Regular temperament: Difference between revisions

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{{Beginner|Mathematical theory of regular temperaments}}

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''.

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)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, [[:File:MiddlePath2015.pdf|''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:

* {{W|Nicola Vicentino}} (1511–1576): [[adaptive JI]], [[31edo|31et]]

* {{W|Hermann von Helmholtz}} (1821–1894): psychoacoustics

* {{W|R. H. M.  Bosanquet}} (1841–1913): regular mapping, generalized keyboard

* [[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

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.  

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.

=== 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.

=== 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.

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.

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.

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.  

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.

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 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]].

* ''[[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]]

=== Key regular temperament concepts ===

* [[Tempering out]]

* [[Tuning map]]

=== Lists of temperaments ===

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

* [[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]]

* [[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 ===

== External links ==

[[Category:Regular temperament theory| ]] <!-- Main article -->