Ed12

Equal divisions of the 12th harmonic (ed12) are tunings obtained by dividing the 12th harmonic in a certain number of equal steps.

The twelfth harmonic, duodecuple, or dodecatave, is particularly wide as far as equivalences go, as there are at absolute most about 3.1 instances of the 12th harmonic within the human hearing range. This width means that the listener probably will not hear the interval as an equivalence, but instead will hear the pseudo-octave or pseudo-tritave or similar as one – this disconnect between period versus equivalence could be used by a composer to surprise their listener, in a similar way that 13edo can be used to make melodies that sound like 12edo, until they suddenly do not.

However, using ed12's does not necessarily imply using the 12th harmonic as an interval of equivalence. The quintessential reason for using a 12th-harmonic based tuning is that it is a compromise between octave and twelfth based tunings, like an ed6 – but ed12 leans more towards octaves than ed6 does. In fact, ed12's optimize for the 1:2:3:4:6:12 chord, with equal magnitudes and opposite signs of error on 3 and 4 and on 2 and 6.

As such, an ed12 sometimes gives you the right amount of stretch for equal temperaments whose 3 is more inaccurate than its higher primes. Here for example, you can choose how much you wish to stretch 31edo depending on your harmonic style: 80ed6 vs 111ed12.

Individual pages for ed12's

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200 and beyond

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