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Survey of efficient temperaments by subgroup: Difference between revisions

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== Which temperaments should I use to make music? ==
== Which temperaments should I use to make music? ==
 
There are many different schools of thought within regular temperament theory (RTT). 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.
There are many different schools of thought within RTT (regular temperament theory).
 
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.
 


For example:  
For example:  
* '''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. They might argue that nobody can hear the harmonic effect of [[prime harmonics]] higher than 11, and they might argue that there is no real reason to use [[subgroup]]s 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 ~5 cents on almost all intervals, anything further out than that sounds out of tune to them. They might argue that it is 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.


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.


'''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.
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. The most common theoretical approach to 12edo is to treat it as a 2.3.5-subgroup temperament, with similar accuracy to [[augmented (temperament)|augmented]]. 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.
 
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 [[subgroup]]s 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 ~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 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.
 
 
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.
 
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).  
 
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 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.


== How to read the tables ==
== 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. If those theorists are correct, then 2.3.5 should encompass all the harmonies that are familiar to most Western listeners.
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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 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*<ref>''*According to the 2.3.5 reading of common practice harmony. Alternate readings are possible.''</ref>, forcing innovation while still keeping some familiarity.
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.
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.
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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.
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 ===
 
'''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 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.
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== Table of temperaments (5 to 45 notes per equave) ==
== 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.  
The temperaments within each cell should be sorted by accuracy, with the lowest [[damage]] (highest accuracy) temperament listed first.  
<!--
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.
-->


