What is "harmonic progression"

Harmonic \Har*mon"ic\ (h[aum]r*m[o^]n"[i^]k), Harmonical \Har*mon"ic*al\ (-[i^]*kal), a. [L. harmonicus, Gr. "armoniko`s; cf. F. harmonique. See Harmony.]

1. Concordant; musical; consonant; as, harmonic sounds.

   Harmonic twang! of leather, horn, and brass.
   --Pope.

2. (Mus.) Relating to harmony, -- as melodic relates to melody; harmonious; esp., relating to the accessory sounds or overtones which accompany the predominant and apparent single tone of any string or sonorous body.

3. (Math.) Having relations or properties bearing some resemblance to those of musical consonances; -- said of certain numbers, ratios, proportions, points, lines, motions, and the like.

   Harmonic interval (Mus.), the distance between two notes of a chord, or two consonant notes.

   Harmonical mean (Arith. & Alg.), certain relations of numbers and quantities, which bear an analogy to musical consonances.

   Harmonic motion, the motion of the point A, of the foot of the perpendicular PA, when P moves uniformly in the circumference of a circle, and PA is drawn perpendicularly upon a fixed diameter of the circle. This is simple harmonic motion. The combinations, in any way, of two or more simple harmonic motions, make other kinds of harmonic motion. The motion of the pendulum bob of a clock is approximately simple harmonic motion.

   Harmonic proportion. See under Proportion.

   Harmonic series or Harmonic progression. See under Progression.

   Spherical harmonic analysis, a mathematical method, sometimes referred to as that of Laplace's Coefficients, which has for its object the expression of an arbitrary, periodic function of two independent variables, in the proper form for a large class of physical problems, involving arbitrary data, over a spherical surface, and the deduction of solutions for every point of space. The functions employed in this method are called spherical harmonic functions.
   --Thomson & Tait.

   Harmonic suture (Anat.), an articulation by simple apposition of comparatively smooth surfaces or edges, as between the two superior maxillary bones in man; -- called also harmonia, and harmony.

   Harmonic triad (Mus.), the chord of a note with its third and fifth; the common chord.

Progression \Pro*gres"sion\, n. [L. progressio: cf. F. progression.]

1. The act of moving forward; a proceeding in a course; motion onward.

2. Course; passage; lapse or process of time.

   I hope, in a short progression, you will be wholly immerged in the delices and joys of religion.
   --Evelyn.

3. (Math.) Regular or proportional advance in increase or decrease of numbers; continued proportion, arithmetical, geometrical, or harmonic.

4. (Mus.) A regular succession of tones or chords; the movement of the parts in harmony; the order of the modulations in a piece from key to key.

   Arithmetical progression, a progression in which the terms increase or decrease by equal differences, as the numbers [lbrace2]2, 4, 6, 8, 1010, 8, 6, 4, 2[rbrace2] by the difference 2.

   Geometrical progression, a progression in which the terms increase or decrease by equal ratios, as the numbers by a continual multiplication or division by 2.

   Harmonic progression, a progression in which the terms are the reciprocals of quantities in arithmetical progression, as 1/2, 1/4, 1/6, 1/8, 1/10.

n. (mathematics) a progression of terms whose reciprocals form an arithmetic progression

Harmonic progression may refer to:

• Chord progression in music
• Harmonic progression (mathematics)

In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. In other words, it is a sequence of the form

$$1/a ,\ \frac{1}{a+d}\ , \frac{1}{a+2d}\ , \frac{1}{a+3d}\ , \cdots, \frac{1}{a+kd},$$

where −a/d is not a natural number and k is a natural number.

(Terms in the form $\frac{x}{y+z}\$ can be expressed as $\frac{\frac{x}{y}}{\frac{y+z}{y}}$ , we can let $\frac{x}{y}=a$ and $\frac{z}{y}=kd$.)

Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.

It is not possible for a harmonic progression (other than the trivial case where a = 1 and k = 0) to sum to an integer. The reason is that, necessarily, at least one denominator of the progression will be divisible by a prime number that does not divide any other denominator.