Regular temperament: Difference between revisions
→See also: Renamed see also to further reading |
|||
| Line 51: | Line 51: | ||
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]] = {{monzo| 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]] = {{monzo| 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 | ; Introductory materials | ||
* [[Dave Keenan & Douglas Blumeyer's guide to RTT]] | * [[Dave Keenan & Douglas Blumeyer's guide to RTT]] | ||