(''Editors:''
=== Full prime limits ===
 
''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
! ~10 notes per equave
! ~10 notes per equave<ref group="note">Number of notes per equave was estimated by multiplying the temperament’s [[graham complexity]] by 2.</ref>
! ~20 notes
! ~20 notes
! ~30 notes
! ~30 notes
! ~40 notes
! ~40 notes
|-
|-
! 5-limit <br>(2 . 3 . 5)
! 5-limit <br>(2.3.5)
| [[hanson]], [[misty]], [[magic]], [[meantone]], [[negri]], [[augmented]], [[porcupine]], [[dimipent]], [[whitewood]], [[blackwood]], [[mavila]]
| [[hanson]], [[magic]], [[meantone]], [[negri]], [[augmented (temperament)|augmented]], [[porcupine]], [[diminished (temperament)|diminished]], [[whitewood]], [[blackwood]], [[mavila]]
| [[helmholtz]], [[orson]], [[wuerschmidt]], [[sensipent]], [[compton]], [[valentine]], [[diaschismic]], [[tetracot]], [[passion]], [[superpyth]], [[ripple]]
| [[helmholtz (temperament)|helmholtz]], [[orson]], [[würschmidt]], [[sensipent]], [[compton]], [[valentine]], [[diaschismic]], [[tetracot]], [[passion]], [[superpyth]], [[ripple]]
| [[kwazy]], [[luna]], [[vishnu]], [[parakleismic]], [[escapade]], [[amity]], [[gravity]], [[rodan]]
| [[kwazy]], [[luna]], [[vishnu]], [[parakleismic]], [[escapade]], [[amity]], [[misty]], [[gravity]], [[rodan]]
| [[enneadecal]], [[gammic]], [[vulture]]
| [[enneadecal]], [[gammic]], [[vulture]]
|-
|-
! 7-limit <br>(2 . 3 . 5 . 7)
! 7-limit <br>(2.3.5.7)
| [[porcupine]], [[pajara]], [[keemun]], [[negri]], [[doublewide]], [[injera]], [[dominant]], [[august]], [[diminished]], [[blacksmith]]
| [[porcupine]], [[pajara]], [[keemun]], [[negri]], [[doublewide]], [[injera]], [[dominant (temperament)|dominant]], [[august]], [[diminished (temperament)|diminished]], [[blackwood]]
| [[orwell]], [[valentine]], [[myna]], [[magic]], [[meantone]], [[mothra]], [[superpyth]], [[flattone]], [[liese]], [[beatles]], [[augene]], [[hedgehog]], [[nautilus]], [[catler]], [[godzilla]],  [[lemba]],
| [[orwell]], [[valentine]], [[myna]], [[magic]], [[meantone]], [[mothra]], [[superpyth]], [[flattone]], [[liese]], [[beatles]], [[augene]], [[hedgehog]], [[nautilus]], [[catler]], [[godzilla]],  [[lemba]]
| [[amity]], [[hemiwuerschmidt]], [[harry]], [[miracle]], [[garibaldi]], [[diaschismic]], [[sensi]]
| [[amity]], [[hemiwürschmidt]], [[harry]], [[miracle]], [[garibaldi]], [[diaschismic]], [[sensi]]
| [[misty]], [[unidec]], [[catakleismic]]
| [[misty]], [[unidec]], [[catakleismic]]
|-
|-
! 11-limit <br>(2 . 3 . 5 . 7 . 11)
! 11-limit <br>(2.3.5.7.11)
| [[triforce]], [[blacksmith]], [[pajaric]], [[negric]]
| [[triforce]], [[blackwood]], [[pajaric]], [[negric]]
| [[orwell]], [[valentine]], [[mohajira]], [[porcupine]], [[hedgehog]], [[astrology]], [[vigintiduo]], [[augene]], [[nautilus]], [[catnip]], [[undevigintone]], [[injera]], [[keemun]], [[progress]], [[dominant]], [[meanenneadecal]], [[duodecim]]
| [[orwell]], [[valentine]], [[mohajira]], [[porcupine]], [[hedgehog]], [[astrology]], [[vigintiduo]], [[augene]], [[nautilus]], [[catnip]], [[undevigintone]], [[injera]], [[keemun]], [[progress]], [[dominant (temperament)|dominant]], [[meanenneadecal]], [[duodecim]]
| [[mothra]], [[nusecond]], [[meantone]], [[squares]], [[quasisupra]], [[pajara]], [[telepathy]], [[suprapyth]], [[negroni]], [[porky]], [[fleetwood]], [[pajarous]], [[sensis]], [[flattone]], [[godzilla]], [[darjeeling]]
| [[mothra]], [[nusecond]], [[meantone]], [[squares]], [[quasisupra]], [[pajara]], [[telepathy]], [[suprapyth]], [[negroni]], [[porky]], [[fleetwood]], [[pajarous]], [[sensis]], [[flattone]], [[godzilla]], [[darjeeling]]
| [[miracle]], [[shrutar]], [[magic]], [[meanpop]], [[migration]], [[andromeda]], [[superpyth]]
| [[miracle]], [[shrutar]], [[magic]], [[meanpop]], [[migration]], [[andromeda]], [[superpyth]]
|-
|-
! 13-limit <br>(2 . 3 . 5 . 7 . 11 . 13)
! 13-limit <br>(2.3.5.7.11.13)
| [[negric]]
| [[negric]]
