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Survey of efficient temperaments by subgroup: Difference between revisions

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→Table of temperaments (5 to 45 notes per equave): 2.3.5.11: por*cu*pine, flattone
 
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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 in the “editor opinions” section. ((This is 100% optional, you can still add temperaments to the table without doing this :) ))<br><br><br>''For the deprecated, archived version of this page see [[User:BudjarnLambeth/Bird’s eye view of rank-2 temperaments]].''}}
This page highlights those [[rank-2 temperament]]s which receive the most discussion among theorists and composers.
 
 
This page highlights those [[rank-2 temperament]]s which get talked about the most among theorists and composers.
 
Composers and theorists disagree about which of these temperaments matter most, but each of these temperaments is valued by at least some sizeable subset of the xenharmonic community.
 
== So, which temperaments should I use to make music? ==


Composers and theorists disagree about which of these temperaments matter most, but all of the temperaments on this page are valued by at least a fair subset of the xenharmonic community.


== Which temperaments should I use to make music? ==
There are many different schools of thought within RTT (regular temperament theory).
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.
Most would agree that a good temperament is ''efficient'', meaning it 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.
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.
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'''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 ~15 [[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.
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'''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 ~5 cents on almost all 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.
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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.
These are not the only possible stances, either: One could imagine a Xenharmonicist C, Xenharmonicist D, etc. Thousands of differing individual perspectives on what traits are important in a temperament.
 
And these are not the only possible stances, either: There is a Xenharmonicist C, Xenharmonicist D, etc. Thousands of differing individual perspectives on what traits see important in a temperament.


To gain more of a grasp on these debates, it may 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).  
To gain more of a grasp on these debates, it may 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.
The most common theoretical approach to 12edo is to treat it as a 2.3.5 subgroup temperament, with similar accuracy to '''augmented'''.


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.
The second most common approach is to interpret 12edo as a 2.3.17.19 subgroup temperament, with similar accuracy to '''semitonic'''. (Such a temperament would go in the “''2.3.other n''” row of the below tables).


So that should provide a helpful point of comparison to measure these other temperaments against.
So that should provide a helpful point of comparison to measure these other temperaments against.


== How to read the table ==
== How to read the tables ==
 
=== Rows ===
'''Rows'''
 
'''The rows categorise temperaments by the [[just intonation subgroup]] they approximate.'''
'''The rows categorise temperaments by the [[just intonation subgroup]] they approximate.'''


 
The 2.3.5 subgroup is what most theorists believe 12 tone equal temperament belongs to. If those theorists are correct, then 2.3.5 should encompass all the harmonies that are familiar to most Western listeners.
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 by xenharmonic composers, being not too complex and including lots of useful harmonies.
The 2.3.5.7 and 2.3.5.7.11 subgroups are the most commonly used by xenharmonic composers, being not too complex and including lots of useful harmonies.


Subgroups with no 2s, e.g. 3.5.7.11, are the biggest and most jarring break away from familiar harmony, may be a good or a bad thing.
Subgroups with no 2s, e.g. 3.5.7.11, are the most jarring break away from familiar harmony, which one may consider a good or a bad thing.
 
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 there.
 
 
You may see the same temperament multiple times on the table. Here’s why:


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.
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<ref group="note">According to the 2.3.5 reading of common practice harmony. Alternate readings are possible.</ref>, 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, perhaps subtly and subconsciously in some instances, but still there.


'''Columns'''
The same temperament may occur multiple times on a table if it is good at approximating multiple different subgroups. For example, magic is good at approximating both the 7-limit and the 11-limit, so it is listed under both.


=== Columns ===
'''The columns categorise temperaments by the approximate number of notes-per-[[equave]] needed to reach all the temperament’s important intervals'''.
'''The columns categorise temperaments by the approximate number of notes-per-[[equave]] needed to reach all the temperament’s important intervals'''.


All of the temperaments listed in these tables 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.


