User:BudjarnLambeth/Bird’s eye view of rank-2 temperaments
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There are at least hundreds, probably thousands, of rank-2 temperaments described. It can be difficult to know where to start.
This page is intended as that starting point. It does not aim to list every temperament. Instead it aims to list only the ones that are of high interest to a sizeable number of composers or theorists.
Composers and theorists disagree amongst themselves about what properties are desirable in a temperament, and you might over time find that you lean more towards one camp or another. This list arranges temperaments by their properties, allowing you the reader to seek out temperaments with whichever properties you value.
So, which temperaments should I use to make music?
Ask 5 xenharmonicists, and you'll get 10 different answers. There are many different schools of thought within RTT (regular temperament theory).
Most would agree that a good temperament 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 15ish 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 5ish cents on most 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.
Neither xenharmonicist can be objectively shown to be right or wrong. There is an amount of science to this, but there is also a lot of personal subjectivity. Ultimately it's up to you to decide what features you think are important in a temperament.
It might 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).
It can be interpreted as a low-to-medium accuracy 5-limit temperament where the most important intervals (the fifth and octave) have an error less than 3 cents, while other notable intervals (like the thirds and sixths) have an error of about 14 cents.
Alternatively, it can be interpreted as a high-accuracy 2.3.17.19 subgroup temperament, where all of the intervals have an error less than 5 cents.
So that should provide a point of comparison to help measure these other temperaments against.
How to read the table
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 (but there is plenty of disagreement about that).
The 2.3.5.7 and 2.3.5.7.11 subgroups are the most commonly used, being not too complex and including lots of useful harmonies.
Subgroups with no 2s, e.g. 3.5.7.11, are the biggest and most jarring break away from familiar harmony, which you 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, 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, maybe subtly and subconsciously in some instances, but still they matter. It is up to you which theorists you believe on that.
Columns
The columns categorise temperaments by the approximate number of notes needed to reach all the important intervals within the practice of that temperament.
All of the temperaments listed in this table 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 the temperaments in this table requiring more notes are also more accurate. The ones requiring less notes are less accurate but are good for their size.
Table of recommended temperaments
(!) = It is unlikely that a useable temperament could exist here because there are too many important intervals to approximate with this number of notes.
Why so many similar temperaments?
Some temperament names represent different but closely related tunings from a shared progenitor temperament, which should be noted when looking at higher-limit behaviours of more simple temperaments.
Why you should use my favorite temperament (editor opinions)
In this section, any editor may create their own subheading, under which they may describe a specific temperament they like and why they think people should use it.
One editor is allowed to add more than one temperament if they like, just keep it within reason.
Editors, please use simple, plain language as much as you can - imagine you're explaining this to a stranger at a bar who has no music theory knowledge at all, but is curious about it.