User:BudjarnLambeth/Bird’s eye view of rank-2 temperaments

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Stuff I'll add next; the numbers are just shorthand instructions for myself to help me know what to do, don't worry about them:

Hanson Me 20 5lim

Valentine H 20/30 5,7,11

Godzilla L 20/30 7, 11

Didacus Me 20 No3

Orgone Me 20 No3

Slendric Me 20 No5

Bleu Me 20 No5

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.

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. 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	magic
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, pajara, porcupine, whitewood
Medium accuracy (7-12c)		magic, meantone, sensi, superpyth
High accuracy (3.5-7c)		mothra, orwell	diaschismic, garibaldi
Very high accuracy (1-3.5c)			miracle, wuerschmidt
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	superpyth
Medium accuracy (7-12c)			magic, meanpop, meantone
High accuracy (3.5-7c)			diaschismic, mothra, orwell
Very high accuracy (1-3.5c)			miracle, wuerschmidt
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
High accuracy (3.5-7c)			diaschismic, mothra, orwell
Very high accuracy (1-3.5c)			wuerschmidt
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

No-2s

	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

	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-5s

	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)
High accuracy (3.5-7c)
Very high accuracy (1-3.5c)
Microtemperament (0-1c)

No-7s

	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

	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)

Higher 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)