Xenharmonic Wiki

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

← Older editNewer edit →
VisualWikitext
Godtone (talk | contribs)
autopatrolled, emailconfirmed
2,654 edits
m finish sentence i accidentally left unfinished
Godtone (talk | contribs)
autopatrolled, emailconfirmed
2,654 edits
invert to put highest accuracy first, considering it looks like noone else is contributing to this page anyway and that i dont recommend the first two categories
Line 33: Line 33:
All temperaments with primes 2 and 3 but no prime 5 go under this category.
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]] [[scale]]s. They are also explored for novelty.


= '''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 ==
== 5-limit focus ==
=== Dicot ===
=== [[Schismic]] ===
[[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.
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]])<sup>4</sup>, 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)<sup>6</sup> overshoots the octave by [[~]][[81/80]] so that the syntonic comma and the [[Pythagorean comma]] are equated.


=== Father ===
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:
[[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 ===
* [[#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.
[[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 ==
* Schismic [[tridecapyth comma|tridecapyth]] (which is the 2.3.5.13 version of [[#Cassandra]]) finds 13/4 as (9/8)<sup>10</sup> and demands an approximately Pythagorean tuning.
=== 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 ===
* [[#Nestoria]] equates [[~]][[19/16]] with [[32/27]] and [[~]][[19/15]] with [[81/64]].
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).


== 7-limit focus ==
== 11-limit focus ==
== 11-limit focus ==
== ~17-limit focus ==
== ~17-limit focus ==
Line 60: Line 59:
== No-5'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.


= '''Very low accuracy (<~18c)''' =
 
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.
= '''High accuracy (<4c)''' =
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.
== 5-limit focus ==
== 5-limit focus ==
== 7-limit focus ==
=== [[Cata]] ===
=== [[Godzilla]] ===
Note count: 15 for {3, 5, 9, 13, 15, 25} ([[4L 7s]])
Note count: 9 for {1, 3, 5, 7, 9(, 21)} ([[5L 4s]])
 
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: {[[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]].
 
=== [[Sensipent]] ===
Note counts:
* 10 for {3, 5, 31} ([[8L 3s]])
* 19 for adding {9, 15, 25} ([[8L 11s]])
 
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 [[961/960|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.


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).
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.


== 11-limit focus ==
=== [[Würschmidt]] ===
== ~17-limit focus ==
Note counts:
=== [[Flattone]] ===
* 10 for {3, 5, 15, 25, 125} ([[3L 7s]])
Note count: 14 for {3, 5, 7, 9, 11, 13}
* 18 for adding {9, 23, 45, 75, 115} ([[3L 16s]])
* 22 (or 23) for adding {11, 55(, 69)} ([[3L 19s]] or [[3L 22s]])


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.
Würschmidt (sometimes written wurschmidt or wuerschmidt for convenience) is a temperament with an approximately 1{{cent}} sharp [[5/4]] as the generator, so that [[6/1]] is reached as (5/4)<sup>8</sup>. The rationale for this is that (5/4)<sup>3</sup> falls short of the octave by [[128/125]], and this is approximately half of [[25/24]], so that if we flatten (5/4)<sup>2</sup> = [[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, [[576/575|S24]], which is favourable as finding interpretations of a variety of intervals that are otherwise given somewhat questionable 5-limit interpretations, [[Würschmidt#Interval chain|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 [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]] so that [[~]][[11/9]][[~]][[27/22]] is the neutral third.


== Higher-limit focus ==
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]].
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 ==


= '''Low accuracy (<12c)''' =
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 ==
== 7-limit focus ==
=== [[Superpyth]] ===
=== [[Garibaldi]] ===
Note counts:
Note count: 18 for {3, 5, 7, 9, 15, 21, 27, 35, 45} ([[12L 5s]] or [[12L 17s]])
* 4 for {3, 7} ([[2L 3s]])
* 12 for {3, 5, 7, 9} ([[5L 7s]] and [[5L 12s]])
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 [[3/2|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.
Bound-violating intervals: [[7/5]], [[21/20]], [[15/14]] (all derived from contrasting odd 7 (sharp) and 5 (flat))


