Also, the triangle wave can be the absolute value of the :
Another definition of the triangle wave, with range from 1 to 1 and period 2 is:
Fourier Series of Triangular Wave.  Homestead
The rate at which a sound wave moves in and out is called the . Frequency is measured in cycles per second. The length of a singal cycle of a waveform is the span of time it takes for that waveform to repeat. People generally hear an increase in the frequency of a sound wave as an increase in pitch. When the frequency of an oscillator is doubled, the pitch of the sound it generates moves an octave up. For example, an oscillator generating a signal that repeats at the rate of 440 cycles per second will have the same pitch as middle A on a piano. An oscillator generating a signal that repeats at 880 cycles per second will have the same pitch as the A an octave above middle A. A common way of saying "cycles per second" is "Hertz," abbreviated "Hz."
Check out Wikipedia's page. Their "Example 1" shows how to derive the Fourier series of a sawtooth wave. All you have to do is normalize the results for your particular time and amplitude values.
Fourier Series of Triangular Wave ..
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triangle, or sawtooth waveform ..
In the second case we just add together the two simple FM spectra, but in the first case we get a more complex mixture involving all the sums and differences of the modulating frequencies. These sum and difference tones ("intermodulation products") are not limited to FM. Anynonlinear synthesis technique produces them. Being nonlinear, it must have something that involvesa power of its input other than 0 or 1; if we feed in sin a + sin b, for example, that term will producenot just (sin a)^n and (sin b)^n, but all sorts of stuff involving sin a * sin b (in various powers),and this produces things like cos(a+b) and cos(ab).For a less impressionistic derivation of the spectrum, see Le Brun,"A Derivation of the Spectrum of FM with a Complex Modulating Wave". The result can be expressed:
A simple equation with a period of 4, with . As this only uses the and , this can be used to simply implement a triangle wave on hardware electronics with less CPU power:
Fourier Series Examples: Triangle Wave  Falstad

Triangle wave : definition of Triangle wave and …
contains no harmonics so no use for filtering Square wave – rich in odd harmonics Triangle wave ..

matlab  The Fourier Series Of This Triangle Wave  …
Triangle wave  enwikimath

The Fourier Series Of This Triangle Wave
I need to work derive the Fourier series of a triangle wave that i ..
A Square and Triangle Wave VCO. Part I  Breadboard …
My question is, how many multiples do I need to calculate for it to sound like a proper square wave using additive synthesis, i.e. calculating a Fourier series? How many harmonics is overkill?
Wavetable Synthesis  The Synthesizer Academy
I need to work derive the Fourier series of a triangle wave that i have generated, I just do not know how to actually go about this problem in Matlab.
Fourier Series Applet  Paul Falstad
Sawtooth waves, also called saw waves, have a very strong, clear, buzzing sound. A sawtooth wave can be made by adding a series of sine waves at different frequencies and volume levels. The frequency of the first, loudest sine wave is what we hear as the frequency of the resulting sawtooth. This is called the frequency. Each of the other, progressively quieter, sine waves that make up a sawtooth have frequencies which are integer multiples of the fundamental frequency. These frequencies are called .
Skill Builder: Advanced Arduino Sound Synthesis  Make:
For example, an ideal sawtooth wave with a fundamental frequency of 100Hz would have harmonics at 200Hz, 300Hz, 400Hz, and so on to infinity, with each harmonic quieter than the last. Because the sawtooth wave contains every integer harmonic of the fundamental frequency, it sounds very rich to our ear. The fundamental frequency defines the pitch of the sound, while the harmonics change the character, or , of the sound without affecting the pitch.
AD9833 Datasheet and Product Info  Analog Devices
Technical note: The amplitude of a given harmonic in a sawtooth wave is equal to the inverse of its harmonic number. For example, a sawtooth wave with a fundamental frequency of 100Hz and an amplitude of 1 would have a harmonic at 200Hz (100*2) with an amplitude of 0.5 (1/2), another harmonic at 300Hz (100*3) with an amplitude of 0.33 (1/3), and so forth. The more harmonics are added in this manner, the more the wave will look like the idealized sawtooth wave depicted in this article.
Synthesis Basics  Beau Sievers
Square waves have a rich sound that's not quite as buzzy as a sawtooth wave, but not as pure as a sine. Old Nintendo game soundtracks were made almost exclusively from square waves. Like sawtooth waves, square waves can be generated by adding a series of sine waves with decreasing volume. However, the square wave contains only the odd numbered harmonics.