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.
That means that for temperaments ''in these tables'', the more notes they require, the more accurate they are. The ones requiring less notes are less accurate, though they are good for their size. (Note that this rule is only true for ''the temperaments in these tables'', it is not true of all temperaments ''in general''.)


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.)
== Table of temperaments (5 to 45 notes per equave) ==
The temperaments within each cell should be sorted by accuracy, with the lowest [[damage]] (highest accuracy) temperament listed first.  


== Table of temperaments some decent number of people would recommend ==
The temperaments within each cell should be sorted by accuracy, with the lowest [[damage]] temperament listed first.


''Editors: If you see any temperaments listed in the wrong order, or see any temperaments in the wrong ‘number of notes recommended’ category, please move them to the correct position.''
<!--
If you see any temperaments listed in the wrong order, or see any temperaments in the wrong ‘approx. number of notes needed’ category, please move them to the correct position.
If you know of a temperament that is recommended by a sizeable subset of the xen community but is not yet included here, please add it.


Please do not add temperaments just for the sake of filling empty cells on the table. It’s okay for some cells to be empty. Only add temperaments if yourself, or at least a few other people, would recommend those temperaments.
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.
-->
{| class="wikitable center-all"
{| class="wikitable center-all"
|+
|-
! JI subgroup
! JI subgroup
! 5 to 15 notes
! ~10 notes per equave<ref group="note">Number of notes per equave was estimated by multiplying the temperament’s [[graham complexity]] by 2.</ref>
! 15 to 25 notes
! ~20 notes
! 25 to 50 notes
! ~30 notes
! 50 to 100 notes
! ~40 notes
! >100 notes
|-
! 5-limit <br>(2.3.5)
| [[hanson]], [[misty]], [[magic]], [[meantone]], [[negri]], [[augmented]], [[porcupine]], [[dimipent]], [[whitewood]], [[blackwood]], [[mavila]]
| [[Helmholtz (temperament)|Helmholtz]], [[orson]], [[wuerschmidt]], [[sensipent]], [[compton]], [[valentine]], [[diaschismic]], [[tetracot]], [[passion]], [[superpyth]], [[ripple]]
| [[kwazy]], [[luna]], [[vishnu]], [[parakleismic]], [[escapade]], [[amity]], [[gravity]], [[rodan]]
| [[enneadecal]], [[gammic]], [[vulture]]
|-
|-
! 5-limit (2.3.5)
! 7-limit <br>(2.3.5.7)
| [[diaschismic]], [[meantone]], [[augmented]], [[porcupine]], [[dimipent]], [[whitewood]], [[blackwood]]
