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{{Editable user page|If you see any temperaments in the wrong category, please move them to the correct category. <br><br>If you know of a temperament that is much-loved by a sizeable subset of the xen community but is not yet included here, please add it. <br><br>If you see a temperament on here that does not have good accuracy for its size in a particular subgroup, please delete that temperament from that subgroup’s row of the table.<br><br>If you see any ways the wording of the page could be improved, please edit it to make those improvements. <br><br>If you see any typos or grammatical or factual errors, please make an edit to correct those. <br><br>Please make the case (to readers) for your favourite temperament(s) in writing at the bottom of the page. (This is 100% optional, you can still add temperaments to the table without doing this :) )}}
<div style="border: 1px green solid; background-color: #efe; padding: 0.5em 1em; margin-bottom: 1em">
'''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>




Line 20: Line 22:




'''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.  
'''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.  


And they might argue that 25 notes per [[equave]] is the most that is practical, any more than that is too cumbersome.
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.  
They might argue that nobody can hear the harmonic effect of prime harmonics higher than 11.  


And they might argue that there's no real reason to use [[subgroup]]s that are missing primes 2 or 3, because those primes are so important to consonance.
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.




'''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.
'''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.
They might argue that it's perfectly possible to learn up to 50 notes per [[equave]].


They might argue that they can hear the subtle, delicate effect of prime harmonics up to 23.
They might argue that they can hear the subtle, delicate effect of prime harmonics up to 23.
Line 46: Line 48:
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.
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 helpful point of comparison to measure these other temperaments against.
So that should provide a point of comparison to help measure these other temperaments against.


== How to read the table ==
== Guide to tables ==


'''Rows'''
'''Rows'''


'''The rows categorise temperaments by the [[just intonation subgroup]] they approximate.'''
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.


The 2.3.5 subgroup is what most theorists believe 12 tone equal temperament belongs to (but there is plenty of disagreement about that).


The 2.3.5.7 and 2.3.5.7.11 subgroups are the most commonly used, being not too complex and including lots of useful harmonies.
'''Columns'''


Subgroups with no 2s, e.g. 3.5.7.11, are the biggest and most jarring break away from familiar harmony, which you may consider a good or a bad thing.
The columns categorise temperaments by [[complexity]].  


Subgroups with 2s and 3s but no 5s, e.g. 2.3.7.11, preserve the most fundamental familiar intervals like the octave and the fifth, but do away with the 5-limit major and minor intervals of common practice harmony, forcing innovation while still keeping some familiarity.
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.


Some theorists believe including 13, 17 or higher in a subgroup is pointless because the brain can't register such complex intervals. Others believe these intervals are registered by the brain, maybe subtly and subconsciously in some instances, but still they matter. It is up to you which theorists you believe on that.


'''Subgroup categorisation'''


'''Columns'''
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).


'''The columns categorise temperaments by the approximate number of notes needed to reach all the important intervals''' within the practice of that temperament.
== 5-limit ==


All of the temperaments listed in this table have low [[badness]] (high relative accuracy), meaning they approximate their target JI subgroup much better than most temperaments with their same amount of needed notes.
{| 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]]
|}
 
== 7-limit ==


That means the temperaments ''in this table'' requiring more notes are also more accurate. The ones requiring less notes are less accurate but are good for their size. (This rule is not true for all temperaments in general, it’s just true for the ones listed in this table.)
{| 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]]
|}


