Equal Temperament

Equal temperament is a tuning system that splits an octave into twelve \((12)\) "equally spaced out" semitones. The frequency, or pitch, of any tone is thus just as much higher- or lower- sounding than either of its two neighboring semitones as any other tone in the scale would be to a neighboring tone thereof. In other words, the proportional distance between any pair of neighboring pitches (such as A4 and B4) is exactly the same as that of any other pair of pitches (say B3 and C3).

This measure of pitchwise distance is found by taking the base frequency and multiplying it by the twelfth root of two \(( \sqrt[12]{2} )\) to find the pitch of the next higher note in the scale. The advantage of equal tempered scales is that every scale can be transposed to any other scale without changing the harmonic proportions of any pair of notes. Equal temperament thus makes every scale have the same proportional distances between its notes as any other scale, unlike other tuning systems such as Pythagorean temperament. This makes key transitions in music less pointed or pronounced and allows for seamless transposition. This article shows how to calculate the frequencies of an equal tempered twelve-note scale.

First Steps

First we must set the frequency for the foundational note in the scale. Let's start with A4, the first A above middle C, and fix its pitch at 440 Hz.

\[ A_4 = 440 \text{ Hz} \]

"Hz" is short for "Hertz", and this means that the actual sound wave of A4 goes through one "cycle" (oscillation) 440 times per second. Here "cycle" means that the sound wave starts out at equilibrium, reaches its peak amplitude, goes back down to equilibrium, drops to its trough amplitude, and then goes back up to equilibrium again. That would be one whole "cycle", and at 440 Hz, the tone goes through 440 of those cycles per second. Recall that "amplitude" is how high (and low) the wave drops from its initial starting point (equilibrium), and "frequency" is how rapidly, that is, how many times per second, a sound wave goes through a full up and down cycle (oscillation) per second. Amplitude also tells how loud or soft the pitch is, and frequency tells how high-pitched or low-pitched it is. For example, blowing a high-pitched whistle sounds a tone of both high amplitude and high frequency, whereas striking the lowest key on a piano sounds a tone of both low amplitude and low frequency. Note that amplitude means power, energy, loudness, and that frequency means pitch.

The octave interval is calculated by doubling (or halving) the base pitch frequency.

\[ \begin{aligned} \text{if } A_4 &= 440 \text{ Hz} \\ \\ \text{and } 440 \text{ Hz} \cdot 2 &= 880 \text{ Hz} \\ \\ \text{then } A_5 &= 880 \text{ Hz} \end{aligned} \]

Likewise, to find the frequency of A3 (an octave below A4), the frequency of A4 is halved.

\[ \begin{aligned} \text{if } A_4 &= 440 \text{ Hz} \\ \\ \text{and } 440 \text{ Hz} \div 2 &= 220 \text{ Hz} \\ \\ \text{then } A_3 &= 220 \text{ Hz} \end{aligned} \]
Next Step

Let's find out the exact frequencies in Hertz of each of the twelve notes between A4 and A5. We already know that the frequencies of those twelve notes lie between 440 Hz and 880 Hz. For A♯4 and B♭4, the next note above A4, we must multiply the frequency of A4 by the twelfth root of two (\( \sqrt[12]{2} \)). However, it may be easier to think in terms of fractional exponents. The twelfth root of two is the same as two raised to the one-over-twelfth power.

\[ \sqrt[12]{2} = 2^{1/12} \]

So the frequency of each note in the scale is simply \( 2^{\frac{1}{12}} \) times higher than the note it comes after.

\[ \begin{aligned} \text{if } A_4 &= 440 \text{ Hz} \\ \\ \text{and } 440 \text{ Hz} \cdot 2^{1/12} &\approx 466.163 \text{ Hz} \\ \\ \text{then } A\sharp_4/B\flat_4 &\approx 466.163 \text{ Hz} \end{aligned} \]

The frequency of A♯4/B♭4 is only approximate because the twelfth root of two is irrational. It cannot be written down either as a whole number or as a rational number (one that can be written as a fraction), so multiplying it by any other number also makes for an irrational number.

\[ \sqrt[12]{2} = 2^{1/12} \approx 1.059463094\text{…} \]

In reality, the series of digits after the decimal point (known as the mantissa) could go on for ever and ever. Here we approximate the value to three digits of precision, which should be more than enough for most applications.

