Bird's eye view of temperaments by accuracy: Difference between revisions
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This page serves to document temperaments people deem to be highly valuable, and organizing them broadly by accuracy preference, and then approximately by subgroup focus, that is, what sort of harmonies, broadly speaking, the temperament is targetting | == Introduction == | ||
This page serves to document temperaments people deem to be highly valuable, and organizing them 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. | ||
== Edit guides == | |||
=== Do <big>not</big> add a temperament if you do not deem it unusually/uniquely valuable for making music. === | |||
Temperaments here should only be ones that one or more people seriously consider to be "cream of the crop". Therefore, when adding a temperament, ''make sure to include a description of how it works'', ideally one that motivates it as to why someone might want to use it. | |||
=== The ''accuracy'' classification of a temperament is the maximum error allowed on (almost*) all intervals. === | |||
<nowiki>*</nowiki>If there are intervals in the corresponding (thr)odd-limit that violate this bound slightly even in an optimized tuning, they should be noted as "'''Bound-violating intervals:'''" under the header for easy comprehension and consideration. | |||
This should not be abused to attempt to reclassify temperaments. There should be ideally zero, at most 2 intervals (more in more complex subgroups). The main exception that justifies more than two bound-violating interval pairs is when a single odd is responsible for all of them (e.g. odd 7 in [[#Buzzard]]). Therefore, most temperaments in a category are more accurate than the bound suggests. | |||
The bounds are: | |||
* [[Bird's eye view of temperaments by accuracy#Microtemperaments (<1c)|Microtemperament]] (~ <1c) | |||
* [[Bird's eye view of temperaments by accuracy#High accuracy (<4c)|High accuracy]] (1 ~ 4c) | |||
* [[Bird's eye view of temperaments by accuracy#Medium accuracy|Medium accuracy]] (4 ~ 7c) | |||
* [[Bird's eye view of temperaments by accuracy#Low accuracy (<12c)|Low accuracy]] (7 ~ 12c) | |||
* [[Bird's eye view of temperaments by accuracy#Very low accuracy (12~18c)|Very low accuracy]] (12 ~ 18c) | |||
* [[Bird's eye view of temperaments by accuracy#Exotemperament|Exotemperament]] (~ >18c) | |||
The two name system is for two reasons: to account for people's varying preferences and terminology for accuracies and to make the system of categories symmetrical, with [[exotemperament]]s and [[microtemperament]]s as the extrema. | The two name system is for two reasons: to account for people's varying preferences and terminology for accuracies and to make the system of categories symmetrical, with [[exotemperament]]s and [[microtemperament]]s as the extrema. | ||
The cent errors are a result of a set of compromises between people of different preference, and being given in cents, are somewhat arbitrary. | The cent errors are a result of a set of compromises between people of different preference, and being given in cents, are somewhat arbitrary. The exotemperament bound is very high because it is not meaningful to pretend that the bound is even remotely precise. The "<~18c" bound was chosen to allow using [[5edo]] and [[7edo]] as the circle of [[~]][[3/2]]'s to barely qualify as not being exotemperaments, due to both being the extrema of [[5L 2s|diatonic]]. | ||
=== Style guide === | |||
The convention for this page in contrast to most Xenharmonic Wiki pages, is that if you should link rank 2 temperaments '''on this page,''' unless you are merely discussing competing extensions or don't intend to log that temperament separately. That way, the page can be fairly self-contained to avoid intimidating or confusing someone using this page as a reference. | |||
The obvious exception is the title referring to the main entry. The purpose of using #Temperament here is to indicate that clicking on the link will send you to the section in '''this''' page, if it exists. If not, feel free to add it. | |||
== Generator tunings == | == Generator tunings == | ||
"Generator tunings" on this page are given in the format '''a\b''', meaning ''a'' steps of ''b'' [[edo]]; the frequency ratio 2<sup>''a''/''b''</sup>. | |||
You can input 44\111 in [[Scale Workshop]] by hovering over "New scale" on the left, then pressing "Rank-2 temperament", then entering 44\111 in the "Generator" input field. | |||
You will then get a list of '''MOS sizes''' to pick from before pressing '''OK''' to generate the scale. The importance of this is that picking from this list gives you a [[MOS scale]] which has only up to two sizes of minimum intervals. The UI shows an incomplete list of MOSes, so you can still add your own amount of notes by modifying the '''Scale size'''. | |||
In some cases, the [[octave]] is split or tempered, in which case you will instead see pairs. | |||
