User:Overthink/Equal-step tunings, bimodular approximants, and the natave

In western music theory, we use 12-EDO, or 12 equal divisions of the octave. The advantage of using an equal step tuning is that one can modulate freely and play the same scales and chords from every key. In this article, we will analyze the math behind why some equal-step tunings work.

Dividing the natave

In an equal-step tuning, the ratio between consecutive notes is the same. Since an interval's size in cents is related exponentially to its frequency ratio, it is natural to consider equal divisions of Euler's number, e. As an interval being divided, it is often called the natave. This may not seem to make sense as e is an irrational number, but I promise that it will make sense. In n-EDe, each step has a frequency ratio of e^(1/n), and every interval in the system has a frequency ratio that is an integer power of this. The m-step interval has a frequency ratio of e^(m/n). This is an irrational number, but there is a formula to give a rational approximation of this interval, which gets more accurate the smaller m/n is. It is as follows: (2n+m)/(2n-m). If we let m/n=1/1, we get e^(1/1)≈(2×1+1)/(2×1-1)=3/1=1901.955¢, which is over 170 cents sharp of e=1731.234¢, not very accurate. However, if we let m/n=1/2, we get e^(1/2)≈(2×2+1)/(2×2-1)=5/3=884.359¢, only 18.7 cents sharp of e^(1/2)=865.617¢. Another example is e^(1/3)=577.078¢≈7/5=582.512¢, only sharp by 5.434¢!

However, our goal is to approximate rational intervals with an equal-step tuning, not the other way around. Luckily, we can use the inverse of the previously stated formula to estimate the width in nataves of a rational interval a/b.