| [[porcupine]], [[pajara]], [[keemun]], [[negri]], [[doublewide]], [[injera]], [[dominant]], [[august]], [[diminished]], [[blacksmith]]
| [[helmholtz]] (aka schismic), [[hanson]], [[magic]]
| [[orwell]], [[valentine]], [[myna]], [[magic]], [[meantone]], [[mothra]], [[superpyth]], [[flattone]], [[liese]], [[beatles]], [[augene]], [[hedgehog]], [[nautilus]], [[catler]], [[godzilla]],  [[lemba]]
| [[wuerschmidt]], [[valentine]]
| [[amity]], [[hemiwuerschmidt]], [[harry]], [[miracle]], [[garibaldi]], [[diaschismic]], [[sensi]]
| [[gravity]]
| [[misty]], [[unidec]], [[catakleismic]]
| [[kwazy]]
|-
|-
! 7-limit (2.3.5.7)
! 11-limit <br>(2.3.5.7.11)
| [[augene]], [[porcupine]], [[blacksmith]]
| [[triforce]], [[blacksmith]], [[pajaric]], [[negric]]
| [[orwell]], [[shrutar]], [[magic]], [[meantone]], [[mothra]], [[superpyth]], [[pajara]], [[godzilla]], [[whitewood]]
| [[orwell]], [[valentine]], [[mohajira]], [[porcupine]], [[hedgehog]], [[astrology]], [[vigintiduo]], [[augene]], [[nautilus]], [[catnip]], [[undevigintone]], [[injera]], [[keemun]], [[progress]], [[dominant]], [[meanenneadecal]], [[duodecim]]
| [[hemiwuerschmidt]], [[miracle]], [[garibaldi]], [[valentine]], [[diaschismic]]
| [[mothra]], [[nusecond]], [[meantone]], [[squares]], [[quasisupra]], [[pajara]], [[telepathy]], [[suprapyth]], [[negroni]], [[porky]], [[fleetwood]], [[pajarous]], [[sensis]], [[flattone]], [[godzilla]], [[darjeeling]]
| [[ennealimmal]], [[harry]]
| [[miracle]], [[shrutar]], [[magic]], [[meanpop]], [[migration]], [[andromeda]], [[superpyth]]
| [[enneadecal]], [[trinity]]
|-
|-
! 11-limit (2.3.5.7.11)
! 13-limit <br>(2.3.5.7.11.13)
| [[negric]]
| [[augene]], [[porcupine]], [[hedgehog]], [[triforce]], [[godzilla]], [[negri]], [[armodue]]
| [[nusecond]], [[modus]], [[lupercalia]], [[Meantone_family#Tridecimal_meantone|meantone]], [[winston]], [[pajara]], [[sensis]], [[ringo]], [[flattone]], [[darjeeling]], [[meanenneadecal]]
| [[miraculous]], [[leapday]], [[andromeda]], [[superkleismic]], [[mothra]], [[mohajira]], [[undevigintone]], [[ogene]], [[nautilus]], [[negroni]], [[injera]]
|-
! 17-limit <br>(2.3.5.7.11.13.17)
|  
|  
| [[pajara]], [[porcupine]], [[augene]], [[blacksmith]], [[whitewood]]
| [[lemba]]+, [[hedgehog]]+
| [[hemiwuerschmidt]], [[miracle]], [[cassandra]], [[diaschismic]], [[valentine]], [[shrutar]], [[orwell]], [[magic]], [[meanpop]], [[andromeda]], [[mothra]], [[meantone]], [[superpyth]], [[nautilus]], [[godzilla]]
| [[nusecond]]+, [[crepuscular]], [[winston]]+, [[pajara]], [[negroni]]+, [[sensis]]+, [[ringo]]+, [[pajarous]], [[augene]]+
| [[ennealimmal]], [[harry]]
| [[miraculous]], [[lupercalia]]+, [[mohajira]], [[superpyth]]+, [[meantoid]], [[injera]], [[meanenneadecal]]
