The sawtooth wave is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle; the convention is that a sawtooth wave ramps upward and sharply drops. However, in a reverse sawtooth wave, the wave ramps downward and sharply rises, it can be considered the extreme case of an asymmetric triangle wave. The piecewise linear function x = t − ⌊ t ⌋ ⏟ floor ⁡ or equivalently x = t based on the floor function of time t is an example of a sawtooth wave with period 1. A more general form, in the range −1 to 1, with period a, is 2 This sawtooth function has the same phase as the sine function. Another function in trigonometric terms with period p and amplitude a: y = − 2 a π arctan ⁡ While a square wave is constructed from only odd harmonics, a sawtooth wave's sound is harsh and clear and its spectrum contains both and odd harmonics of the fundamental frequency; because it contains all the integer harmonics, it is one of the best waveforms to use for subtractive synthesis of musical sounds bowed string instruments like violins and cellos, since the slip-stick behavior of the bow drives the strings with a sawtooth-like motion.

A sawtooth can be constructed using additive synthesis. The infinite Fourier series x reverse sawtooth = 2 A π ∑ k = 1 ∞ k sin ⁡ k converges to a reverse sawtooth wave. A conventional sawtooth can be constructed using x s a w t o o t h = A 2 − A π ∑ k = 1 ∞ k sin ⁡ k where A is amplitude. In digital synthesis, these series are only summed over k such that the highest harmonic, Nmax, is less than the Nyquist frequency; this summation can be more efficiently calculated with a fast Fourier transform. If the waveform is digitally created directly in the time domain using a non-bandlimited form, such as y = x − floor, infinite harmonics are sampled and the resulting tone contains aliasing distortion. An audio demonstration of a sawtooth played at 440 Hz and 880 Hz and 1,760 Hz is available below. Both bandlimited and aliased tones are presented. Sawtooth waves are known for their use in music; the sawtooth and square waves are among the most common waveforms used to create sounds with subtractive analog and virtual analog music synthesizers.

Sawtooth waves are used in switched-mode power supplies. In the regulator chip the feedback signal from the output is continuously compared to a high frequency sawtooth to generate a new duty cycle PWM signal on the output of the comparator; the sawtooth wave is the form of the vertical and horizontal deflection signals used to generate a raster on CRT-based television or monitor screens. Oscilloscopes use a sawtooth wave for their horizontal deflection, though they use electrostatic deflection. On the wave's "ramp", the magnetic field produced by the deflection yoke drags the electron beam across the face of the CRT, creating a scan line. On the wave's "cliff", the magnetic field collapses, causing the electron beam to return to its resting position as as possible; the voltage applied to the deflection yoke is adjusted by various means so that the half-way voltage on the sawtooth's cliff is at the zero mark, meaning that a negative voltage will cause deflection in one direction, a positive voltage deflection in the other.

Frequency is 15.734 kHz on NTSC, 15.625 kHz for PAL and SECAM). The vertical deflection system operates the same w

A non-decimal currency is a currency that has sub-units that are a non-decimal fraction of the main unit, i.e. the number of sub-units in a main unit is not a power of 10. Most currencies were non-decimal, though today all are now decimal. Today, only two countries have non-decimal currencies: Mauritania, where 1 ouguiya = 5 khoums, Madagascar, where 1 ariary = 5 iraimbilanja. However, these are only theoretically non-decimal, as in both cases the value of each sub-unit is too small to be of any practical use and coins of sub-unit denominations are no longer used; the official currency of the Sovereign Military Order of Malta, which retains its claims of sovereignty under international law and has been granted permanent observer status at the United Nations, is the Maltese scudo, subdivided into 12 tarì, each of 20 grani with 6 piccioli to the grano. All other contemporary currencies are either decimal or have no sub-units at all, either because they have been abolished or because they have lost all practical value and are no longer used.

A variety of non-decimal systems have been used. For example, ancient Mesoamerican civilizations common used vigesimal systems. A base 60 was in wide use in ancient Mesopotamia, used in measurements of time, geometry and other fields. Decimal currencies have disadvantages; the principal advantage of most non-decimal currencies is that they are more divided by numbers such as 3 and 8, than decimal currencies, due to being based upon conversion values that have a large number of factors. A currency with a 100:1 ratio is divisible neither into 3 nor into 8. For example, one-third of an Austrian Gulden was 20 Kreuzer; this divisibility is useful when sharing out sums of money. For these reasons, many states chose in the past to adopt non-decimal currencies based on divisions into sub-units such as 12 or 20, sometimes with more than one tier of sub-units. There is a more fortuitous, way in which non-decimal currencies emerged. Multiple currencies would circulate concurrently in an economy, with non-decimal exchange rates between them.

For example, the Reichsthaler/ rixdollar/ riksdaler/ rijksdaalder/ rigsdaler were accepted as a common accounting unit which represented a variety of local coins in Stockholm, Copenhagen and Cologne. Inflation developed locally, with changing subdivisions. For instance the Riksdaler was equivalent to 2 silver dalers in Sweden in 1700, but after the 1715-19 devaluation of the silver daler coin until 1776 one Riksdaler equated to 3 daler silvermint. Most currencies made no distinction between units of accounting and units represented by coins and thus created such shifts. In general, when the major unit was, say, a gold coin and the minor units were silver or copper coins when the relative values of the metals changed because of an increase or decrease in the supply of one of the metals the number of minor units equivalent to one major unit would change, thus the following list does not give a complete picture: it is a list of examples picked from different periods. Many of the subdivisions given below underwent historical changes.

The Russian ruble is said to have become the first decimalized currency when Peter the Great established the ratio 1 ruble = 100 kopecks in 1701. The Japanese were in some sense earlier calculating with the silver momme and its decimal subunits - but the momme was not a coin but a unit of weight equivalent to 3.75 g: accounting was by weight of silver. The British pound sterling was the last major currency to be decimalized, on 15 February 1971; the Maltese waited just one year before following suit and Nigeria followed in 1973. An early proposal for decimalizing the pound in the 19th century envisaged a system of 1 Pound = 10 florins = 100 dimes = 1000 cents; however the only step taken at that time was the introduction in 1849 of a florin coin. A partial listing of former non-decimal currencies: Ancient Greece — 1 drachma = 6 obols Denmark — 1 Krone = 8 Marks = 128 Skillings = 1536 Pfennigs.

Valdese Elementary School known as Rock School, is a historic school building located at Valdese, Burke County, North Carolina. It was built in 1922-1923, is a two-story, fieldstone four square building with a hipped roof, it features a T-shaped clerestory above the auditorium space. It was constructed by Waldensian settlers from Northern Italy, it was listed on the National Register of Historic Places in 1984. Old Rock School is now home to several community organizations and events, including Bluegrass concerts, the Old Colony Players, the Rock School Art Galleries, Old Rock School Railway Museum, Valdese Travel and Tourism; the Old Rock School Railway Museum known as the Piedmont & Western Railroad Museum, is located on the lower level and is operated by the Piedmont and Western Railroad Club. Exhibits include photos, railroad art, railroad lanterns, steam whistles, spikes and china used on trains and other artifacts; the museum features model railroad layouts, including a diorama depicting the original Waldensian settlers arriving by train on May 29, 1893.

The Rock School Art Galleries are located in the Old Rock School. Exhibits are organized by the Rock School Arts Foundation. Town of Valdese: Old Rock School Old Rock School Railway Museum - Piedmont and Western Railroad Club