Triangle wave

Triangle wave sound sample

5 seconds of triangle wave at 220 Hz

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Additive Triangle wave sound sample

After each second, a harmonic is added to a sine wave creating a triangle 220 Hz wave

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A triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.

Like a square wave, the triangle wave contains only odd harmonics. However, the higher harmonics roll off much faster than in a square wave (proportional to the inverse square of the harmonic number as opposed to just the inverse).

Harmonics

It is possible to approximate a triangle wave with additive synthesis by summing odd harmonics of the fundamental while multiplying every other odd harmonic by −1 (or, equivalently, changing its phase by π) and multiplying the amplitude of the harmonics by one over the square of their mode number, $n$ , (which is equivalent to one over the square of their relative frequency to the fundamental).

The above can be summarised mathematically as follows:

{\begin{aligned}x_{\mathrm {triangle} }(t)&{}=\sum _{i=0}^{N}(-1)^{i}n^{-2}\left(\sin[nt]\right)\end{aligned}}

where $N$ is the number of harmonics to include in the approximation, $t$ is the independent variable (e.g. time for sound waves), and $i$ is the harmonic label which is related to its mode number by $n=2i+1$ .

This infinite Fourier series converges to the triangle wave as $N$ tends to infinity, as shown in the animation.

Definitions

Sine, square, triangle, and sawtooth waveforms

Another definition of the triangle wave, with range from -1 to 1 and period $p,$ is:

x(t)={\frac {4}{p}}\left(t-{\frac {p}{2}}\left\lfloor {\frac {2t}{p}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {2t}{p}}+{\frac {1}{2}}\right\rfloor }

where the symbol

\lfloor n\rfloor

represent the floor function of n.

Also, the triangle wave can be the absolute value of the sawtooth wave:

x(t)=2\left|{t \over p}-\left\lfloor {t \over p}+{1 \over 2}\right\rfloor \right|

or, for a range from -1 to +1:

x(t)=2\left|2\left({t \over p}-\left\lfloor {t \over p}+{1 \over 2}\right\rfloor \right)\right|-1

The triangle wave can also be expressed as the integral of the square wave:

\int\sgn(\sin(x))\,dx\,

A simple equation with a period of 4, with $y(0)=1.$ As this only uses the modulo operation and absolute value, this can be used to simply implement a triangle wave on hardware electronics with less CPU power:

y(x) = |x\,\bmod\,4 - 2|-1

or, generalizing the above equation for a period of $p,$ amplitude $a,$ and starting with $y(0)=a/2$ :

y(x)={\frac {2a}{p}}{\Biggl (}{\biggl |}\left(x{\bmod {p}}\right)-{\frac {p}{2}}{\biggr |}-{\frac {p}{4}}{\Biggr )}

The former function is a specialization of the latter for a=2 and p=4:

y(x)={\frac {2\times 2}{4}}{\Biggl (}{\biggl |}\left(x{\bmod {4}}\right)-{\frac {4}{2}}{\biggr |}-{\frac {4}{4}}{\Biggr )}\Leftrightarrow

y(x) = \Biggl( \biggl | \left( x \bmod 4 \right) - 2 \biggr | - 1 \Biggr)

An odd version of the first function can be made, just shifting by one the input value, which will change the phase of the original function:

y(x) = |(x-1)\,\bmod\,4 - 2|-1

Generalizing this to make the function odd for any period and amplitude gives:

y(x) = \frac{4a}{p} \Biggl( \biggl | \left( (x - \frac{p}{4}) \bmod p \right) - \frac{p}{2}\biggr | - \frac{p}{4} \Biggr)

In terms of sine and arcsine with period p and amplitude a:

y(x) = \frac{2a}{\pi}\arcsin\left(\sin\left(\frac{2\pi}{p}x\right)\right)

Arc Length

The arc length per period "s" for a triangle wave, given the amplitude "a" and period length "p":

s={\sqrt {(4a)^{2}+p^{2}}}

References

Weisstein, Eric W. "Fourier Series - Triangle Wave". MathWorld.

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Triangle wave

Harmonics

Definitions

Arc Length

See also

References