| [[enneadecal]], [[trinity]]
|-
|-
! 13-limit
! 19-limit <br>(2.3.5.7.11.13.17.19)
|  
|  
| [[pajara]], [[augene]], [[porcupine]], [[blacksmith]], [[whitewood]]
| [[niner]]++
| [[hemiwuerschmidt]], [[cassandra]], [[diaschismic]], [[orwell]], [[shrutar]], [[magic]], [[mothra]], [[meantone]], [[superpyth]], [[nautilus]]
| [[wilsec]], [[winston]]++, [[augene]]++, [[sensis]]++
| [[ennealimmal]], [[harry]]
| [[roman]]++, [[mohajira]], [[lupercalia]]++, [[superpyth]]++, [[negroni]]++, [[meanenneadecal]], [[ringo]]++, [[injera]], [[meantoid]]
| [[enneadecal]], [[trinity]]
|-
|-
! 17-limit
! Higher prime limits
|  
|  
|  
|  
| [[diaschismic]], [[echidna]], [[shrutar]], [[pajara]]
|
| [[ennealimmal]], [[harry]]
| [[lupercalia]]+++, [[nautilus]]+++, [[negroni]]+++, [[injera]]+
| [[trinity]]
|-
|-
! Higher limits
! 2.3.5.7.other ''n''
| [[negra]]
| no-11 [[godzilla]], no-11 [[pajara]], no-11 [[duodecim]]
| no-11 [[magic]], no-11 [[sensis]], no-11 [[meanpop]]
| no-11 [[catakleismic]]
|-
! 2.3.5.11 <br>and its extensions
| [[porcupine]], [[mavila]], [[dicot]], [[flattone]]
| [[orson]]+, [[tetracot]], [[mohaha]]
| [[larry]] (no-7 [[gravity]]), [[countdown]], no-7 [[catalan]]
| [[escapade]]
|-
! 2.3.5.other ''n''
| [[srutal archagall]], [[stutzel]]
| [[sensipent]], [[nestoria]]
| [[wuerschmidt]]<ref group="note">Subgroup 2.3.5.23 version.</ref>, [[cata]]
|  
|  
|-
! 2.3.7
| [[slendric]], [[archy]], [[bleu]], [[semaphore]]
| [[stearnsmic clan|stearnsmic]], [[skwares]]
| [[hemif]], [[leapfrog]]
|  
|  
| [[shrutar]]
| [[ennealimmal]]
| [[trinity]]
|-
|-
! 2.3.5.7.n
! 2.3.7.11 <br>and its extensions
| [[bleu]], [[supra]], [[semaphore]]+
| [[skwares]], [[stearnsmic clan|stearnsmic]]+, [[suhajira]]
| no-5 [[miracle]], [[radon]]^, [[hemif]]^<ref group="note">^Hemif beats radon in 2.3.7.11, radon beats hemif in 2.3.7.11.13 (in damage not badness).</ref>, [[leapfrog]]
|  
|  
|-
! 2.3.7.other ''n''
| [[baladic]], [[oceanfront]], no-5 [[negra]]
| no-5 no-11 [[liese]], no-11 [[skwares]]
|  
|  
| [[unicorn]]
| [[slendric]]+
|-
! 2.3.11 <br>and its extensions
| [[neutral (temperament)|neutral]] (2.3.11 [[rastmic]]), [[namo]], [[paralimmal]], [[io]]
| [[tribilo]] (2.3.11 [[nexus]]), [[huxley]]
|  
|  
|  
|  
|-
|-
! 2.3.5.11
! 2.3.other ''n''
| [[barbados]], [[boethian]], [[semitonic]], [[superflat]], [[hydrothermal]]
| [[threedic]], [[pepperoni]]
| [[Subgroup temperaments #Historical|historical]]
|
|-
! 2.5.7 <br>and its extensions
| [[didacus]], [[frostburn]], no-3 [[oodako]]
| [[rainy]], [[mercy]], [[huntington]], [[llywelyn]], [[baldy]]
| [[roulette]]
|  
|  
| [[sensible]], [[mohaha]]
|-
! 2.5.other ''n''
