Erac

Eracs, short for error accidentals, are symbols that indicate how much a tempered interval is flat or sharp relative to others in a subgroup. They act as variables representing small pitch differences and have no set size. They are the standard notation for subgroups involving Straddle primes.

Eracs provide a more complete picture of error cancelation than the standard notation of non-prime intervals. For example, 11edo almost perfectly misses primes 3 and 5 present 22edo, which still allows them to cancel out for an accurate 5/3 and 15. 11edo's subgroup might be 2.5/3.15.7.11 in standard notation, or 2.x3.x5.7.11 in erac notation. This becomes even more important in edos like 23 or 29, where several low primes are critically inaccurate and representing all of the error cancelation in standard notation creates a very long subgroup.

Example erac subgroups for edos can be found in EDO#List of Edos.

Symbols

Erac Meanings (intervals represented by an underscore)

<_	Flat by a set arbitrary amount.
>_	Sharp by a set arbitrary amount.
x_	Critically flat/sharp. Shorthand for <_.>_.
<x_ and >x_	Shorthand for <<_.>_ and <_.>>_ respectively.
{ and }	Partial eracs, indicating error that may be ignored.

Eracs are placed before their respective numbers by default (although they may be placed after as well) This is for readability and to remove ambiguity with the denominator, as eracs on the denominator have an inverse effect on the size of a tempered ratio. For example, >5/3 is sharp or 5/3, but 5/>3 is flat.

Erac Temperaments

Temperament optimization algorithms can be built around eracs to create optimal error cancelation, such as GTO. Erac temperaments have the potential to replace exotemperaments by properly representing the large error in approximating certain primes, providing less misleading mappings with systematic accuracy.