User:BudjarnLambeth/Bird’s eye view of rank-2 temperaments
This page has been deprecated, but it is being kept in place for longevity as a reference material. Please see User:BudjarnLambeth/Survey of efficient temperaments by subgroup for the new version.
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.
Guide to tables
Rows
The rows categorise temperaments by accuracy. That is, how closely they approximate just intonation intervals. The categories are:
- Exotemperament: > ~18 cents of error on more than one targeted interval
- Low accuracy: < ~18c error on most targeted intervals
- Medium accuracy: < 12c error on most targeted intervals
- High accuracy: < 7c error on almost all targeted intervals
- Very high accuracy: < 3.5c error on almost all targeted intervals
- Microtemperaments: < 1c on all targeted intervals
The definition of "targeted interval" is left deliberately vague, because some temperaments serve a specific purpose and must be assessed differently. In most cases on this page, it refers to the set of intervals that occur in a tonality diamond of the temperament's subgroup.
Columns
The columns categorise temperaments by complexity.
Rank 2 temperaments can generate scales with any number of notes per equave. However, if they have too few notes, they won't be able approximate enough targeted intervals to be useful, and if they have too many notes, they will be filled with extra notes that don't serve much purpose and get in the way. Just how many notes is about right, varies from temperament to temperament. In layman’s terms: More notes needed = more complexity, less notes needed = less complexity. The real definition of complexity is more involved and rigorous than this, but this is good enough for the purposes of a broad overview page.
Subgroup categorisation
If a temperament fits under multiple subgroup headings (e.g. both No-2s and No-5s) it should be placed only under the lowest numbered heading (in this example, No-2s).
5-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | antitonic, bug, dicot, father | ||||
| Low accuracy (12-18c) | augmented, blackwood, dimipent, porcupine, whitewood | superpyth | |||
| Medium accuracy (7-12c) | meantone | hanson, magic | valentine | ||
| High accuracy (3.5-7c) | diaschismic | ||||
| Very high accuracy (1-3.5c) | wuerschmidt | ||||
| Microtemperament (0-1c) | schismic aka helmholtz | kwazy |
7-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | antitonic, dicot, father | ||||
| Low accuracy (12-18c) | blacksmith | augene, godzilla, pajara, porcupine, whitewood | |||
| Medium accuracy (7-12c) | magic, meantone, mothra, sensi, superpyth | valentine | |||
| High accuracy (3.5-7c) | orwell | diaschismic, garibaldi | |||
| Very high accuracy (1-3.5c) | miracle | ||||
| Microtemperament (0-1c) | ennealimmal | enneadecal, trinity |
11-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | antitonic, dicot, father | ||||
| Low accuracy (12-18c) | augene, blacksmith, pajara, porcupine, whitewood | godzilla, superpyth | |||
| Medium accuracy (7-12c) | magic, meanpop, meantone, mothra, valentine | ||||
| High accuracy (3.5-7c) | diaschismic, orwell | ||||
| Very high accuracy (1-3.5c) | miracle | ||||
| Microtemperament (0-1c) | ennealimmal | enneadecal, trinity |
13-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | augene, blacksmith, pajara, porcupine, whitewood | superpyth | |||
| Medium accuracy (7-12c) | magic, meantone, mothra | ||||
| High accuracy (3.5-7c) | diaschismic, orwell | ||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) | ennealimmal | enneadecal, trinity |
17-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | pajara | ||||
| Medium accuracy (7-12c) | |||||
| High accuracy (3.5-7c) | diaschismic | ||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) | ennealimmal | trinity |
Higher limits
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | |||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) | ennealimmal | trinity |
3.5.7 and its extensions
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | |||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) |
Other no-2s subgroups
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | |||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) |
No-3s subgroups
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | didacus, orgone | ||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) |
2.3.7 and its extensions
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | semaphore | ||||
| Medium accuracy (7-12c) | bleu, slendric | ||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) |
2.3.11 and its extensions
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | neutral (no-5 no-7 rastmic) | ||||
| High accuracy (3.5-7c) | no-5 no-7 pythrabian | ||||
| Very high accuracy (1-3.5c) | tribilo (no-5 no-7 nexus) | ||||
| Microtemperament (0-1c) | no-5 no-7 frameshift |
2.3.13, 2.3.17, etc
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | |||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) |
No-7s subgroups
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | mohaha | ||||
| High accuracy (3.5-7c) | sensible, srutal archagall | ||||
| Very high accuracy (1-3.5c) | cata, nestoria, sensipent | ||||
| Microtemperament (0-1c) |
No-11s subgroups
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | |||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) |
Other subgroups
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | |||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) |