16ed5/3: Difference between revisions

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== Temperaments ==

The 2L 5s scale is generated by a very accurate [[4/3]], such that two of them wind up on a near exact [[16/9]], which period-reduces to [[16/15]] (the minor mossecond). This interval taken 2 times is approximated by an [[8/7]], and taken 4 times is approximated by a [[6/5]] (or [[2/1]] in the next mosoctave). These 2 equivalencies result in two tempered commas: the marvel comma - [[225/224]] ((16/15)2=(8/7)), and the diaschisma - [[2048/2025]] ((16/15)3=(6/5)). The diaschisma can also be tempered by taking 5 generators to mean a [[3/2]] ((4/3)5=(3/2)·(5/3)2), while the marvel comma can also be tempered with a stack of 3 generators, making a [[10/7]] ((4/3)3=(10/7)·(5/3)). The tempered marvel comma also means that the two large [[Tritone|tritones]] ([[64/45|pental]] and [[10/7|septimal]]) are addressed by the same scale step. The tempered diaschisma, on the other hand, means that both pental tritones are also addressed by the same scale step. [[User:Ayceman|I]] propose the name '''tristone''' for the basic temperament, as 3 ~~semitones make a period~~-~~reduced octave~~, ~~and it alludes to the tritone tempering~~.

The 2L 5s scale is generated by a very accurate [[4/3]], such that two of them wind up on a near exact [[16/9]], which period-reduces to [[16/15]] (the minor mossecond). This interval taken 2 times is approximated by an [[8/7]], and taken 4 times is approximated by a [[6/5]] (or [[2/1]] in the next mosoctave). These 2 equivalencies result in two tempered commas: the marvel comma - [[225/224]] ((16/15)2=(8/7)), and the diaschisma - [[2048/2025]] ((16/15)3=(6/5)).

The diaschisma can also be tempered by taking 5 generators to mean a [[3/2]] ((4/3)5=(3/2)·(5/3)2), while the marvel comma can also be tempered with a stack of 3 generators, making a [[10/7]] ((4/3)3=(10/7)·(5/3)).

The tempered marvel comma also means that the two large [[Tritone|tritones]] ([[64/45|pental]] and [[10/7|septimal]]) are addressed by the same scale step. The tempered diaschisma, on the other hand, means that both pental tritones are also addressed by the same scale step.

As 3 semitones make a period-reduced octave, and it alludes to the tritone tempering, [[User:Ayceman|I]] propose the name '''tristone''' for the basic [[Diaschismic family|diaschismic temperament]], based on the 16/15 to 6/5 relationship, as well as the following variants and extensions:

=== Tristone ===

[[Subgroup]]: 5/3.20/9.10/3

[[Comma]] list: 2048/2025

[[POL2]] generator: ~5/4 = 389.8224

[[Mapping]]: [⟨1 2 5], ⟨0 -1 -6]]

TE [[complexity]]: 1.988720

[[RMS temperament measures|RMS]] error: 2.228679 cents

[[Vals]]: 9ed5/3, 16ed5/3, 25ed5/3

==== Tridistone ====

[[Subgroup]]: 5/3.20/9.10/3.1000/189

[[Comma]] list: 2048/2025, 225/224

[[POL2]] generator: ~5/4 = 389.6140

[[Mapping]]: [⟨1 2 5 5], ⟨0 -1 -6 -4]]

TE [[complexity]]: 1.724923

[[RMS temperament measures|RMS]] error: 8.489179 cents

[[Vals]]: 9ed5/3, 16ed5/3

=== Metatristone ===

[[Subgroup]]: 5/3.20/9.15/6

[[Comma]] list: 2048/2025

[[POL2]] generator: ~5/4 = 390.5180

[[Mapping]]: [⟨1 2 4], ⟨0 -1 -5]]

TE [[complexity]]: 2.192193

[[RMS temperament measures|RMS]] error: 2.021819 cents

[[Vals]]: 9ed5/3, 16ed5/3, 25ed5/3

==== Metatridistone ====

[[Subgroup]]: 5/3.20/9.15/6.250/63

[[Comma]] list: 2048/2025, 225/224

[[POL2]] generator: ~5/4 = 390.5430

[[Mapping]]: [⟨1 2 4 4], ⟨0 -1 -5 -3]]

TE [[complexity]]: 1.895168

[[RMS temperament measures|RMS]] error: 7.910273 cents

[[Vals]]: 9ed5/3, 16ed5/3

[[Category:EdVI]]

[[Category:Nonoctave]]

[[Category:Edonoi]]

