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<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>
 


There are at least hundreds, probably thousands, of [[rank-2 temperament]]s described. It can be difficult to know where to start.  
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.
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.
 
== So, which temperaments should I use to make music? ==
 
Ask 5 xenharmonicists, and you'll get 10 different answers. There are many different schools of thought within RTT (regular temperament theory).
 
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.
 
 
For example:
 
 
'''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.
 
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 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.
 
They might argue that it's perfectly possible to learn up to 50 notes per [[equave]].


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.
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.
 
 
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.
 
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.
 
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.


== Guide to tables ==
== Guide to tables ==
Line 14: Line 57:
* Exotemperament: > ~18 [[cents]] of [[error]] on more than one targeted interval
* Exotemperament: > ~18 [[cents]] of [[error]] on more than one targeted interval
* Low accuracy: < ~18c error on most targeted intervals
* Low accuracy: < ~18c error on most targeted intervals
* Medium accuracy: < 12c error on almost all targeted intervals
* Medium accuracy: < 12c error on most targeted intervals
* High accuracy: < 7c 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
* Very high accuracy: < 3.5c error on almost all targeted intervals
Line 43: Line 86:
! Over 100 notes
! Over 100 notes
|-
|-
| Exotemperament (18-∞c)
! Exotemperament (18-∞c)
| [[antitonic]], [[bug]], [[dicot]], [[father]]
| [[antitonic]], [[bug]], [[dicot]], [[father]]
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Low accuracy (12-18c)
! Low accuracy (12-18c)
| [[augmented]], [[blackwood]], [[dimipent]], [[porcupine]], [[whitewood]]
| [[augmented]], [[blackwood]], [[dimipent]], [[porcupine]], [[whitewood]]
| tbd
| [[superpyth]]
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Medium accuracy (7-12c)
! Medium accuracy (7-12c)
| [[meantone]]
| [[meantone]]
| tbd
| [[hanson]], [[magic]]
| tbd
| [[valentine]]
| tbd
|  
| tbd
|  
|-
|-
| High accuracy (3.5-7c)
! High accuracy (3.5-7c)
| [[diaschismic]]
| [[diaschismic]]
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Very high accuracy (1-3.5c)
! Very high accuracy (1-3.5c)
| tbd
|  
| tbd
|  
| [[wuerschmidt]]
| [[wuerschmidt]]
| tbd
|  
| tbd
|  
|-
|-
| Microtemperament (0-1c)
! Microtemperament (0-1c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
| schismic aka [[Helmholtz temperament|Helmholtz]]
| tbd
| [[kwazy]]
|}
|}


Line 97: Line 140:
! Over 100 notes
! Over 100 notes
|-
|-
| Exotemperament (18-∞c)
! Exotemperament (18-∞c)
| [[antitonic]], [[dicot]], [[father]]
| [[antitonic]], [[dicot]], [[father]]
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Low accuracy (12-18c)
! Low accuracy (12-18c)
| [[blacksmith]]
| [[blacksmith]]
| [[pajara]], [[porcupine]], [[whitewood]]
| [[augene]], [[godzilla]], [[pajara]], [[porcupine]], [[whitewood]]
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Medium accuracy (7-12c)
! Medium accuracy (7-12c)
| tbd
|  
| [[meantone]], [[sensi]]
| [[magic]], [[meantone]], [[mothra]], [[sensi]], [[superpyth]]
| tbd
| [[valentine]]
| tbd
|  
| tbd
|  
|-
|-
| High accuracy (3.5-7c)
! High accuracy (3.5-7c)
| tbd
|  
| [[mothra]]
| [[orwell]]
| [[diaschismic]], [[garibaldi]]
| [[diaschismic]], [[garibaldi]]
| tbd
|  
| tbd
|  
|-
|-
| Very high accuracy (1-3.5c)
! Very high accuracy (1-3.5c)
| tbd
|  
| tbd
|  
| [[miracle]], [[wuerschmidt]]
| [[miracle]]
| tbd
|  
| tbd
|  
|-
|-
| Microtemperament (0-1c)
! Microtemperament (0-1c)
| tbd
|  
| tbd
|  
| tbd
|  
| [[ennealimmal]]
| [[ennealimmal]]
| tbd
| [[enneadecal]], [[trinity]]
|}
|}


