The topic of tuning deals extensively with ratios of frequencies. There are often advantages to expressing these ratios logarithmically. For example, log frequency ratios are more perceptually meaningful, and equations involving them are often simpler than with plain ratios. The base two logarithm is particularly useful because of the special perceptual status of a doubling of frequency.
For these reasons, this work will often use the doubling, the base-two logarithm of frequency ratio, instead of frequency ratio itself. The frequency ratio r is equivalent to doublings. For notational brevity, will be notated as , i.e. the log operation will be indicated by underlining.
The term ``doubling'' is used instead of ``octave'' because an octave is an interval between pitches in Western music, not a ratio between frequencies. We will scrupulously observe the distinction between interval and frequency ratio as analogous to the distinction between pitch and frequency. An interval is a discrete element of a musical theory or system (like `P8'). A frequency ratio, on the other hand, is a continuous quantity indicating the ratio between two frequencies. Thus, although in practice an octave usually corresponds to a doubling, an octave is an interval in Western music whereas a ``doubling'' is a logarithmic frequency ratio. One slight problem with ``doubling'' is that it already has a separate musical meaning, as in ``voice doubling,'' but these two meanings can usually be disambiguated from context.
In the tuning literature, a frequency ratio r is often expressed as cents. This unit is useful because it facilitates comparisons with 12TET, a tuning that many people are familiar with. Nonetheless, it will not be used in this work since it imparts a normative status to 12TET that is culturally biased. Where a cent-sized unit is called for, the millidoubling, or mil, (one thousandth of a doubling) works quite nicely since it equals 1.2 cents.