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

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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>




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 to narrow the field a little, by showing you the temperaments that get talked about a lot within the xenharmonic community. 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 theorists or composers.
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 particular camp or another. This list arranges popular temperaments by their properties in a table allowing the views of all ‘camps’ to get a space on the page.
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.
 
'''Tl;dr, you can think of this as a table of what temperaments are popular among theorists and/or composers. These are the ones lots of people are talking about.'''


== So, which temperaments should I use to make music? ==
== So, which temperaments should I use to make music? ==
Line 22: 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 40: Line 40:




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.  
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.
 
And these are not the only possible stances, either! There is a Xenharmonicist C, Xenharmonist D, etc. Thousands of them! Lots of different opinions and preferences out there.
 
So how do you start working it out for yourself? Figuring out your own preferences?


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 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).  


Most theorists interpret 12edo as a 2.3.5 subgroup temperament which is about as accurate as most of the temperaments in the left-most column of the below table. This interpretation is not universal, though.
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.


The second most common approach is to interpret 12edo as a high-accuracy 2.3.17.19 subgroup temperament, which is about as accurate as the temperaments in the middle columns of the table.
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).
'''Columns'''


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.
The columns categorise temperaments by [[complexity]].  


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.
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.


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.


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'''


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).


You may see the same temperament multiple times on the table. Here’s why:
== 5-limit ==


Some temperaments are good at approximating a variety of different subgroups. For example, magic is good at approximating both the 7-limit and the 11-limit, so it is listed under both.
{| 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]]
|}


It would be like if there were a table of pokemon, and one row said “cute pokemon”, and one row said “electric pokemon”: Pikachu would be listed under both.
== 7-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)
| [[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]]
|}


'''Columns'''
== 11-limit ==


'''The columns categorise temperaments by the approximate number of notes needed to reach all the important intervals''' within the practice of that temperament.
{| 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]]
|}


== 13-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"
 
|+
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.)
!
! 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]]
|}


== Table of popular temperaments ==
== 17-limit ==


{| class="wikitable center-all"
{| class="wikitable"
|+
|+
! JI subgroup
!  
! 5 to 15 notes
! Approx. 10 notes
! 15 to 25 notes
! Approx. 20 notes
! 25 to 50 notes
! Approx. 30 notes
! 50 to 100 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]]
|  
| [[hanson]], <br>[[helmholtz]] (aka schismic), <br>[[magic]], [[superpyth]]
|  
| [[valentine]], [[wuerschmidt]]
|  
| [[gravity]]
|  
| [[kwazy]]
|  
|-
|-
! 7-limit (2.3.5.7)
! Low accuracy (12-18c)
| [[augene]], [[blacksmith]], <br>[[porcupine]]
|  
| [[godzilla]], [[magic]], <br>[[meantone]], [[mothra]], <br>[[orwell]], [[pajara]], <br>[[shrutar]], [[whitewood]]
|  
| [[diaschismic]], [[garibaldi]], <br>[[hemiwuerschmidt]], <br>[[miracle]], [[valentine]]
| [[pajara]]
| [[ennealimmal]], [[harry]]
|  
| [[enneadecal]], [[trinity]]
|  
|-
|-
! 11-limit (2.3.5.7.11)
! Medium accuracy (7-12c)
| (!)
|  
| [[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]]
|  
| [[enneadecal]], [[trinity]]
|  
|-
|-
! 13-limit
! High accuracy (3.5-7c)
| (!)
|  
| [[augene]], [[blacksmith]], <br>[[pajara]], [[porcupine]], <br>[[whitewood]]
|  
| [[diaschismic]], [[hemiwuerschmidt]], <br>[[magic]], [[meantone]], <br>[[mothra]], [[nautilus]], <br>[[orwell]], [[shrutar]], <br>[[superpyth]]
| [[diaschismic]]
| [[ennealimmal]], [[harry]]
|  
| [[enneadecal]], [[trinity]]
|  
|-
|-
! 17-limit
! Very high accuracy (1-3.5c)
| (!)
|  
| (!)
|  
| [[diaschismic]], [[echidna]], <br>[[pajara]], [[shrutar]]
|  
| [[ennealimmal]], [[harry]]
|  
| [[trinity]]
|  
|-
|-
! Higher limits
! Microtemperament (0-1c)
| (!)
|  
| (!)
|  
| [[shrutar]]
|  
| [[ennealimmal]]
| [[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)
|
|
|
|
|
|-
|-
! 2.3.5.7.n
! Low accuracy (12-18c)
| (!)
|
|  
|  
|  
| [[unicorn]]
|  
|  
|  
|  
|-
|-
! 2.3.5.11
! Medium accuracy (7-12c)
|
|
|  
|  
| [[mohaha]], [[sensible]]
|  
|  
| [[larry]] (2.3.5.11 [[gravity]])
|  
|  
|-
|-
! 2.3.5.11.n
! High accuracy (3.5-7c)
| (!)
|  
| [[sensible]]
|  
|  
|  
|  
|  
|  
|  
|-
|-
! 2.3.5.n
! Very high accuracy (1-3.5c)
|
|
|  
|  
| [[cata]], [[nestoria]], <br>[[sensipent]], [[srutal archagall]]
| [[wuerschmidt]] (2.3.5.23)
|  
|  
|  
|  
|-
|-
! 2.3.7
! Microtemperament (0-1c)
| [[semaphore]]
| [[bleu]], [[slendric]]
|  
|  
|  
|  
|  
|  
| [[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
|-
|-
! 2.3.7.11
! Exotemperament (18-∞c)
|
|  
|  
| [[bleu]]
|  
|  
|  
|  
|  
|  
|-
|-
! 2.3.7.11.n
! Low accuracy (12-18c)
| (!)
|  
| [[bleu]]
|  
|  
|  
|  
|  
|  
|  
|-
|-
! 2.3.7.n
! Medium accuracy (7-12c)
|  
|  
|  
|  
Line 197: Line 431:
|  
|  
|-
|-
! 2.3.11
! High accuracy (3.5-7c)
|
|
|
|  
|  
| [[neutral]] (2.3.11 [[rastmic]]), <br>2.3.11 [[pythrabian]]
| [[tribilo]] (2.3.11 [[nexus]])
|  
|  
| 2.3.11 [[frameshift]]
|-
|-
! 2.3.11.n
! Very high accuracy (1-3.5c)
|
|  
|
|  
|
|  
|
|  
|
|  
|-
|-
! 2.5.7
! Microtemperament (0-1c)
|
|
|
|
|
|}
 
== Other no-2s 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)
|
|  
|  
| [[didacus]]
|  
|  
|  
|  
|  
|  
|-
|-
! 2.5.7.11
! Medium accuracy (7-12c)
|
|  
|  
| [[didacus]]
|  
|  
|  
|  
|  
|  
|-
|-
! 2.5.7.n
! High accuracy (3.5-7c)
|  
|  
|  
|  
Line 232: Line 492:
|  
|  
|-
|-
! 2.5.11.n
! Very high accuracy (1-3.5c)
|  
|  
|  
|  
Line 239: Line 499:
|  
|  
|-
|-
!2.7.11
! Microtemperament (0-1c)
|
|
| [[orgone]]
|
|
|
|
|  
|
|  
|}
 
== No-3s subgroups ==
 
{| class="wikitable"
|+
!
! Approx. 10 notes
! Approx. 20 notes
! Approx. 30 notes
! Approx. 70 notes
! Over 100 notes
|-
|-
! 3.5.7
! Exotemperament (18-∞c)
| [[arcturus]], [[BPS]], <br>[[canopus]], [[sirius]]
|
|  
|  
|  
|  
|  
|  
|  
|
|-
|-
! 3.5.n
! Low accuracy (12-18c)
|
|  
|  
|  
|  
|  
|  
|  
|  
|
|-
|-
! 3.5.7.n
! Medium accuracy (7-12c)
|  
|  
| [[dubhe]] (3.5.7.17)
| [[didacus]], [[orgone]]
|  
|  
|  
|  
|  
|  
|-
|-
! 3.5.7.11.n
! High accuracy (3.5-7c)
| (!)
|
| [[mintra]] (11-limit)
|  
| [[mintra]] (13-limit)
|  
|  
|  
|  
|
|-
|-
! 3.7.n
! Very high accuracy (1-3.5c)
| [[mintaka]] (3.7.11)
|
|  
|  
|  
|  
|  
|  
|  
|
|-
|-
! Other subgroups
! Microtemperament (0-1c)
|  
|  
|  
|  
Line 289: Line 561:
|}
|}


(!) = ''It is unlikely that a useable temperament could exist here because there are too many important intervals to approximate with this number of notes.''
== 2.3.7 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)
| [[semaphore]]
|
|
|
|
|-
! Medium accuracy (7-12c)
|
| [[bleu]], [[slendric]]
|
|
|
|-
! High accuracy (3.5-7c)
|
|
|
|
|
|-
! Very high accuracy (1-3.5c)
|
|
|
|
|
|-
! Microtemperament (0-1c)
|
|
|
|
|
|}


'''Additional information'''
== 2.3.11 and its extensions ==


Do note that this table doesn’t capture ''all'' of the relationships and commonalities between temperaments. This table ''does'' show when two temperaments share a JI subgroup, which is important information. But another important piece of information this table ''doesn’t'' capture is whether two temperaments share a [[pergen]].
{| 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]]
|}


In short, sometimes, multiple higher limit temperaments are actually different ways of extending the same lower-limit temperament. In this case, they will share a pergen. And this means they will have an overall similar flavor and some musical and mathematical properties in common.
== 2.3.13, 2.3.17, etc ==


If you visit the temperaments’ individual pages, those will usually make their relationships to other temperaments more clear.
{| 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 you should use my favorite temperament (editor opinions) ==
== No-7s 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)
|
| [[mohaha]]
|
|
|
|-
! High accuracy (3.5-7c)
|
| [[sensible]], [[srutal archagall]]
|
|
|
|-
! Very high accuracy (1-3.5c)
|
| [[cata]], [[nestoria]], [[sensipent]]
|
|
|
|-
! Microtemperament (0-1c)
|
|
|
|
|
|}


One editor is allowed to add more than one temperament if they like, just not too many though.
== No-11s 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)
|
|
|
|
|
|}


== I want a simpler, more straightforward overview ==
== Other subgroups ==
For a less complicated list of useful temperaments, see the following pages:
* [[Middle Path table of five-limit rank two temperaments]]
* [[Middle Path table of seven-limit rank two temperaments]]
* [[Middle Path table of eleven-limit rank two temperaments]]


For a description of what the temperaments on the above pages are like, and how they were chosen, read Paul Erlich’s ''Middle Path'' essay:
{| class="wikitable"
* ''[[A Middle Path]]''
|+
!
! 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)
|
|
|
|
|
|}

Latest revision as of 15:28, 16 April 2025

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.


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.

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.

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

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 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.


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 hanson, magic valentine
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, 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

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

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, mothra
High accuracy (3.5-7c) diaschismic, orwell
Very high accuracy (1-3.5c)
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

3.5.7 and its extensions

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

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

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

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

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

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

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

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

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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