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A '''regular temperament''' is a mathematical object that maps an infinite set of intervals in to a smaller, though still infinite, set of [[tempering out|tempered]] intervals. Typically, the source set is assumed to be some [[Harmonic Limit|''p''-limit]] [[just intonation]] — or any [[Just intonation subgroups|subgroup]] thereof — AKA rational numbers, and tempering is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison (it is "tempered out" in the temperament). The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the "dimensionality" of JI, thereby simplifying the pitch relationships.
{{Beginner|Mathematical theory of regular temperaments}}
{{Wikipedia}}


In mathematical terms, it is a function whose domain is a target tuning we wish to approximate, and its range is the intervals of the temperament. In general, this mapping is many-to-one, and two different rational numbers may be mapped to the same tempered interval — in this case we say that the two JI intervals are "tempered together".
A '''regular temperament''' ('''RT''') is an abstract [[tuning system]] that looks the same no matter which pitch you start from (or consider the [[tonic]]). In other words, unlimited free modulation is possible: any [[interval]] can be stacked as many times as you like. A regular temperament is [[generate]]d by a set of generating intervals, usually one of which is considered the [[period]], and any note which is part of the regular temperament can be reached by stacking whole numbers of these generating intervals above a defined root note. For example, [[meantone]] temperament is generated by the [[2/1|octave]] and a tempered (detuned) version of the [[3/2|perfect fifth]], with the octave usually being considered the period, and every interval in meantone can be expressed as an integer number of octaves plus an integer number of fifths. In meantone, a {{W|major second}} is equal to two perfect fifths minus an octave, and a {{W|major third}} is four perfect fifths minus two octaves. Regular temperaments theoretically have an infinite number of notes, and besides [[equal temperament]]s, regular temperaments usually<ref group="note">This is true if there exist two generators such that size in [[cent]]s of one generator divided by that of the other is an {{W|irrational number}}. This is not true for tunings where every generator is a whole number of steps of some [[edo]] or other [[equal-step tuning]].</ref> have an infinite number of notes in between ''any two other notes''.


For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.
In addition to unlimited modulation, regular temperaments are by definition thought of as being approximations of some system of pure or target intervals, very often a [[just intonation]] (JI) [[subgroup]]. Each abstract interval is interpreted as a tempered, or detuned, version of the target interval (more accurately, a set of target intervals). For example, the octave in meantone represents the just ratio [[2/1]], the perfect fifth [[3/2]], and the major third [[5/4]]. Certain intervals are tempered to the [[1/1|unison]], or [[tempering out|tempered out]]; in a regular temperament, these intervals are known as [[comma]]s. In meantone, since stacking up four perfect fifths, down two octaves, and down a major third reaches the unison, we get that {{nowrap|(3/2)<sup>4</sup> / (2/1)<sup>2</sup> / (5/4) {{=}} [[81/80]]}} is tempered out, and thus 81/80 is a comma of meantone. Any two just intervals separated by a comma of a temperament, for example [[9/8]] and [[10/9]] in meantone, are mapped to the same tempered interval in the temperament, in this case a major second. A temperament only qualifies as a regular temperament if this interpretation works in a perfectly consistent way: The product of two tempered intervals must always be the tempered version of the product of the JI intervals; for example, if the ratios 3/2 and 5/4 are in the target interval set, then ~3/2 × ~5/4 = ~[[15/8]] must always be true. ("~" denotes tempered.) In any temperament, each target interval is mapped to a unique tempered interval, though a tempered interval can represent multiple target intervals.


As another example, if [[meantone]] temperament is a function M, then M(6/5) = M(32/27) = "minor third". The difference between these, 81/80 or the "[[syntonic comma]]", is tempered out in meantone temperament. M(81/80) = M(1/1) = "unison".
One particularly simple kind of regular temperaments is equal temperaments, which represent all intervals by multiples of a single step size. JI itself can be considered a [[trivial temperament]] where no tempering is happening: No commas are tempered out, and all of them are preserved as small pitch differences. Another example of a trivial temperament is [[single-pitch tuning]], where there are ''no'' generating intervals, and only a single pitch is available. In between JI and equal temperaments lies the cornucopia of temperaments discussed in [[Paul Erlich]]'s seminal work, [[:File:MiddlePath2015.pdf|''A Middle Path Between Just Intonation and the Equal Temperaments'']].


