) and then the same distance again at a speed y ( e. g. 40 kilometres per hour ), then its average speed is the harmonic mean of x and y ( 48 kilometres per hour ), and its total travel time is the same as if it had traveled the whole distance at that average speed.

The same principle applies to more than two segments: given a series of sub-trips at different speeds, if each sub-trip covers the same distance, then the average speed is the harmonic mean of all the sub-trip speeds, and if each sub-trip takes the same amount of time, then the average speed is the arithmetic mean of all the sub-trip speeds.

The weighted harmonic mean is the correct approach to determine the average specific gravity of a mixture when the composition by weight is known.

In hydrology the harmonic mean is used to average hydraulic conductivity values for flow that is perpendicular to layers ( e. g. geologic or soil ) while flow parallel to layers uses the arithmetic mean.

The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year.

The term weighted average usually refers to a weighted arithmetic mean, but weighted versions of other means can also be calculated, such as the weighted geometric mean and the weighted harmonic mean.

Distortion power factor is a measure of how much the harmonic distortion of a load current decreases the average power transferred to the load.

The distortion power factor describes how the harmonic distortion of a load current decreases the average power transferred to the load.

In order to get these numbers, the nuclear shell model starts from an average potential with a shape something between the square well and the harmonic oscillator.

The ultraviolet catastrophe results from the equipartition theorem of classical statistical mechanics which states that all harmonic oscillator modes ( degrees of freedom ) of a system at equilibrium have an average energy of.

Therefore, M is harmonic, with harmonic mean of divisors k, if and only if the average of its divisors is the product of M with a unit fraction 1 / k.

Therefore, for a perfect number M, τ ( M ) is even and the average of the divisors is the product of M with the unit fraction 2 / τ ( M ); thus, M is a harmonic divisor number.

Since each harmonic oscillator has average energy k < sub > B </ sub > T, the average total energy of the solid is 3Nk < sub > B </ sub > T, and its heat capacity is 3Nk < sub > B </ sub >.

Thus, the average potential energy equals k < sub > B </ sub > T / s, not k < sub > B </ sub > T / 2 as for the quadratic harmonic oscillator ( where s = 2 ).

This expression for undergoes simple harmonic oscillation, and as such is usually expressed as an RMS time average.

Where a function f on a graph is discretely harmonic at a point x if f ( x ) equals the average of f on the neighbors of x.

This reproduces the equipartition theorem of classical thermodynamics --- every harmonic oscillator at temperature T has energy kT on average.

This method of calculating the average power gives the real power regardless of harmonic content of the waveform.