At this point, we have calculated the frequencies of three out of twelve notes in the scale: A4, A♯4/B♭4, and A5.

\[ \boxed{ \begin{array}{|c|c c|} \hline A_4 & 440\phantom{.000} & \text{Hz} \\ \hline A^\sharp_4 / B^\flat_4 & 466.163 & \text{Hz} \\ \hline A_5 & 880\phantom{.000} & \text{Hz} \\ \hline \end{array} } \]

Now it seems there are two ways to calculate the next note. The first way is to change the base frequency to that of A4/B4, which is approximately 466.163, and multiply the same by the twelfth root of two, just like we did with A4. \[ \begin{aligned} \text{if } A^\sharp_4 / B^\flat_4 &\approx 466.163 \text{ Hz} \\ \\ \text{and } 466.163 \text{ Hz} \cdot 2^{1/12} &\approx 494.379 \text{ Hz} \\ \\ \text{then } B_4 &\approx 494.379 \text{ Hz} \end{aligned} \]

However, this way an already rounded number is multiplied by an approximation of the twelfth root of two, another rounded number. So we are dealing with two rounded numbers and losing precision. This loss of precision only accumulates as one goes up the scale, changing the base frequency of the multiplicand to a rounded number for every note. Instead, it is better to keep multiplying the base frequency, 440, (A4) by increasing twelth powers of two. So for the next note at a whole tone interval, the base frequency is multiplied by two to the two-over-twelfth power, and we raise the numerator of the fraction base-two power by one for every additional semitone of separation from the base note.

\[ \begin{aligned} \text{if } A_4 &= 440 \text{ Hz} \\ \\ \text{and } 440 \text{ Hz} \cdot 2^{2/12} &\approx 493.883 \text{ Hz} \\ \\ \text{then } B_4 &\approx 493.883 \text{ Hz} \end{aligned} \]
All Steps

We can keep use the base frequency to calculate the frequency of all the notes in the octave at once.

\[ \boxed{ \begin{array}{|c|r@{}c@{}l c|} \hline A_4 & 440 & \cdot & 2^{0/12} & = 440\phantom{.000}\ \text{Hz} \\ \hline A^\sharp_4 / B^\flat_4 & 440 & \cdot & 2^{1/12} & \approx 466.163\ \text{Hz} \\ \hline B_5 & 440 & \cdot & 2^{2/12} & \approx 493.883\ \text{Hz} \\ \hline C_5 & 440 & \cdot & 2^{3/12} & \approx 523.251\ \text{Hz} \\ \hline C^\sharp_5 / D^\flat_5 & 440 & \cdot & 2^{4/12} & \approx 554.365\ \text{Hz} \\ \hline D_5 & 440 & \cdot & 2^{5/12} & \approx 587.330\ \text{Hz} \\ \hline D^\sharp_5 / E^\flat_5 & 440 & \cdot & 2^{6/12} & \approx 622.254\ \text{Hz} \\ \hline E_5 & 440 & \cdot & 2^{7/12} & \approx 659.255\ \text{Hz} \\ \hline F_5 & 440 & \cdot & 2^{8/12} & \approx 698.456\ \text{Hz} \\ \hline F^\sharp_5 / G^\flat_5 & 440 & \cdot & 2^{9/12} & \approx 739.989\ \text{Hz} \\ \hline G_5 & 440 & \cdot & 2^{10/12} & \approx 783.991\ \text{Hz} \\ \hline G^\sharp_5 / A^\flat_5 & 440 & \cdot & 2^{11/12} & \approx 830.609\ \text{Hz} \\ \hline A_5 & 440 & \cdot & 2^{12/12} & = 880\phantom{.000}\ \text{Hz} \\ \hline \end{array} } \]

Of course, most of the exponents can be reduced and simplified, though this makes no difference in the calculations.