* Taking [[Bird's eye view of temperaments by accuracy#Diaschismic, srutal archagall|#Diaschismic]] as a split example, it is (3\34, 1\2). This means that 3\34 goes in the "Generator" input field and 1\2 goes in the "Period" input field. You must then change '''Number of periods''', to 2 in this case. This will create a series of [[Multi-MOS]] scales that repeat more than once per octave. Notice also that "Generators up/down from 1/1" goes up in pairs. This because you are adding a generator per period, and in diaschismic there are two periods, thus two generators. In some pages like [[Sintel]]'s [https://sintel.pythonanywhere.com/ temperament calculator], this is not displayed. For instance, in diaschismic, the calculator says you need -2 fifths to reach ~5, so to use this in In Scale Workshop you multiply this number by the number of periods to get your mapped in interval: input 4 down generators. | |||
* Taking [[#Godzilla]] as a tempering example, it is (4\30<3>, 19\30<3>). This means that 4\30<3> goes in the "Generator" input field and 19\30<3> goes in the "Period" input field. This reflects tempering the octave sharp so that 3/1 is just in order to get various intervals of 3, 5, 7 and 13 more in tune as they are all tuned flat in [[19edo]]; very significantly so in the case of 7 and 13. | |||
Note that the generator tunings are listed in order of increasing accuracy, with the least accurate being the leftmost; the least accurate tuning is not required to satisfy the maximum cent error, so that a tone-efficient example is included, though the example should be unambiguously representative of the full scope of its harmony within reason. All tunings after it should thus be increasingly accurate so that it satisfies the tuning bounds indicated by the accuracy classification of the temperament (up to any indicated exceptions for the temperament). | Note that the generator tunings are listed in order of increasing accuracy, with the least accurate being the leftmost; the least accurate tuning is not required to satisfy the maximum cent error, so that a tone-efficient example is included, though the example should be unambiguously representative of the full scope of its harmony within reason. All tunings after it should thus be increasingly accurate so that it satisfies the tuning bounds indicated by the accuracy classification of the temperament (up to any indicated exceptions for the temperament). | ||
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==== [[Gariwizmic]] ==== | ==== [[Gariwizmic]] ==== | ||
[[Bird's eye view of temperaments by accuracy#Note counts|Note counts]]: | [[Bird's eye view of temperaments by accuracy#Note counts|Note counts]]: | ||
* 82: see [[#Gary]] | |||
* 176 for {3, 5, 7, 9, 11, 13, 15, 21, 25 | |||
Generator tunings: (55\94, 1\2), (79\176, 1\2), (158\270, 1\2) | Generator tunings: (55\94, 1\2), (79\176, 1\2), (158\270, 1\2) | ||
Gariwizmic has a 1/2-octave period representing [[99/70]], two of them being ~2/1 and tempering out [[9801/9800]]. It is generated by a | Gariwizmic has a 1/2-octave period representing [[99/70]], two of them being ~2/1 and tempering out [[9801/9800]]. It is generated by a [[#Gary]] fifth. As its name implies, it also tempers out the [[wizma]]. | ||
Tempering out the kalisma allows the pythagorean comma to be split into two [[2835/2816|fwiwismas]], and this allows reaching a [[352/351]] ~ [[385/384]] minicomma by 47 fifths plus a semioctave, or alternatively put, a ~[[256/243|limma]] minus 3.5 pythcommas, tempering out [[4096/4095]] and [[1716/1715]]. Primes 5 and 13 are thus reached by a diminished fourth (96/77) + minicomma (39 fifths + 1 period), and a triply augmented fourth (44/27) - minicomma (-27 fifths - 1 period). | |||
It is best represented in 270edo, which is well known for its astoundingly accurate 13-limit, making it one of the best fifth-based rank-2 temperaments. It is very complex mapping-wise despite its great accuracy, but it can be easily thought as cassandra, but with the minicomma as a "generator" for primes 5 and 13 (and 19) which is still reachable within the rank-2 structure (It isn't a true generator; were it independent, the temperament would be [[cassaschismic]]). | It is best represented in 270edo, which is well known for its astoundingly accurate 13-limit, making it one of the best fifth-based rank-2 temperaments. It is very complex mapping-wise despite its great accuracy, but it can be easily thought as cassandra, but with the minicomma as a "generator" for primes 5 and 13 (and 19) which is still reachable within the rank-2 structure (It isn't a true generator; were it independent, the temperament would be [[cassaschismic]]). | ||
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=== No-3's focus === | === No-3's focus === | ||
=== No-5's focus === | === No-5's focus === | ||
== '''High accuracy ( | |||
==== [[Gary]] ==== | |||
Note counts: | |||
* 41 for {3, 9, 7, 11, 21, 27, 33, 77, 81, 99} ([[12L 29s]]) | |||
Generator tunings: 24\41, 55\94, 79\135, 498\851 | |||