A regular temperament is abstract, and has no preferred exact tuning. There are ways to compute an optimal tuning for any given temperament, but there are multiple definitions of "optimal" that disagree with each other, so in general we can consider a regular temperament as having a range of possible tunings of the generators. Once a tuning of each generator is provided the tuning of any interval can be computed as an integer linear combination of generator tunings. This property that all intervals are linear combinations of the generators is in fact what makes a temperament "regular".
== History ==
The roots of '''regular temperament theory''' ('''RTT''') can be traced back for centuries. The practice far predates the theory, and in particular [[meantone]] temperament has been known since the 15th century. Many early pioneers set the stage for the general theory to come:


== Introductions to regular temperament theory ==
* {{W|Nicola Vicentino}} (1511–1576): [[adaptive JI]], [[31edo|31et]]
* [[Mike's Lectures On Regular Temperament Theory|Mike Battaglia's Lectures on RTT]]
* {{W|Leonhard Euler}} (1707–1783): [[5-limit]] tonespace
* [[Dave Keenan's Introduction to RTT]]
* {{W|Hermann von Helmholtz}} (1821–1894): psychoacoustics
* {{W|R. H. M.  Bosanquet}} (1841–1913): regular mapping, generalized keyboard
* {{W|Shohe Tanaka}} (1862–1945): 5-limit tonespace (triangular projection)
* [[Adriaan Fokker]] (1887–1972): [[Fokker block|periodicity blocks]]
* [[Harry Partch]] (1901–1974): [[JI|extended JI]]
* [[Erv Wilson]] (1928–2016): extended tonespace (and projections), [[mos]], scale tree
* [[Easley Blackwood]] (1933–2023): Blackwood[10], syntonic comma vanishing relation as equation
* [[George Secor]] (1943–2020): miracle temperament


== Dimensionality, or rank ==
A significant amount of this theory's early development occurred online via the {{w|Yahoo! Groups}} service. The groundwork was laid by [[Paul Erlich]], [[Graham Breed]], [[Dave Keenan]], [[Herman Miller]], and [[Paul Hahn]] in the late 1990's.  
A rank ''r'' regular temperament in a particular tuning may be defined by giving ''r'' multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank ''r'' temperament will be defined by ''r'' generators, and thus ''r'' [[vals]]. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of commas tempered out by the temperament, or a set of ''r'' independent vals defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[Comma pump examples|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.


=== Rank-1 (equal) temperaments ===
In 2001 [[Gene Ward Smith]] joined Yahoo! Groups and immediately began making major contributions to the conversation, introducing new terminology and higher-level math. He and his closer collaborators such as [[Mike Battaglia]] also did much of the work to document RTT on this wiki.


[[Equal-step tuning|Equal temperaments]] (abbreviated ET or tET) and [[EDO|equal divisions of the octave]] (abbreviated EDO or ED2) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator, making it a rank-1 temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval, let alone the octave, and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO, although one can also use unsupported vals or poorly-supported vals to achieve "fun" results. The familiar 12-note equal temperament, or 12edo, reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-ET.
In 2009 [[Kite Giedraitis]] began developing his own approach to RTT, including some noteworthy innovations.


=== Rank-2 (including linear) temperaments ===
== FAQ ==
=== Why would I want to use a regular temperament? ===
Regular temperaments are of most use to musicians who want their music to sound as much as possible like stacking-based [[just intonation]], but without the difficulties normally associated with it, such as [[wolf interval]]s, [[comma]]s, and [[comma pump]]s. Specifically, if your chord progression [[pump]]s a comma, and you want to avoid pitch shifts, wolf intervals, and/or tonic drift, that comma must be tempered out. Temperaments are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals. Finally, some use temperaments solely for their sound. For example, one might like the sound of neutral thirds, without caring much what ratio they are tuned to. Thus one might use rastmic even though no commas are pumped.