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.
Garibaldi is a very natural and very efficient (for its accuracy) way of extending [[#Schismic]] to prime 7, at the cost of some accuracy so that it is no longer a microtemperament. This is done by interpreting ([[9/8]])<sup>3</sup> as [[~]][[10/7]] by tempering out [[5120/5103|S8/S9]] so that 8/7 and 10/9 are equidistant from 9/8, with the distance being a conveniently general "comma"-sized interval that simultaneously represents not only [[64/63|S8]] and [[81/80|S9]] but also the [[Pythagorean comma]] (as per schismic). [[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has an inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 53 = [[147edo]] also supports it but with yet more inconsistencies, showing a slight preference to 53edo, though which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about prime 11 or prime 13:
* For prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as half of the fifth and as a comma above [[~]][[6/5]] or a comma below [[~]][[5/4]]. This corresponds to being [[cassandra]] + [[andromeda]] (respectively).
* For prime 13, [[53edo]] is better, as it finds [[interseptimal interval]]s distinctly from adjacent [[septimal]] intervals so that [[~]][[15/13]] is half of a practically-just [[4/3]] (tempering out [[676/675|S13/S15]]) and is (resultantly) found as a comma above [[~]][[8/7]] or a comma below [[~]][[7/6]] (which reflects to [[~]][[13/10]] being made the midpoint of [[~]][[9/7]] and [[~]][[21/16]]), and so that [[~]][[16/13]] is a comma below [[~]][[5/4]]. This corresponds to a number of temperaments; the most relevant of which for [[#Schismic]] is the very accurate extension to prime 13 called [[tridecapyth]].


== 11-limit focus ==
== 11-limit focus ==
=== [[Mohaha]] ===
=== [[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-expression]]s by splitting 3/2 into two by tempering out [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], splitting 3/2 into three by tempering out [[1029/1024|S7/S8 = (9/6)/(8/7)<sup>3</sup> = (3/2)/(8/7)<sup>3</sup>]] 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.
 
== ~17-limit focus ==
=== [[Buzzard]] ===
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)
* 31 for {3, 5, ''7'', 9, 13, 15, ''21'', 27, ''35''(, 81)} ([[5L 28s]])
* 20 or 23 for adding {7, 15, 21(, 35)} ([[7L 10s]] and [[7L 17s]])
* 47 for {3, 5, ''7'', 9, 11, 13, 15, ''21'', 27, 33, ''35'', 39(, 81)} (5L 43s (minimum) or 53L 5s)
 
Bound-violating intervals: [[7/4]], [[11/7]] and various intervals made with compound intervals of 7 (corresponding odds italicized), which are the simplest (see [[#2.3.7 Buzzard]])
 
Its [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]], [[8019/8000|S9/S10]], [[676/675|S13/S15]]}, with the structure of its 7-limit implied by the first two equivalences combined with the nontrivial [[JI]] equivalence [[36/35|S6]] = [[64/63|S8]] × [[81/80|S9]]. Tempering out S8/S9 leverages this by splitting [[36/35|S6]] into two syntonic~septimal commas, so buzzard naturally finds an interval between [[6/5]] and [[7/6]] which in the 7-limit is [[32/27]] and in the 13-limit is [[13/11]], while tempering out S6/S7 implies that [[49/48|S7]] is also split into two so that the system also finds an interval between 7/6 and 8/7 which in the 7-limit is 7/6 inflected down by a comma or 8/7 inflected up by a comma, and in the 13-limit is [[15/13]], so that it is clear this system naturally wants to be extended to and interpreted in at least the full 13-limit. Because of the sharp 3 and 5 (in an optimized tuning), [[~]][[16/15]] is tuned quite flat so that a very natural extension to prime 17 exists by equating it with a sharp [[~]][[17/16]] (tempering out [[256/255|S16]]).


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)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], 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.
[[53edo]] and [[58edo]] are good tunings for the 13-limit; though 58edo is more accurate, 53edo supports a variety of structures that might be preferred over the ones 58edo supports, such as [[#Cata]] (and its best extension to prime 7, [[#Catakleismic]], as well as another more complex extension to prime 7 called [[countercata]]), [[#Schismic]] and especially [[marvel]] (so that it supports [[#Garibaldi]]); by contrast, 58edo may be preferred for supporting [[#Echidna]] and [[#Diaschismic]]. [[111edo]] is a very elegant tuning for higher limits, combining it with [[#Sensible]] and [[#Sendai]] (two extensions of [[#Sensipent]].


