Bird's eye view of temperaments by accuracy: Difference between revisions

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 [[exotemperament]]s and [[microtemperament]]s as the extrema.

Note that the generator tunings are listed in order of increasing accuracy, with the least accurate being the leftmost; the least accurate tuning is not required to satisfy the maximum cent error, so that a tone-efficient example is included, though the example should be unambiguously representative of the full scope of its harmony within reason. All tunings after it should thus be increasingly accurate so that it satisfies the tuning bounds indicated by the accuracy classification of the temperament (up to any indicated exceptions for the temperament).

If you're having trouble, don't be afraid to ask in the [[Links#Discord_server|Xenharmonic Alliance Discord server]]! Remember to be persistent and patient.

Wherever you see "Note count(s):" on this page, those are the note counts needed to find all of those harmonics relative to the same note, so that you can build an otonal/harmonic series chord out of all of them.

Because most temperaments on this page have an octave or an ''n''th of an octave (1\''n'') as the period, it suffices in those cases to describe the set of odd harmonics that they target, and how many notes it takes to do so according to their mappings. Because different note counts may allow for different amounts of odd harmonics to be targeted, there is sometimes multiple note counts, to help the reader decide between them.

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]]).

=== No-5's focus ===

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

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

[[#Note counts|Note count]]: 12 for {3, 5, 9, 15, 27, 45(, 81)} ([[5L 7s]] or [[12L 5s]])

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]] = [[64/63|S8]]/[[81/80|S9]], so that it prefers a slightly-sharp or just fifth.

* [[#Nestoria]] equates [[~]][[19/16]] with [[32/27]] and [[~]][[19/15]] with [[81/64]].

[[Bird's eye view of temperaments by accuracy#Note counts|Note counts]]:

72 for {3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 49, 63} ([[27L 45s]])  

The bound is the approximate [[JND|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.

[[#Note counts|Note count]]: 15 for {3, 5, 9, 13, 15, 25} ([[4L 7s]])

This extension can be observed based on an [[S-expression]]-based comma list of: {[[169/168|S13]], [[225/224|S15 = S25*S26*S27]], [[325/324|S10/S12 = S25*S26]](, [[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|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 [[676/675|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]].

[[#Note counts|Note counts]]:

* 10 for {3, 5, 31} ([[8L 3s]])

* the most accurate is [[#Sendai]] which finds primes 23 and 29

* the second most accurate is [[#Sensible]], which finds primes 11, 17 and 23

[[#Note counts|Note counts]]:

* 10 for {3, 5, 15, 25, 125} ([[3L 7s]])

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]].

[[#Note counts|Note count]]: 18 for {3, 5, 7, 9, 15, 21, 27, 35, 45} ([[12L 5s]] or [[12L 17s]])

[[#Generator tunings|Generator tunings]]: 24\31, 31\53, 55\94

[[#Note counts|Note count]]: 23 for {3, 5, 7, 9, 11, 15, 21} ([[10L 11s]] or [[10L 21s]])

[[#Generator tunings|Generator tunings]]: 3\31, 4\41, 7\72

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-expression]]s by splitting 3/2 into two by tempering out {{nowrap| [[243/242|S9/S11 {{=}} (12/8)/(11/9)² {{=}} (3/2)/(11/9)²]] }}, splitting 3/2 into three by tempering out [[1029/1024|S7/S8 = (9/6)/(8/7)³ = (3/2)/(8/7)³]] and then splitting the [[~]][[8/7]] in two by tempering out [[225/224|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.

[[#Note counts|Note counts]]:

* 31 for {3, 5, ''7'', 9, 13, 15, (19*,) ''21'', 27, ''35''(, 81)} ([[5L 28s]])

(<nowiki>***</nowiki> 53edo is uniquely the tuning that uses both the canonic mapping at 41 gens and the simplified higher-damage mapping at -12 gens = ~32/27, as 41 + 12 = 53.)

{{ See also | Sensipent#Sendai interval table }}

[[#Note counts|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).

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.

[[#Note counts|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.

[[#Note counts|Note counts]]:

* 10 for {3, 5, 9, 15, 17} ([[2L 8s]])

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.