| [[augene]], [[porcupine]], [[hedgehog]], [[triforce]], [[godzilla]], [[negri]], [[armodue]]
| [[augene]], [[porcupine]], [[hedgehog]], [[triforce]], [[godzilla]], [[negri]], [[armodue (temperament)|armodue]]
| [[nusecond]], [[modus]], [[lupercalia]], [[Meantone_family#Tridecimal_meantone|meantone]], [[winston]], [[pajara]], [[sensis]], [[ringo]], [[flattone]], [[darjeeling]], [[meanenneadecal]]
| [[nusecond]], [[modus]], [[lupercalia]], [[fokkertone]], [[winston]], [[pajara]], [[sensis]], [[ringo]], [[flattone]], [[darjeeling]], [[meanenneadecal]]
| [[mirkat]], [[diaschismic]], [[miraculous]], [[andromeda]], [[superkleismic]], [[mothra]], [[mohajira]], [[undevigintone]], [[ogene]], [[nautilus]], [[negroni]], [[injera]]
| [[miraculous]], [[leapday]], [[andromeda]], [[superkleismic]], [[mothra]], [[mohajira]], [[undevigintone]], [[ogene]], [[nautilus]], [[negroni]], [[injera]]
|-
|-
! 17-limit <br>(2 . 3 . 5 . 7 . 11 . 13 . 17)
! 17-limit <br>(2.3.5.7.11.13.17)
|  
|  
| [[lemba]]+, [[hedgehog]]+
| [[lemba]]+, [[hedgehog]]+
| [[nusecond]]+, [[crepuscular]], [[winston]]+, [[pajara]], [[negroni]]+, [[sensis]]+, [[ringo]]+, [[pajarous]], [[augene]]+
| [[nusecond]]+, [[crepuscular]], [[winston]]+, [[pajara]], [[negroni]]+, [[sensis]]+, [[ringo]]+, [[pajarous]], [[augene]]+
| [[mirkat]], [[miraculous]], [[lupercalia]]+, [[mohajira]], [[superpyth]]+, [[meantoid]], [[injera]], [[meanenneadecal]]
| [[miraculous]], [[lupercalia]]+, [[mohajira]], [[superpyth]]+, [[fokkertone]]+, [[injera]], [[meanenneadecal]]
|-
|-
! 19-limit <br>(2 . 3 . 5 . 7 . 11 . 13 . 17 . 19)
! 19-limit <br>(2.3.5.7.11.13.17.19)
|  
|  
| [[niner]]++
| [[niner]]++
| [[wilsec]], [[winston]]++, [[augene]]++, [[sensis]]++
| [[wilsec]], [[winston]]++, [[augene]]++, [[sensis]]++
| [[roman]]++, [[mohajira]], [[lupercalia]]++, [[superpyth]]++, [[negroni]]++, [[meanenneadecal]], [[ringo]]++, [[injera]], [[meantoid]]
| [[roman]]++, [[mohajira]], [[lupercalia]]++, [[superpyth]]++, [[negroni]]++, [[meanenneadecal]], [[ringo]]++, [[injera]], [[fokkertone]]++
|-
|-
! Higher prime limits
! Higher prime limits
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|  
|  
| [[lupercalia]]+++, [[nautilus]]+++, [[negroni]]+++, [[injera]]+
| [[lupercalia]]+++, [[nautilus]]+++, [[negroni]]+++, [[injera]]+
|}
=== Other subgroups ===
Subgroups ''without'' a "2" ''don't have'' multiple of 2 intervals (eg the 2/1 octave, the 3/2 perfect fifth, the 5/4 major third).
Subgroups ''without'' a "3" ''don't have'' multiple of 3 intervals (eg the 3/1 perfect twelfth, the 3/2 perfect fifth, the 5/3 major sixth).
Subgroups ''without'' a "5" ''don't have'' multiple of 5 intervals (eg the 5/3 major sixth, the 5/4 major third, the 6/5 minor third).
Subgroups ''with'' a "7", "11", or "other ''n''" include new [[xenharmonic]] consonant intervals that can't be found in common [[12edo]] tuning.
{| class="wikitable center-all"
|-
|-
! 2 . 3 . 5 . 7 . other n
! JI subgroup
! ~10 notes per equave
! ~20 notes
! ~30 notes
! ~40 notes
|-
! 2.3.5.7.other ''n''
| [[negra]]
| [[negra]]
| no-11 [[godzilla]], no-11 [[pajara]], no-11 [[duodecim]]
| no-11 [[godzilla]], no-11 [[pajara]], no-11 [[duodecim]]
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| no-11 [[catakleismic]]
| no-11 [[catakleismic]]
|-
|-
! 2 . 3 . 5 . 11 <br>and its extensions
! 2.3.5.11 <br>and its extensions
| [[porkypine]], [[mavila]], [[dicot]]
| [[porcupine]], [[mavila]], [[dicot]], [[flattone]]
| [[orson]]+, [[tetracot]], [[mohaha]]
| [[orson]]+, [[tetracot]], [[mohaha]]
| [[larry]] (no-7 [[gravity]]), [[countdown]], no-7 [[catalan]]
| [[larry]] (no-7 [[gravity]]), [[countdown]], no-7 [[catalan]]
| [[escapade]]
| [[escapade]]
|-
|-
! 2 . 3 . 5 . other n
! 2.3.5.other ''n''
| [[srutal archagall]], [[stutzel]]
| [[srutal archagall]], [[stutzel]]
| [[sensipent]], [[nestoria]]
| [[sensipent]], [[nestoria]]
| [[wuerschmidt]] (2.3.5.23), [[cata]]
