Survey of efficient temperaments by subgroup: Difference between revisions

{{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 at the bottom of the page. (This is 100% optional, you can still add temperaments to the table without doing this :) )}}

There are at least hundreds, probably thousands, of [[rank-2 temperament]]s described. It can be difficult to know where to start.  

This page is intended to narrow the field a little, by showing you the temperaments that get talked about a lot within the xenharmonic community. It does not aim to list every temperament. Instead it aims to list only the ones that are of high interest to a sizeable number of theorists or composers.

 

 

'''Tl;dr, you can think of this as a table of what temperaments are popular among theorists and/or composers. These are the ones lots of people are talking about.'''

 

== So, which temperaments should I use to make music? ==

 

 

Most would agree that a good temperament approximates some subset of [[just intonation]] relatively accurately with a relatively small number of notes.

'''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 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.

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.  

And these are not the only possible stances, either! There is a Xenharmonicist C, Xenharmonist D, etc. Thousands of differing individual perspectives on what traits see important in a temperament.

So how do you start working it out for yourself? Figuring out your own preferences?

 

 

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.

== How to read the table ==

 

'''Rows'''

 

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, 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, which you 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 they matter. It is up to you which theorists you believe on that.

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.

It would be like if there were a table of pokemon, and one row said “cute pokemon”, and one row said “electric pokemon”: Pikachu would be listed under both.

 

'''The columns categorise temperaments by the approximate number of notes needed to reach all the important intervals''' within the practice of that temperament.

 

 

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 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.)

 

|+

! 5 to 15 notes

! 15 to 25 notes

! 25 to 50 notes

! 50 to 100 notes

! >100 notes

! 5-limit (2.3.5)

| [[diaschismic]], [[meantone]], [[augmented]], [[porcupine]], [[dimipent]], [[whitewood]], [[blackwood]]

| [[helmholtz]] (aka schismic), [[hanson]], [[magic]]

| [[wuerschmidt]], [[valentine]]

| [[gravity]]

| [[kwazy]]

! 7-limit (2.3.5.7)

| [[augene]], [[porcupine]], [[blacksmith]]

| [[orwell]], [[shrutar]], [[magic]], [[meantone]], [[mothra]], [[superpyth]], [[pajara]], [[godzilla]], [[whitewood]]

| [[hemiwuerschmidt]], [[miracle]], [[garibaldi]], [[valentine]], [[diaschismic]]

| [[ennealimmal]], [[harry]]

| [[enneadecal]], [[trinity]]

! 11-limit (2.3.5.7.11)

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

| [[hemiwuerschmidt]], [[miracle]], [[cassandra]], [[diaschismic]], [[valentine]], [[shrutar]], [[orwell]], [[magic]], [[meanpop]], [[andromeda]], [[mothra]], [[meantone]], [[superpyth]], [[nautilus]], [[godzilla]]

| [[enneadecal]], [[trinity]]

! 13-limit

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

| [[hemiwuerschmidt]], [[diaschismic]], [[orwell]], [[shrutar]], [[magic]], [[mothra]], [[meantone]], [[superpyth]], [[nautilus]]

| [[ennealimmal]], [[harry]]

| [[enneadecal]], [[trinity]]

! 17-limit

! Higher limits

| [[shrutar]]

| [[trinity]]

! 2.3.5.7.n

| [[unicorn]]

! 2.3.5.11

| [[larry]] (2.3.5.11 [[gravity]])

! 2.3.5.11.n

! 2.3.5.n

! 2.3.7

| [[archy]], [[semaphore]]

| [[slendric]], [[bleu]]

! 2.3.7.11

! 2.3.7.11.n

| [[bleu]]

! 2.3.7.n

! 2.3.11

! 2.3.11.n

|

| [[namo]]

|

|

|

! 2.5.7

! 2.5.7.11

! 2.5.7.n

! 2.5.11.n

!2.7.11

|

| [[orgone]]

|

|

|

! 3.5.7

| [[canopus]], [[BPS]], [[sirius]], [[arcturus]]

! 3.5.n

! 3.5.7.n

! 3.5.7.11.n

 

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]].

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.

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.

 

 

 

 

 

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.  

One editor is allowed to add more than one temperament if they like, just not too many though.

 

== I want a simpler, more straightforward overview ==

For a less complicated list of useful temperaments, see the following pages:

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:

== I want an overview with written descriptions of each temperament ==

* See [[User:Godtone/Bird's eye view of temperaments by accuracy]]