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

This page serves to document temperaments people deem to be highly valuable, and organizing them 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: ''do not add a temperament if you do not consider it to be unusually/uniquely valuable for making music with''.''' Temperaments here should only be ones that one or more people seriously consider to be "cream of the crop". Therefore, when adding a temperament, ''make sure to include a description of how it works'', ideally one that motivates it as to why someone might want to use it.

The bounds are: '''exotemperament''' (>~18c), '''very low''' (<~18c), '''low''' (<12c), '''medium''' (<7c), '''high''' (<4c), '''microtemperament''' (<1c).

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.

Wherever you see "Generator tunings:" on this page, the tunings are given in the format ''a''\''b'', which means ''a'' steps of ''b'' [[edo]], which means the frequency ratio 2^''a''/''b''.

For example, you can '''input''' 44\111 '''in''' [[Scale Workshop]] '''by hovering over "New scale" on the left, then pressing "Rank-2 temperament", then entering''' (e.g.) 44\111 '''in the "Generator" input field'''.

You will then get a list of "MOS sizes" to pick from before pressing "OK" to generate the scale. The importance of this is that picking from this list gives you a [[MOS scale]] which has certain properties that make its usage more regular and predictable than picking some number of notes not in that list, though ''note the list of MOS sizes is sometimes incomplete on Scale Workshop'' (if the list runs out of space), which is why another part of the standard format of temperaments documented here is ''x''L ''y''s MOS sizes, so that you can manually specify (''x'' + ''y'') notes as the number of notes if the MOS sizes list does not go far enough.

{{ todo | example of an efficient and useful rank 2 temperament with a small generator that the MOS sizes list on Scale Workshop doesn't go far enough for }}

In some cases, the [[octave]] is not the period of the scale or is strongly recommended to be tempered, in which case you will instead see pairs; for example, for [[#Godzilla]] (which is strongly recommended to be tempered), it is (4\30<3>, 19\30<3>). This means that 4\30<3> goes in the "Generator" input field and 19\30<3> goes in the "Period" input field. (In the case of godzilla, this reflects tempering the octave sharp so that 3/1 is just in order to get various intervals of 3, 5, 7 and 13 more in tune as they are all tuned flat in [[19edo]]; very significantly so in the case of 7 and 13).

[[#Generator tunings|Generator tunings]]: 31\53, 69\118, 100\171, 131\224

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.

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

[[Bird's eye view of temperaments by accuracy#Generator tunings|Generator tunings]]: 3\72, 4\99, 7\171, 11\270

Ennealimmal has a 1/9-octave period representing [[27/25]], and two of them represent [[7/6]], tempering out [[4375/4374]]. It is generated by a [[~]][[36/35]] quartertone, with 3/2 being mapped to  6 periods minus 2 generators, 5/4 mapped to 4 periods minus 3 generators, and 7/4 mapped to 8 periods minus 2 generators. It finds a neutral third representing [[49/40]]~[[60/49]] at 3 periods minus 1 generator, tempering out [[2401/2400]]. This temperament therefore tempers out the two smallest superparticular ratios in the [[7-limit]], 2401/2400 and 4375/4374. It is very accurate, with errors of around 0.2 cents in optimized tunings. However, due to the high note count, one may prefer lower accuracy temperaments.

== '''High accuracy (<4c)''' ==

* the simplest but least accurate is [[#Sensor]] (commonly just called "sensi"), which interprets it as a full 17-limit temperament.

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]])³ 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]].

 

 

 

== '''Medium accuracy (<7c)''' ==

==== [[Srutal archagall]] ====

==== [[Bohlen-Pierce-Stearns]] ====

This is arguably the most important temperament of the nonoctave [[3.5.7 subgroup]]. This temperament has a tritave period, and a generator of [[~]][[9/7]]. The tritave-reduced 7th harmonic, [[7/3]], is found at -1 generators, and the tritave reduced 5th harmonic, [[5/3]], is found at +2 generators, tempering out [[245/243]]. It is as simple as a good temperament in its subgroup can be, covering the entire no-evens 7-throdd-limit tonality diamond in 7 notes, with no redundant or missing notes, and any simpler temperament would have to equate simple consonances and have very low accuracy. Its accuracy is quite good, with a no-evens 7-throdd-limit minimax error of 4.73 cents. An excellent scale to explore this temperament is the 9-note mos, or lambda scale, which can be considered the 3.5.7 analog of the diatonic scale.

== '''Low accuracy (<12c)''' ==

== '''Very low accuracy (<~18c)''' ==