| [[würschmidt]]<ref group="note">Subgroup 2.3.5.23 version.</ref>, [[cata]]
|  
|  
|-
|-
! 2 . 3 . 7
! 2.3.7
| [[slendric]], [[archy]], [[bleu]], [[semaphore]]
| [[slendric]], [[archy]], [[bleu]], [[semaphore]]
| [[stearnsmic clan|stearnsmic]], [[skwares]]
| [[stearnsmic clan|stearnsmic]], [[skwares]]
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|  
|  
|-
|-
! 2 . 3 . 7 . 11 <br>and its extensions
! 2.3.7.11 <br>and its extensions
| [[bleu]], [[supra]], [[semaphore]]+
| [[bleu]], [[supra]], [[semaphore]]+
| [[skwares]], [[stearnsmic clan|stearnsmic]]+, [[suhajira]]
| [[skwares]], [[stearnsmic clan|stearnsmic]]+, [[suhajira]]
| no-5 [[miracle]], [[radon]]^, [[hemif]]^<ref>Hemif beats radon in 2.3.7.11, radon beats hemif in 2.3.7.11.13 (in damage not badness).</ref>, [[leapfrog]]
| 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
! 2.3.7.other ''n''
| [[baladic]], [[oceanfront]], no-5 [[negra]]
| [[baladic]], [[oceanfront]], no-5 [[negra]]
|  
| no-5 no-11 [[liese]], no-11 [[skwares]]
|  
|  
| [[slendric]]+
| [[slendric]]+
|-
|-
! 2 . 3 . 11 <br>and its extensions
! 2.3.11 <br>and its extensions
| [[Rastmic_clan#Neutral|neutral]] (2.3.11 [[rastmic]]), [[namo]], [[paralimmal]], [[io]]
| [[neutral (temperament)|neutral]] (2.3.11 [[rastmic]]), [[namo]], [[paralimmal]], [[io]]
| [[tribilo]] (2.3.11 [[nexus]]), [[huxley]]
| [[tribilo]] (2.3.11 [[nexus]]), [[huxley]]
|  
|  
|  
|  
|-
|-
! 2 . 3 . other n
! 2.3.other ''n''
| [[barbados]], [[boethian]], [[semitonic]], [[superflat]], [[hydrothermal]]
| [[barbados]], [[boethian]], [[semitonic]], [[superflat]], [[hydrothermal]]
| [[threedic]], [[pepperoni]]
| [[threedic]], [[pepperoni]]
| [[Subgroup_temperaments#Historical|historical]]
| [[Subgroup temperaments #Historical|historical]]
|  
|  
|-
|-
! 2 . 5 . 7 <br>and its extensions
! 2.5.7 <br>and its extensions
| [[didacus]], [[frostburn]], no-3 [[oodako]]
| [[didacus]], [[frostburn]], no-3 [[oodako]], [[augment]]
| [[rainy]], [[mercy]], [[llywelyn]], [[baldy]]
| [[rainy]], [[mercy]], [[huntington]]
|  
| [[llywelyn]], [[silver]], [[baldy]]
|
| [[roulette]]
|-
|-
! 2 . 5 . other n
! 2.5.other ''n''
| [[insect]], [[sulis]], [[wizz]], [[vengeance]], [[marveltri]], [[superquintal]], [[movila]]
| [[insect]], [[sulis]], [[wizz]], [[vengeance]], [[marveltri]], [[superquintal]], [[movila]]
|  
| no-3 no-7 [[emka]], [[wizz]]+
|  
|  
|  
|  
|-
|-
! 2 . 7 <br>and its extensions
! 2.7 <br>and its extensions
| [[orgone]], [[shipwreck]], [[stacks]], [[ultrakleismic]], [[machine]]
| [[orgone]], [[shipwreck]], [[stacks]], [[ultrakleismic]], [[machine]]
| [[counterultrakleismic]], [[mechanism]], [[mabon]]
| [[counterultrakleismic]], [[mechanism]], [[mabon]]
Line 204: Line 182:
| [[machine|apparatus]]
| [[machine|apparatus]]
|-
|-
! 3 . 5 . 7 <br>and its extensions
! 3.5.7 <br>and its extensions
| [[canopus]], [[BPS]], [[sirius]], [[arcturus]], [[dubhe]] (3.5.7.17), [[Catalog of 3.5.7 subgroup rank two temperaments|vega]]  
| [[canopus]], [[BPS]], [[sirius]], [[arcturus]], [[dubhe]], [[Catalog of 3.5.7 subgroup rank two temperaments|vega]]  
| [[izar]], [[mintra]] (3.5.7.11)
| [[izar]], [[mintra]]
| [[No-twos_subgroup_temperaments#Alhena|alhena]], [https://sintel.pythonanywhere.com/result?edos=39+%26+7&submit_edo=submit&subgroup=3.5.7.11.13 remus] (3.5.7.11.13)
| [[alhena]], [[remus]]
| [https://sintel.pythonanywhere.com/result?edos=39+%26+131&submit_edo=submit&subgroup=3.5.7.11.13 erigone] (3.5.7.11.13)
| [[erigone]]
|-
|-
! 3 . 5 . other n
! 3.5.other ''n''
| [[aldebaran]], [[deneb]], [[polaris]]
| [[aldebaran]], [[deneb]], [[polaris]]
|  
|  
Line 216: Line 194:
| [[alnilam]], [[fomalhaut]]
| [[alnilam]], [[fomalhaut]]
|-
|-
! 3 . 7 <br>and its extensions
! 3.7 <br>and its extensions
| [[mintaka]] (3.7.11), [[keladic]]
| [[mintaka]] (no-13), [[keladic]]
| [[mebsuta]], [[minalzidar]]
| [[mebsuta]], [[minalzidar]]
| [[mintaka]] (3.7.11.13)
| [[mintaka]] (with 13)
|  
|  
|-
|-
! 4 . n <br>and its extensions
! 4.''n'' <br>and its extensions
| [[meanquad]], [[tetrahanson]], [[tetrameantone]], [[quarchy]], [[tetrominant]]
| [[meanquad]], [[tetrahanson]], [[tetrameantone]], [[quarchy]], [[tetrominant]]
|  
|  
Line 228: Line 206:
|  
|  
|-
|-
! 5 . n <br>and its extensions
! 5.''n'' <br>and its extensions
| [[antipyth]], [[juggernaut]]
| [[antipyth]], [[juggernaut]]
|  
|  
Line 252: Line 230:
! >90 notes
! >90 notes
|-
|-
! 5-limit <br>(2 . 3 . 5)
! 5-limit <br>(2.3.5)
| [[ennealimmal]], [[qintosec]], [[counterhanson]], [[undim]]
| [[ennealimmal]], [[quintosec]], [[counterhanson]], [[undim]]
| [[tricot]], [[pental]]
| [[alphatricot]], [[quintile]]
| [[minortone]], [[vavoom]]
| [[minortone]], [[vavoom]]
|  
|  
|  
|  
|-
|-
! 7-limit <br>(2 . 3 . 5 . 7)
! 7-limit <br>(2.3.5.7)
| [[ennealimmal]], [[tertiaseptal]], [[hemififths]], [[quadritikleismic]], [[grendel]], [[unidec]]
| [[ennealimmal]], [[tertiaseptal]], [[hemififths]], [[quadritikleismic]], [[grendel]], [[unidec]]
| [[hendecatonic]]
| [[hendecatonic (temperament)|hendecatonic]]
| [[sesquiquartififths]], [[quinmite]], [[parakleismic]]
| [[sesquiquartififths]], [[quinmite]], [[parakleismic]]
| [[neptune]], [[gamera]], [[nessafof]], [[octoid]], [[septiquarter]]
| [[neptune]], [[gamera]], [[nessafof]], [[octoid]], [[septiquarter]]
| [[supermajor]], [[enneadecal]], [[term]]
| [[supermajor (temperament)|supermajor]], [[enneadecal]], [[term]]
|-
|-
! 11-limit <br>(2 . 3 . 5 . 7 . 11)
! 11-limit <br>(2.3.5.7.11)
| [[unidec]], [[tritikleismic]], [[hemithirds]], [[wizard]], [[diaschismic]]
| [[unidec]], [[tritikleismic]], [[hemithirds]], [[wizard]], [[diaschismic]]
|  
|  
| [[quadritikleismic]], [[harry]]
| [[quadritikleismic]], [[harry]]
| [[quasiorwell]], [[octoid]], [[sqrtphi]], [[hemiwuerschmidt]], [[ennealimnic]], [[catakleismic]]
| [[quasiorwell]], [[octoid]], [[sqrtphi]], [[hemiwürschmidt]], [[ennealimnic]], [[catakleismic]]
| [[hemienneadecal]], [[hemiennealimmal]], [[abigail]], [[hemitert]], [[newt]], [[decoid]], [[vishnu]]
| [[hemienneadecal]], [[hemiennealimmal]], [[abigail]], [[hemitert]], [[newt]], [[decoid]], [[vishnu]]
|-
|-
! 13-limit <br>(2 . 3 . 5 . 7 . 11 . 13)
! 13-limit <br>(2.3.5.7.11.13)
| [[hemififths]], [[orwell]], [[magic]]
| [[hemififths]], [[orwell]], [[magic]]
| [[mystery]], [[myna]], [[sensus]], [[meanpop]]
| [[diaschismic]], [[myna]], [[sensus]], [[meanpop]]
| [[cassandra]]
| [[cassandra]], [[grosstone]]
| [[widefourth]], [[octopus]], [[buzzard]], [[catakleismic]], [[rodan]], [[shrutar]]
| [[widefourth]], [[octopus]], [[mystery]], [[buzzard]], [[catakleismic]], [[rodan]], [[shrutar]]
| [[abigail]], [[satin]], [[trinity]], [[newt]], [[acyuta]], [[deca]], [[quasiorwell]], [[decoid]], [[vulture]], [[hemiennealimmal]], [[Horwell_temperaments#Emkay|emkay]], [[countercata]]
| [[abigail]], [[satin]], [[trinity]], [[newt]], [[acyuta]], [[deca]], [[quasiorwell]], [[decoid]], [[vulture]], [[hemiennealimmal]], [[emkay]], [[countercata]]
|-
|-
! 17-limit <br>(2 . 3 . 5 . 7 . 11 . 13 . 17)
! 17-limit <br>(2.3.5.7.11.13.17)
| [[leapday]], [[superkleismic]]+, [[meantonic]], [[huygens]]
| [[leapday]], [[superkleismic]]+, [[fokkertone]]+, [[fokkertone]]
| [[hendec]], [[marvolo]], [[diaschismic]], [[sensus]], [[andromeda]], [[modus]]
| [[hendec]], [[marvolo]], [[diaschismic]], [[sensus]], [[andromeda]], [[modus]]
| [[comptone]]
| [[comptone]]
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| [[satin]], [[trinity]], [[octoid]], [[quincy]], [[mirkat]], [[ekadash]]+, [[quadritikleismic]], [[neominor]]+, [[ennealimnic]], [[sqrtphi]]
| [[satin]], [[trinity]], [[octoid]], [[quincy]], [[mirkat]], [[ekadash]]+, [[quadritikleismic]], [[neominor]]+, [[ennealimnic]], [[sqrtphi]]
|-
|-
! 19-limit <br>(2 . 3 . 5 . 7 . 11 . 13 . 17 . 19)
! 19-limit <br>(2.3.5.7.11.13.17.19)
| [[octacot]], [[andromeda]], [[meantonic]], [[huygens]]
| [[octacot]], [[andromeda]], [[fokkertone]]+, [[fokkertone]]
| [[hendec]]+, [[sensus]]+, [[crepuscular]]+, [[hitchcock]], [[modus]]
| [[hendec]]+, [[sensus]]+, [[crepuscular]]+, [[hitchcock]], [[modus]]
| [[marvolo]], [[miraculous]]+
| [[marvolo]], [[miraculous]]+
| [[ennealim]], [[bikleismic]], [[srutal]], [[valentino]]+
| [[octopus]], [[ennealim]], [[bikleismic]], [[srutal]], [[valentino]]+
| [[newt|neonewt]]+, [[deca]]+, [[vulture]], [[satin]], [[stearnscape]]+, [[hemiennealimmal]], [[trinity]], [[quincy]], [[octoid]], [[sqrtphi]]
| [[newt|neonewt]]+, [[deca]]+, [[vulture]], [[satin]], [[stearnscape]]+, [[hemiennealimmal]], [[trinity]], [[quincy]], [[octoid]], [[sqrtphi]]
|-
|-
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| [[satin]], [[trinity]], [[sqrtphi]]+, [[gizzard]]+, [[ketchup]], [[semisept]], [[cassandric]]
| [[satin]], [[trinity]], [[sqrtphi]]+, [[gizzard]]+, [[ketchup]], [[semisept]], [[cassandric]]
|-
|-
! 2 . 3 . 5 . 7 . other n
! 2.3.5.7.other ''n''
| no-11 [[harry]], [[unicorn]]
| no-11 [[harry]], [[unicorn]]
| no-11 [[buzzard]], no-11 [[orwell]]
| no-11 [[buzzard]], no-11 [[orwell]]
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| no-11 [[ennealimmal]], no-11 [[decoid]], no-11 [[ennealimnic]], no-11 [[quadritikleismic]]
| no-11 [[ennealimmal]], no-11 [[decoid]], no-11 [[ennealimnic]], no-11 [[quadritikleismic]]
|-
|-
! 2 . 3 . 5 . 11 <br>and its extensions
! 2.3.5.11 <br>and its extensions
| no-7 [[qintosec]], [[pental]]+, no-7 [[maquila]], [[sensible]], [[ampersand]]+
| 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]]
| [[twentcufo]], no-7 [[emka]], no-7 [[gwazy]], no-7 [[rodan]], no-7 [[cataclysmic]], no-7 [[sfourth]]
|  
|  
|  
| [[majvam]]+
| no-7 [[vulture]]
| no-7 [[hemienneadecal]], no-7 [[vulture]]
|-
|-
! 2 . 3 . 5 . other n
! 2.3.5.other ''n''
|
|  
|  
|  
|  
|  
|  
| [[majvam]]
|  
|  
|-
|-
! 2 . 3 . 7
! 2.3.7
|  
|  
|  
|  
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|  
|  
|-
|-
! 2 . 3 . 7 . 11 <br>and its extensions
! 2.3.7.11 <br>and its extensions
| no-5 [[quanic]]
| no-5 [[quanic]]
|  
|  
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|  
|  
|-
|-
! 2 . 3 . 7 . other n
! 2.3.7.other ''n''
| [[hypnosis]]
| [[hypnosis]]
|  
|  
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|  
|  
|-
|-
! 2 . 3 . 11 <br>and its extensions
! 2.3.11 <br>and its extensions
| 2.3.11 [[pythrabian]]
| rank-2 [[pythrabian]]
| 2.3.11 [[frameshift]]
| rank-2 [[frameshift]]
|  
|  
|  
|  
|  
|  
|-
|-
! 2 . 3 . other n
! 2.3.other ''n''
|  
|  
|  
|  
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|  
|  
|-
|-
! 2 . 5 . 7 <br>and its extensions
! 2.5.7 <br>and its extensions
|  
|  
|  
|  
|  
|  
|  
|  
| [[Subgroup temperaments#Daemotertiaschis|daemotertiaschis]]
| [[Subgroup temperaments #Daemotertiaschis|daemotertiaschis]]
|-
|-
! 2 . 5 . other n
! 2.5.other ''n''
|  
|  
|  
|  
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|  
|  
|-
|-
! 2 . 7 <br>and its extensions
! 2.7 <br>and its extensions
|  
|  
|  
|  
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|  
|  
|-
|-
! 3 . 5 . 7 <br>and its extensions
! 3.5.7 <br>and its extensions
|  
|  
|  
|  
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|  
|  
|-
|-
! 3 . 5 . other n
! 3.5.other ''n''
|  
|  
|  
|  
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|  
|  
|-
|-
! 3 . 7 <br>and its extensions
! 3.7 <br>and its extensions
|  
|  
|  
|  
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|  
|  
|-
|-
! 4 . n <br>and its extensions
! 4.''n'' <br>and its extensions
|  
|  
|  
|  
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|  
|  
|-
|-
! 5 . n <br>and its extensions
! 5.''n'' <br>and its extensions
|  
|  
|  
|  
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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.
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.


