Survey of efficient temperaments by subgroup
This page highlights those rank-2 temperaments which receive the most discussion among theorists and composers.
Composers and theorists disagree about which of these temperaments matter most, but all of the temperaments on this page are valued by at least a fair subset of the xenharmonic community.
Which temperaments should I use to make music?
There are many different schools of thought within RTT (regular temperament theory).
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.
For example:
Xenharmonicist A might argue that an error less than ~15 cents on most intervals, and less than 5 cents on the really important ones (like the perfect fifth and the octave), is accurate enough.
And they might argue that 25 notes per equave is the most that is practical, any more than that is too cumbersome.
They might argue that nobody can hear the harmonic effect of prime harmonics higher than 11.
And they might argue that there's no real reason to use subgroups that are missing primes 2 or 3, because those primes are so important to consonance.
Xenharmonicist B might argue that the error must be less than ~5 cents on almost all intervals, anything further out than that sounds out of tune to them.
They might argue that it's perfectly possible to learn up to 50 notes per equave.
They might argue that they can hear the subtle, delicate effect of prime harmonics up to 23.
And they might argue that subgroups like 3.5.7.11 and 2.5.7.11 are the most fertile ground for new and exciting musical exploration.
These are not the only possible stances, either: One could imagine a Xenharmonicist C, Xenharmonicist D, etc. Thousands of differing individual perspectives on what traits are important in a temperament.
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.
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.
How to read the tables
Rows
The rows categorise temperaments by the just intonation subgroup they approximate.
The 2.3.5 subgroup is what most theorists believe 12 tone equal temperament belongs to. If those theorists are correct, then 2.3.5 should encompass all the harmonies that are familiar to most Western listeners.
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[note 1], 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.
Columns
The columns categorise temperaments by the approximate number of notes-per-equave needed to reach all the temperament’s important intervals.
All of the temperaments listed in these tables have low badness (high relative accuracy), meaning they approximate their target JI subgroup much better than most temperaments with their same amount of needed notes.
That means that for temperaments in these tables, the more notes they require, the more accurate they are. The ones requiring less notes are less accurate, though they are good for their size. (Note that this rule is only true for the temperaments in these tables, it is not true of all temperaments in general.)
Table of temperaments (5 to 45 notes per equave)
The temperaments within each cell should be sorted by accuracy, with the lowest damage (highest accuracy) temperament listed first.
(Editors:
If you see any temperaments listed in the wrong order, or see any temperaments in the wrong ‘approx. number of notes needed’ category, please move them to the correct position.
If you know of a temperament that is recommended by a sizeable subset of the xen community but is not yet included here, please add it.
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.)
Table of temperaments (more notes per equave)
Pergens and temperament relationships
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.
If you visit the temperaments’ individual pages, those will usually make their relationships to other temperaments more clear.
Schismic/helmholtz/garibaldi/nestoria/andromeda/cassandra, and kleismic/hanson/cata are two prominent examples of temperaments on these tables sharing a pergen. There are other examples on the tables also.
Most linked-to rank-2 temperaments
These were the top 100 rank-2 temperament pages with the most incoming links on the wiki on 27 Oct 2024. (When this section was written.)
- Meantone (313 links)
- Porcupine (144)
- Superpyth (108)
- Magic (107)
- Mavila (97)
- Orwell (81)
- Miracle (78)
- Pajara (76)
- Sensi (71)
- Flattone (64)
- Amity (59)
- Mohajira (59)
- Negri (59)
- Blackwood (58)
- Tetracot (56)
- Valentine (53)
- Wuerschmidt (53)
- Slendric (52)
- Compton (51)
- Ennealimmal (50)
- Helmholtz (49)
- Dicot (47)
- Garibaldi (47)
- Hanson (45)
- Catakleismic (44)
- Diaschismic (43)
- Hemififths (42)
- Myna (41)
- Father (40)
- Squares (40)
- Rodan (39)
- Semaphore (39)
- Augmented (38)
- Diminished (38)
- Srutal (38)
- Godzilla (37)
- Harry (37)
- Injera (37)
- Diasem (36)
- Enneadecal (35)
- Orgone (34)
- Parakleismic (34)
- Hedgehog (33)
- Luna (33)
- Octacot (33)
- Augene (32)
- Dominant (32)
- Hemithirds (32)
- Keemun (32)
- Lemba (32)
- Mothra (32)
- Whitewood (32)
- Archy (31)
- Liese (31)
- Bleu (29)
- Vishnu (29)
- Hemiwuerschmidt (28)
- Superkleismic (27)
- Echidna (26)
- Orson (26)
- Tertiaseptal (26)
- Triforce (26)
- Passion (25)
- Tritonic (25)
- Unidec (25)
- Wizard (25)
- Buzzard (24)
- Cassandra (24)
- Ripple (24)
- Vulture (24)
- Armodue (23) (disambiguation page)
- Atomic (23)
- Bug (23)
- Escapade (23)
- Pontiac (23)
- Ampersand (22)
- Bohpier (22)
- Mohaha (22)
- Parapyth (22)
- August (21)
- Blacksmith (21)
- Kwazy (21)
- Octoid (21)
- Tritikleismic (21)
- Kleismic (20)
- Misty (20)
- Schismatic (20) (already listed as “Helmholtz”)
- Shrutar (20)
- Sqrtphi (20)
- Beatles (19)
- Didacus (19)
- Meanpop (19)
- Arcturus (18)
- Gorgo (18)
- Guiron (18)
- Leapday (18)
- Mitonic (18)
- Nautilus (18)
- Sensipent (18)
A simpler overview
For a more streamlined, strictly curated list of useful temperaments, see the following pages:
- Middle Path table of five-limit rank two temperaments
- Middle Path table of seven-limit rank two temperaments
- Middle Path table of eleven-limit rank two 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
Advanced reading
- Tour of regular temperaments: a huge list of temperament families, many of which remain rarely-used and unexplored
- x31eq Catalog of Regular Temperaments: an even huger list by Graham Breed
- Rank-3 and rank-4 temperaments: these are more complicated, rarely-used, types of temperaments
- Equal-step tunings (i.e. rank-1 temperaments)
- More lists of temperaments:
- A deprecated, archived in-development version of this page: User:BudjarnLambeth/Bird’s eye view of rank-2 temperaments
Notes
- ↑ According to the 2.3.5 reading of common practice harmony. Alternate readings are possible.
- ↑ Number of notes per equave was estimated by multiplying the temperament’s graham complexity by 2.
- ↑ Subgroup 2.3.5.23 version.
- ↑ ^Hemif beats radon in 2.3.7.11, radon beats hemif in 2.3.7.11.13 (in damage not badness).