Line 151: Line 194:
! Over 100 notes
! Over 100 notes
|-
|-
| Exotemperament (18-∞c)
! Exotemperament (18-∞c)
| [[antitonic]], [[dicot]], [[father]]
| [[antitonic]], [[dicot]], [[father]]
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Low accuracy (12-18c)
! Low accuracy (12-18c)
| tbd
|  
| [[blacksmith]], [[pajara]], [[porcupine]], [[whitewood]]
| [[augene]], [[blacksmith]], [[pajara]], [[porcupine]], [[whitewood]]
| tbd
| [[godzilla]], [[superpyth]]
| tbd
|  
| tbd
|  
|-
|-
| Medium accuracy (7-12c)
! Medium accuracy (7-12c)
| tbd
|  
| tbd
|  
| [[meanpop]], [[meantone]]
| [[magic]], [[meanpop]], [[meantone]], [[mothra]], [[valentine]]
| tbd
|  
| tbd
|  
|-
|-
| High accuracy (3.5-7c)
! High accuracy (3.5-7c)
| tbd
|  
| tbd
|  
| [[diaschismic]], [[mothra]]
| [[diaschismic]], [[orwell]]
| tbd
|  
| tbd
|  
|-
|-
| Very high accuracy (1-3.5c)
! Very high accuracy (1-3.5c)
| tbd
|  
| tbd
|  
| [[miracle]], [[wuerschmidt]]
| [[miracle]]
| tbd
|  
| tbd
|  
|-
|-
| Microtemperament (0-1c)
! Microtemperament (0-1c)
| tbd
|  
| tbd
|  
| tbd
|  
| [[ennealimmal]]
| [[ennealimmal]]
| tbd
| [[enneadecal]], [[trinity]]
|}
|}


Line 205: Line 248:
! Over 100 notes
! Over 100 notes
|-
|-
| Exotemperament (18-∞c)
! Exotemperament (18-∞c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Low accuracy (12-18c)
! Low accuracy (12-18c)
| tbd
|  
| [[blacksmith]], [[pajara]], [[porcupine]], [[whitewood]]
| [[augene]], [[blacksmith]], [[pajara]], [[porcupine]], [[whitewood]]
| tbd
| [[superpyth]]
| tbd
|  
| tbd
|  
|-
|-
| Medium accuracy (7-12c)
! Medium accuracy (7-12c)
| tbd
|  
| tbd
|  
| [[meantone]]
| [[magic]], [[meantone]], [[mothra]]
| tbd
|  
| tbd
|  
|-
|-
| High accuracy (3.5-7c)
! High accuracy (3.5-7c)
| tbd
|  
| tbd
|  
| [[diaschismic]], [[mothra]]
| [[diaschismic]], [[orwell]]
| tbd
|  
| tbd
|  
|-
|-
| Very high accuracy (1-3.5c)
! Very high accuracy (1-3.5c)
| tbd
|  
| tbd
|  
| [[wuerschmidt]]
|  
| tbd
|  
| tbd
|  
|-
|-
| Microtemperament (0-1c)
! Microtemperament (0-1c)
| tbd
|  
| tbd
|  
| tbd
|  
| [[ennealimmal]]
| [[ennealimmal]]
| tbd
| [[enneadecal]], [[trinity]]
|}
|}


Line 259: Line 302:
! Over 100 notes
! Over 100 notes
|-
|-
| Exotemperament (18-∞c)
! Exotemperament (18-∞c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Low accuracy (12-18c)
! Low accuracy (12-18c)
| tbd
|  
| tbd
|  
| [[pajara]]
| [[pajara]]
| tbd
|  
| tbd
|  
|-
|-
| Medium accuracy (7-12c)
! Medium accuracy (7-12c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| High accuracy (3.5-7c)
! High accuracy (3.5-7c)
| tbd
|  
| tbd
|  
| [[diaschismic]]
| [[diaschismic]]
| tbd
|  
| tbd
|  
|-
|-
| Very high accuracy (1-3.5c)
! Very high accuracy (1-3.5c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Microtemperament (0-1c)
! Microtemperament (0-1c)
| tbd
|  
| tbd
|  
| tbd
|  
| [[ennealimmal]]
| [[ennealimmal]]
| tbd
| [[trinity]]
|}
|}


Line 313: Line 356:
! Over 100 notes
! Over 100 notes
|-
|-
| Exotemperament (18-∞c)
! Exotemperament (18-∞c)
| tbd
|
| tbd
|
| tbd
|
| tbd
|
| tbd
|
|-
! 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 ==
 
{| class="wikitable"
|+
!
! Approx. 10 notes
! Approx. 20 notes
! Approx. 30 notes
! Approx. 70 notes
! Over 100 notes
|-
! Exotemperament (18-∞c)
|  
|  
|  
|  
|  
|-
|-
| Low accuracy (12-18c)
! Low accuracy (12-18c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Medium accuracy (7-12c)
! Medium accuracy (7-12c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| High accuracy (3.5-7c)
! High accuracy (3.5-7c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Very high accuracy (1-3.5c)
! Very high accuracy (1-3.5c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Microtemperament (0-1c)
! Microtemperament (0-1c)
| tbd
|  
| tbd
|  
| tbd
|  
| [[ennealimmal]] (19lim)
|  
| tbd
|  
|}
|}


== No-2s ==
== Other no-2s subgroups ==


{| class="wikitable"
{| class="wikitable"
Line 367: Line 464:
! Over 100 notes
! Over 100 notes
|-
|-
| Exotemperament (18-∞c)
! Exotemperament (18-∞c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Low accuracy (12-18c)
! Low accuracy (12-18c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Medium accuracy (7-12c)
! Medium accuracy (7-12c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| High accuracy (3.5-7c)
! High accuracy (3.5-7c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Very high accuracy (1-3.5c)
! Very high accuracy (1-3.5c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Microtemperament (0-1c)
! Microtemperament (0-1c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|}
|}


