Tetracot: Difference between revisions
→Tunings: + norm-based tunings |
m Fix punctuation |
||
| (4 intermediate revisions by 2 users not shown) | |||
| Line 24: | Line 24: | ||
'''Tetracot''', in this article, is the [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] in the 2.3.5.11.13 [[subgroup]] [[generator|generated]] by a submajor second of about 174–178{{cent}} which represents both [[10/9]] and [[11/10]]. It is so named because the generator is a quarter of fifth: four such generators make a perfect fifth which approximates [[3/2]], which cannot occur in [[12edo]], resulting in [[100/99]], [[144/143]], and [[243/242]] being [[tempering out|tempered out]]. This is in contrast to [[meantone]], where 10/9 is tuned sharper than or equal to just in order to be equated with [[9/8]]. | '''Tetracot''', in this article, is the [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] in the 2.3.5.11.13 [[subgroup]] [[generator|generated]] by a submajor second of about 174–178{{cent}} which represents both [[10/9]] and [[11/10]]. It is so named because the generator is a quarter of fifth: four such generators make a perfect fifth which approximates [[3/2]], which cannot occur in [[12edo]], resulting in [[100/99]], [[144/143]], and [[243/242]] being [[tempering out|tempered out]]. This is in contrast to [[meantone]], where 10/9 is tuned sharper than or equal to just in order to be equated with [[9/8]]. | ||
[[Equal temperament]]s that [[support]] tetracot include {{EDOs| 27, 34, and 41 }}. | |||
See [[Tetracot family]] for | Tetracot has four strong [[extension]]s for the 7-, 11-, and 13-limit, which use the same methods of obtaining the [[11/1|11th]] and [[13/1|13th]] harmonics (10 generators up and 2 generators down, respectively) but differ in their methods of obtaining the [[7/1|7th harmonic]]: | ||
* [[Monkey]] (34 & 41) obtains the 7th harmonic at 15 generators down, tempering out [[875/864]] and thereby equating [[7/4]] with ([[6/5]])<sup>3</sup>; | |||
* [[Bunya]] (34d & 41) obtains the 7th harmonic at 26 generators up, tempering out [[225/224]] and thereby equating [[7/2]] with ([[15/8]])<sup>2</sup>; | |||
* [[Modus]] (27e & 34d) obtains the 7th harmonic at 8 generators down, tempering out [[64/63]] and thereby equating 7/4 with [[16/9]]; | |||
* [[Wollemia]] (27e & 34) obtains the 7th harmonic at 19 generators up, tempering out [[126/125]] and thereby equating [[7/1]] with ([[5/3]])<sup>3</sup>([[3/2]]). | |||
See [[Tetracot family]] for technical data. | |||
== Intervals == | == Intervals == | ||
| Line 272: | Line 278: | ||
| | | | ||
| 177.778 | | 177.778 | ||
| | | 27e val, upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone | ||
|- | |- | ||
| | | | ||
| Line 297: | Line 303: | ||
| | | | ||
| 180.000 | | 180.000 | ||
| | | 20ce val, upper bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone | ||
|- | |- | ||
| | | | ||