Survey of efficient temperaments by subgroup: Difference between revisions

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.

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.

| [[hanson]], [[magic]], [[meantone]], [[negri]], [[augmented (temperament)|augmented]], [[porcupine]], [[diminished (temperament)|diminished]], [[whitewood]], [[blackwood]], [[mavila]]

| [[helmholtz (temperament)|helmholtz]], [[orson]], [[würschmidt]], [[sensipent]], [[compton]], [[valentine]], [[diaschismic]], [[tetracot]], [[passion]], [[superpyth]], [[ripple]]

| [[kwazy]], [[luna]], [[vishnu]], [[parakleismic]], [[escapade]], [[amity]], [[misty]], [[gravity]], [[rodan]]

| [[porcupine]], [[pajara]], [[keemun]], [[negri]], [[doublewide]], [[injera]], [[dominant (temperament)|dominant]], [[august]], [[diminished (temperament)|diminished]], [[blackwood]]

| [[amity]], [[hemiwürschmidt]], [[harry]], [[miracle]], [[garibaldi]], [[diaschismic]], [[sensi]]

| [[triforce]], [[blackwood]], [[pajaric]], [[negric]]

| [[orwell]], [[valentine]], [[mohajira]], [[porcupine]], [[hedgehog]], [[astrology]], [[vigintiduo]], [[augene]], [[nautilus]], [[catnip]], [[undevigintone]], [[injera]], [[keemun]], [[progress]], [[dominant (temperament)|dominant]], [[meanenneadecal]], [[duodecim]]

| [[augene]], [[porcupine]], [[hedgehog]], [[triforce]], [[godzilla]], [[negri]], [[armodue (temperament)|armodue]]

| [[nusecond]], [[modus]], [[lupercalia]], [[fokkertone]], [[winston]], [[pajara]], [[sensis]], [[ringo]], [[flattone]], [[darjeeling]], [[meanenneadecal]]

| [[miraculous]], [[leapday]], [[andromeda]], [[superkleismic]], [[mothra]], [[mohajira]], [[undevigintone]], [[ogene]], [[nautilus]], [[negroni]], [[injera]]

| [[miraculous]], [[lupercalia]]+, [[mohajira]], [[superpyth]]+, [[fokkertone]]+, [[injera]], [[meanenneadecal]]

| [[roman]]++, [[mohajira]], [[lupercalia]]++, [[superpyth]]++, [[negroni]]++, [[meanenneadecal]], [[ringo]]++, [[injera]], [[fokkertone]]++

| [[porcupine]], [[mavila]], [[dicot]], [[flattone]]

| [[würschmidt]]<ref group="note">Subgroup 2.3.5.23 version.</ref>, [[cata]]

| [[didacus]], [[frostburn]], no-3 [[oodako]], [[augment]]

| [[rainy]], [[mercy]], [[huntington]]

| [[llywelyn]], [[silver]], [[baldy]]

| [[hendecatonic (temperament)|hendecatonic]]

| [[supermajor (temperament)|supermajor]], [[enneadecal]], [[term]]

| [[quasiorwell]], [[octoid]], [[sqrtphi]], [[hemiwürschmidt]], [[ennealimnic]], [[catakleismic]]

| [[diaschismic]], [[myna]], [[sensus]], [[meanpop]]

| [[cassandra]], [[grosstone]]

| [[widefourth]], [[octopus]], [[mystery]], [[buzzard]], [[catakleismic]], [[rodan]], [[shrutar]]

| [[leapday]], [[superkleismic]]+, [[fokkertone]]+, [[fokkertone]]

| [[octacot]], [[andromeda]], [[fokkertone]]+, [[fokkertone]]

| [[octopus]], [[ennealim]], [[bikleismic]], [[srutal]], [[valentino]]+

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

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]] (479)'''

