Revision as of 13:46, 23 April 2021

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 microtemperaments 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 crepuscular, flattone, hystrix, sensi, unidec, quartonic, catakleismic, modus, pontiac, whirrschmidt, zarvo, vishnu, and vulture.

Ennealimmal

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 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||.

Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 EDOs, though its hardly likely anyone could tell the difference.

If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example.) In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS.

Comma list: 2401/2400, 4375/4374

Tuning ranges:

valid range: [26.667, 66.667] (1\45 to 1\18)
nice range: [48.920, 49.179]
strict range: [48.920, 49.179]

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

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

Mapping generators: ~27/25, ~5/3

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

@@ Line 847: / Line 847: @@
 = Parakleismic =
-In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13&gt;, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being &lt;&lt;13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie &lt;&lt;13 14 35 -36 ...|| adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.
+In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13&gt;, with the [[118edo|118EDO]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being &lt;&lt;13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie &lt;&lt;13 14 35 -36 -8 19 -102 42 -132 -222|| adding 385/384. For the 7-limit [[99edo|99EDO]] may be preferred, but in the 11-limit it is best to stick with 118.
-Comma: 124440064/1220703125
+[[Comma]]: 124440064/1220703125
-POTE generator: ~6/5 = 315.240
+[[Mapping]]: [&lt;1 5 6|, &lt;0 -13 -14|]
-Map: [&lt;1 5 6|, &lt;0 -13 -14|]
+[[POTE tuning|POTE generator]]: ~6/5 = 315.240
-EDOs: 19, 61, 80, 99, 118, 453, 571, 689, 1496
+[[EDO|Vals]]: {{Val list| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}
-Badness: 0.0433
+[[Badness]]: 0.043279
-==7-limit==
+== 7-limit ==
-Commas: 3136/3125, 4375/4374
+[[Comma list]]: 3136/3125, 4375/4374
-POTE generator: ~6/5 = 315.181
+[[Mapping]]: [&lt;1 5 6 12|, &lt;0 -13 -14 -35|]
+[[Wedgie]]: &lt;&lt;13 14 35 -8 19 42||
+[[POTE tuning|POTE generator]]: ~6/5 = 315.181
-Map: [&lt;1 5 6 12|, &lt;0 -13 -14 -35|]
+[[EDO|Vals]]: {{Val list| 19, 80, 99, 217, 316, 415 }}
-EDOs: 19, 80, 99, 217, 316, 415
+[[Badness]]: 0.027431
-Badness: 0.0274
+== 11-limit ==
+Comma list: 385/384, 3136/3125, 4375/4374
-==11-limit==
+Mapping: [&lt;1 5 6 12 -6|, &lt;0 -13 -14 -35 36|]
-Commas: 385/384, 3136/3125, 4375/4374
 POTE generator: ~6/5 = 315.251
-Map: [&lt;1 5 6 12 -6|, &lt;0 -13 -14 -35 36|]
+Vals: {{Val list| 19, 99, 118 }}
-EDOs: 19, 99, 118
+Badness: 0.049711
-Badness: 0.0497
+== Parkleismic ==
+Comma list: 176/175, 1375/1372, 2200/2187
-==Parkleismic==
+Mapping: [&lt;1 5 6 12 20|, &lt;0 -13 -14 -35 -63|]
-Commas: 176/175, 1375/1372, 2200/2187
 POTE generator: ~6/5 = 315.060
-Map: [&lt;1 5 6 12 20|, &lt;0 -13 -14 -35 -63|]
+Vals: {{Val list| 19e, 80, 179, 259cd }}
-EDOs: 80, 179, 259cd
+Badness: 0.055884
-Badness: 0.0559
+=== 13-limit ===
