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 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. 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)	tbd	blacksmith, pajara, porcupine, whitewood	tbd	tbd	tbd
Medium accuracy (7-12c)	tbd	tbd	meantone	tbd	tbd
High accuracy (3.5-7c)	tbd	tbd	diaschismic, mothra	tbd	tbd
Very high accuracy (1-3.5c)	tbd	tbd	wuerschmidt	tbd	tbd
Microtemperament (0-1c)	tbd	tbd	tbd	ennealimmal	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)	tbd	tbd	pajara	tbd	tbd
Medium accuracy (7-12c)	tbd	tbd	tbd	tbd	tbd
High accuracy (3.5-7c)	tbd	tbd	diaschismic	tbd	tbd
Very high accuracy (1-3.5c)	tbd	tbd	tbd	tbd	tbd
Microtemperament (0-1c)	tbd	tbd	tbd	ennealimmal	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	ennealimmal (19lim)	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)	tbd	mohaha	tbd	tbd	tbd
High accuracy (3.5-7c)	tbd	sensible, srutal archagall	tbd	tbd	tbd
Very high accuracy (1-3.5c)	tbd	cata, nestoria, [[sensipent]	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