| [[insect]], [[sulis]], [[wizz]], [[vengeance]], [[marveltri]], [[superquintal]], [[movila]]
| no-3 no-7 [[emka]], [[wizz]]+
|  
|  
| [[larry]] (2.3.5.11 [[gravity]])
|  
|  
|-
|-
! 2.3.5.11.n
! 2.7 <br>and its extensions
| [[orgone]], [[shipwreck]], [[stacks]], [[ultrakleismic]], [[machine]]
| [[counterultrakleismic]], [[mechanism]], [[mabon]]
|  
|  
| [[sensible]]
| [[machine|apparatus]]
|-
! 3.5.7 <br>and its extensions
| [[canopus]], [[BPS]], [[sirius]], [[arcturus]], [[dubhe]], [[Catalog of 3.5.7 subgroup rank two temperaments|vega]]
| [[izar]], [[mintra]]
| [[alhena]], [[remus]]
| [[erigone]]
|-
! 3.5.other ''n''
| [[aldebaran]], [[deneb]], [[polaris]]
|  
|  
|  
|  
| [[alnilam]], [[fomalhaut]]
|-
! 3.7 <br>and its extensions
| [[mintaka]] (no-13), [[keladic]]
| [[mebsuta]], [[minalzidar]]
| [[mintaka]] (with 13)
|  
|  
|-
|-
! 2.3.5.n
! 4.''n'' <br>and its extensions
| [[meanquad]], [[tetrahanson]], [[tetrameantone]], [[quarchy]], [[tetrominant]]
|
| [[fourwar]]
|
|-
! 5.''n'' <br>and its extensions
| [[antipyth]], [[juggernaut]]
|  
|  
| [[nestoria]], [[cata]], [[sensipent]], [[srutal archagall]]
| [[wuerschmidt]] (2.3.5.23)
|  
|  
|  
|  
|-
|-
! 2.3.7
! Other subgroups
| [[archy]], [[semaphore]]
| [[greeley]], [[halftone]], [[semiwolf]], [[auk]]
| [[slendric]], [[bleu]]
|
|
|  
|  
|}
== Table of temperaments (more notes per equave) ==
{| class="wikitable mw-collapsible mw-collapsed center-all mw-collapsed”"
|+
! JI subgroup
! ~50 notes
! ~60 notes
! ~70 notes
! ~80 notes
! >90 notes
|-
! 5-limit <br>(2.3.5)
| [[ennealimmal]], [[quintosec]], [[counterhanson]], [[undim]]
| [[alphatricot]], [[quintile]]
| [[minortone]], [[vavoom]]
|  
|  
|  
|  
|-
|-
! 2.3.7.11
! 7-limit <br>(2.3.5.7)
| [[ennealimmal]], [[tertiaseptal]], [[hemififths]], [[quadritikleismic]], [[grendel]], [[unidec]]
| [[hendecatonic]]
| [[sesquiquartififths]], [[quinmite]], [[parakleismic]]
| [[neptune]], [[gamera]], [[nessafof]], [[octoid]], [[septiquarter]]
| [[supermajor]], [[enneadecal]], [[term]]
|-
! 11-limit <br>(2.3.5.7.11)
| [[unidec]], [[tritikleismic]], [[hemithirds]], [[wizard]], [[diaschismic]]
|  
|  
| [[bleu]]
| [[quadritikleismic]], [[harry]]
| [[quasiorwell]], [[octoid]], [[sqrtphi]], [[hemiwuerschmidt]], [[ennealimnic]], [[catakleismic]]
| [[hemienneadecal]], [[hemiennealimmal]], [[abigail]], [[hemitert]], [[newt]], [[decoid]], [[vishnu]]
|-
! 13-limit <br>(2.3.5.7.11.13)
| [[hemififths]], [[orwell]], [[magic]]
| [[diaschismic]], [[myna]], [[sensus]], [[meanpop]]
| [[cassandra]], [[grosstone]]
| [[widefourth]], [[octopus]], [[mystery]], [[buzzard]], [[catakleismic]], [[rodan]], [[shrutar]]
