Just versus Equal Temperament

Classical Mechanics Level 4

The primary notes in music are A, B, C, D, E, F, and G. Each note can either be sharp, meaning it is slightly higher in pitch, or flat, meaning it is slightly lower in pitch. The distance between a note and its sharp/flat is known as a half-step, also known as a semitone. There are 12 half-steps between a note and its octave. An interval is the distance between notes and is given all sorts of names depending on the number of half-steps between the notes. Common intervals are shown in the image below, relative to the note C:

When tuning instruments, especially the piano, there are two main types of tuning: Just temperament and Equal Temperament.

In just temperament, one tunes an instrument based upon the whole number ratios between notes, determined based on the harmonics of the instrument. The most common ratios are 2:1 for an octave, 3:2 for a perfect fifth, and 4:3 for a perfect fourth. For example, if one was to tune the string D on a violin using just temperament, since it is known that the A string has a frequency of 440 Hz and is a perfect fifth above D, one would tune the D and A strings such that the ratio of the frequencies matches that of a perfect fifth. The A string would then have a frequency of $(\frac{3}{2})^{-1} \times 440 Hz = 293.33 Hz$ .

There are many problems with this method. Notably, this method of tuning creates some "awkward" frequency ratios. The frequency ratio between a C and an E, a major third, is 81:64. Yikes! In addition, if one was to tune using fifths up to some octave (octaves have the ratio of $2^{n}:1$ where $n$ is some octave above the chosen note). According to the circle of fifths , if we were to continually apply perfect fifths, we would eventually reach some number of octaves above the starting note, or a frequency ratio that is some multiple of two. But $(\frac{3}{2})^{n}$ can never equal a multiple of two!

But maybe equal temperament can solve the problem. Since there are twelve half-steps between a note and its perfect octave, each half step should be equal to $\sqrt[12]{2}$ . That way, half-steps are uniform and we have made sure that octaves are in tune.

Starting on the note A4 (440 Hz), compare using just and equal temperament to tune the note F#5. The number next to a note indicates what octave the note is in and it changes every time it is equal to or passes a C (e.g: A#3, B3, C4, C#5). By what percentage will F#5, tuned using just temperament, be sharper than F#5, tuned using equal temperament?

Keep in mind that this is an introductory music theory problem. I am well aware that there are other tuning methods that may be more practical.

The answer is 0.33935.

2 solutions

Alexander McDowell
Apr 2, 2020

Firstly, we need to determine the interval between A4 and F#5. The notes between A4 and F#5 (inclusive) are A4, A#4, B4, C5, C#5, D5, D#5, E5, F5, F#5; therefore there are nine half-steps between A4 and F#5 which corresponds to a major sixth. We are given the ratios for the intervals perfect fourth, perfect fifth, and octave as 4:3, 3:2, and 2:1 respectively. The distance between a perfect fourth and a perfect fifth is two half-steps (otherwise known as a whole step), which is the same distance between a perfect fifth and a major sixth. Under just temperament, a whole step corresponds to a ratio of $\frac{3}{2} \times (\frac{4}{3})^{-1} = \frac{9}{8}$ , meaning that a major sixth is a ratio of $\frac{9}{8} \times \frac{3}{2} = \frac{27}{16}$ .

Under equal temperament, the ratio between the frequencies A4 and F#5 is $\sqrt[12]{2^{9}}:1$ .

Thus, the just temperament tuning is sharp, compared to equal temperament, by $100 * (\frac{\frac{27}{16}}{\sqrt[12]{2^{9}}} - 1) = 0.33935\%$

I am a stranger to music theory but had a go at the problem anyway. I found a reference saying that the octave number changes between B and C so I had F#4 as lower than A4 with the interval then being a minor third.

Justin Travers - 1 year, 2 months ago

You are correct. I should have written F#5 instead of F#4. The problem has been changed to reflect that and I apologize for the confusion.

Alexander McDowell - 1 year, 2 months ago

$\sqrt[12]{2^9}:1$ is the ratio of frequencies F#5 and A5 (or F#4 and A4), not F#5 and A4. It is less than two so it cannot span more than one octave.

Hypergeo H. - 1 year, 2 months ago

Hypergeo H.
Apr 11, 2020

Nice question!

Just temperament : F#5 is three major fifths higher than A4, so its frequency relative to A4 is $\large \left(\frac 32\right)^3$ .

Equal temperament : F#5 is one octave and nine semitones higher than A4, so its frequency relative to A4 is $\large 2\cdot 2^{\frac {9}{12}} =2^\frac 74$ .

Ratio : The percentage difference is $\large \displaystyle \frac {\left(\frac 32\right)^3} { 2^{\frac {7}{4}}} -1 = 0.0033935 \approx 0.339\%$ .

Just versus Equal Temperament

The answer is 0.33935.

2 solutions

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