@@ Line 407: / Line 407: @@
 == Temperaments ==
-The 2L 5s scale is generated by a very accurate [[4/3]], such that two of them wind up on a near exact [[16/9]], which period-reduces to [[16/15]] (the minor mossecond). This interval taken 2 times is approximated by an [[8/7]], and taken 4 times is approximated by a [[6/5]] (or [[2/1]] in the next mosoctave). These 2 equivalencies result in two tempered commas: the marvel comma - [[225/224]] ((<sup>16</sup>/<sub>15</sub>)<sup>2</sup>=(<sup>8</sup>/<sub>7</sub>)), and the diaschisma - [[2048/2025]] ((<sup>16</sup>/<sub>15</sub>)<sup>3</sup>=(<sup>6</sup>/<sub>5</sub>)). The diaschisma can also be tempered by taking 5 generators to mean a [[3/2]] ((<sup>4</sup>/<sub>3</sub>)<sup>5</sup>=(<sup>3</sup>/<sub>2</sub>)·(<sup>5</sup>/<sub>3</sub>)<sup>2</sup>), while the marvel comma can also be tempered with a stack of 3 generators, making a [[10/7]] ((<sup>4</sup>/<sub>3</sub>)<sup>3</sup>=(<sup>10</sup>/<sub>7</sub>)·(<sup>5</sup>/<sub>3</sub>)). The tempered marvel comma also means that the two large [[Tritone|tritones]] ([[64/45|pental]] and [[10/7|septimal]]) are addressed by the same scale step. The tempered diaschisma, on the other hand, means that both pental tritones are also addressed by the same scale step. [[User:Ayceman|I]] propose the name '''tristone''' for the basic temperament, as 3 semitones make a period-reduced octave, and it alludes to the tritone tempering.
+The 2L 5s scale is generated by a very accurate [[4/3]], such that two of them wind up on a near exact [[16/9]], which period-reduces to [[16/15]] (the minor mossecond). This interval taken 2 times is approximated by an [[8/7]], and taken 4 times is approximated by a [[6/5]] (or [[2/1]] in the next mosoctave). These 2 equivalencies result in two tempered commas: the marvel comma - [[225/224]] ((<sup>16</sup>/<sub>15</sub>)<sup>2</sup>=(<sup>8</sup>/<sub>7</sub>)), and the diaschisma - [[2048/2025]] ((<sup>16</sup>/<sub>15</sub>)<sup>3</sup>=(<sup>6</sup>/<sub>5</sub>)).
+The diaschisma can also be tempered by taking 5 generators to mean a [[3/2]] ((<sup>4</sup>/<sub>3</sub>)<sup>5</sup>=(<sup>3</sup>/<sub>2</sub>)·(<sup>5</sup>/<sub>3</sub>)<sup>2</sup>), while the marvel comma can also be tempered with a stack of 3 generators, making a [[10/7]] ((<sup>4</sup>/<sub>3</sub>)<sup>3</sup>=(<sup>10</sup>/<sub>7</sub>)·(<sup>5</sup>/<sub>3</sub>)).
+The tempered marvel comma also means that the two large [[Tritone|tritones]] ([[64/45|pental]] and [[10/7|septimal]]) are addressed by the same scale step. The tempered diaschisma, on the other hand, means that both pental tritones are also addressed by the same scale step.
+As 3 semitones make a period-reduced octave, and it alludes to the tritone tempering, [[User:Ayceman|I]] propose the name '''tristone''' for the basic [[Diaschismic family|diaschismic temperament]], based on the 16/15 to 6/5 relationship, as well as the following variants and extensions:
+=== Tristone ===
+[[Subgroup]]: 5/3.20/9.10/3
+[[Comma]] list: 2048/2025
+[[POL2]] generator: ~5/4 = 389.8224
+[[Mapping]]: [⟨1 2 5], ⟨0 -1 -6]]
+TE [[complexity]]: 1.988720
+[[RMS temperament measures|RMS]] error: 2.228679 cents
+[[Vals]]: 9ed5/3, 16ed5/3, 25ed5/3
+==== Tridistone ====
+[[Subgroup]]: 5/3.20/9.10/3.1000/189
+[[Comma]] list: 2048/2025, 225/224
+[[POL2]] generator: ~5/4 = 389.6140
+[[Mapping]]: [⟨1 2 5 5], ⟨0 -1 -6 -4]]
+TE [[complexity]]: 1.724923
+[[RMS temperament measures|RMS]] error: 8.489179 cents
+[[Vals]]: 9ed5/3, 16ed5/3
+=== Metatristone ===
+[[Subgroup]]: 5/3.20/9.15/6
+[[Comma]] list: 2048/2025
+[[POL2]] generator: ~5/4 = 390.5180
+[[Mapping]]: [⟨1 2 4], ⟨0 -1 -5]]
+TE [[complexity]]: 2.192193
+[[RMS temperament measures|RMS]] error: 2.021819 cents
+[[Vals]]: 9ed5/3, 16ed5/3, 25ed5/3
+==== Metatridistone ====
+[[Subgroup]]: 5/3.20/9.15/6.250/63
+[[Comma]] list: 2048/2025, 225/224
+[[POL2]] generator: ~5/4 = 390.5430
+[[Mapping]]: [⟨1 2 4 4], ⟨0 -1 -5 -3]]
+TE [[complexity]]: 1.895168
+[[RMS temperament measures|RMS]] error: 7.910273 cents
+[[Vals]]: 9ed5/3, 16ed5/3
 [[Category:EdVI]]
 [[Category:Nonoctave]]
 [[Category:Edonoi]]