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

Triangle wave | |
---|---|

General information | |

General definition | |

Fields of application | Electronics, synthesizers |

Domain, codomain and image | |

Domain | |

Codomain | |

Basic features | |

Parity | Odd |

Period | 1 |

Specific features | |

Root | |

Derivative | Square wave |

Fourier series |

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).

A triangle wave of period *p* that spans the range [0, 1] is defined as

For a triangle wave spanning the range [−1, 1] the expression becomes

A more general equation for a triangle wave with amplitude and period using the modulo operation and absolute value is

For example, for a triangle wave with amplitude 5 and period 4:

A phase shift can be obtained by altering the value of the term, and the vertical offset can be adjusted by altering the value of the term.

As this only uses the modulo operation and absolute value, it can be used to simply implement a triangle wave on hardware electronics.

Note that in many programming languages, the `%`

operator is a remainder operator (with result the same sign as the dividend), not a modulo operator; the modulo operation can be obtained by using `((x % p) + p) % p`

in place of `x % p`

. In e.g. JavaScript, this results in an equation of the form `4*a/p * Math.abs((((x - p/4) % p) + p) % p - p/2) - a`

.

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

A triangle wave with period *p* and amplitude *a* can be expressed in terms of sine and arcsine (whose value ranges from −*π*/2 to *π*/2):

Another definition of the triangle wave, with range from −1 to 1 and period *p*, is

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:

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

The arc length per period for a triangle wave, denoted by *s*, is given in terms of the amplitude *a* and period length *p* by

**^**Kraft, Sebastian; Zölzer, Udo (5 September 2017). "LP-BLIT: Bandlimited Impulse Train Synthesis of Lowpass-filtered Waveforms".*Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17)*. 20th International Conference on Digital Audio Effects (DAFx-17). Edinburgh. pp. 255–259.

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