A p-limit rank-2 temperament maps all intervals of p-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a '''linear temperament'''. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-ET by tempering out the Pythagorean comma.
=== How does regular temperament theory help me compose music? ===
The skill of music composition is acquired by studying the disciplines such as {{w|harmony}}, {{w|musical form|form}}, {{w|orchestration}}, in addition to extensive listening. One common misconception is that learning regular temperament theory can be a substitute for any of those. Regular temperament theory does indeed present you with numerous tuning systems, and provide the tools to help you compare and choose between them based on some common goals. It also tells you how harmonic resources are available in each tuning system, though the question of putting them together to a piece of work is really up to you to experiment with. In other words, one may think of the relationship between regular temperament theory and composition as this: regular temperament theory tells you how to ''choose'' a tuning, while composition regards how to ''use'' a chosen tuning.  


Regular temperaments of ranks two and three are cataloged on the [[Optimal patent val]] page. Rank-2 temperaments are also listed at [[Proposed names for rank 2 temperaments]] by their generator mappings, and at [[Map of rank-2 temperaments]] by their generator size. See also the [[pergen]]s page. There is also [[Graham Breed]]'s [http://x31eq.com/catalog2.html giant list of regular temperaments].
=== What do I need to know to understand all the numbers on the pages for individual regular temperaments? ===
Although the concept of regular temperament is centuries old and predates much of modern mathematics, members of the Yahoo! Alternative Tuning List have developed a particular form of numerical shorthand for describing the properties of temperaments. The most important of these are [[val]]s ([[mapping]]s), [[monzo]]s and [[tempering out|tempering out comma]]s, which any student of the modern regular temperament paradigm should become familiar with. These concepts are rather straightforward and require little math to understand.


== Why would I want to use a regular temperament? ==
The [[rank]] of a temperament is its dimension. It equals the number of generators in the [[Just intonation subgroup|subgroup]] being used minus the number of independent commas that are tempered out.
Regular temperaments are of most use to musicians who want their music to sound as much as possible like low-overtone just intonation, but without the difficulties normally associated with [[low-overtone JI]], such as wolf intervals, commas, and comma pumps. Specifically, if your chord progression pumps a comma, and you want to avoid pitch shifts, wolf intervals, and/or tonic drift, that comma must be tempered out. Temperaments are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals. Finally, some use temperaments solely for their sound. For example, one might like the sound of neutral 3rds, without caring much what ratio they are tuned to. Thus one might use Rastmic even though no commas are pumped.


== What do I need to know to understand all the numbers on the pages for individual regular temperaments? ==
Another recent contribution to the field of temperament is the concept of [[optimization]], which can take many forms. The point of optimization is to minimize the difference between a temperament and JI by finding an optimal tuning for the generator. The most frequently used forms of optimization are [[POTE tuning|POTE]] ("Pure-Octave Tenney–Euclidean"), [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please") and more recently [[CWE]] ("Constained Weil–Euclidean"), which has become the new standard instead of POTE since POTE is meant to be an approximation. Optimization is rather intensive mathematically, but it is seldom left as an exercise to the reader; most temperaments are presented here in their optimal forms in terms of POTE and CTE generators. In addition, for each temperament there is a [[optimal ET sequence|sequence of equal temperaments]] showing possible [[equal-step tuning]]s in the order of better absolute accuracy to JI.  
Although the concept of regular temperament is centuries old and predates much of modern mathematics, members of the Yahoo! Alternative Tuning List have developed a particular form of numerical shorthand for describing the properties of temperaments. The most important of these are [[val]]s (aka mappings) and [[tempering out|tempering out comma]]s, which any student of the modern regular temperament paradigm should become familiar with. These concepts are rather straightforward and require little math to understand.


The rank of a temperament equals the number of primes in the subgroup minus the number of linearly independent (i.e. non-redundant) commas that are tempered out.
The most common browser tools used for finding optimal tunings (useful for investigating new temperaments) are [[Graham Breed]]'s [http://x31eq.com/temper/ Temperament Finder] and [[Sintel]]'s [https://sintel.pythonanywhere.com/ Temperament Calculator]; the former gives temperament names (usually consistent with the wiki) and implements a wide variety of features like finding related temperaments while the latter implements CWE and more complex types of subgroups (like allowing ratios as generators) and supports an alternative notation to [[warts]] that is more convenient for arbitrary subgroups.