| [[abigail]], [[satin]], [[trinity]], [[newt]], [[acyuta]], [[deca]], [[quasiorwell]], [[decoid]], [[vulture]], [[hemiennealimmal]], [[emkay]], [[countercata]]
|-
! 17-limit <br>(2.3.5.7.11.13.17)
| [[leapday]], [[superkleismic]]+, [[meantonic]], [[huygens]]
| [[hendec]], [[marvolo]], [[diaschismic]], [[sensus]], [[andromeda]], [[modus]]
| [[comptone]]
| [[heinz]], [[octopus]], [[lizard]], [[rodan]], [[echidna]], [[shrutar]]
| [[satin]], [[trinity]], [[octoid]], [[quincy]], [[mirkat]], [[ekadash]]+, [[quadritikleismic]], [[neominor]]+, [[ennealimnic]], [[sqrtphi]]
|-
! 19-limit <br>(2.3.5.7.11.13.17.19)
| [[octacot]], [[andromeda]], [[meantonic]], [[huygens]]
| [[hendec]]+, [[sensus]]+, [[crepuscular]]+, [[hitchcock]], [[modus]]
| [[marvolo]], [[miraculous]]+
| [[octopus]], [[ennealim]], [[bikleismic]], [[srutal]], [[valentino]]+
| [[newt|neonewt]]+, [[deca]]+, [[vulture]], [[satin]], [[stearnscape]]+, [[hemiennealimmal]], [[trinity]], [[quincy]], [[octoid]], [[sqrtphi]]
|-
! Higher prime limits
| [[winston]]+++, [[porky]]+++
|  
|  
|  
|  
| [[srutaloo]], [[shrutar]]
| [[satin]], [[trinity]], [[sqrtphi]]+, [[gizzard]]+, [[ketchup]], [[semisept]], [[cassandric]]
|-
! 2.3.5.7.other ''n''
| no-11 [[harry]], [[unicorn]]
| no-11 [[buzzard]], no-11 [[orwell]]
| no-11 [[cassandra]]
| no-11 [[hemischis]]
| no-11 [[ennealimmal]], no-11 [[decoid]], no-11 [[ennealimnic]], no-11 [[quadritikleismic]]
|-
! 2.3.5.11 <br>and its extensions
| no-7 [[quintosec]], [[quintile]]+, no-7 [[maquila]], [[sensible]], [[ampersand]]+
| [[twentcufo]], no-7 [[emka]], no-7 [[gwazy]], no-7 [[rodan]], no-7 [[cataclysmic]], no-7 [[sfourth]]
|  
|  
| [[majvam]]+
| no-7 [[hemienneadecal]], no-7 [[vulture]]
|-
|-
! 2.3.7.11.n
! 2.3.5.other ''n''
|  
|  
| [[bleu]]
|  
|  
|  
|  
| [[majvam]]
|  
|  
|-
|-
! 2.3.7.n
! 2.3.7
|
|
|
|
|  
|  
|-
! 2.3.7.11 <br>and its extensions
| no-5 [[quanic]]
|  
|  
|  
|  
Line 193: Line 324:
|  
|  
|-
|-
! 2.3.11
! 2.3.7.other ''n''
| [[hypnosis]]
|
|
|  
|  
| 2.3.11 [[pythrabian]], [[neutral]] (2.3.11 [[rastmic]])
| [[tribilo]] (2.3.11 [[nexus]])
|  
|  
| 2.3.11 [[frameshift]]
|-
|-
! 2.3.11.n
! 2.3.11 <br>and its extensions
|
| rank-2 [[pythrabian]]
| [[namo]]
| rank-2 [[frameshift]]
|
|  
|
|  
|
|  
|-
|-
! 2.5.7
! 2.3.other ''n''
|
|  
|  
| [[didacus]]
|  
|  
|  
|  
|  
|  
|-
|-
! 2.5.7.11
! 2.5.7 <br>and its extensions
|  
|  
| [[didacus]]
|  
|  
|  
|  
|  
|  
| [[Subgroup temperaments #Daemotertiaschis|daemotertiaschis]]
|-
|-
! 2.5.7.n
! 2.5.other ''n''
|  
|  
|  
|  
Line 228: Line 359:
|  
|  
|-
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! 2.5.11.n
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== Pergens and temperament relationships ==