\[ \begin{aligned} 2^{0/12} &= 2^{0} = 1 \\ \\ 2^{2/12} &= 2^{1/6} \\ \\ 2^{3/12} &= 2^{1/4} \\ \\ 2^{4/12} &= 2^{1/3} \\ \\ 2^{6/12} &= 2^{1/2} \\ \\ 2^{8/12} &= 2^{2/3} \\ \\ 2^{9/12} &= 2^{3/4} \\ \\ 2^{10/12} &= 2^{5/6} \\ \\ 2^{12/12} &= 2^{1} = 2 \\ \\ \end{aligned} \]

So a simplified chart shows as follows.

\[ \boxed{ \begin{array}{|c|r@{}c@{}l c|} \hline A_4 & 440 & \cdot & 1 & = 440\phantom{.000}\ \text{Hz} \\ \hline A^\sharp_4 / B^\flat_4 & 440 & \cdot & 2^{1/12} & \approx 466.163\ \text{Hz} \\ \hline B_5 & 440 & \cdot & 2^{1/6} & \approx 493.883\ \text{Hz} \\ \hline C_5 & 440 & \cdot & 2^{1/4} & \approx 523.251\ \text{Hz} \\ \hline C^\sharp_5 / D^\flat_5 & 440 & \cdot & 2^{4/12} & \approx 554.365\ \text{Hz} \\ \hline D_5 & 440 & \cdot & 2^{5/12} & \approx 587.330\ \text{Hz} \\ \hline D^\sharp_5 / E^\flat_5 & 440 & \cdot & 2^{1/2} & \approx 622.254\ \text{Hz} \\ \hline E_5 & 440 & \cdot & 2^{7/12} & \approx 659.255\ \text{Hz} \\ \hline F_5 & 440 & \cdot & 2^{2/3} & \approx 698.456\ \text{Hz} \\ \hline F^\sharp_5 / G^\flat_5 & 440 & \cdot & 2^{3/4} & \approx 739.989\ \text{Hz} \\ \hline G_5 & 440 & \cdot & 2^{5/6} & \approx 783.991\ \text{Hz} \\ \hline G^\sharp_5 / A^\flat_5 & 440 & \cdot & 2^{11/12} & \approx 830.609\ \text{Hz} \\ \hline A_5 & 440 & \cdot & 2 & = 880\phantom{.000}\ \text{Hz} \\ \hline \end{array} } \]

Now to keep going we would take A5 at 880 Hz and start over by multiplying it by increasing twelfth powers of two until we got to A6 at 1760 Hz, and so on. To get the pitches of all the notes of the piano this way, we can just start with A0 at 27.5 Hz, the lowest note on a piano.