Another recent contribution to the field of temperament is the concept of optimization, which can take many forms. The point of optimization is to minimize the difference between a temperament and JI by finding an optimal tuning for the generator. The two most frequently used forms of optimization are [[POTE tuning|POTE]] ("Pure-Octave Tenney-Euclidean") and [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please"). Optimization is rather intensive mathematically, but it is seldom left as an exercise to the reader; most temperaments are presented here in their optimal forms in terms of POTE generators. In addition, for each temperament there is a list of EDOs showing possible EDO tunings in the order of better accuracy.  
Usually, temperaments have names coming from a wide array of [[temperament names|sources]], but they can also have systematic and rigorous names, from which the comma(s) can be deduced. The most common systematic temperament naming system on the wiki is [[Kite's color notation]]: {{nowrap|wa {{=}} 3-limit|yo {{=}} 5-over|gu {{=}} 5-under|zo {{=}} 7-over|and ru {{=}} 7-under}} (see also [[Kite's color notation/Temperament names]]).


Yet another recent development is the concept of a [[pergen]], appearing here as (P8, P5/2) or somesuch. Every rank-2, rank-3, rank-4, etc. temperament has a pergen, which specifies the period and the generator(s). Assuming the prime subgroup includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a 5th or some fraction of some 3-limit interval. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every strong extension of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but don't uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.
Yet another recent development is the concept of a [[pergen]], appearing in our [[Tour of regular temperaments]] as (P8, P5/2) or somesuch, which classifies temperaments by their period and the generator(s), giving ideas of how to notate these temperaments. For rank-2 temperaments we have developed a similar classification system called [[ploidacot]].


Each temperament has two names: a traditional name and a [[Color notation|color name]]. The traditional names are [[Temperament Names|arbitrary]], but the color names are systematic and rigorous, and the comma can be deduced from the color name. Wa = 3-limit, yo = 5-over, gu = 5-under, zo = 7-over, and ru = 7-under. See also [[Color notation/Temperament Names|Color Notation/Temperament Names]].
== Further reading ==
=== Introductory materials ===
* ''[[A Middle Path]]'': this is [[Paul Erlich]]'s guide to RTT (regular temperament theory)
* [[Dave Keenan & Douglas Blumeyer's guide to RTT]]
* [[Keenan Pepper's explanation of vals]]


== Links ==
=== Key regular temperament concepts ===
* [[Tour of Regular Temperaments]]
These topics are covered in the introductory materials above, but you can read about them here in more depth:
* [[Mathematical theory of regular temperaments]]
* [[Monzo]]
* [[Wikipedia: Regular temperament]]
* [[Val]]
* [[Mapping]]
* [[Comma basis]]
* [[Patent val]]
* [[Tempering out]]
* [[Rank and codimension]]
* [[Tuning map]]
 
=== Lists of temperaments ===
Temperaments that approximate important harmonies relatively well with a small number of notes:
* [[Low harmonic entropy linear temperaments]]
* [[Middle Path table of 5-limit rank-2 temperaments]]
* [[Middle Path table of 7-limit rank-2 temperaments]]
* [[Middle Path table of 11-limit rank-2 temperaments]]
 
More comprehensive lists:
* [[Bird's eye view of temperaments by accuracy]] (article): temperaments the Xen Wiki contributors find most useful for approximating JI - with edo tunings and note counts for the harmonies they target, and explanations of their structure
* [[Survey of efficient temperaments by subgroup]] (table): good general-purpose temperaments, sorted by size (notes per equave) and by JI subgroup
* [[Map of rank-2 temperaments]] (table): temperaments (some general, some niche) sorted by the size of their period and generator
* [[Temperaments for MOS shapes]] (table): temperaments (some general, some niche) sorted by the scale shape they generate
* [[Tour of regular temperaments]] (article): huge gallery of the dozens of families of temperaments that have been described; ''very technical - not for the faint of heart''
 