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 these tables sharing a pergen. There are other examples on the tables 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 ==
== 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.)
These are the top 105 rank-2 temperament pages with the most incoming links on the wiki as of 23 Oct 2025, about one year after this section was first written.


# '''[[Meantone]] (313 links)'''
# '''[[Meantone]] (479)'''
# '''[[Porcupine]] (144)'''
# '''[[Porcupine]] (195)'''
# '''[[Superpyth]] (108)'''
# '''[[Superpyth]] (162)'''
# '''[[Magic]] (107)'''
# '''[[Magic]] (153)'''
# '''[[Mavila]] (97)'''
# '''[[Mavila]] (127)'''
# '''[[Orwell]] (81)'''
# '''[[Miracle]] (112)'''
# '''[[Miracle]] (78)'''
# '''[[Orwell]] (111)'''
# '''[[Pajara]] (76)'''
# '''[[Blackwood]] (95)'''
# '''[[Sensi]] (71)'''
# '''[[Flattone]] (95)'''
# '''[[Flattone]] (64)'''
# '''[[Sensi]] (92)'''
# '''[[Amity]] (59)'''
# '''[[Pajara]] (91)'''
# '''[[Mohajira]] (59)'''
# '''[[Slendric]] (89)'''
# '''[[Negri]] (59)'''
# '''[[Valentine]] (87)'''
# '''[[Blackwood]] (58)'''
# '''[[Würschmidt]] (86)'''
# '''[[Tetracot]] (56)'''
# '''[[Negri]] (84)'''
# '''[[Valentine]] (53)'''
# '''[[Tetracot]] (82)'''
# '''[[Wuerschmidt]] (53)'''
# '''[[Mohajira]] (79)'''
# '''[[Slendric]] (52)'''
# '''[[Compton]] (70)'''
# '''[[Compton]] (51)'''
# '''[[Amity]] (69)'''
# [[Ennealimmal]] (50)
# '''[[Diaschismic]] (69)'''
# [[Helmholtz]] (49)
# [[Garibaldi]] (68)
# [[Dicot]] (47)
# [[Dicot]] (64)
# [[Garibaldi]] (47)
# [[Ennealimmal]] (60)
# [[Hanson]] (45)
# [[Father]] (60)
# [[Catakleismic]] (44)
# [[Diminished (temperament)|Diminished]] (59)
# [[Diaschismic]] (43)
# [[Mothra]] (58)
# [[Hemififths]] (42)
# [[Myna]] (57)
# [[Myna]] (41)
# [[Rodan]] (57)
# [[Father]] (40)
# [[Catakleismic]] (57)
# [[Squares]] (40)
# [[Godzilla]] (56)
# [[Rodan]] (39)
# [[Hanson]] (56)
# [[Semaphore]] (39)
# [[Squares]] (55)
# [[Augmented]] (38)
# [[Hemififths]] (54)
# [[Diminished]] (38)
# [[Injera]] (53)
# [[Srutal]] (38)
# [[Octacot]] (53)
# [[Godzilla]] (37)
# [[Semaphore]] (52)
# [[Harry]] (37)
# [[Harry]] (52)
# [[Injera]] (37)
# [[Archy]] (50)
# [[Diasem]] (36)
# [[Augmented (temperament)|Augmented]] (50)
# [[Enneadecal]] (35)
# [[Helmholtz (temperament)|Helmholtz]] (48)
# [[Orgone]] (34)
# [[Superkleismic]] (46)
# [[Parakleismic]] (34)
# [[Augene]] (46)
# [[Hedgehog]] (33)
# [[Keemun]] (46)
# [[Luna]] (33)
# [[Whitewood]] (43)
# [[Octacot]] (33)
# [[Buzzard]] (43)
# [[Augene]] (32)
# [[Orgone]] (42)
# [[Dominant]] (32)
# [[Dominant (temperament)|Dominant]] (42)
# [[Hemithirds]] (32)
# [[Kleismic]] (42) (already listed as ''hanson'')
# [[Keemun]] (32)
# [[Liese]] (42)
# [[Lemba]] (32)
# [[Didacus]] (41)
# [[Mothra]] (32)
# [[Hemiwürschmidt]] (41)
# [[Whitewood]] (32)
# [[Parakleismic]] (41)
# [[Archy]] (31)
# [[Vishnu]] (40)
# [[Liese]] (31)
# [[Enneadecal]] (40)
# [[Bleu]] (29)
# [[Hemithirds]] (40)
# [[Vishnu]] (29)
# [[Lemba]] (40)
# [[Hemiwuerschmidt]] (28)
# [[Srutal]] (39)
# [[Superkleismic]] (27)
# [[Hedgehog]] (39)
# [[Echidna]] (26)
# [[Luna]] (38)
# [[Orson]] (26)
# [[Triforce]] (38)
# [[Tertiaseptal]] (26)
# [[Echidna]] (37)
