What are the advantages to using the 12th root of 2 over a simple fraction of 1/12 between keys/semitones when setting pitch frequencies on a piano?

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The 12th root of 2 tuning is called equal temperament tuning and is the “modern” tuning. It is fairly close (but not exact) to “natural” sounds. This gets into a bit of physics.

The simplest model is a plucked string (which is also very similar to how notes are produced in a tube – eg a flute). The main frequency is derived from twice the length of the string. Then the next frequency is twice and the next is three times etc. These are called harmonics (2nd harmonic, 3rd harmonic etc) and is always an integer multiple of the fundamental frequency. So if the string has a fundamental note of 220Hz, then the 2nd harmonic is 440Hz, the third harmonic is 660Hz.

The note you get by the ratio of the harmonics against the fundamental note is the octave. The note that you get by the ratio of the harmonics to the 2nd harmonic – the 3rd to the 2nd is called the perfect fifth or dominant harmonic ie 3/2. Because these ratios are naturally produced, humans also appear to consider them harmonious (ie notes with frequencies of these ratios sound “good” played together or after one another). By building on these ratios, the Pythagorean system of tuning is developed. But there are some problems with this kind of tuning.

Then building on the idea of simple ratios a form of tuning called just intonation was developed (using simple ratios 3:2, 4:3, 5:4) But the problem with this tuning is that chords only sound good when produced on the root key. Music or harmonies played on a different root sounds rather bad.

Finally the modern system which “divides” the octave into equal ratios of 12th root of 2 was developed. Although it is “off” slightly compared to just intonation, its close enough and is easy to tune to and music can be transposed to different keys without sounding “awful”.

Music tuned by arithmetic rather than geometric division of the octave would sound REALLY bad. There would be large jumps at the low end and hardly any differences of tonality at the high end of the scale.

When looking at the *logarithm base 2* of the frequency, that’s actually exactly what we are doing: Each semitone increases the logarithm by 1/12, until, after 12 semitones, the logarithm has increased by 1, and the frequency thus doubled.

The reason we’re looking at the logarithm is because our ear doesn’t hear frequency differences, it hears frequency ratios. Going up a certain interval (like a semitone) means *multiplying* the frequency with the ratio of that interval. Mathematical phenomena that feature a lot of multiplications (not only music, but also e.g. exponential growth) are often better understood when looking at their logarithms, where the multiplications become simple additions.

It will be completely out of tune (in any tuning system) if the frequencies are 1/12.

Just calculate it for 2 octaves and you’ll see. An octave is double the frequency. Let’s pick the notes 120, 240, 480 Hz. That’s 2 octaves.

For the 120-240 Hz octave, each semitone would be +10 Hz. For the 240-480 octave, each semitone would be +20 Hz.

Clearly, that is wrong since they are different steps. If you use 1/12th root increments, it will be the same (geometric increment) for both octaves.

Read this for more info: https://en.m.wikipedia.org/wiki/Musical_tuning#Tuning_systems