== Table of recommended temperaments ==
== 11-limit ==


{| class="wikitable center-all"
{| class="wikitable"
|+
|+
! JI subgroup
!  
! ~10 notes
! Approx. 10 notes
! ~20 notes
! Approx. 20 notes
! ~30 notes
! Approx. 30 notes
! ~70 notes
! Approx. 70 notes
! >100 notes
! Over 100 notes
|-
|-
! 5-limit (2.3.5)
! Exotemperament (18-∞c)
| [[augmented]], [[blackwood]], <br>[[diaschismic]], [[dimipent]], <br>[[meantone]], [[porcupine]], <br>[[whitewood]]
| [[antitonic]], [[dicot]], [[father]]
| [[hanson]], <br>[[helmholtz]] (aka schismic), <br>[[magic]], [[superpyth]]
|
| [[valentine]], [[wuerschmidt]]
|
| [[gravity]]
|
| [[kwazy]]
|
|-
! 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]]
|
|
|-
|-
! 7-limit (2.3.5.7)
! Microtemperament (0-1c)
| [[augene]], [[blacksmith]], <br>[[porcupine]]
|  
| [[godzilla]], [[magic]], <br>[[meantone]], [[mothra]], <br>[[orwell]], [[pajara]], <br>[[shrutar]], [[whitewood]]
|  
| [[diaschismic]], [[garibaldi]], <br>[[hemiwuerschmidt]], <br>[[miracle]], [[valentine]]
|  
| [[ennealimmal]], [[harry]]
| [[ennealimmal]]
| [[enneadecal]], [[trinity]]
| [[enneadecal]], [[trinity]]
|}
== 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)
|
|
|
|
|
|-
|-
! 11-limit (2.3.5.7.11)
! Microtemperament (0-1c)
| (!)
|  
| [[augene]], [[blacksmith]], <br>[[pajara]], [[porcupine]], <br>[[whitewood]]
|  
| [[diaschismic]], [[godzilla]], <br>[[hemiwuerschmidt]], [[magic]], <br>[[meanpop]], [[meantone]], <br>[[miracle]], [[mothra]], <br>[[nautilus]], [[orwell]], <br>[[shrutar]], [[superpyth]], <br>[[valentine]]
|  
| [[ennealimmal]], [[harry]]
| [[ennealimmal]]
| [[enneadecal]], [[trinity]]
| [[enneadecal]], [[trinity]]
|}
== 17-limit ==
{| class="wikitable"
|+
!
! Approx. 10 notes
! Approx. 20 notes
! Approx. 30 notes
! Approx. 70 notes
! Over 100 notes
|-
|-
! 13-limit
! Exotemperament (18-∞c)
| (!)
|
| [[augene]], [[blacksmith]], <br>[[pajara]], [[porcupine]], <br>[[whitewood]]
|
| [[diaschismic]], [[hemiwuerschmidt]], <br>[[magic]], [[meantone]], <br>[[mothra]], [[nautilus]], <br>[[orwell]], [[shrutar]], <br>[[superpyth]]
|
| [[ennealimmal]], [[harry]]
|
| [[enneadecal]], [[trinity]]
|  
|-
! Low accuracy (12-18c)
|  
|
| [[pajara]]
|
|
|-
! Medium accuracy (7-12c)
|
|
|
|
|
|-
! High accuracy (3.5-7c)
|
|
| [[diaschismic]]
|  
|  
|-
! Very high accuracy (1-3.5c)
|
|
|
|
|
|-
|-
! 17-limit
! Microtemperament (0-1c)
| (!)
|  
| (!)
|  
| [[diaschismic]], [[echidna]], <br>[[pajara]], [[shrutar]]
|  
| [[ennealimmal]], [[harry]]
| [[ennealimmal]]
| [[trinity]]
| [[trinity]]
|}
== Higher limits ==
{| class="wikitable"
|+
!
! Approx. 10 notes
! Approx. 20 notes
! Approx. 30 notes
! Approx. 70 notes
! Over 100 notes
|-
! Exotemperament (18-∞c)
|
|
|
|
|
|-
! Low accuracy (12-18c)
|
|
|
|
|
|-
|-
! Higher limits
! Medium accuracy (7-12c)
| (!)
|
| (!)
|
| [[shrutar]]
|
|
|  
|-
! High accuracy (3.5-7c)
|
|
|
|
|
|-
! Very high accuracy (1-3.5c)
|  
|
|
|
|
|-
! Microtemperament (0-1c)
|  
|
|
| [[ennealimmal]]
| [[ennealimmal]]
| [[trinity]]
| [[trinity]]
|}
== 3.5.7 and its extensions ==
{| 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)
|
|
|
|
|
|-
|-
! 2.3.5.7.n
! Microtemperament (0-1c)
| (!)
|
|  
|  
|  
| [[unicorn]]
|  
|  
|  
|  
|}
== Other no-2s subgroups ==
{| class="wikitable"
|+
!
! Approx. 10 notes
! Approx. 20 notes
! Approx. 30 notes
! Approx. 70 notes
! Over 100 notes
|-
|-
! 2.3.5.11
! Exotemperament (18-∞c)
|
|
|  
|  
| [[mohaha]], [[sensible]]
|  
|  
| [[larry]] (2.3.5.11 [[gravity]])
|  
|  
|-
|-
! 2.3.5.11.n
! Low accuracy (12-18c)
| (!)
|  
| [[sensible]]
|  
|  
|  
|  
|  
|  
|  
|-
|-
! 2.3.5.n
! Medium accuracy (7-12c)
|
|
|  
|  
| [[cata]], [[nestoria]], <br>[[sensipent]], [[srutal archagall]]
| [[wuerschmidt]] (2.3.5.23)
|  
|  
|  
|  
|-
|-
! 2.3.7
! High accuracy (3.5-7c)
| [[semaphore]]
|  
| [[bleu]], [[slendric]]
|  
|  
|  
|  
|  
|  
|  
|-
|-
! 2.3.7.11
! Very high accuracy (1-3.5c)
|
|  
|  
| [[bleu]]
|  
|  
|  
|  
|  
|  
|-
|-
! 2.3.7.11.n
! Microtemperament (0-1c)
| (!)
|  
| [[bleu]]
|  
|  
|  
|  
|  
|  
|  
|}
== No-3s subgroups ==
{| class="wikitable"
|+
!
! Approx. 10 notes
! Approx. 20 notes
! Approx. 30 notes
! Approx. 70 notes
! Over 100 notes
|-
|-
! 2.3.7.n
! Exotemperament (18-∞c)
|  
|  
|  
|  
Line 182: Line 525:
|  
|  
|-
|-
! 2.3.11
! Low accuracy (12-18c)
|
|
|
|  
|  
| [[neutral]] (2.3.11 [[rastmic]]), <br>2.3.11 [[pythrabian]]
| [[tribilo]] (2.3.11 [[nexus]])
|  
|  
| 2.3.11 [[frameshift]]
|-
|-
! 2.3.11.n
! Medium accuracy (7-12c)
|
|
|
| [[didacus]], [[orgone]]
|
|
|
|
|
|
|-
! High accuracy (3.5-7c)
|
|
|
|
|
|-
! Very high accuracy (1-3.5c)
|  
|  
|  
|  
|  
|-
|-
! 2.5.7
! Microtemperament (0-1c)
|
|  
|  
| [[didacus]]
|  
|  
|  
|  
|  
|  
|}
== 2.3.7 and its extensions ==
{| class="wikitable"
|+
!
! Approx. 10 notes
! Approx. 20 notes
! Approx. 30 notes
! Approx. 70 notes
! Over 100 notes
|-
|-
! 2.5.7.11
! Exotemperament (18-∞c)
|
|  
|  
| [[didacus]]
|  
|  
|  
|  
|  
|  
|-
|-
! 2.5.7.n
! Low accuracy (12-18c)
| [[semaphore]]
|
|  
|  
|  
|  
|
|-
! Medium accuracy (7-12c)
|
| [[bleu]], [[slendric]]
|  
|  
|  
|  
|  
|  
|-
|-
! 2.5.11.n
! High accuracy (3.5-7c)
|  
|  
|  
|  
Line 224: Line 600:
|  
|  
|-
|-
!2.7.11
! Very high accuracy (1-3.5c)
|
|  
| [[orgone]]
|  
|
|  
|
|  
|
|  
|-
|-
! 3.5.7
! Microtemperament (0-1c)
| [[arcturus]], [[BPS]], <br>[[canopus]], [[sirius]]
|
|  
|  
|  
|  
|  
|  
|  
|
|}
 