\[ \boxed{ \begin{array}{|c|r@{}c@{}l c|} \hline A_0 & 27.5 & \cdot & 2^{0/12} & = \phantom{0}27.5\phantom{00}\ \text{Hz} \\ \hline A^\sharp_0/B^\flat_0 & 27.5 & \cdot & 2^{1/12} & \approx \phantom{0}29.135\ \text{Hz} \\ \hline B_0 & 27.5 & \cdot & 2^{2/12} & \approx \phantom{0}30.868\ \text{Hz} \\ \hline C_0 & 27.5 & \cdot & 2^{3/12} & \approx \phantom{0}32.703\ \text{Hz} \\ \hline C^\sharp_0/D^\flat_0 & 27.5 & \cdot & 2^{4/12} & \approx \phantom{0}34.648\ \text{Hz} \\ \hline D_0 & 27.5 & \cdot & 2^{5/12} & \approx \phantom{0}36.708\ \text{Hz} \\ \hline D^\sharp_0/E^\flat_0 & 27.5 & \cdot & 2^{6/12} & \approx \phantom{0}38.891\ \text{Hz} \\ \hline E_0 & 27.5 & \cdot & 2^{7/12} & \approx \phantom{0}41.203\ \text{Hz} \\ \hline F_0 & 27.5 & \cdot & 2^{8/12} & \approx \phantom{0}43.654\ \text{Hz} \\ \hline F^\sharp_0/G^\flat_0 & 27.5 & \cdot & 2^{9/12} & \approx \phantom{0}46.249\ \text{Hz} \\ \hline G_0 & 27.5 & \cdot & 2^{10/12} & \approx \phantom{0}48.999\ \text{Hz} \\ \hline G^\sharp_0/A^\flat_0 & 27.5 & \cdot & 2^{11/12} & \approx \phantom{0}51.913\ \text{Hz} \\ \hline \end{array} } \]
\[ \boxed{ \begin{array}{|c|r@{}c@{}l c|} \hline A_1 & 55 & \cdot & 2^{0/12} & = \phantom{0}55\phantom{.000}\ \text{Hz} \\ \hline A^\sharp_1/B^\flat_1 & 55 & \cdot & 2^{1/12} & \approx \phantom{0}58.270\ \text{Hz} \\ \hline B_1 & 55 & \cdot & 2^{2/12} & \approx \phantom{0}61.736\ \text{Hz} \\ \hline C_2 & 55 & \cdot & 2^{3/12} & \approx \phantom{0}65.406\ \text{Hz} \\ \hline C^\sharp_2/D^\flat_2 & 55 & \cdot & 2^{4/12} & \approx \phantom{0}69.296\ \text{Hz} \\ \hline D_2 & 55 & \cdot & 2^{5/12} & \approx \phantom{0}73.416\ \text{Hz} \\ \hline D^\sharp_2/E^\flat_2 & 55 & \cdot & 2^{6/12} & \approx \phantom{0}77.782\ \text{Hz} \\ \hline E_2 & 55 & \cdot & 2^{7/12} & \approx \phantom{0}82.407\ \text{Hz} \\ \hline F_2 & 55 & \cdot & 2^{8/12} & \approx \phantom{0}87.307\ \text{Hz} \\ \hline F^\sharp_2/G^\flat_2 & 55 & \cdot & 2^{9/12} & \approx \phantom{0}92.499\ \text{Hz} \\ \hline G_2 & 55 & \cdot & 2^{10/12} & \approx \phantom{0}97.999\ \text{Hz} \\ \hline G^\sharp_2/A^\flat_2 & 55 & \cdot & 2^{11/12} & \approx 103.826\ \text{Hz} \\ \hline \end{array} } \]
\[ \boxed{ \begin{array}{|c|r@{}c@{}l c|} \hline A_2 & 110 & \cdot & 2^{0/12} & = 110\phantom{.000}\ \text{Hz} \\ \hline A^\sharp_2/B^\flat_2 & 110 & \cdot & 2^{1/12} & \approx 116.541\ \text{Hz} \\ \hline B_2 & 110 & \cdot & 2^{2/12} & \approx 123.471\ \text{Hz} \\ \hline C_3 & 110 & \cdot & 2^{3/12} & \approx 130.813\ \text{Hz} \\ \hline C^\sharp_3/D^\flat_3 & 110 & \cdot & 2^{4/12} & \approx 138.591\ \text{Hz} \\ \hline D_3 & 110 & \cdot & 2^{5/12} & \approx 146.832\ \text{Hz} \\ \hline D^\sharp_3/E^\flat_3 & 110 & \cdot & 2^{6/12} & \approx 155.563\ \text{Hz} \\ \hline E_3 & 110 & \cdot & 2^{7/12} & \approx 164.814\ \text{Hz} \\ \hline F_3 & 110 & \cdot & 2^{8/12} & \approx 174.614\ \text{Hz} \\ \hline F^\sharp_3/G^\flat_3 & 110 & \cdot & 2^{9/12} & \approx 184.997\ \text{Hz} \\ \hline G_3 & 110 & \cdot & 2^{10/12} & \approx 195.998\ \text{Hz} \\ \hline G^\sharp_3/A^\flat_3 & 110 & \cdot & 2^{11/12} & \approx 207.652\ \text{Hz} \\ \hline \end{array} } \]
\[ \boxed{ \begin{array}{|c|r@{}c@{}l c|} \hline A_3 & 220 & \cdot & 2^{0/12} & = 220\phantom{.000}\ \text{Hz} \\ \hline A^\sharp_3/B^\flat_3 & 220 & \cdot & 2^{1/12} & \approx 233.082\ \text{Hz} \\ \hline B_3 & 220 & \cdot & 2^{2/12} & \approx 246.942\ \text{Hz} \\ \hline C_4 & 220 & \cdot & 2^{3/12} & \approx 261.626\ \text{Hz} \\ \hline C^\sharp_4/D^\flat_4 & 220 & \cdot & 2^{4/12} & \approx 277.183\ \text{Hz} \\ \hline D_4 & 220 & \cdot & 2^{5/12} & \approx 293.665\ \text{Hz} \\ \hline D^\sharp_4/E^\flat_4 & 220 & \cdot & 2^{6/12} & \approx 311.127\ \text{Hz} \\ \hline E_4 & 220 & \cdot & 2^{7/12} & \approx 329.628\ \text{Hz} \\ \hline F_4 & 220 & \cdot & 2^{8/12} & \approx 349.228\ \text{Hz} \\ \hline F^\sharp_4/G^\flat_4 & 220 & \cdot & 2^{9/12} & \approx 369.994\ \text{Hz} \\ \hline G_4 & 220 & \cdot & 2^{10/12} & \approx 391.995\ \text{Hz} \\ \hline G^\sharp_4/A^\flat_4 & 220 & \cdot & 2^{11/12} & \approx 415.305\ \text{Hz} \\ \hline \end{array} } \]
