User:Godtone/Bird's eye view of temperaments by accuracy
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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. The main exception that justifies more than two bound-violating interval pairs is when a single odd is responsible for all of them. 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.
Also: the convention for this page, in contrast to most xen wiki pages, is that if you want to refer to another rank 2 temperament while discussing a rank 2 temperament, you should link to that rank 2 temperament's entry on this page (unless you are merely discussing competing extensions/don't intend to log that temperament separately on the page) so that the page can be fairly self-contained to avoid intimidating and confusing someone using this page as a reference. The obvious exception is every temperament here may link to its own entry on other pages. Also note that the purpose of using #Temperament to link to something on this page is to indicate that clicking on the link will not take you away from the page and that #Temperament is intended to be logged on the page.
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.
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.
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.
~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 to 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.
Bug
Bug equates 5/3 with 9/5, so that two semi-twelfth intervals can be made into a twelfth. It can be useful for creating 5-limit pentatonic scales, along with father. In addition, it is tempered out by 14edo, which in a sense might be the largest 5-limit exotemperament EDO.
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.
Decimal
Decimal makes many semi-closely-related intervals equivalent, which can be useful at times when one wants to create low-cardinality scales or use it as an analysis system. For example, the generator can be an 8/7~7/6 (tempering out 49/48) or 6/5~5/4 (tempering out 25/24). The period is treated as 7/5~10/7 (tempering out 50/49).
11-limit focus
~17-limit focus
Higher-limit focus
No-2's focus
No-3's focus
No-5's focus
Semaphore
Semaphore and bug are quite similar in that they have semi-twelfths (or semi-fourths). However, unlike bug, semaphore's semi-twelfths have a ratio of 12/7~7/4.
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.
5-limit focus
7-limit focus
Godzilla
Note count: 9 for {1, 3, 5, 7, 9(, 21)} (5L 4s)
Godzilla is a very coarse temperament, where we use context to suggest ~4:5:6:7(:8). It admits a natural extension to prime 13 based on interpreting its semifourth of ~8/7~7/6 much more accurately as 15/13, though if you specifically want 15/13 as the semifourth, there is much more accurate temperaments available that don't require interpreting it inaccurately as ~8/7~7/6, such as #Immunity, or if you don't need a semifourth as the generator, #Cata. Nonetheless, insofar as it makes sense, it's notable for providing a usefully-small 9-note scale for the entire 9-odd-limit (insofar as it is capable of approximating its sound with context).
11-limit focus
~17-limit focus
Flattone
Note count: 14 for {3, 5, 7, 9, 11, 13}
Flattone is a low-accuracy 11- or 13-limit temperament. It is an alternative extension of #Meantone of interest because it maps 7/4 to the arguably more intuitive diminished seventh and 11/8 to the similarly simple augmented fourth (aka tritone). If one maps the 13-limit, the best way is by equating a sharpened ~16/13 with the already-very-flat ~5/4, continuing the strong flat tendency. It tunes meantone much flatter than usual so that the whole tone is much closer to 10/9 than it is to 9/8, and is maybe most notable as being supported by 26edo, the smallest edo consistent in the 13-odd-limit. Maybe surprisingly, it is one of the most accurate temperaments in this accuracy category; its most off primes are 5 and 13, which are the only ones to meaningfully transgress the 12 ¢ bound, along with odd 9 being tuned very flat which has the benefit of causing the tuning of 6/5 to be relatively accurate.
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.
No-2's focus
No-3's focus
No-5's focus
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 commonly relevant as targets for detempering.
5-limit focus
7-limit focus
Superpyth
Note counts:
Bound-violating intervals: 9/8 (and 8/7 in flatter tunings like 22edo)
Superpyth is the natural "opposite" of #Septimal meantone in a surprisingly large number of surprisingly exact senses; the main ones of note are that the fifth is mistuned in opposite directions (sharp in superpyth), and that superpyth makes prime 7 most immediately accessible on the chain of fifths with prime 5 requiring more complex movements (when measured in number of fifths), while septimal meantone does the opposite. Superpyth makes the major third ~9/7, the minor third ~7/6, the major second a blend between a sharp ~9/8 and a flat ~8/7 and the minor second ~28/27, which is tuned very flat so that it becomes a quarter-tone in any good tuning of superpyth. Superpyth finds ~5/4 as the augmented second and ~6/5 correspondingly as the diminished fourth.