== No-3s ==
== No-3s subgroups ==


{| class="wikitable"
{| class="wikitable"
Line 421: Line 518:
! Over 100 notes
! Over 100 notes
|-
|-
| Exotemperament (18-∞c)
! Exotemperament (18-∞c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Low accuracy (12-18c)
! Low accuracy (12-18c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Medium accuracy (7-12c)
! Medium accuracy (7-12c)
| tbd
|  
| tbd
| [[didacus]], [[orgone]]
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| High accuracy (3.5-7c)
! High accuracy (3.5-7c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Very high accuracy (1-3.5c)
! Very high accuracy (1-3.5c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Microtemperament (0-1c)
! Microtemperament (0-1c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|}
|}


== No-5s ==
== 2.3.7 and its extensions ==


{| class="wikitable"
{| class="wikitable"
Line 475: Line 572:
! Over 100 notes
! Over 100 notes
|-
|-
| Exotemperament (18-∞c)
! Exotemperament (18-∞c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Low accuracy (12-18c)
! Low accuracy (12-18c)
| [[semaphore]]
| [[semaphore]]
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Medium accuracy (7-12c)
! Medium accuracy (7-12c)
| tbd
|  
| tbd
| [[bleu]], [[slendric]]
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| High accuracy (3.5-7c)
! High accuracy (3.5-7c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Very high accuracy (1-3.5c)
! Very high accuracy (1-3.5c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Microtemperament (0-1c)
! Microtemperament (0-1c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|}
|}


== No-7s ==
== 2.3.11 and its extensions ==


{| class="wikitable"
{| class="wikitable"
Line 529: Line 626:
! Over 100 notes
! Over 100 notes
|-
|-
| Exotemperament (18-∞c)
! Exotemperament (18-∞c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
! 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 ==
 
{| class="wikitable"
|+
!
! Approx. 10 notes
! Approx. 20 notes
! Approx. 30 notes
! Approx. 70 notes
! Over 100 notes
|-
|-
| Low accuracy (12-18c)
! Exotemperament (18-∞c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Medium accuracy (7-12c)
! Low accuracy (12-18c)
| tbd
|
|
|
|
|
|-
! Medium accuracy (7-12c)
|
|
|
|
|
|-
! High accuracy (3.5-7c)
|
|
|
|
|
|-
! Very high accuracy (1-3.5c)
|
|
|
|
|
|-
! Microtemperament (0-1c)
|
|
|
|
|
|}
 
== No-7s subgroups ==
 
{| 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]]
| [[mohaha]]
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| High accuracy (3.5-7c)
! High accuracy (3.5-7c)
| tbd
|  
| [[sensible]], [[srutal archagall]]
| [[sensible]], [[srutal archagall]]
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Very high accuracy (1-3.5c)
! Very high accuracy (1-3.5c)
| tbd
|  
| [[cata]], [[nestoria]], [[sensipent]]
| [[cata]], [[nestoria]], [[sensipent]]
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Microtemperament (0-1c)
! Microtemperament (0-1c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|}
|}


== No-11s ==
== No-11s subgroups ==


{| class="wikitable"
{| class="wikitable"
Line 583: Line 788:
! Over 100 notes
! Over 100 notes
|-
|-
| Exotemperament (18-∞c)
! Exotemperament (18-∞c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Low accuracy (12-18c)
! Low accuracy (12-18c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Medium accuracy (7-12c)
! Medium accuracy (7-12c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| High accuracy (3.5-7c)
! High accuracy (3.5-7c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Very high accuracy (1-3.5c)
! Very high accuracy (1-3.5c)
| tbd
|  
| tbd
|  
| tbd
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| tbd
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| tbd
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|-
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| Microtemperament (0-1c)
! Microtemperament (0-1c)
| tbd
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| tbd
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| tbd
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| tbd
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| tbd
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|}
|}


== Higher subgroups ==
== Other subgroups ==


{| class="wikitable"
{| class="wikitable"
Line 637: Line 842:
! Over 100 notes
! Over 100 notes
|-
|-
| Exotemperament (18-∞c)
! Exotemperament (18-∞c)
| tbd
|  
| tbd
|  
| tbd
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| tbd
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| tbd
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|-
|-
| Low accuracy (12-18c)
! Low accuracy (12-18c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Medium accuracy (7-12c)
! Medium accuracy (7-12c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| High accuracy (3.5-7c)
! High accuracy (3.5-7c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Very high accuracy (1-3.5c)
! Very high accuracy (1-3.5c)
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
| tbd
|  
|-
|-
| Microtemperament (0-1c)
! 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
Retrieved from "https://en.xen.wiki/w/User:BudjarnLambeth/Bird’s_eye_view_of_rank-2_temperaments"