Survey of efficient temperaments by subgroup: Difference between revisions

This page highlights those [[rank-2 temperament]]s which receive the most discussion among theorists and composers.

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).

== How to read the tables ==

=== Rows ===

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.

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.

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''.)

== Table of temperaments (5 to 45 notes per equave) ==

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.

|-

! ~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

! ~30 notes

! ~40 notes

! 5-limit <br>(2.3.5)

! 3.7 <br>and its extensions

| [[mintaka]] (no-13), [[keladic]]

| [[mebsuta]], [[minalzidar]]

| [[mintaka]] (with 13)

|  

! 4.''n'' <br>and its extensions

| [[meanquad]], [[tetrahanson]], [[tetrameantone]], [[quarchy]], [[tetrominant]]

|  

| [[fourwar]]

|  

! 5.''n'' <br>and its extensions

| [[antipyth]], [[juggernaut]]

|  

|  

|  

! Other subgroups

|}

 

 

{| class="wikitable mw-collapsible mw-collapsed center-all mw-collapsed”"

|+

! 5-limit <br>(2.3.5)

! 7-limit <br>(2.3.5.7)

! 11-limit <br>(2.3.5.7.11)

| [[quadritikleismic]], [[harry]]

| [[leapday]], [[superkleismic]]+, [[meantonic]], [[huygens]]

| [[hendec]], [[marvolo]], [[diaschismic]], [[sensus]], [[andromeda]], [[modus]]

| [[comptone]]

| [[heinz]], [[octopus]], [[lizard]], [[rodan]], [[echidna]], [[shrutar]]

| [[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]]

| [[srutaloo]], [[shrutar]]

| [[satin]], [[trinity]], [[sqrtphi]]+, [[gizzard]]+, [[ketchup]], [[semisept]], [[cassandric]]

|-

| 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]]+

| [[majvam]]+

| no-7 [[hemienneadecal]], no-7 [[vulture]]

! 2.3.5.other ''n''

|  

|  

|  

| [[majvam]]

! 2.3.7

|  

|  

|  

! 2.3.7.11 <br>and its extensions

| no-5 [[quanic]]

! 2.3.7.other ''n''

! 2.3.11 <br>and its extensions

| rank-2 [[pythrabian]]

| rank-2 [[frameshift]]

|  

! 2.3.other ''n''

|  

|  

|  

|  

|  

! 2.5.7 <br>and its extensions

|  

|  

|  

| [[Subgroup temperaments #Daemotertiaschis|daemotertiaschis]]

! 2.5.other ''n''

|  

|  

|  

! 2.7 <br>and its extensions

|  

|  

|  

|  

! 3.5.7 <br>and its extensions

|  

|  

|  

|  

! 3.5.other ''n''

|  

|  

|  

|  

|  

! 3.7 <br>and its extensions

|  

|  

|  

! 4.''n'' <br>and its extensions

|  

|  

|  

! 5.''n'' <br>and its extensions

|  

|  

|  

|  

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

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.

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.)

# [[Helmholtz (temperament)|Helmholtz]] (49)

* 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

* [[Tour of regular temperaments]]: a huge list of temperament families, many of which remain rarely-used and unexplored

 

[[Category:Lists of temperaments]]