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 particular interest to composers and theorists.
Of course, composers and theorists disagree amongst themselves about what properties are desirable in a temperament, so this list arranges temperaments by their properties, allowing the reader to choose which properties to prioritise.
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 almost all 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 dead weight notes that don't approximate anything and just get in the way. There is a kind of 'goldilocks zone' in the middle, an amount of notes that is 'just right', and this goldilocks zone varies from temperament to temperament. In an informal, non-rigorous sense, that is what complexity measures.
On this page, the approximate 'goldilocks number' of notes per equate is estimated by subtracting the smallest number in the second row of a temperament's mapping from the largest. Extra leniency is afforded to temperaments with multiple periods per octave as they are disadvantaged by this method.
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. 40 notes | Approx. 80 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | antitonic, bug, dicot, father | tbd | tbd | tbd | tbd |
| Low accuracy (12-18c) | augmented, blackwood, dimipent, porcupine, whitewood | tbd | tbd | tbd | tbd |
| Medium accuracy (7-12c) | meantone | tbd | tbd | tbd | tbd |
| High accuracy (3.5-7c) | diaschismic | tbd | tbd | tbd | tbd |
| Very high accuracy (1-3.5c) | tbd | tbd | wuerschmidt | tbd | tbd |
| Microtemperament (0-1c) | tbd | tbd | tbd | tbd | tbd |
7-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 40 notes | Approx. 80 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | antitonic, dicot, father | tbd | tbd | tbd | tbd |
| Low accuracy (12-18c) | blacksmith, pajara, porcupine, whitewood | tbd | tbd | tbd | tbd |
| Medium accuracy (7-12c) | tbd | meantone, sensi | tbd | tbd | tbd |
| High accuracy (3.5-7c) | tbd | tbd | diaschismic, garibaldi, mothra | tbd | tbd |
| Very high accuracy (1-3.5c) | tbd | tbd | miracle, wuerschmidt | tbd | tbd |
| Microtemperament (0-1c) | tbd | tbd | tbd | ennealimmal | tbd |
11-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 40 notes | Approx. 80 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | antitonic, dicot, father | tbd | tbd | tbd | tbd |
| Low accuracy (12-18c) | tbd | blacksmith, pajara, porcupine, whitewood | tbd | tbd | tbd |
| Medium accuracy (7-12c) | tbd | tbd | meanpop, meantone | tbd | tbd |
| High accuracy (3.5-7c) | tbd | tbd | diaschismic, mothra | tbd | tbd |
| Very high accuracy (1-3.5c) | tbd | tbd | miracle, wuerschmidt | tbd | tbd |
| Microtemperament (0-1c) | tbd | tbd | tbd | ennealimmal | tbd |
13-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 40 notes | Approx. 80 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | tbd | tbd | tbd | tbd | tbd |
| Low accuracy (12-18c) | blacksmith, pajara, porcupine, whitewood | tbd | tbd | tbd | tbd |
| Medium accuracy (7-12c) | tbd | meantone | tbd | tbd | tbd |
| High accuracy (3.5-7c) | mothra | diaschismic | tbd | tbd | tbd |
| Very high accuracy (1-3.5c) | tbd | wuerschmidt | tbd | tbd | tbd |
| Microtemperament (0-1c) | tbd | ennealimmal | tbd | tbd | tbd |
17-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 40 notes | Approx. 80 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | tbd | tbd | tbd | tbd | tbd |
| Low accuracy (12-18c) | pajara | tbd | tbd | tbd | tbd |
| Medium accuracy (7-12c) | tbd | tbd | tbd | tbd | tbd |
| High accuracy (3.5-7c) | tbd | diaschismic | tbd | tbd | tbd |
| Very high accuracy (1-3.5c) | tbd | tbd | tbd | tbd | tbd |
| Microtemperament (0-1c) | tbd | ennealimmal | tbd | tbd | tbd |