Gary is an extremely good 2.3.7.11 temperament generated by a slightly sharp fifth which maps the pythagorean comma to 64/63, and two of those to 33/32, tempering out [[19712/19683]] and [[131072/130977]], making it an [[Olympic clan|olympic]] and [[Symbiotic comma|symbiotic]] temperament. It can be seen as the no-5, no-13 restriction of [[#Cassandra]]. The lack of 5 and 13 makes these two mappings be optimal, reaching errors well below a cent. | |||
It also provides an astonighingly accurate approximation to [[19/15]], which then mapped to the major third tempers out 1540/1539 '''and''' 1216/1215, whose subgroup is 2.3.7.11.19/5, which has no independent prime 19 or 5; only the specific compound interval of dividing the two. | |||
41edo is the simplest and easiest to use, [[135edo]] provides an essentially perfect tuning with manageable fineness, to which no other tuning is as good until [[851edo]] makes scene. | |||
Prime 31 can also be easily included by tempering out [[1024/1023]], essentially equating [[64/63]] with [[63/62]] and [[33/32]] with [[32/31]]. It accrues more error here than the rest of primes, wanting to tune the fifth a tenth of a cent sharper to get everything within subcent values, which makes everything ever so slightly worsely tuned but still within microtemperament range. | |||
There are multiple ways of incorporating prime 5 and 13: | |||
* [[#Cassandra]] is the simplest, tempering out the [[225/224|marvel]] and [[325/324|marveltwin]] commas. High accuracy. | |||
* [[#Cotoneum]] is a good one, though with high complexity, tempering out the [[quince comma]] and the [[minisma]]. Very high accuracy. | |||
* [[#Gariwizmic]] is one of the best, also with high complexity, tempering out the [[1716/1715|lumma]] and the minisma. Extremely high accuracy. | |||
== '''High accuracy (1~4c)''' == | |||
The bound is the approximate [[JND|melodic JND (Just-Noticeable-Difference)]], though note that this doesn't mean that damage/mistuning is ''imperceptible'' in these temperaments as the harmonic JND can often be significantly smaller, depending largely on context, timbre and who is listening/who you ask. | The bound is the approximate [[JND|melodic JND (Just-Noticeable-Difference)]], though note that this doesn't mean that damage/mistuning is ''imperceptible'' in these temperaments as the harmonic JND can often be significantly smaller, depending largely on context, timbre and who is listening/who you ask. | ||
=== 5-limit focus === | === 5-limit focus === | ||
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=== No-3's focus === | === No-3's focus === | ||
=== No-5's focus === | === No-5's focus === | ||
== '''Medium accuracy (4~7c)''' == | |||
== '''Medium accuracy ( | |||
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 === | ||
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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. | 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]] ==== | ==== [[Srutal archagall|Diaschismic, Srutal archagall]] ==== | ||
[[#Note counts|Note counts]]: | [[#Note counts|Note counts]]: | ||
* 10 for {3, 5, 9, 15, 17} ([[2L 8s]]) | * 10 for {3, 5, 9, 15, 17} ([[2L 8s]]) | ||
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The 2.3.7 part of [[#Buzzard]] is not as accurate as everything else in the 13-limit; specifically, its interval of 7 barely violates the 4{{cent}} bound, however it makes up for it by being much simpler (mapping-wise) so that it is interesting as a 2.3.7-subgroup temperament that splits [[3/1]] into four equal parts, each representing a sharp [[~]][[21/16]], which defines it in the 2.3.7 subgroup. [[53edo]], [[58edo]] and [[111edo]] are good tunings. | The 2.3.7 part of [[#Buzzard]] is not as accurate as everything else in the 13-limit; specifically, its interval of 7 barely violates the 4{{cent}} bound, however it makes up for it by being much simpler (mapping-wise) so that it is interesting as a 2.3.7-subgroup temperament that splits [[3/1]] into four equal parts, each representing a sharp [[~]][[21/16]], which defines it in the 2.3.7 subgroup. [[53edo]], [[58edo]] and [[111edo]] are good tunings. | ||
== '''Low accuracy ( | == '''Low accuracy (7~12c)''' == | ||
Low accuracy temperaments in small prime limits are commonly considered due to their simplicity. As a result, "higher-limit focus" tends to not be focused on at this accuracy, as the error involved on intervals beyond the [[17-limit]] is potentially too much depending on the context and who you ask, though again such temperaments are commonly relevant as targets for detempering. | Low accuracy temperaments in small prime limits are commonly considered due to their simplicity. As a result, "higher-limit focus" tends to not be focused on at this accuracy, as the error involved on intervals beyond the [[17-limit]] is potentially too much depending on the context and who you ask, though again such temperaments are commonly relevant as targets for detempering. | ||
=== 5-limit focus === | === 5-limit focus === | ||
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=== No-5's focus === | === No-5's focus === | ||
== '''Very low accuracy ( | == '''Very low accuracy (12~18c)''' == | ||
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. | 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 === | ||