== ~17-limit focus ==
== Higher-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.
=== [[Sendai]] ===
{{ See also | Sensipent#Sendai interval table }}
Note counts:
* 13 for {3, 5, 23, 31, 69, 115} ([[8L 11s]])
* 29 for adding {9, 15, 25, 29, 87, 145} ([[19L 8s]])
 
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-2's focus ==
== No-3's focus ==
== No-3's focus ==
== No-5's focus ==
== No-5's focus ==


= '''Medium accuracy (<7c)''' =
= '''Medium accuracy (<7c)''' =
Line 224: Line 250:
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.
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.


= '''High accuracy (<4c)''' =
 
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.
 
= '''Low accuracy (<12c)''' =
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 ==
== 5-limit focus ==
=== [[Cata]] ===
== 7-limit focus ==
Note count: 15 for {3, 5, 9, 13, 15, 25} ([[4L 7s]])
=== [[Superpyth]] ===
Note counts:
* 4 for {3, 7} ([[2L 3s]])
* 12 for {3, 5, 7, 9} ([[5L 7s]] and [[5L 12s]])
Bound-violating intervals: [[9/8]] (and [[8/7]] in flatter tunings like 22edo)


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).
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 [[3/2|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.


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.
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.


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]].
== 11-limit focus ==
 
=== [[Mohaha]] ===
=== [[Sensipent]] ===
Note counts:
Note counts:
* 10 for {3, 5, 31} ([[8L 3s]])
* 9 for {3, 5, 9, 11(, 33, 35)} ([[7L 3s]], note odd 35 comes from the [[mohajira]] mapping of 7 specifically)
* 19 for adding {9, 15, 25} ([[8L 11s]])
* 20 or 23 for adding {7, 15, 21(, 35)} ([[7L 10s]] and [[7L 17s]])


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 [[961/960|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]].
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)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], 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.


Sensipent finds [[6/1]] (the fifth plus two octaves) at 7 generators.
== ~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 ==


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{{cent}} sharp [[5/4]] as the generator, so that [[6/1]] is reached as (5/4)<sup>8</sup>. The rationale for this is that (5/4)<sup>3</sup> falls short of the octave by [[128/125]], and this is approximately half of [[25/24]], so that if we flatten (5/4)<sup>2</sup> = [[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, [[576/575|S24]], which is favourable as finding interpretations of a variety of intervals that are otherwise given somewhat questionable 5-limit interpretations, [[Würschmidt#Interval chain|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 [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]] 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]].


= '''Very low accuracy (<~18c)''' =
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 ==
== 7-limit focus ==
=== [[Garibaldi]] ===
=== [[Godzilla]] ===
Note count: 18 for {3, 5, 7, 9, 15, 21, 27, 35, 45} ([[12L 5s]] or [[12L 17s]])
Note count: 9 for {1, 3, 5, 7, 9(, 21)} ([[5L 4s]])


Bound-violating intervals: [[7/5]], [[21/20]], [[15/14]] (all derived from contrasting odd 7 (sharp) and 5 (flat))
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.
 
Garibaldi is a very natural and very efficient (for its accuracy) way of extending [[#Schismic]] to prime 7, at the cost of some accuracy so that it is no longer a microtemperament. This is done by interpreting ([[9/8]])<sup>3</sup> as [[~]][[10/7]] by tempering out [[5120/5103|S8/S9]] so that 8/7 and 10/9 are equidistant from 9/8, with the distance being a conveniently general "comma"-sized interval that simultaneously represents not only [[64/63|S8]] and [[81/80|S9]] but also the [[Pythagorean comma]] (as per schismic). [[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has an inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 53 = [[147edo]] also supports it but with yet more inconsistencies, showing a slight preference to 53edo, though which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about prime 11 or prime 13:
* For prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as half of the fifth and as a comma above [[~]][[6/5]] or a comma below [[~]][[5/4]]. This corresponds to being [[cassandra]] + [[andromeda]] (respectively).
* For prime 13, [[53edo]] is better, as it finds [[interseptimal interval]]s distinctly from adjacent [[septimal]] intervals so that [[~]][[15/13]] is half of a practically-just [[4/3]] (tempering out [[676/675|S13/S15]]) and is (resultantly) found as a comma above [[~]][[8/7]] or a comma below [[~]][[7/6]] (which reflects to [[~]][[13/10]] being made the midpoint of [[~]][[9/7]] and [[~]][[21/16]]), and so that [[~]][[16/13]] is a comma below [[~]][[5/4]]. This corresponds to a number of temperaments; the most relevant of which for [[#Schismic]] is the very accurate extension to prime 13 called [[tridecapyth]].


