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m BudjarnLambeth moved page User:BudjarnLambeth/Regular temperament safari guide to User:BudjarnLambeth/Bird’s eye view of rank-2 temperaments: Sounds more professional, more accurate description of intended contents
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Began major rewrite, still need to add in the actual temperaments themselves and pass all the text through a few rounds of review and revision
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Major rewrite of this page coming in the next few days. Big thank you to all the editors on Discord who have helped and are helping me figure this out. Page will be renamed to “bird’s eye view of rank 2 temperaments” on the day of the rewrite.
{{todo|complete table|comment=At the moment, all or almost all of the temperaments are placed in the "medium accuracy" column. This is not correct. I don't know how to assess accuracy, so I need someone else to please place them in the correct column.}}


See #wiki and the thread #wiki>bird’s eye view… in the XA Discord for more details.
There are at least hundreds, probably thousands, of [[rank-2 temperament]]s 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 second largest number in a temperament's mapping from the first 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 ==
 
{| class="wikitable"
|+
!
! 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
|}
 
== 7-limit ==
 
{| class="wikitable"
|+
!
! 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
|}
 
== 11-limit ==
 
{| class="wikitable"
|+
!
! 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
|}
 
== 13-limit ==
 
{| class="wikitable"
|+
!
! 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 limits ==
 
{| class="wikitable"
|+
!
! 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 ==
 
{| class="wikitable"
|+
!
! 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 ==
 
{| class="wikitable"
|+
!
! 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 ==
 
{| class="wikitable"
|+
!
! 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-7s ==
 
{| class="wikitable"
|+
!
! 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-11s ==
 
{| class="wikitable"
|+
!
! 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 ==
 
{| class="wikitable"
|+
!
! 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
|}
Retrieved from "https://en.xen.wiki/w/User:BudjarnLambeth/Bird’s_eye_view_of_rank-2_temperaments"