\[ \boxed{ \begin{array}{|c|r@{}c@{}l c|} \hline A_4 & 440 & \cdot & 2^{0/12} & = 440\phantom{.000}\ \text{Hz} \\ \hline A^\sharp_4/B^\flat_4 & 440 & \cdot & 2^{1/12} & \approx 466.163\ \text{Hz} \\ \hline B_4 & 440 & \cdot & 2^{2/12} & \approx 493.883\ \text{Hz} \\ \hline C_5 & 440 & \cdot & 2^{3/12} & \approx 523.251\ \text{Hz} \\ \hline C^\sharp_5/D^\flat_5 & 440 & \cdot & 2^{4/12} & \approx 554.365\ \text{Hz} \\ \hline D_5 & 440 & \cdot & 2^{5/12} & \approx 587.330\ \text{Hz} \\ \hline D^\sharp_5/E^\flat_5 & 440 & \cdot & 2^{6/12} & \approx 622.254\ \text{Hz} \\ \hline E_5 & 440 & \cdot & 2^{7/12} & \approx 659.255\ \text{Hz} \\ \hline F_5 & 440 & \cdot & 2^{8/12} & \approx 698.456\ \text{Hz} \\ \hline F^\sharp_5/G^\flat_5 & 440 & \cdot & 2^{9/12} & \approx 739.989\ \text{Hz} \\ \hline G_5 & 440 & \cdot & 2^{10/12} & \approx 783.991\ \text{Hz} \\ \hline G^\sharp_5/A^\flat_5 & 440 & \cdot & 2^{11/12} & \approx 830.609\ \text{Hz} \\ \hline \end{array} } \]
\[ \boxed{ \begin{array}{|c|r@{}c@{}l c|} \hline A_5 & 880 & \cdot & 2^{0/12} & = 880\phantom{.000}\ \text{Hz} \\ \hline A^\sharp_5/B^\flat_5 & 880 & \cdot & 2^{1/12} & \approx 932.328\ \text{Hz} \\ \hline B_5 & 880 & \cdot & 2^{2/12} & \approx 987.767\ \text{Hz} \\ \hline C_6 & 880 & \cdot & 2^{3/12} & \approx 1046.502\ \text{Hz} \\ \hline C^\sharp_6/D^\flat_6 & 880 & \cdot & 2^{4/12} & \approx 1108.731\ \text{Hz} \\ \hline D_6 & 880 & \cdot & 2^{5/12} & \approx 1174.659\ \text{Hz} \\ \hline D^\sharp_6/E^\flat_6 & 880 & \cdot & 2^{6/12} & \approx 1244.508\ \text{Hz} \\ \hline E_6 & 880 & \cdot & 2^{7/12} & \approx 1318.510\ \text{Hz} \\ \hline F_6 & 880 & \cdot & 2^{8/12} & \approx 1396.913\ \text{Hz} \\ \hline F^\sharp_6/G^\flat_6 & 880 & \cdot & 2^{9/12} & \approx 1479.978\ \text{Hz} \\ \hline G_6 & 880 & \cdot & 2^{10/12} & \approx 1567.982\ \text{Hz} \\ \hline G^\sharp_6/A^\flat_6 & 880 & \cdot & 2^{11/12} & \approx 1661.219\ \text{Hz} \\ \hline \end{array} } \]
\[ \boxed{ \begin{array}{|c|r@{}c@{}l c|} \hline A_6 & 1760 & \cdot & 2^{0/12} & = 1760\phantom{.000}\ \text{Hz} \\ \hline A^\sharp_6/B^\flat_6 & 1760 & \cdot & 2^{1/12} & \approx 1864.655\ \text{Hz} \\ \hline B_6 & 1760 & \cdot & 2^{2/12} & \approx 1975.533\ \text{Hz} \\ \hline C_7 & 1760 & \cdot & 2^{3/12} & \approx 2093.005\ \text{Hz} \\ \hline C^\sharp_7/D^\flat_7 & 1760 & \cdot & 2^{4/12} & \approx 2217.461\ \text{Hz} \\ \hline D_7 & 1760 & \cdot & 2^{5/12} & \approx 2349.318\ \text{Hz} \\ \hline D^\sharp_7/E^\flat_7 & 1760 & \cdot & 2^{6/12} & \approx 2489.016\ \text{Hz} \\ \hline E_7 & 1760 & \cdot & 2^{7/12} & \approx 2637.020\ \text{Hz} \\ \hline F_7 & 1760 & \cdot & 2^{8/12} & \approx 2793.826\ \text{Hz} \\ \hline F^\sharp_7/G^\flat_7 & 1760 & \cdot & 2^{9/12} & \approx 2959.955\ \text{Hz} \\ \hline G_7 & 1760 & \cdot & 2^{10/12} & \approx 3135.963\ \text{Hz} \\ \hline G^\sharp_7/A^\flat_7 & 1760 & \cdot & 2^{11/12} & \approx 3322.438\ \text{Hz} \\ \hline \end{array} } \]
\[ \boxed{ \begin{array}{|c|r@{}c@{}l c|} \hline A_7 & 3520 & \cdot & 2^{0/12} & = 3520\phantom{.000}\ \text{Hz} \\ \hline A^\sharp_7/B^\flat_7 & 3520 & \cdot & 2^{1/12} & \approx 3729.310\ \text{Hz} \\ \hline B_7 & 3520 & \cdot & 2^{2/12} & \approx 3951.066\ \text{Hz} \\ \hline C_8 & 3520 & \cdot & 2^{3/12} & \approx 4186.009\ \text{Hz} \\ \hline \end{array} } \]


