Ragismic microtemperaments: Difference between revisions

The ragisma is [[4375/4374]] with a [[monzo]] of |-1 -7 4 1>, the smallest 7-limit [[superparticular]] ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.

Temperaments not discussed here include [[Jubilismic clan #Crepuscular|crepuscular]], [[Meantone family #Flattone|flattone]], [[Porcupine family #Hystrix|hystrix]], [[Starling temperaments #Sensi|sensi]], [[Gamelismic clan #Unidec|unidec]], [[Orwellismic temperaments #Quartonic|quartonic]], [[Kleismic family #Catakleismic|catakleismic]], [[Tetracot family #Modus|modus]], [[Schismatic family #Pontiac|pontiac]], [[Tricot family|trillium]], [[Würschmidt family #Whirrschmidt|whirrschmidt]],  [[Gravity family #Zarvo|zarvo]], [[Vishnuzmic family #Vishnu|vishnu]], and [[Vulture family #Vulture|vulture]].  

[[Ennealimmal]] temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the [[ennealimma|ennealimmal comma]], |1 -27 18>, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two period equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is <<18 27 18 1 -22 -34||.

[[Mapping]]: [<9 1 1 12|, <0 2 3 2|]

[[Wedgie]]: <<18 27 18 1 -22 -34||

[[POTE tuning|POTE generators]]: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980

[[EDO|Vals]]: {{Val list| 27, 45, 72, 99, 171, 441, 612 }}

Mapping: [<9 1 1 12 -75|, <0 2 3 2 16|]

Mapping: [<9 1 1 12 -75 93|, <0 2 3 2 16 -9|]

Mapping: [<9 1 1 12 124|, <0 2 3 2 -14|]

Mapping: [<9 1 1 12 124 93|, <0 2 3 2 -14 -9|]

Mapping: [<9 1 1 12 -2|, <0 2 3 2 5|]

Mapping: [<9 1 1 12 -2 -33|, <0 2 3 2 5 10|]

Mapping: [<9 1 1 12 -2 -33 -3|, <0 2 3 2 5 10 6|]

Mapping: [<9 1 1 12 -2 20|, <0 2 3 2 5 2|]

Mapping: [<9 1 1 12 51|, <0 2 3 2 -3|]

Mapping: [<9 1 1 12 51 20|, <0 2 3 2 -3 2|]

Mapping: [<27 1 0 34 177|, <0 2 3 2 -4|]

[[Mapping]]: [<1 6 10 3|, <0 -23 -40 -1|]

[[Wedgie]]: <<23 40 1 10 -63 -110||

[[POTE tuning|POTE generator]] ~8/7 = 230.336

[[EDO|Vals]]: {{Val list| 26, 73, 99, 224, 323, 422, 745d }}

Mapping: [<2 12 20 6 5|, <0 -23 -40 -1 5|]

Mapping: [<2 12 20 6 5 17|, <0 -23 -40 -1 5 -25|]

The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.0002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of <<37 46 75 -13 15 45||. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.

[[Mapping]]: [<1 15 19 30|, <0 -37 -46 -75|]

[[Wedgie]]: <<37 46 75 -13 15 45||

[[POTE tuning|POTE generator]]: ~9/7 = 435.082

[[EDO|Vals]]: {{Val list| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}

Mapping: [<2 30 38 60 41|, <0 -37 -46 -75 -47|]

Enneadecal temperament tempers out the enneadeca, |-14 -19 19>, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5 or 7 limits, and [[494edo]] shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.

[[Mapping]]: [<19 0 14 -37|, <0 1 1 3|]

[[Wedgie]]: <<19 19 57 -14 37 79||

[[POTE tuning|POTE generator]]: ~3/2 = 701.880

[[EDO|Vals]]: {{Val list| 19, 152, 171, 665, 836, 1007, 2185 }}

Mapping: [<38 0 28 -74 11|, <0 1 1 3 2|]

Mapping: [<38 0 28 -74 11 502|, <0 1 1 3 2 -6|]