== 2.3.11 and its extensions ==
 
{| class="wikitable"
|+
!
! Approx. 10 notes
! Approx. 20 notes
! Approx. 30 notes
! Approx. 70 notes
! Over 100 notes
|-
|-
! 3.5.n
! Exotemperament (18-∞c)
|
|  
|  
|  
|  
|  
|  
|  
|  
|
|-
|-
! 3.5.7.n
! Low accuracy (12-18c)
|
|  
|  
| [[dubhe]] (3.5.7.17)
|  
|  
|  
|  
|  
|  
|-
|-
! 3.5.7.11.n
! Medium accuracy (7-12c)
| (!)
|  
| [[mintra]] (11-limit)
| [[neutral]] (no-5 no-7 [[rastmic]])
| [[mintra]] (13-limit)
|
|
|  
|  
|
|-
|-
! 3.7.n
! High accuracy (3.5-7c)
| [[mintaka]] (3.7.11)
|
| no-5 no-7 [[pythrabian]]
|  
|  
|  
|  
|  
|  
|
|-
|-
! Other subgroups
! Very high accuracy (1-3.5c)
|
|  
|  
| [[tribilo]] (no-5 no-7 [[nexus]])
|  
|  
|  
|  
|-
! Microtemperament (0-1c)
|  
|  
|  
|  
|
|
| no-5 no-7 [[frameshift]]
|}
|}


(!) = ''It is unlikely that a useable temperament could exist here because there are too many important intervals to approximate with this number of notes.''
== 2.3.13, 2.3.17, etc ==
 
{| 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)
|
|
|
|
|
|}


'''Why so many similar temperaments?'''
== No-7s subgroups ==


Some temperament names represent different but closely related tunings from a shared progenitor temperament, which should be noted when looking at higher-limit behaviours of more simple temperaments.
{| 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)
|
|
|
|
|
|}


== Why you should use my favorite temperament (editor opinions) ==
== No-11s subgroups ==


In this section, any editor may create their own subheading, under which they may describe a specific temperament they like and why they think people should use it.  
{| 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)
|
|
|
|
|
|}


One editor is allowed to add more than one temperament if they like, just keep it within reason.
== Other subgroups ==


Editors, please use simple, plain language as much as you can - imagine you're explaining this to a stranger at a bar who has no music theory knowledge at all, but is curious about it.
{| 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)
|
|
|
|
|
|}
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