Superpyth is a common choice for a beginner, with 22edo and 27edo having different advantages and 22edo the most explored by far, though the number of unique advantages and opportunities in 27edo make it formidable as a competitor. 22edo is approximately the pure-9/7's tuning while 27edo is approximately the pure-7/6's tuning. For 5 more notes, 27edo has the advantage of not equating 7/5 and 10/7 and having a more accurate 8/7 and 7/4. A more optimized edo tuning is 49edo but that comes at the cost of a lot of notes if you aren't merely looking to take a MOS scale subset of it.
11-limit focus
Mohaha
Note counts:
- 9 for {3, 5, 9, 11(, 33, 35)} (7L 3s, note odd 35 comes from the mohajira mapping of 7 specifically)
- 20 or 23 for adding {7, 15, 21(, 35)} (7L 10s and 7L 17s)
Mohaha is a 2.3.5.11 (no-7's 11-limit) "hemi-meantone" temperament that splits #Meantone's fifth into two ~11/9's by tempering out S9/S11 = (12/8)/(11/9)2 = (3/2)/(11/9)2, which as S9 = (9/8)/(10/9) is tempered implies tempering out S11 = (11/10)/(12/11) = (11/8)/(15/11) as well. It has two main extensions to the full 11-limit; if you accept the #Septimal meantone mapping of 7 you get migration, but maybe more natural is if you instead equate the flat ~33/32 interval with S6 = 36/35 = (6/5)/(7/6), which results in mohajira, which finds 7 at a negative number of gens so that composite harmonics of 7 are simpler to find (as primes 3, 5 and 11 are all found at a positive number of gens). Because of this, mohajira is usually the preferred extension as it is more note-efficient, but both extensions merge in 31edo, which is a good tuning for both.
~17-limit focus
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.
No-2's focus
No-3's focus
No-5's focus
High or medium accuracy (<7c)
Many temperaments that people consider theoretically tend to fall into this category, due to its balance of simplicity and accuracy and due to the common usage of meantone temperaments, though plenty of simple temperaments exist that are even more accurate, documented in higher-accuracy categories.
5-limit focus
Meantone
Note count: 5 for {1, 3, 5}
Meantone is an incredibly efficient temperament for targetting 5-odd-limit harmony whose characteristic is flattening 3/2 (the generator) by a few cents. Perhaps unsurprisingly, it was historically the most commonly used temperament. It does this by sacrificing a distinction between 9/8 and 10/9 so that two "tones" makes 5/4, hence its name.
Srutal archagall
Note counts:
Srutal archagall is the natural extension of 5-limit diaschismic to prime 17 by interpreting the generator as a near-just 17/16 and the period as ~24/17~17/12.
It is notable as preserving a lot of intuitions of 12edo like 9/8 as two semitones, 6/5 as three semitones, 5/4 as a semitone less than 4/3 which is itself a semitone less than half an octave. It essentially "doubles up" on familiar categories by: two minor seconds, ~25/24 and ~18/17~17/16~16/15, two major seconds, ~10/9 and ~9/8~17/15, two minor thirds, ~20/17 and ~6/5, and two major thirds, ~5/4 and ~51/40~32/25. Furthermore, major and minor intervals are separated by a semitone of 17/16, though interestingly this results in alternating the tuning of the otherwise-familiar category, e.g. 20/17 = (5/4)/(17/16)). The distance between these pairs of familiar intervals is an exaggerated syntonic comma (81/80) which is also equal to a flattened diesis (128/125), making it useful as a more accurate alternative to meantone. Notably the intervals of 5 require using the period offset to reach, so that the minor third reached by the circle of fifths is actually 20/17.
For abundant options, one might prefer a 22-note MOS over a 12-note one, so that srutal archagall can be seen as a detempering of 22edo, but the 12-note MOS is likely the easiest and most intuitive to approach for a beginner. 34edo is a good tuning for optimizing the 2.3.5.17 subgroup.
7-limit focus
Septimal meantone
Note count: 11 for {3, 5, 7, 9, 15, 21, 25} (7L 5s)
Bound-violating intervals: 9/8, 10/9, 36/25 (none if odd 9 is omitted)
Septimal meantone is an extension of #meantone that finds 8/7 as the diminished third and 7/6 as the augmented second, so that using an augmented sixth with a major triad forms a harmonic seventh chord. Though 12 notes is about sufficient for achieving its harmony, often one wants to use a 19-note MOS (12L 7s) for more freedom and availability.
It has two main extensions to prime 11, both similarly complex, discussed in meantone vs meanpop, though the one called undecimal meantone is arguably more elegant as being the merge of septimal meantone and the no-3's 11-limit temperament #Didacus, which can be seen as every other gen of undecimal meantone. An alternative extension that splits the generator is by interpreting ~11/9 as half of the meantone fifth, by tempering out S9/S11 = (12/8)/(11/9)2 = (3/2)/(11/9)2, which as S9 = (9/8)/(10/9) is tempered implies tempering out S11 = (11/10)/(12/11) = (11/8)/(15/11) as well. This leads to #Migration if you accept the septimal meantone mapping of 7 (which becomes double as complex), or mohaha if you interpret it as no-7's.
Mothra
Note count: 14 for {3, 5, 7, 9, 21, 35, 49} (5L 11s, or 5L 16s for odd 15)
Bound-violating intervals: 9/8, 10/9 (none in 7-odd-limit or if 9 is omitted)
Mothra makes a near-just 8/7 equal to a third of a meantone fifth (~3/2) and is most notable for being a surprisingly elegant extension of meantone to the 7-limit (though it splits the generator), as tempering out 81/80 makes S6 = 36/35 (the distance between 6/5 and 7/6) and S8 = 64/63 (the distance between 8/7 and 9/8) equivalent, so it seems natural to want to equate S6 = S7 = S8, where S7 = 49/48 (the distance between 7/6 and 8/7), so that 9/8, 8/7, 7/6, 6/5 are made equidistant. As a result, not only is 8/7 a third of 3/2, but also, because of tempering out S6/S7, we have that 7/6 is a third of 8/5. Combining it with #Septimal meantone (among other things) results in 31edo, where it is quite accurate, while combining it with the less accurate #Flattone results in 26edo, where it is quite damaged.
The only real drawback of mothra is that because it splits the meantone fifth in three, it takes 12 generators to reach prime 5.
11-limit focus
~17-limit focus
Echidna
Note counts:
- 24 for {3, 5, 7, 11, 15, 17, 33} (14L 8s)
- 26 for adding {9} (14L 8s or 22L 14s)
- 46 for adding {13} (22L 14s)
Echidna has a generator of 11/10 or equivalently 9/7 because 11/10 * 9/7 = 99/70 is its period of half an octave, and can be seen as splitting the fourth of srutal archagall into three 11/10's by tempering out S10/S11 = (12/9)/(11/10)3 = (4/3)/(11/10)3 so that 12/11 and 10/9 are made equidistant from 11/10. It can be seen as a high-accuracy version of hedgehog and as a mild detempering of 22edo that achieves an accurate and distinctly consistent 11-odd-limit. In fact, of the three smallest edos that are distinctly consistent in the 11-odd-limit, which are 58edo, 72edo and 80edo, echidna is supported by the smallest and third-smallest (so 72edo is in a sense the odd one out, being the one that doesn't support echidna). The smallest edo consistent in the 11-odd-limit, 22edo, is in fact a trivial tuning of echidna, where the generator is conflated with 12/11 and 10/9. 58edo and 80edo are both interesting options, being the merge of echidna and a variety of other notable temperaments, so depending on preference and tuning needs, though 80edo is the more optimal tuning for it (especially in the full 17-limit).
Echidna is notable as achieving no-13's 17-limit harmony with accuracy in a surprisingly small number of notes. 13/8 can be found too but is the most complex, being found at (11/10)16 plus a half-octave period, octave-reduced. However, as primes 5 and 11 are also found in the same direction, intervals of 13 are common even in the 22-note MOS, so the 36-note MOS is more useful than might be suspected, despite not finding every odd from the same position.
Catakleismic
Note counts:
- 23 for {3, 5, 7, 9, 15, 25, 27, 39, 45, 65, 75(, 125)} (15L 4s or 19L 15s)
- 29 for adding {21, 35, 63} (19L 15s)
As catakleismic is largely just a certain extension of cata to prime 7, see #Cata. It admits a number of possible extensions to prime 11 depending on user preference and the tuning used, hence its listing here as a ~13-limit temperament.
Diaschismic
Note count: 36 for {3, 5, 7, 9, 11, 13, 15, 17, 21, 25, 33, 35, 39, 51} (12L 22s)
Diaschismic is an extension of #Srutal archagall to the full 17-limit of similar complexity to #Srutal (with which it merges in 46edo so that they're complimentary) but which damages the 2.3.5.17 subgroup slightly more. Familiarizing oneself with the structure of srutal archagall is recommendable, even if the ideal tunings differ slightly, as navigation will be similar.