== 11-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-expression]]s by splitting 3/2 into two by tempering out [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], splitting 3/2 into three by tempering out [[1029/1024|S7/S8 = (9/6)/(8/7)<sup>3</sup> = (3/2)/(8/7)<sup>3</sup>]] 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.
== ~17-limit focus ==
== ~17-limit focus ==
=== [[Buzzard]] ===
=== [[Flattone]] ===
Note counts:
Note count: 14 for {3, 5, 7, 9, 11, 13}
* 31 for {3, 5, ''7'', 9, 13, 15, ''21'', 27, ''35''(, 81)} ([[5L 28s]])
* 47 for {3, 5, ''7'', 9, 11, 13, 15, ''21'', 27, 33, ''35'', 39(, 81)} (5L 43s (minimum) or 53L 5s)


Bound-violating intervals: [[7/4]], [[11/7]] and various intervals made with compound intervals of 7 (corresponding odds italicized), which are the simplest (see [[#2.3.7 Buzzard]])
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.
 
Its [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]], [[8019/8000|S9/S10]], [[676/675|S13/S15]]}, with the structure of its 7-limit implied by the first two equivalences combined with the nontrivial [[JI]] equivalence [[36/35|S6]] = [[64/63|S8]] × [[81/80|S9]]. Tempering out S8/S9 leverages this by splitting [[36/35|S6]] into two syntonic~septimal commas, so buzzard naturally finds an interval between [[6/5]] and [[7/6]] which in the 7-limit is [[32/27]] and in the 13-limit is [[13/11]], while tempering out S6/S7 implies that [[49/48|S7]] is also split into two so that the system also finds an interval between 7/6 and 8/7 which in the 7-limit is 7/6 inflected down by a comma or 8/7 inflected up by a comma, and in the 13-limit is [[15/13]], so that it is clear this system naturally wants to be extended to and interpreted in at least the full 13-limit. Because of the sharp 3 and 5 (in an optimized tuning), [[~]][[16/15]] is tuned quite flat so that a very natural extension to prime 17 exists by equating it with a sharp [[~]][[17/16]] (tempering out [[256/255|S16]]).
 
[[53edo]] and [[58edo]] are good tunings for the 13-limit; though 58edo is more accurate, 53edo supports a variety of structures that might be preferred over the ones 58edo supports, such as [[#Cata]] (and its best extension to prime 7, [[#Catakleismic]], as well as another more complex extension to prime 7 called [[countercata]]), [[#Schismic]] and especially [[marvel]] (so that it supports [[#Garibaldi]]); by contrast, 58edo may be preferred for supporting [[#Echidna]] and [[#Diaschismic]]. [[111edo]] is a very elegant tuning for higher limits, combining it with [[#Sensible]] and [[#Sendai]] (two extensions of [[#Sensipent]].


== Higher-limit focus ==
== Higher-limit focus ==
=== [[Sendai]] ===
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.
{{ See also | Sensipent#Sendai interval table }}
Note counts:
* 13 for {3, 5, 23, 31, 69, 115} ([[8L 11s]])
* 29 for adding {9, 15, 25, 29, 87, 145} ([[19L 8s]])
 
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-2's focus ==
== No-3's focus ==
== No-3's focus ==
== No-5'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]])<sup>4</sup>, 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)<sup>6</sup> 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:
= '''Exotemperaments (>~18c)''' =
Exotemperaments are useful as targets for [[detempering]], as they often underly the logic of various [[JI]] [[scale]]s. They are also explored for novelty.


* [[#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.
== 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.


* Schismic [[tridecapyth comma|tridecapyth]] (which is the 2.3.5.13 version of [[#Cassandra]]) finds 13/4 as (9/8)<sup>10</sup> and demands an approximately Pythagorean tuning.
=== 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.


* [[#Nestoria]] equates [[~]][[19/16]] with [[32/27]] and [[~]][[19/15]] with [[81/64]].
=== 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 ==
== 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 ==
== 11-limit focus ==
== ~17-limit focus ==
== ~17-limit focus ==
Line 324: Line 333:
== No-3's focus ==
== No-3's focus ==
== No-5'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.
Retrieved from "https://en.xen.wiki/w/Bird%27s_eye_view_of_temperaments_by_accuracy"