Piano key frequencies

This is a list of the <a href="/facts/Fundamental_frequencies/TJBJKHow">fundamental frequencies</a> in <a href="/facts/Hertz/6Wshj5qm">hertz</a> (cycles per second) of the keys of a modern 88-key standard or 108-key extended <a href="/facts/Piano/lRI60lQK">piano</a> in <a href="/facts/Equal_temperament/8AwSblW3">twelve-tone equal temperament</a>, with the 49th key, the fifth A (called A4), tuned to 440 Hz (referred to as <a href="/facts/A440_(pitch_standard)/WIUBW1x2">A440</a>). Every octave is made of twelve steps called <a href="/facts/Semitones/lVBsLYSA">semitones</a>. A jump from the lowest semitone to the highest semitone in one octave doubles the <a href="/facts/Frequency/QUSbPaW8">frequency</a> (for example, the fifth A is 440 Hz and the sixth A is 880 Hz). The frequency of a pitch is derived by multiplying (ascending) or dividing (descending) the frequency of the previous pitch by the <a href="/facts/Twelfth_root_of_two/1X4Qt28H">twelfth root of two</a> (approximately 1.059463). For example, to get the frequency one semitone up from A4 (A♯4), multiply 440 Hz by the twelfth root of two. To go from A4 up two semitones (one <a href="/facts/Whole_tone/x94d53Fo">whole tone</a>) to B4, multiply 440 twice by the twelfth root of two (or once by the sixth root of two, approximately 1.122462). To go from A4 up three semitones to C5 (a <a href="/facts/Minor_third/4y8zBxwY">minor third</a>), multiply 440 Hz three times by the twelfth root of two (or once by the fourth root of two, approximately 1.189207). For other tuning schemes, refer to <a href="/facts/Musical_tuning/v5Q1kYxC">musical tuning</a>.
This list of frequencies is for a theoretically ideal piano. On an actual piano, the ratio between semitones is slightly larger, especially at the high and low ends, where string stiffness causes <a href="/facts/Inharmonicity/xDSTCe95">inharmonicity</a>, i.e., the tendency for the <a href="/facts/Harmonic/9U9REzeR">harmonic</a> makeup of each note to run <a href="/facts/Sharp_(music)/dUgIJtHX">sharp</a>. To compensate for this, octaves are tuned slightly wide, <a href="/facts/Stretched_octave/I5LDJE2S">stretched</a> according to the inharmonic characteristics of each instrument. This deviation from equal temperament is called the <a href="/facts/Railsback_curve/1ywcUXep">Railsback curve</a>.
The following equation gives the frequency f (Hz) of the nth key on the idealized standard piano with the 49th key tuned to A4 at 440 Hz:

f
        (
        n
        )
        =
        
          
            (
            
              
                
                  2
                  
                    12
                  
                
              
              
            
            )
          
          
            n
            −
            49
          
        
        ×
        440
        
        
          Hz
        
        
        =
        
          2
          
            
              
                n
                −
                49
              
              12
            
          
        
        ×
        440
        
        
          Hz
        
        
      
    
    {\displaystyle f(n)=\left({\sqrt[{12}]{2}}\,\right)^{n-49}\times 440\,{\text{Hz}}\,=2^{\frac {n-49}{12}}\times 440\,{\text{Hz}}\,}

where n is shown in the table below.
Conversely, the key number of a pitch with a frequency f (Hz) on the idealized standard piano is:

n
        =
        12
        
        
          log
          
            2
          
        
        ⁡
        
          (
          
            
              f
              
                440
                
                
                  Hz
                
              
            
          
          )
        
        +
        49
      
    
    {\displaystyle n=12\,\log _{2}\left({\frac {f}{440\,{\text{Hz}}}}\right)+49}

Piano key frequencies open-in-new

Piano key frequencies