Bird's eye view of temperaments by accuracy

This page is a work in progress, serving to document temperaments broadly by accuracy preference, and then approximately by subgroup focus, that is, what sort of harmonies, broadly speaking, the temperament is targetting. Under each accuracy and subgroup focus is found an incomplete list of temperaments, organized approximately by complexity (how many notes per octave are required). The complexity given is the note count per octave (or for no-2's, per tritave), with the set of odds used for deriving the complexity given. Sometimes two complexities are given and the average is taken for the purpose of ranking.

Importantly: the "accuracy" classification of a temperament is the maximum error allowed on all intervals. If there is some intervals in the corresponding (thr)odd-limit that violate this bound slightly even in an optimized tuning, they should be noted as "Bound-violating intervals:" under the header for easy comprehension and consideration, but this should not be abused to attempt to reclassify temperaments, as there should only be one or two such intervals at most, and ideally zero. Therefore, most temperaments in a category are more accurate than the bound suggests.

The bounds are: exotemperament (>~18c), low or very low (<~18c), medium or low (<12c), high or medium (<7c), very high or high (<3.5c), microtemperament (<1c).

The two name system is for two reasons: to account for people's varying preferences and terminology for accuracies and to make the system of categories symmetrical, with exotemperaments and microtemperaments as the extrema.

The cent errors are a result of a set of compromises between people of different preference, and being given in cents, are somewhat arbitrary. It should be noted that because the exotemperament bound is very high, it is not meaningful to pretend that the bound is even remotely precise, so that the "<~18c" bound was chosen to allow using 5edo and 7edo as the circle of ~3/2's to barely qualify as not being exotemperaments, both as such a valuation is potentially contentious and as they may yield interesting simplified logics taking advantage of the damage for error cancellations.

Explanation of subgroup focuses

For temperaments with prime 2, complexity is judged by the odd-limit of the temperament's subgroup, potentially plus some composite odds that the temperament can reasonably be said to target if it doesn't hurt complexity a lot. For temperaments without prime 2, we use the analogous concept of throdd-limit, where the equave is the tritave (3/1).

Each accuracy category is split into the following subgroup focuses, which are enumerated here so as to explain what is meant by them:

5-limit focus

The main purpose of the temperament is 5-limit harmony, and may admit one or two "sporadic primes" > 11 if they don't damage the 5-limit more than it already is, that is, if they correspond to a strong extension.

7-limit focus

The main purpose of the temperament is 7-limit harmony, and may admit one or two "sporadic primes" > 13 if they don't damage the 7-limit more than it already is, that is, if they correspond to a strong extension.

11-limit focus

The main purpose of the temperament is 11-limit harmony, and may admit one or two "sporadic primes" > 17 if they don't damage the 7-limit more than it already is, that is, if they correspond to a strong extension.

~17-limit focus

The main purpose of the temperament is approximately 17-limit harmony, potentially minus one prime (hence the ~). The omitted prime could be prime 17, so pure 13-limit temperaments are documented under this category. However, such a temperament may not omit primes 2, 3 or 5, due no-2's focus, no-3's focus and no-5's focus categories. If such a temperament admits a "sporadic prime" that is mapped somewhat simply relative to the temperament's lower-limit complexity, then it should instead be classified under higher-limit focus.

Higher-limit focus

The main purpose of the temperament is subgroup harmonies of the 19-limit, 23-limit, 29-limit, 31-limit, etc. Subgroup temperaments with multiple primes > 17 should usually go here, unless one of those primes is very complex to reach relative to the temperament's complexity and only included because it's essentially "free" in the sense of not damaging the temperament.

No-2's focus

All no-2's temperaments go under this category, which includes all tritave temperaments. If prime 3 is not present, it must be clearly noted as "(no-3's)" so that those looking for no-2's and no-3's temperaments can find them easily. Even harmonics are allowed as long as they don't implicate the existence of prime 2 in the subgroup, but judging their complexity becomes more difficult as a result, so all such temperaments must be clearly noted as "(

No-3's focus

All temperaments with prime 2 but no prime 3 go under this category.

No-5's focus

All temperaments with primes 2 and 3 but no prime 5 go under this category.

Exotemperaments (>~18c)

Exotemperaments are useful as targets for detempering, as they often underly the logic of various JI scales. They are also explored for novelty.

5-limit focus

Dicot

Dicot equates 5/4 with 6/5 into a generic neutral third, so that 3/2 is found at 2 generators. It is most notable as appearing commonly as an underlying logic of JI scales which do not find both 5/4 and 6/5 relative to the same scale degree anywhere, but for which 5/4 and 6/5 subtend the same number of scalesteps.

Father

Father equates 4/3 with 5/4, so that 3/2 and 5/4 are made into octave-complements. Thus it is extremely simple (and extremely high damage). If one detempers father into a JI scale, one must ensure 5/4 and 4/3 do not appear relative to the same scale degree anywhere and that they subtend the same number of scalesteps, which rules out most 5-limit JI scales people usually consider.

7-limit focus

Dominant

Dominant makes a generic minor third of 7/6~6/5 and major third of 5/4~9/7. It is the result of attempting to temper both 64/63 and 81/80. An especially notable detempering is 14:16:18:20:21:24:27:28, a low-complexity, 7-note, 7-limit, over-7 diatonic JI scale, which has 10/9, 9/8 and 8/7 as whole tones, and 21/20 and 28/27 as small semitones, thus fulfilling the 5L 2s diatonic pattern.

Low or very low accuracy (<~18c)

Low or very low accuracy temperaments are of interest to people wanting simple scales and who are fine with high-damage. As a result, they tend not to have "higher-limit focus", as the error involved on intervals beyond the 17-limit is too much. A variety of people consider this category to largely or even entirely be composed of exotemperaments, while others argue for various entries in this category being reasonable to consider harmonically based on the temperability of the simplest LCJI intervals.

Higher-limit focus

Temperaments in the higher-limit focus category imparting more than 12 cents of damage are rare, but are most common as implicitly being the targets of detempering of various JI scales.

Medium or low accuracy (<12c)

Medium or low accuracy temperaments in small prime limits are commonly considered due to their simplicity. As a result, "higher-limit focus" tends to not be focused on at this accuracy, as the error involved on intervals beyond the 17-limit is potentially too much depending on the context and who you ask, though again such temperaments are

Higher-limit focus

Temperaments in the higher-limit focus category imparting more than 7 cents of damage tend not to be considered, but are most common as implicitly being the targets of detempering of various JI scales.

High or medium accuracy (<7c)

Most temperaments people consider tend to fall into this category, due to its balance of simplicity and accuracy, though plenty of simple temperaments exist that are even more accurate, documented in higher-accuracy categories.

Very high or high accuracy (<3.5c)

The bound is the approximate melodic JND (Just-Noticeable-Difference), though note that this doesn't mean that damage/mistuning is imperceptible in these temperaments as the harmonic JND can often be significantly smaller, depending largely on context, timbre and who you ask.

Microtemperaments (<1c)

These temperaments essentially serve as ways of simultaneously simplifying and imparting new structure onto JI with minimal to unnoticeable tuning damage.