'''Additional information'''
One important piece of information these tables do not capture is whether two temperaments share a [[pergen]].


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]].
Sometimes, multiple higher limit temperaments are actually different ways of extending the same lower-limit temperament. In this case, they will share a pergen. This means they will have an overall similar flavor and some musical and mathematical properties in common.
 
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.


If you visit the temperaments’ individual pages, those will usually make their relationships to other temperaments more clear.
If you visit the temperaments’ individual pages, those will usually make their relationships to other temperaments more clear.


Schismic/helmholtz/garibaldi/nestoria/andromeda/cassandra, and kleismic/hanson/cata are two prominent examples of temperaments on this table sharing a pergen. There are other examples on the table also.
Schismic/helmholtz/garibaldi/nestoria/andromeda/cassandra, and kleismic/hanson/cata are two prominent examples of temperaments on these tables sharing a pergen. There are other examples on the tables also.


== Most linked-to rank-2 temperaments ==
These were the top 100 rank-2 temperament pages with the most incoming links on the wiki on 27 Oct 2024. (When this section was written.)


'''Note to editors'''
# '''[[Meantone]] (313 links)'''
# '''[[Porcupine]] (144)'''
# '''[[Superpyth]] (108)'''
# '''[[Magic]] (107)'''
# '''[[Mavila]] (97)'''
# '''[[Orwell]] (81)'''
# '''[[Miracle]] (78)'''
# '''[[Pajara]] (76)'''
# '''[[Sensi]] (71)'''
# '''[[Flattone]] (64)'''
# '''[[Amity]] (59)'''
# '''[[Mohajira]] (59)'''
# '''[[Negri]] (59)'''
# '''[[Blackwood]] (58)'''
# '''[[Tetracot]] (56)'''
# '''[[Valentine]] (53)'''
# '''[[Wuerschmidt]] (53)'''
# '''[[Slendric]] (52)'''
# '''[[Compton]] (51)'''
# [[Ennealimmal]] (50)
# [[Helmholtz (temperament)|Helmholtz]] (49)
# [[Dicot]] (47)
# [[Garibaldi]] (47)
# [[Hanson]] (45)
# [[Catakleismic]] (44)
# [[Diaschismic]] (43)
# [[Hemififths]] (42)
# [[Myna]] (41)
# [[Father]] (40)
# [[Squares]] (40)
# [[Rodan]] (39)
# [[Semaphore]] (39)
# [[Augmented]] (38)
# [[Diminished]] (38)
# [[Srutal]] (38)
# [[Godzilla]] (37)
# [[Harry]] (37)
# [[Injera]] (37)
# [[Diasem]] (36)
# [[Enneadecal]] (35)
# [[Orgone]] (34)
# [[Parakleismic]] (34)
# [[Hedgehog]] (33)
# [[Luna]] (33)
# [[Octacot]] (33)
# [[Augene]] (32)
# [[Dominant]] (32)
# [[Hemithirds]] (32)
# [[Keemun]] (32)
# [[Lemba]] (32)
# [[Mothra]] (32)
# [[Whitewood]] (32)
# [[Archy]] (31)
# [[Liese]] (31)
# [[Bleu]] (29)
# [[Vishnu]] (29)
# [[Hemiwuerschmidt]] (28)
# [[Superkleismic]] (27)
# [[Echidna]] (26)
# [[Orson]] (26)
# [[Tertiaseptal]] (26)
# [[Triforce]] (26)
# [[Passion]] (25)
# [[Tritonic]] (25)
# [[Unidec]] (25)
# [[Wizard]] (25)
# [[Buzzard]] (24)
# [[Cassandra]] (24)
# [[Ripple]] (24)
# [[Vulture]] (24)
# [[Armodue]] (23) (''disambiguation page'')
# [[Atomic]] (23)
# [[Bug]] (23)
# [[Escapade]] (23)
# [[Pontiac]] (23)
# [[Ampersand]] (22)
# [[Bohpier]] (22)
# [[Mohaha]] (22)
# [[Parapyth]] (22)
# [[August]] (21)
# [[Blacksmith]] (21)
# [[Kwazy]] (21)
# [[Octoid]] (21)
# [[Tritikleismic]] (21)
# [[Kleismic]] (20)
# [[Misty]] (20)
# [[Schismatic]] (20) (''already listed as “Helmholtz”'')
# [[Shrutar]] (20)
# [[Sqrtphi]] (20)
# [[Beatles]] (19)
# [[Didacus]] (19)
# [[Meanpop]] (19)
# [[Arcturus]] (18)
# [[Gorgo]] (18)
# [[Guiron]] (18)
# [[Leapday]] (18)
# [[Mitonic]] (18)
# [[Nautilus]] (18)
# [[Sensipent]] (18)


