Regular temperament: Difference between revisions

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".

* [[Mike's Lectures On Regular Temperament Theory|Mike Battaglia's Lectures on RTT]]

* [[Dave Keenan's Introduction to RTT]]

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.

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].

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.

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.

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]].

* [[Tour of Regular Temperaments]]

* [[Mathematical theory of regular temperaments]]

* [[Wikipedia: Regular temperament]]