Survey of efficient temperaments by subgroup: Difference between revisions

This page highlights those [[rank-2 temperament]]s which receive the most discussion among theorists and composers.

== Which temperaments should I use to make music? ==

There are many different schools of thought within regular temperament theory (RTT). Most would agree that a good temperament is ''efficient'', meaning it approximates some subset of [[just intonation]] relatively accurately with a relatively small number of notes. What they disagree on is ''how'' accurate is "relatively accurate", ''how'' small is "relatively small", and ''which'' JI subsets are interesting enough to be worth approximating.

To gain more of a grasp on these debates, it may help to compare these temperaments to [[12edo]], a.k.a. the familiar 12-tone equal temperament which most modern music is tuned to by default. 12edo has, of course, 12 notes per equave, which makes it fairly small by temperament standards but not abnormally so. The most common theoretical approach to 12edo is to treat it as a 2.3.5-subgroup temperament, with similar accuracy to [[augmented (temperament)|augmented]]. The second most common approach is to interpret 12edo as a 2.3.17.19-subgroup temperament, with similar accuracy to [[semitonic]]. (Such a temperament would go in the ''2.3.other n'' row of the below tables). So that should provide a helpful point of comparison to measure these other temperaments against.

The 2.3.5.7 and 2.3.5.7.11 subgroups are the most commonly used by xenharmonic composers, being not too complex and including lots of useful harmonies.

Subgroups with no 2s, e.g. 3.5.7.11, are the most jarring break away from familiar harmony, which one may consider a good or a bad thing.

Subgroups with 2s and 3s but no 5s, e.g. 2.3.7.11, preserve the most fundamental familiar intervals like the octave and the fifth, but do away with the 5-limit major and minor intervals of common practice harmony<ref group="note">According to the 2.3.5 reading of common practice harmony. Alternate readings are possible.</ref>, forcing innovation while still keeping some familiarity.

Some theorists believe including 13, 17 or higher in a subgroup is pointless because the brain can't register such complex intervals. Others believe these intervals are registered by the brain, perhaps subtly and subconsciously in some instances, but still there.

The same temperament may occur multiple times on a table if it is good at approximating multiple different subgroups. For example, magic is good at approximating both the 7-limit and the 11-limit, so it is listed under both.

'''The columns categorise temperaments by the approximate number of notes-per-[[equave]] needed to reach all the temperament’s important intervals'''.

Please do not add temperaments just for the sake of filling empty cells on the table. It's okay for some cells to be empty. Only add temperaments if yourself, or at least a few other people, would recommend those temperaments.

If you see a temperament on here that does not have good accuracy for its size in a particular subgroup, please delete that temperament from that subgroup’s row of the table.

Subgroups ''without'' a "2" ''don't have'' multiple of 2 intervals (eg the 2/1 octave, the 3/2 perfect fifth, the 5/4 major third).

Subgroups ''without'' a "3" ''don't have'' multiple of 3 intervals (eg the 3/1 perfect twelfth, the 3/2 perfect fifth, the 5/3 major sixth).

Subgroups ''without'' a "5" ''don't have'' multiple of 5 intervals (eg the 5/3 major sixth, the 5/4 major third, the 6/5 minor third).

Subgroups ''with'' a "7", "11", or "other ''n''" include new [[xenharmonic]] consonant intervals that can't be found in common [[12edo]] tuning.

|-

! ~10 notes per equave

! ~20 notes

! ~30 notes

! ~40 notes

! 2.3.5.7.other ''n''

| [[negra]]

| no-11 [[godzilla]], no-11 [[pajara]], no-11 [[duodecim]]

| no-11 [[magic]], no-11 [[sensis]], no-11 [[meanpop]]

| no-11 [[catakleismic]]

! 2.3.5.11 <br>and its extensions

| [[porcupine]], [[mavila]], [[dicot]], [[flattone]]

| [[orson]]+, [[tetracot]], [[mohaha]]

| [[larry]] (no-7 [[gravity]]), [[countdown]], no-7 [[catalan]]

| [[escapade]]

! 2.3.5.other ''n''

| [[würschmidt]]<ref group="note">Subgroup 2.3.5.23 version.</ref>, [[cata]]

! 2.3.7

! 2.3.7.11 <br>and its extensions

| no-5 [[miracle]], [[radon]]^, [[hemif]]^<ref group="note">^Hemif beats radon in 2.3.7.11, radon beats hemif in 2.3.7.11.13 (in damage not badness).</ref>, [[leapfrog]]

| [[slendric]]+

! 2.3.11 <br>and its extensions

! 2.5.7 <br>and its extensions

| [[didacus]], [[frostburn]], no-3 [[oodako]], [[augment]]

! 2.5.other ''n''

| no-3 no-7 [[emka]], [[wizz]]+

| [[machine|apparatus]]

| [[aldebaran]], [[deneb]], [[polaris]]

! 3.7 <br>and its extensions

| [[meanquad]], [[tetrahanson]], [[tetrameantone]], [[quarchy]], [[tetrominant]]

| [[fourwar]]

! 5.''n'' <br>and its extensions

| [[greeley]], [[halftone]], [[semiwolf]], [[auk]]

! 5-limit <br>(2.3.5)

! Higher prime limits

| [[winston]]+++, [[porky]]+++

! 2.3.5.other ''n''

! 2.3.7

! 2.3.7.11 <br>and its extensions

! 2.3.11 <br>and its extensions

! 2.3.other ''n''

|  

! 2.5.7 <br>and its extensions

! 2.5.other ''n''

! 2.7 <br>and its extensions

! 3.5.7 <br>and its extensions

|  

! 3.5.other ''n''

! 3.7 <br>and its extensions

! 4.''n'' <br>and its extensions

! 5.''n'' <br>and its extensions

|  

One important piece of information these tables do not capture is whether two temperaments share a [[pergen]].

Sometimes, multiple higher limit temperaments are actually different ways of extending the same lower-limit temperament. In this case, they will share a pergen. This means they will have an overall similar flavor and some musical and mathematical properties in common.

== Most linked-to rank-2 temperaments ==

These are the top 105 rank-2 temperament pages with the most incoming links on the wiki as of 23 Oct 2025, about one year after this section was first written.

== A simpler overview ==

For a more streamlined, strictly curated list of useful temperaments, see the following pages:

* [[Middle Path table of 5-limit rank-2 temperaments]]

* [[Middle Path table of 7-limit rank-2 temperaments]]

* [[Middle Path table of 11-limit rank-2 temperaments]]

For a description of what the temperaments on the above pages are like, and how those temperaments were chosen, read Paul Erlich's ''Middle Path'' essay:

== A more descriptive overview ==

* See [[User:Godtone/Bird's eye view of temperaments by accuracy]] which includes written descriptions of the temperaments, and is more mathematically rigorous than this survey

* [[Equal-step tuning]]s (i.e. rank-1 temperaments)

** [[Ordered lists of ET rank two temperaments]]

** and everything else in [[:Category:Lists of temperaments]]

* A deprecated, archived in-development version of this page: [[User:BudjarnLambeth/Bird’s eye view of rank-2 temperaments]]