# Regular temperament

*This article gives an introduction to regular temperaments. For a formal mathematical discussion, see Mathematical theory of regular temperaments.*

A **regular temperament** 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. Regular temperaments generally have an infinite number of notes; and other than equal temperaments, every regular temperament actually has an infinite number of notes in between *any two other notes*.

In addition to unlimited modulation, regular temperaments by definition are thought of as being approximations of some more complicated 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). A temperament only qualifies as a regular temperament if this interpretation works in a perfectly consistent way. For example, the sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. Multiple pure intervals may be represented by the same tempered interval (so they are tempered together), but a single pure interval must never be represented by different tempered intervals; if so, the temperament is irregular.

One particularly simple kind of regular temperaments is the equal temperaments, which represent all intervals by multiples of a single smallest step. At the other extreme, JI itself can be considered a trivial temperament where no tempering is happening: no commas are tempered out, but all are preserved as small pitch differences. In between 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:

- Nicola Vicentino (1511–1576): adaptive JI, 31et
- Leonhard Euler (1707–1783): tonespace (5-limit)
- Hermann von Helmholtz (1821–1894): psychoacoustics
- RHM Bosanquet (1841–1913): regular mapping, generalized keyboard
- Shohe Tanaka (1862–1945): 5-limit tonespace (triangular projection)
- Adriaan Fokker (1887–1972): periodicity blocks
- Harry Partch (1901–1974): 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 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 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?

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 formal primes in the subgroup 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 two most frequently used forms of optimization are POTE ("Pure-Octave Tenney-Euclidean"), TOP ("Tenney OPtimal", or "Tempered Octaves, Please") and more recently CTE ("Constained Tenney-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 CTE 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.

Each temperament has two names: a traditional name and a color name. The traditional names are diverse in terms of sources, whereas the color names are systematic and rigorous, and the comma(s) 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.

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.

Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma Porcupine aka Triyo has the 5th sharpened by one-fifth of 250/243 = [1 -5 3⟩. Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Trigu tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name.

## 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

### 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 harmonies relatively well with a small number of notes:

- Low harmonic entropy linear temperaments
- Middle Path table of five-limit rank two temperaments
- Middle Path table of seven-limit rank two temperaments
- Middle Path table of eleven-limit rank two temperaments

More comprehensive lists:

- Map of rank-2 temperaments: a visual map of temperaments based on the size of their period and generator
- Tour of regular temperaments: a huge gallery of the dozens of families of temperaments that have been described; not for the faint of heart