Straddle primes
Straddle primes are functional primes in a temperament that straddle their respective just interval by having at least one flat and one sharp approximation available. Dual-fifth systems are usually straddle-3.
The straddle ratio m:n denotes the ratio of the flatter approximation (m) to the sharper approximation (n). A straddle ratio of approximately 1:1 is called equistraddle.
Eracs
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 groups 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 JI group might be 2.5/3.15.7.11 in standard notation, which is a subgroup of the erac group 2.x3.x5.7.11. 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 JI group.
Example erac groups for edos can be found in EDO#List of Edos.
Symbols
| <_ | 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. |
| ~_ | Approximate, tempered equivalent. A widely accepted symbol used in this context to indicate that no eracs apply. |
Eracs are placed before their respective numbers. 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 of 5/3, but 5/>3 is flat and may simplify to <5/3.
Erac simplification
Deciding how eracs should be assigned to elements of a subgroup is not an exact science. In addition to the amount of precision the composer finds most useful, the exact ways the errors cancel are important to consider.
Assume a 2.3.5 subgroup with a slightly inaccurate 3 which is still good enough to not need eracs, but a straddle 5 that's somewhere between 1:1 and 2:1 error. The direction errors tend in determines the best erac subgroup. The following calculations assume that the most important part of 2.3.5 is 4:5:6 and its inversions and retroversions, with the accuracy of 15 being less of a concern.
Just: 3/2 702 0 5/4 386 0 5/3 884 0
Effectively just flat 3, straddle-5 tends flat: 3/2 699 -3 <5/4 377 -9 >5/4 392 +6 <5/3 878 -6 >5/3 893 +9
Here, it makes the most sense to call the subgroup 2.3.<5.>5. The 3 is flat, but not by enough to be worth keeping track of in many cases. Even though the sharp 5 is more accurate, the slightly flat 3 adds to this error and makes the sharp 5/3 less accurate than the flat one. Therefore on average, the two versions of 5 result in about equal errors.
Effectively just sharp 3, straddle-5 tends flat: 3/2 705 +3 <5/4 377 -9 >5/4 392 +6 <5/3 872 -12 >5/3 887 +3
This is the same as above, but now the 3 is slightly sharp. In this case, the subgroup should be 2.3.<<5.>5. The 3 cancels out some error on the sharp 5, so the sharp 5 results in about half the error of the flat 5 on average.
Straddle-prime temperaments
Temperament optimization algorithms can be built around eracs to create optimal error cancelation, such as GTO. Straddle-prime temperaments have the potential to replace exotemperaments by properly representing the large error in approximating certain primes, providing less misleading mappings with systematic accuracy. Straddle-prime temperaments may also be found as (strong or weak) extensions of subgroup temperaments with prime powers (such as 2.9.21).
An example of a straddle-prime temperament is dual-3 Meantone, where >3/2 * <3/2 = 9/4, and (9/4)^2 = 5/1. This is supported by 18edo and 37edo.
Erac-derived temperaments
These were a concept derived by Cole to be able to quickly create conventional temperaments.
His procedure:
- For a flat harmonic and a sharp harmonic:
- take the flat harmonic and take it to the power of how many eracs the sharp harmonic has
- take the sharp harmonic and take it to the power of how many eracs the flat harmonic has
- multiply them together
- take it to the power of how many steps there are in an EDO
- octave-reduce (if the interval is greater than 600c octave reduced, take the inverse)