# '''[[Porcupine]] (195)'''

# '''[[Superpyth]] (162)'''

# '''[[Magic]] (153)'''

# '''[[Mavila]] (127)'''

# '''[[Miracle]] (112)'''

# '''[[Orwell]] (111)'''

# '''[[Blackwood]] (95)'''

# '''[[Flattone]] (95)'''

# '''[[Sensi]] (92)'''

# '''[[Pajara]] (91)'''

# '''[[Slendric]] (89)'''

# '''[[Valentine]] (87)'''

# '''[[Würschmidt]] (86)'''

# '''[[Negri]] (84)'''

# '''[[Tetracot]] (82)'''

# '''[[Mohajira]] (79)'''

# '''[[Compton]] (70)'''

# '''[[Amity]] (69)'''

# '''[[Diaschismic]] (69)'''

# [[Garibaldi]] (68)

# [[Dicot]] (64)

# [[Ennealimmal]] (60)

# [[Father]] (60)

# [[Diminished (temperament)|Diminished]] (59)

# [[Mothra]] (58)

# [[Myna]] (57)

# [[Rodan]] (57)

# [[Catakleismic]] (57)

# [[Godzilla]] (56)

# [[Hanson]] (56)

# [[Squares]] (55)

# [[Hemififths]] (54)

# [[Injera]] (53)

# [[Octacot]] (53)

# [[Semaphore]] (52)

# [[Harry]] (52)

# [[Archy]] (50)

# [[Augmented (temperament)|Augmented]] (50)

# [[Helmholtz (temperament)|Helmholtz]] (48)

# [[Superkleismic]] (46)

# [[Augene]] (46)

# [[Keemun]] (46)

# [[Whitewood]] (43)

# [[Buzzard]] (43)

# [[Orgone]] (42)

# [[Dominant (temperament)|Dominant]] (42)

# [[Kleismic]] (42) (already listed as ''hanson'')

# [[Liese]] (42)

# [[Didacus]] (41)

# [[Hemiwürschmidt]] (41)

# [[Parakleismic]] (41)

# [[Vishnu]] (40)

# [[Enneadecal]] (40)

# [[Hemithirds]] (40)

# [[Lemba]] (40)

# [[Srutal]] (39)

# [[Hedgehog]] (39)

# [[Luna]] (38)

# [[Triforce]] (38)

# [[Echidna]] (37)

# [[Ripple]] (36)

# [[Tritonic]] (36)

# [[Escapade]] (36)

# [[Passion]] (35)

# [[Schismic]] (35) (already listed as ''helmholtz'')

# [[Nautilus]] (34)

# [[Bleu]] (34)

# [[Vulture]] (33)

# [[Wizard]] (33)

# [[Orson]] (32)

# [[Schismatic]] (32) (already listed as ''helmholtz'')

# [[Unidec]] (31)

# [[Muggles]] (30)

# [[Sensipent]] (30)

# [[Tritikleismic]] (30)

# [[Unicorn]] (30)

# [[Beatles]] (29)

# [[Bohpier]] (29)

# [[Machine]] (29)

# [[Shrutar]] (29)

# [[Tertiaseptal]] (29)

# [[Misty]] (28)

# [[Mohaha]] (28)

# [[Pontiac]] (28)

# [[Porky]] (28)

# [[Semisept]] (28)

# [[August]] (28)

# [[Bug]] (28)

# [[Doublewide]] (28)

# [[Cassandra]] (27)

# [[Decimal]] (27)

# [[Immunity]] (26)

# [[Octoid]] (26)

# [[Supermajor]] (26) (disambiguation page)

# [[Quadritikleismic]] (25)

# [[Ultrapyth]] (25)

# [[Gorgo]] (25)

# [[Leapday]] (25)

# [[Decoid]] (24)

# [[Gravity]] (24)

# [[Guiron]] (24)

# [[Quartonic]] (24)

# [[Sqrtphi]] (24)

* [[Middle Path table of 5-limit rank-2 temperaments]]

* [[Middle Path table of 7-limit rank-2 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: