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

'''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.'''

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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.  

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.

They might argue that it's perfectly possible to learn up to 50 notes per [[equave]].

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

== Guide to tables ==

The rows categorise temperaments by accuracy. That is, how closely they approximate [[just intonation]] intervals. The categories are:

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 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.

! Exotemperament (18-∞c)

! Medium accuracy (7-12c)

| [[hanson]], [[magic]]

! Very high accuracy (1-3.5c)

 

! Approx. 20 notes

! Approx. 30 notes

! Very high accuracy (1-3.5c)

== 11-limit ==

{| class="wikitable"

!  

! Approx. 10 notes

! Approx. 20 notes

! Approx. 30 notes

! Approx. 70 notes

! Over 100 notes

! Exotemperament (18-∞c)

| [[antitonic]], [[dicot]], [[father]]

| [[augene]], [[blacksmith]], [[pajara]], [[porcupine]], [[whitewood]]

| [[godzilla]], [[superpyth]]

! Medium accuracy (7-12c)

| [[magic]], [[meanpop]], [[meantone]], [[mothra]], [[valentine]]

| [[diaschismic]], [[orwell]]

| [[miracle]]

! Microtemperament (0-1c)

|  

|  

|  

| [[ennealimmal]]

! Microtemperament (0-1c)

|  

|  

|  

| [[ennealimmal]]

! Exotemperament (18-∞c)

|  

! Low accuracy (12-18c)

|  

| [[pajara]]

| [[diaschismic]]

|  

|  

! Microtemperament (0-1c)

|  

|  

|  

| [[ennealimmal]]

! Medium accuracy (7-12c)

|  

! High accuracy (3.5-7c)

! Very high accuracy (1-3.5c)

|  

! Microtemperament (0-1c)

|  

! Microtemperament (0-1c)

|  

! 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)

|  

|  

! 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)

! Exotemperament (18-∞c)

! Low accuracy (12-18c)

! High accuracy (3.5-7c)

! Very high accuracy (1-3.5c)

|  

|  

|  

|  

|  

! Microtemperament (0-1c)

|  

|}

 

 

! 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)

 

! Exotemperament (18-∞c)

! Low accuracy (12-18c)

! Very high accuracy (1-3.5c)

| [[sensible]], [[srutal archagall]]

|-

! Very high accuracy (1-3.5c)

== No-11s subgroups ==

! Approx. 10 notes

! Very high accuracy (1-3.5c)