Higher-limit focus
Sensible
Note counts:
- 26 for {3, 5, 11, 17, 23, 31, 33, 51, 55, 69, 85, 99(, 115)} (19L 8s)
- 35 for adding {9, 15, 25, 93(, 155)} (19L 8s or 19L 27s)
Sensible is a fairly accurate (though quite complex) subgroup interpretation of sensi (or extension of #Sensipent) which is especially accurate in the CEE tuning (443.115 ¢) if avoiding composite odds and intervals of 23 so that under such restrictions it classifies as #Very high accuracy, meaning it's one of the most accurate temperaments in this category, especially if you use error-cancellations of sharp harmonics to your advantage to construct complex harmonic series chords in order to justify the generally-higher errors of composite harmonics. 46edo is a somewhat reasonable but trivial tuning of it, 65edo is somewhat better, and 111edo is even better, but it works especially well in a more optimized tuning, hence its significance as a rank 2 temperament. The 27-note MOS, though not achieving all of its odd harmonics from a single note, is sufficient, because the missing odds {9, 15, 25, 93(, 155)} also do reasonably occur within the span of the 27-note MOS. Therefore, sensible can be seen as a detempering of 27edo to a more accurate rank 2 temperament on a mostly-unrelated subgroup.
Srutal
Note count: 36 for {3, 5, 7, 9, 11, 13, 15, 17, 23, 33, 35, 51} (12L 22s)
Bound-violating intervals: 13/9
Srutal is an at least no-19's 23-limit temperament, being an extension of #Srutal archagall to the full 17-limit and finding 23/16 as an augmented fourth, that is, as a tritone of (~9/8)3. Therefore, familiarizing oneself with srutal archagall is recommendable as the structures and tunings are nearly identical (and their tunings merge meaningfully in 80edo), with the main difference being number of notes and breadth of harmonies targetted. Srutal can find more primes than just those in the no-19's 23-limit but they are more complex so more likely to be used opportunistically in a 34-note MOS, so that this temperament can be seen as a detemperament of 34edo. 80edo is a good tuning for it, though 46edo deals well enough with the no-19's 23-limit part (potentially add-31) at the cost of a variety of distinctions. An alternative extension of srutal archagall to just the full 17-limit which is damages the 2.3.5.17 subgroup slightly more is #Diaschismic.
No-2's focus
No-3's focus
No-5's focus
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 is listening/who you ask.
5-limit focus
Cata
Note count: 15 for {3, 5, 9, 13, 15, 25} (4L 7s)
Cata is a very efficient 5-limit and 2.3.5.13-subgroup temperament with a generator of a very slightly sharpened 6/5, two of which make 13/9 and thus three of which make 26/15 which is made into half of 3/1 so that its octave complement of 15/13 is half of 4/3. It is amazing for its combination of accuracy and simplicity, because making six ~6/5 generators equal to a fourth or fifth (up to octave-reduction) is the simplest equivalence possible without incurring a lot of damage. Its 7-note scale of 4L 3s is usable, and its interpretation is accurately {25/24, 6/5, 5/4, 36/25~13/9, 3/2, 26/15, 2/1} so that it is at the simplest structural level well-supplied with plausible harmony, as this structure will persist and be duplicated in every superset/derived MOS scale, such as the likely more useful 15-note one, whose tuning range is at broadest in the 15edo to 19edo range, corresponding to the small step being at least half the size of the large step so that it has Rothenberg propriety (for those that care about this property).
Cata admits an elegant extension to prime 7 called Catakleismic, at the cost of some accuracy, a higher complexity and a smaller valid tuning range.
This extension can be observed based on an S-expression-based comma list of: {S13, S15 = S25*S26*S27, S10/S12 = S25*S26(, S25, S26 = S13/S15, S27)}, which is notable as making use of the record prime gap between 23 and 29 for an opportune no-11's 13-limit tempering opportunity ~28/27~27/26~26/25~25/24, which as shown, implies tempering many notable commas, the most accurate of which is the ragisma (S25/S27), corresponding here to having an interval ~14/13~27/25~13/12, and (arguably) the most interesting of which is making use of the exceptional numerical coincidence that S13/S15 = S26. The tuning range for catakleismic is approximately 53edo to 72edo - which are both reasonable tunings for it, with 53edo more accurate on the full subgroup and 72edo more accurate in the 7-limit.
Sensipent
Note counts:
Sensipent is an accurate 2.3.5.31 temperament with a generator of ~31/24~40/31, where the two interpretations of the generator differ by S31 = (31/30)/(32/31), which is the best extension of 5-limit sensipent as its generator serves as half of 40/24 = 5/3, so that the generator is the midpoint of 4/3 and 5/4, whose difference is 16/15, hence the relevance of making ~32/31~31/30.
Sensipent finds 6/1 (the fifth plus two octaves) at 7 generators.
It admits a number of extensions of varying accuracy:
- the most accurate is #Sendai which finds primes 23 and 29
- the second most accurate is #Sensible, which finds primes 11, 17 and 23
- the simplest but least accurate is #Sensor (commonly just called "sensi"), which interprets it as a full 17-limit temperament.
