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Survey of efficient temperaments by subgroup: Difference between revisions

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There are at least hundreds, probably thousands, of [[rank-2 temperament]]s described. It can be difficult to know where to start.  
This page highlights those [[rank-2 temperaments]] which get talked about the most among theorists and composers.


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.
Composers and theorists disagree about which of these temperaments matter most, but each of these temperaments is valued by at least some sizeable subset of the xenharmonic community.


Composers and theorists disagree amongst themselves about what properties are desirable in a temperament, and you might over time find that you lean more towards one particular camp or another. This list arranges popular temperaments by their properties in a table, allowing the views of all ‘camps’ to get a space on the page.
== So, which temperaments should I use to make music? ==


'''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? ==
There are many different schools of thought within RTT (regular temperament theory).
 
Ask 5 xenharmonicists, and you'll get 10 different answers. There are many different schools of thought within RTT (regular temperament theory).


Most would agree that a good temperament approximates some subset of [[just intonation]] relatively accurately with a relatively small number of notes.
Most would agree that a good temperament approximates some subset of [[just intonation]] relatively accurately with a relatively small number of notes.
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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.  
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, Xenharmonicist D, etc. Thousands of differing individual perspectives on what traits see important in a temperament.
And these are not the only possible stances, either: There is a Xenharmonicist C, Xenharmonicist D, etc. Thousands of differing individual perspectives on what traits see important in a temperament.


So how do you get your head around all these wide ranging perspectives?
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).  
 
It might 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).  


Most theorists interpret 12edo as a 2.3.5 subgroup temperament which is about as accurate as most of the temperaments in the left-most column of the below table. This interpretation is not universal, though.
Most theorists interpret 12edo as a 2.3.5 subgroup temperament which is about as accurate as most of the temperaments in the left-most column of the below table. This interpretation is not universal, though.
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The 2.3.5.7 and 2.3.5.7.11 subgroups are the most commonly used by xenharmonic composers, being not too complex and including lots of useful harmonies.
The 2.3.5.7 and 2.3.5.7.11 subgroups are the most commonly used by xenharmonic composers, 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 no 2s, e.g. 3.5.7.11, are the biggest and most jarring break away from familiar harmony, may be 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.
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.
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 there.




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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.
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.




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