Straddle primes

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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.

Motivation

Straddle primes were created for the purpose of error cancellation and flexibility of tempering. For example:

• 37edo has a remarkably accurate 19-limit, with the exception of prime 3. It works, but it doesn't stack predictably. In order to represent intervals involving prime 3 more accurately, powers of 3 can involve stacking the more accurate sharp 3 with an occasional flat 3 to offset error.
• 86edo is larger and has two diatonic fifths, Meantone and Archy. This gives it two diatonic scales with different qualities useful as a base for quasi-diatonic ternary scales. It also clearly straddles the whole 11-limit, providing both the option to choose a version of a prime that results in a better approximation of a ratio, and doubling the probability of encountering a target prime in a given scale. In practice, this probability is lower than double due to preferring approximations with good error cancellation.

With a sufficiently large edo, it's possible to have accurate approximations of primes while also using versions altered by an edostep to generate more interesting scales or result from stacking a different interval. This may be seen as better due to the lower error in using a different JI ratio to represent an altered prime, which begs the question, why use straddle primes at all? The answer is twofold:

• Straddling primes without using a more accurate approximation in the middle widens the smallest step size a tuning must have to represent the straddle. 37edo would have to be tripled to 111edo in order to get an accurate 3, which is totally different in character and size. Larger tuning systems are less practically convenient and make decisions harder.
• Using the same ratio to describe two slightly different intervals drastically simplifies the ratios produced by stacking it, and less care is needed to ensure that error cancellation works out mathematically.

Straddle-primes differ significantly from the typical way ratios are understood in RTT. They do not work for everyone, but their high level of flexibility makes more unusual approaches possible.

g_

I have a philosophy where I limit edo size to the minimum required to do whatever I'm trying to do, in order to make composition easier. When using a tuning with two diatonic fifths, the smallest interval I actually need always ending up being the difference between them. 3.5¢ is an often mentioned value for the just-noticeable difference (JND), but I find that it can be much larger depending on context. My lower bound for the typical size of an aberrisma, around 25¢, is the limit where it stops sounding like a clear melodic step in many contexts of the music I write, like a fast melody competing with other instruments, slightly detuned synths, and reverb. This also makes error less noticeable, limiting the effect of extreme concordance. Most of the time in my music, there's hardly a difference between a just 4:5:6 and one tempered with Meantone. In more ideal contexts where notes are more clear, my lower bound is about half that size, not coincidentally the step size of the largest edos I use. Even then, being audible enough as a step is a genuine concern.

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 cancellation 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 cancellation in standard notation creates a very long JI group.

Example erac groups 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.
~_	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 temperaments

Straddle temperaments use inaccuracy to distort intervals in interesting ways while maintaining their lattice structure.

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).

Examples of straddle-prime temperaments include:

• Straddle-3 Meantone (sometimes called dual-3 Meantone or Greentone), where >3/2 * <3/2 = 9/4, and (9/4)^2 = 5/1. This is supported by 18edo and 37edo.
• Monarch and Interarch
• Straddle cluster temperaments

SRTT

SRTT, short for Straddle Regular Temperament Theory, is a generalization of straddle temperaments with eracs that uses variables to represent small alterations of primes. Using the variable r to represent the erac alteration, the erac subgroup 2.<3.>3.5.7 may be written as 2.3/r.3r.5.7 or simplified to 2.3r.5.7.r^2. If error cancellation is not assumed, more variables may be used, such as 3m/2 for a Meantone fifth and 3a/2 for an Archy fifth as found in Monarch.

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)