Logarithmic intonation

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Logarithmic intonation (LI) is a form of intonation that is similar to just intonation but rather than using primes as the basis elements, it uses the natural logarithms of integers (ln(2), ln(3), ln(4) and so on). Logarithmic intonation is a superset of just intonation, because the interval n/d can be expressed as ln(xⁿ)/ln(x^d) for any integer x, but the majority of it consists of irrational intervals.

The simplest subgroup of logarithmic intonation is ln(2).ln(3), but this is a nonoctave system–if the octave is desired, the simplest subgroup is ln(2).ln(4) which contains the octave as ln(4)/ln(2) and can be rewritten as 2.ln(4). The subgroup ln(2).ln(3).ln(4) or equivalently 2.ln(3).ln(4) can be viewed as analogous to the 5-limit of just intonation. The interval ln(2) itself is a descending wide tritone of -635 cents, with it's octave-equivalent ln(4) being a narrow tritone of 565 cents that can be approximated by 25/18 and 18/13.