=== Other writings on temperaments ===
* [[Mike's lectures on regular temperament theory|Mike Battaglia's lectures on RTT]]
 
== Notes ==
<references group="note"/>
 
== External links ==
* [http://x31eq.com/paradigm.html Graham Breed's "The Regular Mapping Paradigm"]
* [https://youtu.be/ZoAuVgndmbU John Moriarty – Tuning Theory 2: Temperament ("Microtonal" Theory)], a video lecture
 
[[Category:Regular temperament theory| ]] <!-- Main article -->

Latest revision as of 19:45, 8 June 2026

This is a beginner page. It is written to allow new readers to learn about the basics of the topic easily.
The corresponding expert page for this topic is Mathematical theory of regular temperaments.
English Wikipedia has an article on:

A regular temperament (RT) is an abstract tuning system that looks the same no matter which pitch you start from (or consider the tonic). In other words, unlimited free modulation is possible: any interval can be stacked as many times as you like. A regular temperament is generated by a set of generating intervals, usually one of which is considered the period, and any note which is part of the regular temperament can be reached by stacking whole numbers of these generating intervals above a defined root note. For example, meantone temperament is generated by the octave and a tempered (detuned) version of the perfect fifth, with the octave usually being considered the period, and every interval in meantone can be expressed as an integer number of octaves plus an integer number of fifths. In meantone, a major second is equal to two perfect fifths minus an octave, and a major third is four perfect fifths minus two octaves. Regular temperaments theoretically have an infinite number of notes, and besides equal temperaments, regular temperaments usually[note 1] have an infinite number of notes in between any two other notes.

In addition to unlimited modulation, regular temperaments are by definition thought of as being approximations of some system of pure or target intervals, very often a just intonation (JI) subgroup. Each abstract interval is interpreted as a tempered, or detuned, version of the target interval (more accurately, a set of target intervals). For example, the octave in meantone represents the just ratio 2/1, the perfect fifth 3/2, and the major third 5/4. Certain intervals are tempered to the unison, or tempered out; in a regular temperament, these intervals are known as commas. In meantone, since stacking up four perfect fifths, down two octaves, and down a major third reaches the unison, we get that (3/2)4 / (2/1)2 / (5/4) = 81/80 is tempered out, and thus 81/80 is a comma of meantone. Any two just intervals separated by a comma of a temperament, for example 9/8 and 10/9 in meantone, are mapped to the same tempered interval in the temperament, in this case a major second. A temperament only qualifies as a regular temperament if this interpretation works in a perfectly consistent way: The product of two tempered intervals must always be the tempered version of the product of the JI intervals; for example, if the ratios 3/2 and 5/4 are in the target interval set, then ~3/2 × ~5/4 = ~15/8 must always be true. ("~" denotes tempered.) In any temperament, each target interval is mapped to a unique tempered interval, though a tempered interval can represent multiple target intervals.

One particularly simple kind of regular temperaments is equal temperaments, which represent all intervals by multiples of a single step size. JI itself can be considered a trivial temperament where no tempering is happening: No commas are tempered out, and all of them are preserved as small pitch differences. Another example of a trivial temperament is single-pitch tuning, where there are no generating intervals, and only a single pitch is available. In between JI and equal temperaments lies the cornucopia of temperaments discussed in Paul Erlich's seminal work, A Middle Path Between Just Intonation and the Equal Temperaments.

History

The roots of regular temperament theory (RTT) can be traced back for centuries. The practice far predates the theory, and in particular meantone temperament has been known since the 15th century. Many early pioneers set the stage for the general theory to come:

A significant amount of this theory's early development occurred online via the Yahoo! Groups service. The groundwork was laid by Paul Erlich, Graham Breed, Dave Keenan, Herman Miller, and Paul Hahn in the late 1990's.

In 2001 Gene Ward Smith joined Yahoo! Groups and immediately began making major contributions to the conversation, introducing new terminology and higher-level math. He and his closer collaborators such as Mike Battaglia also did much of the work to document RTT on this wiki.