+Comma list: 169/168, 176/175, 325/324, 1375/1372
-===13-limit===
+Mapping: [&lt;1 5 6 12 20 10|, &lt;0 -13 -14 -35 -63 -24|]
-Commas: 169/168, 176/175, 325/324, 1375/1372
 POTE generator: ~6/5 = 315.075
-Map: [&lt;1 5 6 12 20 10|, &lt;0 -13 -14 -35 -63 -24|]
+Vals: {{Val list| 19e, 80, 179 }}
-EDOs: 15, 19, 80, 179
+Badness: 0.036559
-Badness: 0.0366
+== Paradigmic ==
+Comma list: 540/539, 896/891, 3136/3125
-==Paradigmic==
+Mapping: [&lt;1 5 6 12 -1|, &lt;0 -13 -14 -35 17|]
-Commas: 540/539, 896/891, 3136/3125
 POTE generator: ~6/5 = 315.096
-Map: [&lt;1 5 6 12 -1|, &lt;0 -13 -14 -35 17|]
+Vals: {{Val list| 19, 61d, 80, 99e, 179e }}
-EDOs: 19, 80, 99e, 179e
+Badness: 0.041720
-Badness: 0.0417
+=== 13-limit ===
+Comma list: 169/168, 325/324, 540/539, 832/825
-===13-limit===
+Mapping: [&lt;1 5 6 12 -1 10|, &lt;0 -13 -14 -35 17 -24|]
-Commas: 169/168, 325/324, 540/539, 832/825
 POTE generator: ~6/5 = 315.080
-Map: [&lt;1 5 6 12 -1 10|, &lt;0 -13 -14 -35 17 -24|]
+Vals: {{Val list| 19, 61d, 80, 99e, 179e }}
-EDOs: 19, 80, 99e, 179e
+Badness: 0.035781
-Badness: 0.0358
+== Semiparakleismic ==
+Comma list: 3025/3024, 3136/3125, 4375/4374
-== Semiparakleismic ==
+Mapping: [&lt;2 10 12 24 19|, &lt;0 -13 -14 -35 -23|]
-Commas: 3025/3024, 3136/3125, 4375/4374
 POTE generator: ~6/5 = 315.181
-Map: [&lt;2 10 12 24 19|, &lt;0 -13 -14 -35 -23|]
+Vals: {{Val list| 80, 118, 198, 316, 514c, 830c }}
-EDOs: 80, 118, 198, 316, 514c, 830c
-Badness: 0.0342
+Badness: 0.034208
 === 13-limit ===
-Commas: 352/351, 1001/1000, 3025/3024, 4375/4374
+Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
-POTE generator: ~6/5 = 315.1563
+Mapping: [&lt;2 10 12 24 19 -1|, &lt;0 -13 -14 -35 -23 16|]
-Map: [<2 10 12 24 19 -1|, <0 -13 -14 -35 -23 16|]
+POTE generator: ~6/5 = 315.156
-EDOs: {{EDOs|80, 118, 198}}
+Vals: {{Val list| 80, 118, 198 }}
-Badness: 0.0338
+Badness: 0.033775
 === Gentsemiparakleismic ===
-Commas: 169/168, 325/324, 364/363, 3136/3125
+Comma list: 169/168, 325/324, 364/363, 3136/3125
-POTE generator: ~6/5 = 315.1839
+Mapping: [&lt;2 10 12 24 19 20|, &lt;0 -13 -14 -35 -23 -24|]
-Map: [<2 10 12 24 19 20|, <0 -13 -14 -35 -23 -24|]
+POTE generator: ~6/5 = 315.184
-EDOs: {{EDOs|80, 118f, 198f}}
+Vals: {{Val list| 80, 118f, 198f }}
-Badness: 0.0405
+Badness: 0.040467
 =Quincy=

Ragismic microtemperaments: Difference between revisions

Revision as of 13:46, 23 April 2021

Ennealimmal

11-limit

Hemiennealimmal

13-limit

Semihemiennealimmal

Semiennealimmal

13-limit

Quadraennealimmal

Ennealimnic

13-limit

17-limit

Ennealim

Ennealiminal

13-limit

Trinealimmal

Gamera

Hemigamera

13-limit

Supermajor

Semisupermajor

Enneadecal

Hemienneadecal

13-limit

Deca

11-limit

13-limit

Mitonic

Abigail

11-limit

13-limit

Semidimi

7-limit

Brahmagupta

11-limit

13-limit

Quasithird

7-limit

11-limit

13-limit

Semidimfourth

7-limit

Neusec

13-limit

Acrokleismic

11-limit

13-limit

Counteracro

13-limit

Seniority

Orga

11-limit

13-limit

Quatracot

11-limit

13-limit

Octoid

11-limit

13-limit

Music

Octopus

Amity

7-limit

11-limit

13-limit

Hitchcock

13-limit

Hemiamity

Parakleismic

7-limit

11-limit

Parkleismic

13-limit

Paradigmic

13-limit

Semiparakleismic

13-limit

Gentsemiparakleismic

Quincy