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Encyclopedia of Microtonal Music Theory

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diminished-5th / dim5 / -5 / b5

[Joe Monzo]

A diatonic interval one chromatic semitone smaller than the perfect-5th. The diminished-5th is composed of two tones (i.e., "whole-tones") and two diatonic-semitones. The diminished 5th occurs naturally in every diatonic scale, between the "leading-tone" (7th degree or "VII") and the "subdominant" (4th degree or "IV"). Example, in the key of C ("t" = tone, "s" = semitone) :

 M2  m2  M2  M2  m2  M2  M2
  t   s   t   t   s   t   t
A   B   C   D   E   F   G   A
      diminished 5th

diminished 5th  =  2t    + 2s     =  2(t-s) + 4s
                =  2(M2) + 2(m2)  =  2(+1)  + 4(m2)

The diminished-5th thus contains 2 chromatic semitones and 4 diatonic semitones (or equivalently, 2 augmented-primes and 4 minor-2nds). In 12-edo these two types of semitones are equivalent, thus in that tuning the diminished-5th contains 6 equal semitones.

The diminished-5th is not properly the same as the "tritone", but because the usual 12-edo scale makes the true tritone (augmented-4th) and the diminished-5th enharmonically equivalent (i.e., exactly the same pitch), the diminished-5th is commonly also called "tritone".

In pythagorean tuning, the diminished-5th is mathematically (9/8)2(256/243)2 = ratio 1024/729, expressed in decimal form as exactly 1.404663923182441700960219478737997256515775034293552812071330589849108367626886145... (the decimal part repeats after the 81st place), with a logarithmic interval size of 5.88 Semitones = ~588.269994807675 cents.

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