Bird's eye view of temperaments by accuracy: Difference between revisions
remove subgroups, add basic meantone, add srutal |
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{{Editable user page|Please add any valuable temperaments you can think of. Make sure to follow the format and consider carefully the number of notes needed for what set of odds and what the damage of the most off intervals is to classify it correctly.}} | |||
This page is a work in progress, serving to document temperaments broadly by accuracy preference, and then approximately by subgroup focus, that is, what sort of harmonies, broadly speaking, the temperament is targetting. Under each accuracy and subgroup focus is found an incomplete list of temperaments, organized ''approximately'' by complexity (how many notes per octave are required). The complexity given is the note count per octave (or for no-2's, per tritave), with the set of odds used for deriving the complexity given. Sometimes two complexities are given and the average is taken for the purpose of ranking. | This page is a work in progress, serving to document temperaments broadly by accuracy preference, and then approximately by subgroup focus, that is, what sort of harmonies, broadly speaking, the temperament is targetting. Under each accuracy and subgroup focus is found an incomplete list of temperaments, organized ''approximately'' by complexity (how many notes per octave are required). The complexity given is the note count per octave (or for no-2's, per tritave), with the set of odds used for deriving the complexity given. Sometimes two complexities are given and the average is taken for the purpose of ranking. | ||
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Low or very low accuracy temperaments are of interest to people wanting simple scales and who are fine with high damage. As a result, they tend not to have "higher-limit focus", as the error involved on intervals beyond the [[17-limit]] is too much. A variety of people consider this category to largely or even entirely be composed of exotemperaments, while others argue for various entries in this category being reasonable to consider harmonically based on the temperability of the simplest [[LCJI]] intervals. | Low or very low accuracy temperaments are of interest to people wanting simple scales and who are fine with high damage. As a result, they tend not to have "higher-limit focus", as the error involved on intervals beyond the [[17-limit]] is too much. A variety of people consider this category to largely or even entirely be composed of exotemperaments, while others argue for various entries in this category being reasonable to consider harmonically based on the temperability of the simplest [[LCJI]] intervals. | ||
== 5-limit focus == | == 5-limit focus == | ||
== 7-limit focus == | == 7-limit focus == | ||
== 11-limit focus == | == 11-limit focus == | ||
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Many temperaments that people consider theoretically tend to fall into this category, due to its balance of simplicity and accuracy and due to the common usage of [[meantone]] temperaments, though plenty of simple temperaments exist that are even more accurate, documented in higher-accuracy categories. | Many temperaments that people consider theoretically tend to fall into this category, due to its balance of simplicity and accuracy and due to the common usage of [[meantone]] temperaments, though plenty of simple temperaments exist that are even more accurate, documented in higher-accuracy categories. | ||
== 5-limit focus == | == 5-limit focus == | ||
=== [[Meantone]] === | |||
Note count: 5 for {1, 3, 5} | |||
Meantone is an incredibly efficient temperament for targetting [[5-odd-limit]] harmony whose characteristic is flattening [[3/2]] (the generator) by a few cents. Perhaps unsurprisingly, it was historically the most commonly used temperament. It does this by sacrificing a distinction between [[9/8]] and [[10/9]] so that two "tones" makes [[5/4]], hence its name. | |||
=== [[Srutal archagall]] === | |||
Note counts: | |||
* 10 for {3, 5, 9, 15, 17} ([[2L 8s]]) | |||
* 12 for adding {25} ([[10L 2s]]) | |||