In 2009 Kite Giedraitis began developing his own approach to RTT, including some noteworthy innovations.

FAQ

Why would I want to use a regular temperament?

Regular temperaments are of most use to musicians who want their music to sound as much as possible like stacking-based just intonation, but without the difficulties normally associated with it, such as wolf intervals, commas, and comma pumps. Specifically, if your chord progression pumps a comma, and you want to avoid pitch shifts, wolf intervals, and/or tonic drift, that comma must be tempered out. Temperaments are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals. Finally, some use temperaments solely for their sound. For example, one might like the sound of neutral thirds, without caring much what ratio they are tuned to. Thus one might use rastmic even though no commas are pumped.

How does regular temperament theory help me compose music?

The skill of music composition is acquired by studying the disciplines such as harmony, form, orchestration, in addition to extensive listening. One common misconception is that learning regular temperament theory can be a substitute for any of those. Regular temperament theory does indeed present you with numerous tuning systems, and provide the tools to help you compare and choose between them based on some common goals. It also tells you how harmonic resources are available in each tuning system, though the question of putting them together to a piece of work is really up to you to experiment with. In other words, one may think of the relationship between regular temperament theory and composition as this: regular temperament theory tells you how to choose a tuning, while composition regards how to use a chosen tuning.

What do I need to know to understand all the numbers on the pages for individual regular temperaments?

Although the concept of regular temperament is centuries old and predates much of modern mathematics, members of the Yahoo! Alternative Tuning List have developed a particular form of numerical shorthand for describing the properties of temperaments. The most important of these are vals (mappings), monzos and tempering out commas, which any student of the modern regular temperament paradigm should become familiar with. These concepts are rather straightforward and require little math to understand.

The rank of a temperament is its dimension. It equals the number of generators in the subgroup being used minus the number of independent commas that are tempered out.

Another recent contribution to the field of temperament is the concept of optimization, which can take many forms. The point of optimization is to minimize the difference between a temperament and JI by finding an optimal tuning for the generator. The most frequently used forms of optimization are POTE ("Pure-Octave Tenney–Euclidean"), TOP ("Tenney OPtimal", or "Tempered Octaves, Please") and more recently CWE ("Constained Weil–Euclidean"), which has become the new standard instead of POTE since POTE is meant to be an approximation. Optimization is rather intensive mathematically, but it is seldom left as an exercise to the reader; most temperaments are presented here in their optimal forms in terms of POTE and CTE generators. In addition, for each temperament there is a sequence of equal temperaments showing possible equal-step tunings in the order of better absolute accuracy to JI.

The most common browser tools used for finding optimal tunings (useful for investigating new temperaments) are Graham Breed's Temperament Finder and Sintel's Temperament Calculator; the former gives temperament names (usually consistent with the wiki) and implements a wide variety of features like finding related temperaments while the latter implements CWE and more complex types of subgroups (like allowing ratios as generators) and supports an alternative notation to warts that is more convenient for arbitrary subgroups.

Usually, temperaments have names coming from a wide array of sources, but they can also have systematic and rigorous names, from which the comma(s) can be deduced. The most common systematic temperament naming system on the wiki is Kite's color notation: wa = 3-limit, yo = 5-over, gu = 5-under, zo = 7-over, and ru = 7-under (see also Kite's color notation/Temperament names).

Yet another recent development is the concept of a pergen, appearing in our Tour of regular temperaments as (P8, P5/2) or somesuch, which classifies temperaments by their period and the generator(s), giving ideas of how to notate these temperaments. For rank-2 temperaments we have developed a similar classification system called ploidacot.

Further reading

Introductory materials

Key regular temperament concepts

These topics are covered in the introductory materials above, but you can read about them here in more depth:

Lists of temperaments

Temperaments that approximate important harmonies relatively well with a small number of notes:

More comprehensive lists:

Other writings on temperaments

Notes

  1. ↑ This is true if there exist two generators such that size in cents of one generator divided by that of the other is an irrational number. This is not true for tunings where every generator is a whole number of steps of some edo or other equal-step tuning.

External links

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