Survey of efficient temperaments by subgroup: Difference between revisions
mNo edit summary |
Made body text more formal |
||
| Line 2: | Line 2: | ||
This page highlights those [[rank-2 temperament]]s which | This page highlights those [[rank-2 temperament]]s which recieve the most discussion among theorists and composers. | ||
Composers and theorists disagree about which of these temperaments matter most, but | Composers and theorists disagree about which of these temperaments matter most, but all of these temperaments are valued by at least a large subset of the xenharmonic community. | ||
== So, which temperaments should I use to make music? == | == So, which temperaments should I use to make music? == | ||
There are many different schools of thought within RTT (regular temperament theory). | There are many different schools of thought within RTT (regular temperament theory). | ||
Most would agree that a good temperament is | 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. | 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. | ||
| Line 37: | Line 36: | ||
And these are not the only possible stances, either: One could imagine a Xenharmonicist C, Xenharmonicist D, etc: Thousands of differing individual perspectives on what traits are important in a temperament. | |||
And these are not the only possible stances, either: | |||
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). | 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 those temperaments in the left-most column of the below table. | |||
The second most common approach is to interpret 12edo as a | The second most common approach is to interpret 12edo as a 2.3.17.19 subgroup temperament, with similar accuracy to those temperaments in the middle columns. | ||
So that should provide a helpful point of comparison to measure these other temperaments against. | So that should provide a helpful point of comparison to measure these other temperaments against. | ||
| Line 56: | Line 53: | ||
The 2.3.5 subgroup is what most theorists believe 12 tone equal temperament belongs to | The 2.3.5 subgroup is what most theorists believe 12 tone equal temperament belongs to. If those theorists are correct, then 2.3.5 should encompass all the harmonies that are familiar to most Western listeners. | ||
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 | Subgroups with no 2s, e.g. 3.5.7.11, are the most jarring break away from familiar harmony, which one 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. | |||
(''*According to the 2.3.5 interpretation of common practice harmony.'') | |||
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, | 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, perhaps subtly and subconsciously in some instances, but still there. | ||
The same temperament may occur multiple times on the 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. | |||
| Line 77: | Line 76: | ||
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. | 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 | That means that for temperaments ''in this table'', 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 this table'', it is not true of all temperaments in general.) | ||
== Table of temperaments some decent number of people would recommend == | == Table of temperaments some decent number of people would recommend == | ||
The temperaments within each cell should be sorted by accuracy, with the lowest [[damage]] temperament listed first. | The temperaments within each cell should be sorted by accuracy, with the lowest [[damage]] temperament listed first. | ||
''Editors: If you see any temperaments listed in the wrong order, or see any temperaments in the wrong | ''Editors: 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.'' | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
| Line 244: | Line 243: | ||
'''Additional information''' | '''Additional information''' | ||
One important piece of information this table doesn’t capture is whether two temperaments share a [[pergen]]. | |||
Sometimes, multiple higher limit temperaments are actually different ways of extending the same lower-limit temperament. In this case, they will share a pergen. This means they will have an overall similar flavor and some musical and mathematical properties in common. | |||
If you visit the temperaments’ individual pages, those will usually make their relationships to other temperaments more clear. | If you visit the temperaments’ individual pages, those will usually make their relationships to other temperaments more clear. | ||