Alternatively, one could also begin with a base pitch multiplicand and just keep adding on to the numerator in the twelfth power of two multiplier, without ever changing the base.

\[ \text{if } A_4 = 440 \text{ Hz} \] \[ \text{then} \] \[ \begin{aligned} A^\sharp_5/B^\flat_5 &= 440 \cdot 2^{13/12} \text{ Hz} \\ B_5 &= 440 \cdot 2^{14/12} \text{ Hz} \\ C_5 &= 440 \cdot 2^{15/12} \text{ Hz} \\ C^\sharp_5/D^\flat_5 &= 440 \cdot 2^{16/12} \text{ Hz} \\ D_5 &= 440 \cdot 2^{17/12} \text{ Hz} \\ D^\sharp_5/E^\flat_5 &= 440 \cdot 2^{18/12} \text{ Hz} \\ E_5 &= 440 \cdot 2^{19/12} \text{ Hz} \end{aligned} \] \[ \text{and so on...} \]
Any Step

Thus the formula can be generalized to find the frequency of any note \( x \) semitones above a base frequency by multiplying the base frequency \( f_0 \) by two raised to the power of \( x \) twelfths.

\[ f(x) = f_0 \cdot 2^{x/12} \]

And going in the other direction, to find the frequency of a note \( x \) semitones below a base frequency, the base frequency \( f_0 \) is multiplied by two raised to the power of negative \( x \) twelfths.

\[ f(x) = f_0 \cdot 2^{-x/12} \]