[[#Note counts|Note count]]: 11 for {3, 5, 7, 9, 15, 21, 25} ([[7L 5s]], [[12L 7s]])

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 in half is by interpreting [[~]][[11/9]] as half of the meantone fifth, by tempering out [[243/242|S9/S11 = (12/8)/(11/9)² = (3/2)/(11/9)²]], which as [[81/80|S9 = (9/8)/(10/9)]] is tempered implies tempering out [[121/120|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.

[[#Note counts|Note count]]: 15 for {3, 5, 7, 9, 21, 35, 49} ([[5L 11s]], or [[5L 16s]] for odd 15)

The only real drawback of mothra is that because it splits the meantone fifth in three, it takes 12 generators to reach prime 5.

[[#Note counts|Note count]]: 13 for {3, 5, 7, 9, 15, 25} ([[3L 10s]], [[3L 13s]], or [[3L 16s]])

Magic is generated by a major third of around 380 cents, five of which make a perfect twelft ([[3/1]]). This results in the magic comma, [[3125/3072]], being tempered out. Since the major third is tuned quite flat, it makes sense to equate two of them to 14/9, tempering out [[225/224]] and mapping 7/4 to +12 generators, which also tempers out [[245/243]]. While magic has slightly higher complexity and error than [[#Septimal meantone]], it doesn't temper out [[81/80]] and therefore can distinguish [[9/8]] and [[10/9]], and is in fact one of the simplest temperaments capable of mapping the [[9-odd-limit]] distinctly. The canonical extension to the 11- and 13-limits tempers out 100/99 and 105/104, but that increases complexity and lowers accuracy.

[[#Note counts|Note count]]: 22 for {3, 5, 7, 9, 15, 21, 25, 35, 45, 75} ([[9L 13s]])

Orwell is a tempermant generated by a slightly sharp subminor third representing [[7/6]], seven of which reach [[6/1]]. Its name stems from George Orwell's book 1984, as a good generator is [[84edo|19\84]]. Three subminor thirds reach 8/5, the [[5th]] subharmonic. This temperament is supported by notable edos [[22edo|22]], [[31edo|31]], and [[53edo|53]], as well as the less known though still notable [[84edo]]. This temperament represents the 7-limit with good accuracy and relatively low complexity. However, the mapping for 11 is very simple, so this temperament really comes into its own in the 11-limit (see [[#Undecimal Orwell]]). Important commas tempered out by orwell include [[225/224]], [[1728/1715]], [[2430/2401]], [[6144/6125]], [[65625/65536]], and [[2109375/2097152]].

[[#Note counts|Note counts]]:

* 24 for {3, 5, 7, 11, 15, 17, 33} ([[14L 8s]])

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.

[[#Note counts|Note counts]]:

* 23 for {3, 5, 7, 9, 13, 15, 25, 27, 39, 45, 65, 75(, 125)} ([[15L 4s]] or 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.

[[#Note counts|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.

{{ See also | Sensipent#Sensible interval table }}

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{{cent}}) if avoiding composite odds and intervals of 23 so that under such restrictions it classifies as [[#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.

[[#Note counts|Note count]]: 36 for {3, 5, 7, 9, 11, 13, 15, 17, 23, 33, 35, 51} (12L 22s)

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]].

[[Bird's eye view of temperaments by accuracy#Note counts|Note count]]:

* 7 for {5, 7, 25, 35, 49} ([[4L 5s (3/1-equivalent)|4L 5s<3/1>]])  

[[#Generator tunings|Generator tuning]]: 10\13edt

[[#Note counts|Note counts]]:

* 10 for {5, 7, 11, 25, 35, 55} ([[6L 7s]])  

[[37edo]] is a good tuning for its size and is practically equivalent to the pure-11's tuning, though for less damage on 5 and 7, [[68edo]] is a good tuning that much better targets composite harmonies, so might be advised as the first edo tuning to try.

[[#Note counts|Note count]]: 29 for {5, 7, 11, 13, 17, 19, 25, 55, 65, 91} ([[6L 25s]])

Though not quite as accurate as [[#Didacus]], mediantone is notable for extending didacus to the full no-3's 19-limit. It does well in an optimized tuning, though if one wants to use an edo tuning, the main tuning of interest is [[80edo]], which finds primes 3 and 23 as well.

