Regular temperament: Difference between revisions
+link to a video lecture; cleanup |
Reorganized sections and links, replaced "abstract musical system" with "abstract temperament" (following the update of the Temperament page), misc. edits |
||
| Line 6: | Line 6: | ||
}} | }} | ||
{{Wikipedia}} | {{Wikipedia}} | ||
:''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 [[temperament]] 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 temperament]]s, every regular temperament actually has an infinite number of notes in between ''any two other notes''. | |||
A | In addition to unlimited modulation, regular temperaments are usually thought of as being tempered versions of some more complicated system of pure or target intervals, very often [[just intonation]] (JI). A temperament only qualifies as a regular temperament if this approximation 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 [[Wikipedia:Triviality (mathematics)|trivial]] temperament where no tempering is happening: no [[comma]]s are tempered out, but all are preserved as small pitch differences. | |||
At the other extreme, JI itself can be considered a | |||
In between 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]]''. | In between 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 [[Pioneers of the Regular Mapping Paradigm|early pioneers]] set the stage for the general theory to come: | |||
The roots of ''' | |||
* Nicola Vicentino (1511–1576): [[adaptive JI]], [[31edo|31-ET]] | * Nicola Vicentino (1511–1576): [[adaptive JI]], [[31edo|31-ET]] | ||
| Line 70: | Line 53: | ||
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|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. | 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|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. | ||
== See also == | |||
'''Introductory materials''' | |||
* [[Mike's Lectures On Regular Temperament Theory|Mike Battaglia's Lectures on RTT]] | |||
* [[Keenan's explanation of vals|Keenan Pepper's explanation of vals]] | |||
* [[Dave Keenan's Introduction to RTT]] | |||
* [[Douglas Blumeyer's RTT How-To]] | |||
'''Lists''' | |||
* [[Tour of regular temperaments]] | |||
'''Related topics''' | |||
* [[Monzo]] | |||
* [[Val]] | |||
* [[Mapping]] | |||
* [[Comma basis]] | |||
* [[Patent val]] | |||
* [[Tempering out]] | |||
* [[Rank and codimension]] | |||
* [[Tuning map]] | |||
== External links == | == External links == | ||