Srutal archagall is the natural extension of 5-limit [[diaschismic]] to prime 17 by interpreting the generator as a near-just [[17/16]] and the period as [[~]][[24/17]][[~]][[17/12]]. | |||
It is notable as preserving a lot of intuitions of 12edo like [[9/8]] as two semitones, [[6/5]] as three semitones, [[5/4]] as a semitone less than [[4/3]] which is itself a semitone less than half an octave. It essentially "doubles up" on familiar categories by: two major seconds, [[~]][[10/9]] and [[~]][[9/8]][[~]][[17/15]], two minor thirds, [[~]][[20/17]] and [[~]][[6/5]], and two major thirds, [[~]][[5/4]] and [[~]][[51/40]][[~]][[32/25]]. The distance between these pairs is an exaggerated syntonic comma ([[81/80]]) making it useful as a slightly more accurate alternative to meantone. Notably the intervals of 5 require using the period offset to reach, so that the minor third reached by the circle of fifths is actually [[20/17]]. | |||
For abundant options, one might prefer a 22-note MOS over a 12-note one, so that srutal archagall can be seen as a detempering of [[22edo]], but the 12-note MOS is likely the easiest and most intuitive to approach for a beginner. [[34edo]] is a good tuning for optimizing the 2.3.5.17 subgroup. | |||
== 7-limit focus == | == 7-limit focus == | ||
== 11-limit focus == | == 11-limit focus == | ||
== ~17-limit focus == | == ~17-limit focus == | ||
== Higher-limit focus == | == Higher-limit focus == | ||
=== [[Srutal]] === | |||
Note count: 36 for {3, 5, 7, 9, 11, 13, 15, 17, 23, 33, 35, 51} (12L 22s) | |||
Bound-violating intervals: [[13/9]] | |||
Srutal is an at least no-19's [[23-limit]] temperament, being an extension of [[srutal archagall]] to the full 17-limit and finding [[23/16]] as an augmented fourth, that is, as a tritone of ([[~]][[9/8]])<sup>3</sup>. Therefore, familiarizing oneself with srutal archagall is recommendable as the structures and tunings are nearly identical (and merge meaningfully in [[80edo]]), with the main difference being number of notes and breadth of harmonies targetted. Srutal can find more primes than just those in the no-19's 23-limit but they are more complex so more likely to be used opportunistically in a 34-note MOS, so that this temperament can be seen as a detemperament of [[34edo]]. [[80edo]] is a good tuning for it, though [[46edo]] deals well enough with the no-19's 23-limit part (potentially add-31) at the cost of a variety of distinctions. | |||
== No-2's focus == | == No-2's focus == | ||
== No-3's focus == | == No-3's focus == | ||
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=== [[Miracle]] === | === [[Miracle]] === | ||
Note count: 23 for {3, 5, 7, 9, 11, 15, 21} ([[10L 11s]] or [[10L 21s]]) | Note count: 23 for {3, 5, 7, 9, 11, 15, 21} ([[10L 11s]] or [[10L 21s]]) | ||
Miracle is an elegant temperament that splits [[3/2]] into six equal parts that can be derived as the most natural and efficient way of doing so through [[S-expression]]s by splitting 3/2 into two by tempering [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], splitting 3/2 into three by tempering [[1029/1024|S7/S8 = (9/6)/(8/7)<sup>3</sup> = (3/2)/(8/7)<sup>3</sup>]] and then splitting the [[~]][[8/7]] in two by tempering [[225/224|S15 = (15/14)/(16/15)]] so that [[15/14]] and [[16/15]] are equated. [[72edo]] is a very good tuning of miracle, though [[31edo]] and [[41edo]] may be preferred for smaller note counts and for the various things they support, EG [[meantone]] for 31edo and [[garibaldi]] for 41edo. | Miracle is an elegant temperament that splits [[3/2]] into six equal parts that can be derived as the most natural and efficient way of doing so through [[S-expression]]s by splitting 3/2 into two by tempering [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], splitting 3/2 into three by tempering [[1029/1024|S7/S8 = (9/6)/(8/7)<sup>3</sup> = (3/2)/(8/7)<sup>3</sup>]] and then splitting the [[~]][[8/7]] in two by tempering [[225/224|S15 = (15/14)/(16/15)]] so that [[15/14]] and [[16/15]] are equated. [[72edo]] is a very good tuning of miracle, though [[31edo]] and [[41edo]] may be preferred for smaller note counts and for the various things they support, EG [[meantone]] for 31edo and [[garibaldi]] for 41edo. | ||