Würschmidt
Note counts:
- 10 for {3, 5, 15, 25, 125} (3L 7s)
- 18 for adding {9, 23, 45, 75, 115} (3L 16s)
- 22 (or 23) for adding {11, 55(, 69)} (3L 19s or 3L 22s)
Würschmidt (sometimes written wurschmidt or wuerschmidt for convenience) is a temperament with an approximately 1 ¢ sharp 5/4 as the generator, so that 6/1 is reached as (5/4)8. The rationale for this is that (5/4)3 falls short of the octave by 128/125, and this is approximately half of 25/24, so that if we flatten (5/4)2 = 25/16 by 128/125 twice we get ~3/2. Therefore, in an optimized tuning, we can expect the fifth to be slightly flat, so that 25/24 is sharpened so that it makes sense to equate with a slightly flattened ~24/23 by tempering out their difference, S24, which is favourable as finding interpretations of a variety of intervals that are otherwise given somewhat questionable 5-limit interpretations, as documented in its interval chain. Because of dividing 6/1 into eight, it admits a neutral third at 4 generators so that an extension to prime 11 is also natural by tempering out S9/S11 = (12/8)/(11/9)2 = (3/2)/(11/9)2 so that ~11/9~27/22 is the neutral third.
Würschmidt can be seen as something like a cluster temperament with 3 main clusters, and with ~128/125~46/45 as the interval separating intervals in a given cluster. A notable extension to prime 7 is #Hemiwürschmidt by splitting the generator into two ~28/25's, which is thus the result of combining würschmidt with #Didacus.
7-limit focus
11-limit focus
Miracle
Note count: 23 for {3, 5, 7, 9, 11, 15, 21} (10L 11s or 10L 21s)
Miracle is an elegant temperament that splits 3/2 into six equal parts that can be derived as the most natural and efficient way of doing so through S-expressions by splitting 3/2 into two by tempering out S9/S11 = (12/8)/(11/9)2 = (3/2)/(11/9)2, splitting 3/2 into three by tempering out S7/S8 = (9/6)/(8/7)3 = (3/2)/(8/7)3 and then splitting the ~8/7 in two by tempering out S15 = (15/14)/(16/15) so that 15/14 and 16/15 are equated. 72edo is a very good tuning of miracle, though 31edo and 41edo may be preferred for smaller note counts and for the various things they support, EG meantone for 31edo and garibaldi for 41edo.
~17-limit focus
Higher-limit focus
Sendai
Note counts:
Sendai is an accuracy-focused extension of #Sensipent to primes 23 and 29. If one is fine with lowering the accuracy but increasing the number of interpretations of harmony, it can merge meaningfully with #Sensible, giving access to primes 11 and 17, and this has the benefit that combining them does not force an edo (or more generally a rank 1) tuning, though if one wants to use an edo/rank 1 tuning, the obvious choice is 65edo which gets you prime 19 too (though that could be added as a more complex extension of either).
No-2's focus
No-3's focus
No-5's focus
Microtemperaments (<1c)
These temperaments essentially serve as ways of simultaneously simplifying and imparting new structure onto JI with minimal to unnoticeable tuning damage.
5-limit focus
Schismic
Note count: 12 for {3, 5, 9, 15, 27, 45(, 81)} (5L 7s or 12L 5s)
Schismic is a very accurate and efficient 5-limit temperament which is almost identical to Pythagorean tuning except that it tempers the perfect fifth very slightly flat so as to find 8/5 accurately at (9/8)4, that is, as the Pythagorean augmented fifth, or equivalently, finding 5/4 as the Pythagorean diminished fourth. Note that the smallest edo that validates its status as a microtemperament is 118edo, as 53edo, though a tone-efficient tuning, doesn't temper the fifth flat enough, being approximately the Pythagorean tuning of schismic. In schismic, (9/8)6 overshoots the octave by ~81/80 so that the syntonic comma and the Pythagorean comma are equated.
Many extensions to other primes exist, but most are not accurate enough to be microtemperaments, except for the extension to prime 41 by tempering out 1025/1024 = (41/32)/(32/25). However, as it is common to want to extend schismic, we will note common extensions here:
- #Garibaldi finds ~8/7 as 9/8 * 81/80 by tempering out 5120/5103 = S8/S9, so that it prefers a slightly-sharp or just fifth.
- Schismic tridecapyth (which is the 2.3.5.13 version of #Cassandra) finds 13/4 as (9/8)10 and demands an approximately Pythagorean tuning.