Logarithmic intonation: Difference between revisions

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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(xd) for any integer x, but the majority of it consists of irrational intervals.

'''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(xd) for any integer x, but the majority of it consists of irrational intervals. It can be viewed as using the e-[[logharmonic series]] instead of the [[harmonic series]].

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. [[Regular temperament]]s can be defined with these subgroups as they can with prime subgroups. 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]].

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 {{Novelty}}
-'''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(x<sup>n</sup>)/ln(x<sup>d</sup>) for any integer x, but the majority of it consists of irrational intervals.
+'''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(x<sup>n</sup>)/ln(x<sup>d</sup>) for any integer x, but the majority of it consists of irrational intervals. It can be viewed as using the e-[[logharmonic series]] instead of the [[harmonic series]].
 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. [[Regular temperament]]s can be defined with these subgroups as they can with prime subgroups. 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]].