Sawtooth wave

The sawtooth wave (or saw wave) is a kind of <a href="/facts/Non-sinusoidal_waveform/mD1CB09M">non-sinusoidal waveform</a>. It is so named based on its resemblance to the teeth of a plain-toothed <a href="/facts/Saw/mh8PG7xy">saw</a> with a zero <a href="/facts/Rake_angle/qNDylLV2">rake angle</a>. A single sawtooth, or an intermittently triggered sawtooth, is called a ramp waveform.
The convention is that a sawtooth wave ramps upward and then sharply drops. In a reverse (or inverse) sawtooth wave, the wave ramps downward and then sharply rises. It can also be considered the extreme case of an asymmetric <a href="/facts/Triangle_wave/72BUtbRl">triangle wave</a>.
The equivalent <a href="/facts/Piecewise_linear_function/6bKlCbEa">piecewise linear functions</a>

x
        (
        t
        )
        =
        t
        −
        ⌊
        t
        ⌋
      
    
    {\displaystyle x(t)=t-\lfloor t\rfloor }

x
        (
        t
        )
        =
        t
        
          mod
          
            1
          
        
      
    
    {\displaystyle x(t)=t{\bmod {1}}}

based on the <a href="/facts/Floor_function/ssehyMMf">floor function</a> of time t is an example of a sawtooth wave with <a href="/facts/Wave_period/QUSbPaW8">period</a> 1.
A more general form, in the range −1 to 1, and with period p, is

2
        
          (
          
            
              
                t
                p
              
            
            −
            
              ⌊
              
                
                  
                    1
                    2
                  
                
                +
                
                  
                    t
                    p
                  
                
              
              ⌋
            
          
          )
        
      
    
    {\displaystyle 2\left({\frac {t}{p}}-\left\lfloor {\frac {1}{2}}+{\frac {t}{p}}\right\rfloor \right)}

This sawtooth function has the same <a href="/facts/Phase_(waves)/cpGtuVll">phase</a> as the <a href="/facts/Sine/gQ6LXK8z">sine</a> function.
While a <a href="/facts/Square_wave_(waveform)/h1cWR1KM">square wave</a> is constructed from only odd harmonics, a sawtooth wave's sound is harsh and clear and its spectrum contains both even and odd <a href="/facts/Harmonic/9U9REzeR">harmonics</a> of the <a href="/facts/Fundamental_frequency/TJBJKHow">fundamental frequency</a>. Because it contains all the integer harmonics, it is one of the best waveforms to use for <a href="/facts/Subtractive_synthesis/2Kk3TqUE">subtractive synthesis</a> of musical sounds, particularly bowed string instruments like violins and cellos, since the <a href="/facts/Stick-slip_phenomenon/a1crawoc">slip-stick behavior</a> of the bow drives the strings with a sawtooth-like motion.

A sawtooth can be constructed using <a href="/facts/Additive_synthesis/N8bbwbEg">additive synthesis</a>. For period p and amplitude a, the following infinite <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> converge to a sawtooth and a reverse (inverse) sawtooth wave:

 
 
 
 f
 =
 
 
 1
 p
 
 
 
 
 {\displaystyle f={\frac {1}{p}}}

x
          
            sawtooth
          
        
        (
        t
        )
        =
        −
        
          
            
              2
              a
            
            π
          
        
        
          ∑
          
            k
            =
            1
          
          
            ∞
          
        
        
          
            (
            −
            1
            )
          
          
            k
          
        
        
          
            
              sin
              ⁡
              (
              2
              π
              k
              f
              t
              )
            
            k
          
        
      
    
    {\displaystyle x_{\text{sawtooth}}(t)=-{\frac {2a}{\pi }}\sum _{k=1}^{\infty }{(-1)}^{k}{\frac {\sin(2\pi kft)}{k}}}

x
          
            reverse sawtooth
          
        
        (
        t
        )
        =
        
          
            
              2
              a
            
            π
          
        
        
          ∑
          
            k
            =
            1
          
          
            ∞
          
        
        
          
            (
            −
            1
            )
          
          
            k
          
        
        
          
            
              sin
              ⁡
              (
              2
              π
              k
              f
              t
              )
            
            k
          
        
      
    
    {\displaystyle x_{\text{reverse sawtooth}}(t)={\frac {2a}{\pi }}\sum _{k=1}^{\infty }{(-1)}^{k}{\frac {\sin(2\pi kft)}{k}}}

In <a href="/facts/Digital_data/JChckAvr">digital</a> synthesis, these series are only summed over k such that the highest harmonic, Nmax, is less than the <a href="/facts/Nyquist_frequency/zwFUKdVR">Nyquist frequency</a> (half the <a href="/facts/Sampling_frequency/xxck7aLN">sampling frequency</a>). This summation can generally be more efficiently calculated with a <a href="/facts/Fast_Fourier_transform/caT7UUCh">fast Fourier transform</a>. If the waveform is digitally created directly in the time domain using a non-<a href="/facts/Bandlimited/LrYmxzlA">bandlimited</a> form, such as y = x − <a href="/facts/Floor_function/ssehyMMf">floor</a>(x), infinite harmonics are sampled and the resulting tone contains <a href="/facts/Aliasing/Pzg84HL0">aliasing</a> distortion.

An audio demonstration of a sawtooth played at <a href="/facts/440_Hz/WIUBW1x2">440 Hz</a> (A4) and 880 Hz (A5) and 1,760 Hz (A6) is available below. Both bandlimited (non-aliased) and aliased tones are presented.

Sawtooth wave open-in-new

Sawtooth wave