19-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 40 notes | Approx. 80 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | tbd | tbd | tbd | tbd | tbd |
| Low accuracy (12-18c) | tbd | tbd | tbd | tbd | tbd |
| Medium accuracy (7-12c) | tbd | tbd | tbd | tbd | tbd |
| High accuracy (3.5-7c) | tbd | tbd | tbd | tbd | tbd |
| Very high accuracy (1-3.5c) | tbd | tbd | tbd | tbd | tbd |
| Microtemperament (0-1c) | tbd | ennealimmal | tbd | tbd | tbd |
Higher limits
| Approx. 10 notes | Approx. 20 notes | Approx. 40 notes | Approx. 80 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | tbd | tbd | tbd | tbd | tbd |
| Low accuracy (12-18c) | tbd | tbd | tbd | tbd | tbd |
| Medium accuracy (7-12c) | tbd | tbd | tbd | tbd | tbd |
| High accuracy (3.5-7c) | tbd | tbd | tbd | tbd | tbd |
| Very high accuracy (1-3.5c) | tbd | tbd | tbd | tbd | tbd |
| Microtemperament (0-1c) | tbd | tbd | tbd | tbd | tbd |
No-2s
| Approx. 10 notes | Approx. 20 notes | Approx. 40 notes | Approx. 80 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | tbd | tbd | tbd | tbd | tbd |
| Low accuracy (12-18c) | tbd | tbd | tbd | tbd | tbd |
| Medium accuracy (7-12c) | tbd | tbd | tbd | tbd | tbd |
| High accuracy (3.5-7c) | tbd | tbd | tbd | tbd | tbd |
| Very high accuracy (1-3.5c) | tbd | tbd | tbd | tbd | tbd |
| Microtemperament (0-1c) | tbd | tbd | tbd | tbd | tbd |
No-3s
| Approx. 10 notes | Approx. 20 notes | Approx. 40 notes | Approx. 80 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | tbd | tbd | tbd | tbd | tbd |
| Low accuracy (12-18c) | tbd | tbd | tbd | tbd | tbd |
| Medium accuracy (7-12c) | tbd | tbd | tbd | tbd | tbd |
| High accuracy (3.5-7c) | tbd | tbd | tbd | tbd | tbd |
| Very high accuracy (1-3.5c) | tbd | tbd | tbd | tbd | tbd |
| Microtemperament (0-1c) | tbd | tbd | tbd | tbd | tbd |
No-5s
| Approx. 10 notes | Approx. 20 notes | Approx. 40 notes | Approx. 80 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | tbd | tbd | tbd | tbd | tbd |
| Low accuracy (12-18c) | semaphore | tbd | tbd | tbd | tbd |
| Medium accuracy (7-12c) | tbd | tbd | tbd | tbd | tbd |
| High accuracy (3.5-7c) | tbd | tbd | tbd | tbd | tbd |
| Very high accuracy (1-3.5c) | tbd | tbd | tbd | tbd | tbd |
| Microtemperament (0-1c) | tbd | tbd | tbd | tbd | tbd |
No-7s
| Approx. 10 notes | Approx. 20 notes | Approx. 40 notes | Approx. 80 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | tbd | tbd | tbd | tbd | tbd |
| Low accuracy (12-18c) | tbd | tbd | tbd | tbd | tbd |
| Medium accuracy (7-12c) | mohaha | tbd | tbd | tbd | tbd |
| High accuracy (3.5-7c) | srutal archagall | sensible | tbd | tbd | tbd |
| Very high accuracy (1-3.5c) | cata, nestoria, sensipent | tbd | tbd | tbd | tbd |
| Microtemperament (0-1c) | tbd | tbd | tbd | tbd | tbd |
No-11s
| Approx. 10 notes | Approx. 20 notes | Approx. 40 notes | Approx. 80 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | tbd | tbd | tbd | tbd | tbd |
| Low accuracy (12-18c) | tbd | tbd | tbd | tbd | tbd |
| Medium accuracy (7-12c) | tbd | tbd | tbd | tbd | tbd |
| High accuracy (3.5-7c) | tbd | tbd | tbd | tbd | tbd |
| Very high accuracy (1-3.5c) | tbd | tbd | tbd | tbd | tbd |
| Microtemperament (0-1c) | tbd | tbd | tbd | tbd | tbd |
Higher subgroups
| Approx. 10 notes | Approx. 20 notes | Approx. 40 notes | Approx. 80 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | tbd | tbd | tbd | tbd | tbd |
| Low accuracy (12-18c) | tbd | tbd | tbd | tbd | tbd |
| Medium accuracy (7-12c) | tbd | tbd | tbd | tbd | tbd |
| High accuracy (3.5-7c) | tbd | tbd | tbd | tbd | tbd |
| Very high accuracy (1-3.5c) | tbd | tbd | tbd | tbd | tbd |
| Microtemperament (0-1c) | tbd | tbd | tbd | tbd | tbd |
TODO: Add to the appropriate micro temperament categories: ennealimmal, enneadecal, kwazy, schismic, trinity Recheck the accuracy of the note numbers, a lot of them don’t feel right