Kleismic: Difference between revisions

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| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[15L 4s]]

| Mapping = 1; 6 5 14

| Pergen = (P8, P12/6)

| Odd limit 1 = 5 | Mistuning 1 = 1.35 | Complexity 1 = 15

| Odd limit 2 = (2.3.5.13) 15 | Mistuning 2 = 2.35 | Complexity 2 = 34

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: ''"Kleismic" redirects here. For the temperament families, see [[Kleismic family]] and [[Kleismic rank three family]].''

'''Kleismic''', known in the [[5-limit]] as either '''hanson''' or simply "kleismic", is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[kleismic family]], characterized by the vanishing of the kleisma ([[15625/15552]]). It is [[generator|generated]] by a [[6/5|classical minor third (6/5)]], six of which make a [[3/1|twelfth (3/1)]].

'''Kleismic''', known in the [[5-limit]] as either '''hanson''' or '''tribiyoti''' or simply "kleismic", is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[kleismic family]], characterized by the vanishing of the kleisma [[15625/15552]] = [-6 -5 6⟩. It is [[generator|generated]] by a [[6/5|classical minor third (6/5)]], six of which make a [[3/1|twelfth (3/1)]].

Another useful interpretation of the kleisma as a comma is that it makes the classical chromatic semitone, [[25/24]], into a third-tone by equating three of this interval to [[9/8]]. As 9/8 = (25/24)(26/25)(27/26), it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the marveltwin comma (S25 × S26 = [[325/324]]), and the tunbarsma (S25 = [[625/624]]), resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]], sometimes known as '''cata'''. As the chain of generators naturally gives us hemitwelfths at only 3 generator steps, this also corresponds directly to an interpretation of these as [[26/15]] (and thus hemifourths as [[15/13]]) by tempering out S26 = [[676/675]].

@@ Line 7: / Line 7: @@
 | MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[15L 4s]]
 | Mapping = 1; 6 5 14
+| Pergen = (P8, P12/6)
 | Odd limit 1 = 5 | Mistuning 1 = 1.35 | Complexity 1 = 15
 | Odd limit 2 = (2.3.5.13) 15 | Mistuning 2 = 2.35 | Complexity 2 = 34
@@ Line 12: / Line 13: @@
 : ''"Kleismic" redirects here. For the temperament families, see [[Kleismic family]] and [[Kleismic rank three family]].''
-'''Kleismic''', known in the [[5-limit]] as either '''hanson''' or simply "kleismic", is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[kleismic family]], characterized by the vanishing of the kleisma ([[15625/15552]]). It is [[generator|generated]] by a [[6/5|classical minor third (6/5)]], six of which make a [[3/1|twelfth (3/1)]].
+'''Kleismic''', known in the [[5-limit]] as either '''hanson''' or '''tribiyoti''' or simply "kleismic", is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[kleismic family]], characterized by the vanishing of the kleisma [[15625/15552]] = [-6 -5 6⟩. It is [[generator|generated]] by a [[6/5|classical minor third (6/5)]], six of which make a [[3/1|twelfth (3/1)]].
 Another useful interpretation of the kleisma as a comma is that it makes the classical chromatic semitone, [[25/24]], into a third-tone by equating three of this interval to [[9/8]]. As 9/8 = (25/24)(26/25)(27/26), it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the marveltwin comma (S25 × S26 = [[325/324]]), and the tunbarsma (S25 = [[625/624]]), resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]], sometimes known as '''cata'''. As the chain of generators naturally gives us hemitwelfths at only 3 generator steps, this also corresponds directly to an interpretation of these as [[26/15]] (and thus hemifourths as [[15/13]]) by tempering out S26 = [[676/675]].