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This is a page to easily navigate to temperaments that fit your interests.  
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'''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.'''
</div>


Low complexity temperaments are likely to be of most interest to artists new to music tuning theory. That is because they tend to approximate all of the important intervals within less than 30 notes (occasionally even less than 12!), so they are the most practical to map onto a physical instrument.


Medium and higher complexity temperaments provide a bigger range of new intervals, and they approximate those intervals more accurately, but they may be unwieldy, requiring dozens or even hundreds of notes to approximate all the important intervals.
There are at least hundreds, probably thousands, of [[rank-2 temperament]]s described. It can be difficult to know where to start.  


Feel free to add temperaments to the appropriate category if they are not already here. (If you're not sure which category to put them in, put them in honorable mentions).
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.


== 1. Low complexity temperaments ==
== So, which temperaments should I use to make music? ==
Temperaments which approximate most of their consonant intervals with less than 30 notes, and where those intervals are mostly less than 20 cents out of tune.


Ask 5 xenharmonicists, and you'll get 10 different answers. There are many different schools of thought within RTT (regular temperament theory).


(For comparison, in 12edo, most intervals are about 10 cents out of tune, with better fifths but less good thirds.)
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.


1.1 5-limit


[[Meantone]], [[augmented]], [[mavila]], [[porcupine]], [[blackwood]], [[diminished]], [[srutal]]
For example:




1.2 7-limit
'''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.  


[[Blacksmith]], septimal [[diminished]], [[dominant]], [[august]], [[pajara]], [[semaphore]], [[septimal meantone]], [[injera]], septimal [[negri]], [[augene]]
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.


1.3 No-2s subgroups of 7-limit
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.




1.4 Other subgroups of 7-limit
'''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]].


1.5 11-limit
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.


1.6 No-2s subgroups of 11-limit


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.


1.7 Other subgroups of 11-limit
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.


1.8 13-limit
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.


1.9 No-2s of 13-limit
== Guide to tables ==


'''Rows'''


1.10 No-2s subgroups of 13-limit
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.




1.11 Other subgroups of 13-limit
'''Columns'''


The columns categorise temperaments by [[complexity]].


1.12 Higher limits
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.




1.13 No-2s higher limit subgroups
'''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).


1.13 Other higher limit subgroups
== 5-limit ==


{| class="wikitable"
|+
!
! 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 temperament|Helmholtz]]
| [[kwazy]]
|}


== 2. Medium complexity temperaments ==
== 7-limit ==


Temperaments which approximate most of their consonant intervals with less than 60 notes, and where those intervals are mostly less than 5 cents out of tune.
{| class="wikitable"
|+
!
! 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 ==


2.1 5-limit
{| class="wikitable"
|+
!
! 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]]
|}


[[Magic]], [[ripple]], [[hanson]], [[negri]], [[tetracot]], [[superpyth]], [[helmholtz]], [[sensi]], [[passion]], [[wuerschmidt]], [[compton]], [[amity]], [[orson]]
== 13-limit ==


{| class="wikitable"
|+
!
! 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]]
|}


2.2 7-limit
== 17-limit ==


[[keemun]], [[catler]], [[hedgehog]], septimal [[superpyth]], septimal [[sensi]], [[lemba]], septimal [[porcupine]], [[flattone]], septimal [[magic]], [[doublewide]], [[nautilus]], [[beatles]], [[liese]], [[cynder]], [[orwell]], [[garibaldi]], [[myna]], [[miracle]]
{| class="wikitable"
|+
!
! 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 ==


2.3 No-2s subgroups of 7-limit
{| class="wikitable"
|+
!
! 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 ==


2.4 Other subgroups of 7-limit
{| class="wikitable"
|+
!
! 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 ==


2.5 11-limit
{| class="wikitable"
|+
!
! 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 ==


2.6 No-2s subgroups of 11-limit
{| class="wikitable"
|+
!
! 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 ==


2.7 Other subgroups of 11-limit
{| class="wikitable"
|+
!
! 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 ==


2.8 13-limit
{| class="wikitable"
|+
!
! 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 ==


2.9 No-2s of 13-limit
{| class="wikitable"
|+
!
! 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 ==


2.10 No-2s subgroups of 13-limit
{| class="wikitable"
|+
!
! 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 ==


2.11 Other subgroups of 13-limit
{| class="wikitable"
|+
!
! 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 ==


2.12 Higher limits
{| class="wikitable"
 
|+
 
!
2.13 No-2s higher limit subgroups
! Approx. 10 notes
 
! Approx. 20 notes
 
! Approx. 30 notes
2.14 Other higher limit subgroups
! Approx. 70 notes
 
! Over 100 notes
 
|-
== 3. High complexity temperaments ==
! Exotemperament (18-∞c)
 
|
Temperaments where most of their consonant intervals are less than 1 cent out of tune.
|
 
|
 
|
3.1 5-limit
|
 
|-
[[Vishnu]], [[luna]]
! Low accuracy (12-18c)
 
|
 
|
3.2 7-limit
|
 
|
[[Ennealimmal]]
|
 
|-
 
! Medium accuracy (7-12c)
3.3 No-2s subgroups of 7-limit
|
 
|
 
|
3.4 Other subgroups of 7-limit
|
 
|
 
|-
3.5 11-limit
! High accuracy (3.5-7c)
 
|
 
|
3.6 No-2s subgroups of 11-limit
|
 
|
 
|
3.7 Other subgroups of 11-limit
|-
 
! Very high accuracy (1-3.5c)
 
|
3.8 13-limit
|
 
|
 
|
3.9 No-2s of 13-limit
|
 
|-
 
! Microtemperament (0-1c)
3.10 No-2s subgroups of 13-limit
|
 
|
 
|
3.11 Other subgroups of 13-limit
|
 
|
 
|}
3.12 Higher limits
 
 
3.13 No-2s higher limit subgroups
 
 
3.14 Other higher limit subgroups
 
 
4. Honorable mentions
 
Temperaments which are interesting in some way, but might not meet the criteria for the other lists.
 
 
4.1 5-limit
 
 
4.2 7-limit
 
 
4.3 No-2s subgroups of 7-limit
 
 
4.4 Other subgroups of 7-limit
 
 
4.5 11-limit
 
 
4.6 No-2s subgroups of 11-limit
 
 
4.7 Other subgroups of 11-limit
 
 
4.8 13-limit
 
 
4.9 No-2s of 13-limit
 
 
4.10 No-2s subgroups of 13-limit
 
 
4.11 Other subgroups of 13-limit
 
 
4.12 Higher limits
 
 
4.13 No-2s higher limit subgroups
 
 
4.14 Other higher limit subgroups
 
 
5. Exoemperaments
 
Temperaments where most intervals are at least 20 cents out of tune.
 
 
5.1 5-limit
 
[[Father]], [[bug]]
 
5.2 7-limit
 
 
5.3 No-2s subgroups of 7-limit
 
 
5.4 Other subgroups of 7-limit
 
 
5.5 11-limit
 
 
5.6 No-2s subgroups of 11-limit
 
 
5.7 Other subgroups of 11-limit
 
 
5.8 13-limit
 
 
5.9 No-2s of 13-limit
 
 
5.10 No-2s subgroups of 13-limit
 
 
5.11 Other subgroups of 13-limit
 
 
5.12 Higher limits
 
 
5.13 No-2s higher limit subgroups
 
 
5.14 Other higher limit subgroups
Retrieved from "https://en.xen.wiki/w/User:BudjarnLambeth/Bird’s_eye_view_of_rank-2_temperaments"