Survey of efficient temperaments by subgroup
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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 to narrow the field a little, by showing you the temperaments that get talked about a lot within the xenharmonic community. 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 theorists or composers.
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 particular camp or another. This list arranges popular temperaments by their properties in a table allowing the views of all ‘camps’ to get a space on the page.
Tl;dr, you can think of this as a table of what temperaments are popular among theorists and/or composers. These are the ones lots of people are talking about.
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.
And these are not the only possible stances, either! There is a Xenharmonicist C, Xenharmonist D, etc. Thousands of differing individual perspectives on what traits see important in a temperament.
So how do you start working it out for yourself? Figuring out your own preferences?
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).
Most theorists interpret 12edo as a 2.3.5 subgroup temperament which is about as accurate as most of the temperaments in the left-most column of the below table. This interpretation is not universal, though.
The second most common approach is to interpret 12edo as a high-accuracy 2.3.17.19 subgroup temperament, which is about as accurate as the temperaments in the middle columns of the table.
So that should provide a helpful point of comparison to 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.
You may see the same temperament multiple times on the table. Here’s why:
Some temperaments are good at approximating a variety of different subgroups. For example, magic is good at approximating both the 7-limit and the 11-limit, so it is listed under both.
It would be like if there were a table of pokemon, and one row said “cute pokemon”, and one row said “electric pokemon”: Pikachu would be listed under both.
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. (This rule is not true for all temperaments in general, it’s just true for the ones listed in this table.)
Table of temperaments some decent number of people would recommend
The temperaments within each cell should be sorted by accuracy, with the lowest damage temperament listed first
(!) = It is unlikely that a useable temperament could exist here because there are too many important intervals to approximate with this number of notes.
Additional information
Do note that this table doesn’t capture all of the relationships and commonalities between temperaments. This table does show when two temperaments share a JI subgroup, which is important information. But another important piece of information this table doesn’t capture is whether two temperaments share a pergen.
In short, sometimes, multiple higher limit temperaments are actually different ways of extending the same lower-limit temperament. In this case, they will share a pergen. And 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.
Of particular note: schismic/helmholtz/garibaldi/nestoria, and kleismic/hanson/cata are two prominent examples of temperaments on this table sharing a pergen.
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 not too many though.
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.
I want a simpler, more straightforward overview
For a less complicated 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 they were chosen, read Paul Erlich’s Middle Path essay: