

A116474


Equal divisions of the octave with progressively increasing consistency levels.


3



1, 3, 4, 5, 22, 26, 29, 58, 80, 94, 282, 311, 17461
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OFFSET

3,2


COMMENTS

An equal temperament is consistent at level N (odd integer) if all the intervals in the Nlimit tonality diamond (set of ratios with odd factors of numerator and denominator not exceeding N) are approximated consistently, i.e. the composition of the approximations is the closest approximation of the composition.
These EDOs are not necessarily any good for musical purposes. Even though 4EDO is consistent through the 7 limit, no one would seriously consider using it for 7limit music because the approximations are so bad.
While for the smallest values these EDOs are not directly usable, their consistency is even so a valuable feature. For example, 4EDO is consistent through the 7 limit, but is not usable directly for 7limit music. However, indirectly, by means of subsequently adjusting the harmony, it can be and has been useful as a compositional tool for composing music in the 7limit. The same comment applies to 3 in the 5limit and 5 in the 9limit. Any of the values above 5 are usable directly as equal temperament approximations to the corresponding limit.  Gene Ward Smith, Mar 29 2006


LINKS

Table of n, a(n) for n=3..15.
Tables of consistency levels.
Entry in Tonalsoft encyclopedia of microtonal music theory.


EXAMPLE

3EDO is consistent through the 5 limit because 6/5, 5/4 and 4/3 map to 1 step and 3/2, 8/5 and 5/3 map to 2 steps and all the compositions work out, for example 6/5 * 5/4 = 3/2 and 1 step + 1 step = 2 steps. It is not consistent through the 7 limit because 8/7 and 7/6 both map to 1 step, but 8/7 * 7/6 = 4/3 also maps to 1 step.


CROSSREFS

Cf. A116475, A117577, A117578.
Sequence in context: A334638 A039573 A037347 * A208807 A126896 A334225
Adjacent sequences: A116471 A116472 A116473 * A116475 A116476 A116477


KEYWORD

nonn,more


AUTHOR

Keenan Pepper, Mar 17 2006


EXTENSIONS

More terms from Gene Ward Smith, Mar 29 2006


STATUS

approved