[[#Note counts|Note counts]]:

* 8 for {3, 7, 21} ([[5L 3s]])

The 2.3.7 part of [[#Buzzard]] is not as accurate as everything else in the 13-limit; specifically, its interval of 7 barely violates the 4{{cent}} bound, however it makes up for it by being much simpler (mapping-wise) so that it is interesting as a 2.3.7-subgroup temperament that splits [[3/1]] into four equal parts, each representing a sharp [[~]][[21/16]], which defines it in the 2.3.7 subgroup. [[53edo]], [[58edo]] and [[111edo]] are good tunings.

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.

[[#Note counts|Note counts]]:

* 4 for {3, 7} ([[2L 3s]])

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.

[[#Note counts|Note counts]]:

* 9 for {3, 5, 9, 11(, 33, 35)} ([[7L 3s]], note odd 35 comes from the [[mohajira]] mapping of 7 specifically)

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 [[243/242|S9/S11 = (12/8)/(11/9)² = (3/2)/(11/9)²]], which as [[81/80|S9 = (9/8)/(10/9)]] is tempered implies tempering out [[121/120|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.

[[#Note counts|Note count]]: 22 for {3, 5, 7, 9, 11, 15, 21, 25, 33, 35, 45, 55, 75, 77} ([[9L 13s]])

Orwell is a tempermant generated by a slightly sharp subminor third representing [[7/6]], seven of which reach [[6/1]]. Three subminor thirds reach [[8/5]], and two reach [[11/8]]. This temperament is arguably one of the simplest 11-limit temperaments with decent accuracy, and is supported by the highly notable edos {{edos|22, 31, and 53.}} This temperament is more accurate in lower limits, very nearly being a microtemperament in the 5-limit (5-odd-limit minimax error 1.006{{c}}!), and quite good accuracy in the 7-limit, with the least accurate prime being 11, so this temperament can only barely be considered "low accuracy". However, the mapping for 11 is very simple, so this temperament really comes into its own in the 11-limit. Important commas tempered out by orwell include [[99/98]], [[121/120]], [[176/175]], [[225/224]], [[385/384]], [[540/539]], and [[1728/1715]].

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.

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.

[[#Note counts|Note count]]: 9 for {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). Due to its inaccuracy, it is recommended to use a sharp octave-tempering for this temperament, such as [[30edt]] instead of [[19edo]], in which case you also improve various intervals of 13 as well. Doing this means that you can use voicing across octaves to improve the accuracy and hence psychoacoustic convincingness of godzilla.

[[#Note counts|Note count]]: 9 for {3, 5, 7(, 13, 15)} ([[9L 1s]])

Like [[#Godzilla]], negri is a temperament that trades accuracy for simplicity. It divides the 4/3 into four equal semitone-like steps that resemble steps of 10edo, so that it can be seen as halving the generator of godzilla, which it uses to find prime 5 more quickly. As a result, odd 9 is found at 8 generators, so you can't find odd 9 from the same position as odd 5 unless you use at least 12 notes, which corresponds to the 19-note MOS scale of [[10L 9s]], at which point (like godzilla) you may as well use [[19edo]] with octave-tempering (e.g. [[30edt]]), though [[29edo]] also supports a stranger tuning of it if you want to try a subset of that with sharp-octave-tempering. Negri can also be interpreted as a temperament of the 2.3.5.13 subgroup, for which an interesting tuning is [[48edo]] by using the 48f val also used by its strange rendition of [[#Buzzard]], and as a 5-limit temperament, it is supported by yet more tunings (generally: ones using a flat mapping for 5/4 and 3/2 which split 4/3 into four of the implied ~16/15).

[[#Note counts|Note counts]]:

* 6 for {3, 5(, 15)} ([[3L 3s]])

Augene is the result of using 1\3 as an approximation of ~5/4 so that the period is a third of 2/1, and then using a sharp 3/2 or equivalently a flat 6/5 as the generator. Because the approximation of 5/4 is so sharp, any good tuning of augene will have a noticeably sharp ~3/2 so that ~6/5 becomes more in-tune (which is also important as it's ''also'' the generator, being equal to ~3/2 minus a period). Because the amount required to get ~6/5 reasonably in-tune is quite significant, it makes sense to lean into this and take advantage of it by tempering out the difference between the very sharp ~9/8 and a flat ~8/7, so that the minor third becomes ~7/6 as in [[#Superpyth]]. Augene merges with superpyth in [[27edo]], which is a recommendable tuning for also being the smallest edo to have all intervals of the [[7-odd-limit]] tuned distinctly (not equated), in which case you get harmonies of the no-11's [[13-odd-limit]] too. It's unclear whether a psychoacoustically optimal tuning of augene would have the fifth sharper or flatter than the 27edo tuning; if you think having the fifth more in-tune is preferable, you could try the [[39edo]] tuning {{nowrap| (23\39, 1\3) }}, which uses a very sharp mapping for ~7/4 so that 7 is barely sharper than 5; by some metrics this tuning is more optimal than 27edo. If you want 6/5 more in-tune, you could try the [[42edo]] tuning {{nowrap| (25\42, 1\3) }}, which may be preferred for having a potentially more convincing approximation of ~4:5:6 than all the other options discussed, as well as for being distinctly consistent in the 7-odd-limit like 27edo. Some even prefer the [[15edo]] tuning, as though it damages the 7-limit even more than augene does so that the 7-odd-limit is no longer tuned distinctly (because of ~8/7 = 1\5 = ~7/6), it includes an approximation of ~11/8, and can be thought of as xenmelodically/structurally interesting for a version of augene where the 5edo fifth is the generator and the 3edo major third is the period.

[[Bird's eye view of temperaments by accuracy#Note counts|Note count]]: 10 for {3, 5, 7, 9} ([[2L 8s]])

This temperament was discovered by [[Paul Erlich]], and its scales are known for having properties similar to diatonic in the 5-limit. This temperament tempers out [[50/49]], setting [[7/5]] and [[10/7]] to the half-octave. It is generated by a fifth, or alternatively a semitone that is the fifth minus the half-octave. It tempers out [[64/63]] and [[2048/2025]], meaning harmonics 5 is mapped to a half-octave minus two semitones, and harmonic 7 is an octave minus two semitones. This temperament is best used with a decatonic interval classification, so 3/2 is a Perfect 7th₁₀, 5/4 is a Major 4th₁₀, and 7/4 is a Major 9th₁₀. The [[4:5:6:7]] otonal tetrad is written P1-M4-P7-M9. Changing both major intervals to minor ones gives us the [[70:84:105:120|1/(7:8:10:12)]] utonal tetrad. Since 50/49 is tempered out, [[25/24]] and [[49/48]] are equated, both to the augmented unison of the decatonic scale. It works just like in diatonic, where changing the major third of the [[4:5:6]] triad to a minor one gives the [[10:12:15|1/(4:5:6)]] triad. While this temperament has poor accuracy overall, since 12edo's accuracy is accepted by many, this temperament's accuracy can still be considered reasonable.

[[Bird's eye view of temperaments by accuracy#Note counts|Note count]]: 15 for {3, 5, 7, 9, 11, 15} ([[7L 8s]])

This temperament is generated by a submajor second of around 163 cents representing all of [[10/9]], [[11/10]], and [[12/11]]. It maps prime 3 to -3 generators, and prime 5 to -5 generators, tempering out [[250/243]]. This means two 10/9s are equated to 6/5, and three 10/9s are equated to 4/3. The 10/9 generator is then naturally equated to 11/10 and 12/11, mapping prime 11 to -5 generators. An important high-damage equivalence of this mapping is 11/9~6/5 via tempering of [[55/54]], a comma of over 31 cents. The 2.3.5.11 version of this temperament is arguably one of the simplest temperaments in this subgroup with acceptable accuracy, even if barely so. Even if one doesn't accept its accuracy, it is still useful as a mapping to organise intervals. The canonical mapping for prime 7 finds it at +6 generators, tempering out [[64/63]], which makes sense as the fifth is already very sharp. In that regard, it is notable in the full 11-limit as well, and is one of the best ways to analyse the 11-limit of [[22edo]].

[[#Note counts|Note count]]: 14 for {3, 5, 7, 9, 11, 13}

Flattone is a low-accuracy [[11-limit|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{{cent}} bound, along with odd 9 being tuned very flat which has the benefit of causing the tuning of [[6/5]] to be relatively accurate.

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.