# [[Triforce]] (26)
# [[Ripple]] (36)
# [[Passion]] (25)
# [[Tritonic]] (36)
# [[Tritonic]] (25)
# [[Escapade]] (36)
# [[Unidec]] (25)
# [[Passion]] (35)
# [[Wizard]] (25)
# [[Schismic]] (35) (already listed as ''helmholtz'')
# [[Buzzard]] (24)
# [[Nautilus]] (34)
# [[Cassandra]] (24)
# [[Bleu]] (34)
# [[Ripple]] (24)
# [[Vulture]] (33)
# [[Vulture]] (24)
# [[Wizard]] (33)
# [[Armodue]] (23) (''disambiguation page'')
# [[Orson]] (32)
# [[Atomic]] (23)
# [[Schismatic]] (32) (already listed as ''helmholtz'')
# [[Bug]] (23)
# [[Unidec]] (31)
# [[Escapade]] (23)
# [[Muggles]] (30)
# [[Pontiac]] (23)
# [[Sensipent]] (30)
# [[Ampersand]] (22)
# [[Tritikleismic]] (30)
# [[Bohpier]] (22)
# [[Unicorn]] (30)
# [[Mohaha]] (22)
# [[Beatles]] (29)
# [[Parapyth]] (22)
# [[Bohpier]] (29)
# [[August]] (21)
# [[Machine]] (29)
# [[Blacksmith]] (21)
# [[Shrutar]] (29)
# [[Kwazy]] (21)
# [[Tertiaseptal]] (29)
# [[Octoid]] (21)
# [[Misty]] (28)
# [[Tritikleismic]] (21)
# [[Mohaha]] (28)
# [[Kleismic]] (20)
# [[Pontiac]] (28)
# [[Misty]] (20)
# [[Porky]] (28)
# [[Schismatic]] (20) (''already listed as “Helmholtz”'')
# [[Semisept]] (28)
# [[Shrutar]] (20)
# [[August]] (28)
# [[Sqrtphi]] (20)
# [[Bug]] (28)
# [[Beatles]] (19)
# [[Doublewide]] (28)
# [[Didacus]] (19)
# [[Cassandra]] (27)
# [[Meanpop]] (19)
# [[Decimal]] (27)
# [[Arcturus]] (18)
# [[Immunity]] (26)
# [[Gorgo]] (18)
# [[Octoid]] (26)
# [[Guiron]] (18)
# [[Supermajor]] (26) (disambiguation page)
# [[Leapday]] (18)
# [[Quadritikleismic]] (25)
# [[Mitonic]] (18)
# [[Ultrapyth]] (25)
# [[Nautilus]] (18)
# [[Gorgo]] (25)
# [[Sensipent]] (18)
# [[Leapday]] (25)
# [[Atomic]] (24)
# [[Decoid]] (24)
# [[Gravity]] (24)
# [[Guiron]] (24)
# [[Quartonic]] (24)
# [[Sqrtphi]] (24)


== A simpler overview ==
== A simpler overview ==
For a more streamlined, strictly curated list of useful temperaments, see the following pages:
For a more streamlined, strictly curated list of useful temperaments, see the following pages:
* [[Middle Path table of five-limit rank two temperaments]]
* [[Middle Path table of 5-limit rank-2 temperaments]]
* [[Middle Path table of seven-limit rank two temperaments]]
* [[Middle Path table of 7-limit rank-2 temperaments]]
* [[Middle Path table of eleven-limit rank two temperaments]]
* [[Middle Path table of 11-limit rank-2 temperaments]]


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:
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]]''


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== Advanced reading ==
== Advanced reading ==
* [[Tour of regular temperaments]]: a huge list of temperament families, many of which remain rarely-used and unexplored
* [[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
* [[Equal-step tuning]]s (i.e. rank-1 temperaments)
* [[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]]
* A deprecated, archived in-development version of this page: [[User:BudjarnLambeth/Bird’s eye view of rank-2 temperaments]]


== Notes ==
== Notes ==
<references group="note" />


[[Category:Tables]]
[[Category:Lists of temperaments]]
Retrieved from "https://en.xen.wiki/w/Survey_of_efficient_temperaments_by_subgroup"