**The first law:**When a string fixed at each end like the piano string, is struck at one end, it vibrates in a complex form, most strongly in its full length but also perceptibly in segments of that length such as 1/2, 1/3, 1/4, and 1/5.

**The second law**is equally important. It may be stated as follows:

**Length and Frequency:**The frequency of a string - that is to say, the number of vibrations per unit of time (c.p.s.) which it can perform, is proportional inversely to its length. Thus, since an octave above a given tone has twice as many cycles per second as the original tone, it follows that to obtain a sound an octave above a given sound we must have a string one-half as long.

**Weight, Tension and Thickness:**Similar laws exist with regard to the influence upon string vibrations, of weight, tension and thickness. Without undertaking to prove these completely, we may state them briefly as follows:

**Weight:**The frequency of a string's vibration is inversely proportional to the square root of its weight. In other words, if the weight be divided by 4 (the square of 2) the frequency will be multiplied by 2. To produce a tone one octave above the original tone the weight of the string must be only 1/4 its original weight.

**Tension:**The frequency of string vibrations is directly proportional to the square root of their tension. In other words, to get twice as many cycles per second you must multiply the tension by 2^{2}=4. To get 4 times as many cycles per second you must multiply the tension by 4^{2}=16. So if a string be stretched with a weight of 10 lbs. and it is desired to make it sound an octave higher, this can be done by making the stretching weight (4 X 10 =) 40 lbs.

**Thickness:**The frequency of string vibrations is inversely proportional to the thickness of the string. If a string of a given length and weight produces a sound of a given frequency, a string of the same length and of twice the thickness will give a sound one octave lower; that is, of half the frequency.

**Mechanical Variable Factors:**All these laws, be it remembered, are based upon the assumption of mathematical strings, in which weight and stiffness remain constant throughout all changes in length. In the case of the actual piano string, in which the weight and tension do vary with the length, some compensation must therefore be made. Thus, to illustrate, it is found that whereas the acoustical law of frequency requires a doubling of the string length at each octave downwards, or halving at each octave upwards, the practical string, where weight and tension vary with length, is better served by a proportion of 1:1.875 or 1:1.9375, instead of 1:2. This difference must be kept in mind by designers of "scales" for pianos. [Piano Tuning and Allied Arts, White, 1938]

(NOTE: We have also to consider volume, density and pressure as influenses on frequency. DP 11/18/03)

These laws do not apply in all instances when considering volume.

See Also

**Compression Wave**

**Compression Wave Velocity**

**Curved Wave Universe of Motion**

**Dissociating Water with Microwave**

**Figure 6.9 - Russell depicts his waves in two ways**

**Figure 6.10 - Wave Dynamics between Cube Corners**

**Figure 7.1 - Step 1 - Wave Vortex Crests at Maximum Polarization**

**Figure 8.1 - Russells Painting of Wave Form Dynamics**

**Figure 8.10 - Each Phase of a Wave as Discrete Steps**

**Figure 8.11 - Four Fundamental Phases of a Wave**

**Figure 8.14 - Some Basic Waveforms and their constituent Aliquot Parts**

**Figure 8.2 - Compression Wave Phase Illustration**

**Figure 8.3 - Coiled Spring showing Longitudinal Wave**

**Figure 8.4 - Transverse Wave**

**Figure 9.10 - Phases of a Wave as series of Expansions and Contractions**

**Figure 9.11 - Compression Wave with expanded and contracted Orbits**

**Figure 9.13 - Wave Flow as function of Periodic Attraction and Dispersion**

**Figure 9.14 - Wave Flow and Phase as function of Particle Rotation**

**Figure 9.15 - Wave Flow and Wave Length as function of Particle Oscillatory Rotation**

**Figure 9.5 - Phases of a Wave as series of Expansions and Contractions**

**Figure 9.9 - Wave Disturbance from 0 Center to 0 Center**

**Figure 12.10 - Russells Locked Potential Wave**

**Figure 12.12 - Russells Multiple Octave Waves as Fibonacci Spirals**

**Figure 13.13 - Gravity Syntropic and Radiative Entropic Waves**

**Figure 14.07 - Love Principle: Two sympathetic waves expanding from two points have one coincident centering locus**

**In the Wave lies the Secret of Creation**

**Laws of Vibration**

**Longitudinal Wave**

**Longitudinal Waves in Vacuum**

**Matter Waves and Electricity**

**Nodal Waves**

**One More Step Toward Building The Cube-Sphere Wave-Field**

**Part 11 - SVP Music Model**

**Part 12 - Russells Locked Potentials**

**Part 26 - Science of Sound Vibration Acoustics and Music**

**Quantum Entanglement**

**Raleigh Wave**

**Shock Wave**

**Sympathetic Oscillation**

**Sympathetic Vibration**

**Table 12.02.01 - Wavelengths and Frequencies**

**Three Main Parts of a Wave**

**Transverse Wave**

**wave**

**Wave Field**

**Wave Fields - Summarize and Simplify**

**wave number**

**WaveLength**

**3.8 - There are no Waves**

**3.9 - Nodes Travel Faster Than Waves or Light**

**8.3 - Conventional View of Wave Motion**

**8.4 - Wave types and metaphors**

**8.5 - Wave Motion Observables**

**8.6 - Wave Form Components**

**8.8 - Water Wave Model**

**9.2 - Wave Velocity Propagation Questions**

**9.30 - Eighteen Attributes of a Wave**

**9.31 - Oscillatory Motion creating Waveforms**

**9.34 - Wave Propagation**

**9.35 - Wave Flow**

**12.05 - Three Main Parts of a Wave**

**16.06 - Electric Waves are Sound Waves**