Please do not add temperaments just for the sake of filling empty cells on the table. It’s okay for some cells to be empty.
== A simpler overview ==
 
For a more streamlined, strictly curated list of useful temperaments, see the following pages:
Only add temperaments if yourself, or at least a few other people, would recommend those temperaments.
 
== Why you should use my favorite temperament (individual wiki editor opinions) ==
 
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'''.
 
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.
 
Sort the explanations in alphabetical order (e.g. meantone, orwell, valentine).
 
A single editor is allowed to add more than one temperament if they like. Multiple editors are also allowed to recommend the same temperament - it may be useful to readers to see multiple different editors’ perspectives on the same temperament to get a more full understanding of it.
 
 
'''Note for readers:''' The following section of the page is a '''gallery of individual personal opinions'''. It is only here '''to give you a sampling of some of the many views''' of composers and theorists about specific temperaments. It is all opinion, not fact, you are free to take or discard any of it.
 
 
;[[Orwell]] - ''recommended by [[User:BudjarnLambeth]] (2024)''
“I recommend orwell temperament because it has a good approximation of the full 11-limit, which includes all the JI intervals that I can easily hear and recognise, and it does so with a relatively small number of notes, about 22. (Though you can also go up to 31 notes to approximate the 11-limit even better.)”
 
 
(My entry in this section was just a placeholder to show how to format an entry. Once someone else adds an entry I will delete mine.)
 
== I want a simpler, more straightforward overview ==
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 five-limit rank two temperaments]]
* [[Middle Path table of seven-limit rank two temperaments]]
* [[Middle Path table of seven-limit rank two temperaments]]
* [[Middle Path table of eleven-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:
For a description of what the temperaments on the above pages are like, and how those temperaments were chosen, read Paul Erlich’s ''Middle Path'' essay:
* ''[[A Middle Path]]''
* ''[[A Middle Path]]''


== I want an overview with written descriptions of each temperament ==
== A more descriptive overview ==
* See [[User:Godtone/Bird's eye view of temperaments by accuracy]]
* See [[User:Godtone/Bird's eye view of temperaments by accuracy]] which includes written descriptions of the temperaments, and is more mathematically rigorous than this survey


== Advanced reading ==
== Advanced reading ==
* [[Tour of regular temperaments]]: a huge list of temperament families, many of which remain rarely-used and unexplored
* [https://x31eq.com/catalog2.html x31eq Catalog of Regular Temperaments]: an even huger list by [[Graham Breed]]
* [[Rank-3]] and [[rank-4]] temperaments: these are more complicated, rarely-used, types of temperaments
* [[Rank-3]] and [[rank-4]] temperaments: these are more complicated, rarely-used, types of temperaments
* [[Tour of regular temperaments]]: a huge list of temperament families, many of which remain rarely-used
* [[Equal-step tuning]]s (i.e. rank-1 temperaments)
* More lists of temperaments:
** [[Low harmonic entropy linear temperaments]]
** [[Map of rank-2 temperaments]]
** [[Ordered lists of ET rank two temperaments]]
** and everything else in [[:Category:Lists of temperaments]]
* A deprecated, archived in-development version of this page: [[User:BudjarnLambeth/Bird’s eye view of rank-2 temperaments]]
 
== Notes ==
<references group="note" />
 
[[Category:Lists of temperaments]]
Retrieved from "https://en.xen.wiki/w/Survey_of_efficient_temperaments_by_subgroup"