Ragismic microtemperaments: Difference between revisions

This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the ragisma, [[4375/4374]] ({{monzo| -1 -7 4 1 }}). The ragisma is the smallest [[7-limit]] [[superparticular ratio]].  

Since {{nowrap|(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 {{nowrap| 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.

Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:  

* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]

* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]

* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]]

* ''[[Chlorine]]'' (+{{monzo| -52 -17 34}}) → [[17th-octave temperaments #Chlorine|17th-octave temperaments]]

== Supermajor ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 52734375/52706752

{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082

{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}

[[Badness]]: 0.010836

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 35156250/35153041

Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082

{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}

Badness: 0.012773

Enneadecal temperament tempers out the [[enneadeca]], {{monzo| -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-limit, 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.

''For the 5-limit temperament, see [[19th-octave temperaments#(5-limit) enneadecal]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 703125/702464

{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}

: mapping generators: ~28/27, ~3

[[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)

{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 16384/16335

Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}

Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)

{{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }}

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 2205/2197

Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}

Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)

{{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }}

=== Hemienneadecal ===

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 234375/234256

Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}

: mapping generators: ~55/54, ~3

Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)

{{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}

Badness: 0.009985

Subgroup: 2.3.5.7.11.13

 

Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}

Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)

Badness: 0.020782

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213

Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}

Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444)

{{Optimal ET sequence|legend=1| 152, 342, 494, 1330, 1824, 2318d }}

Badness: 0.030391

==== Semihemienneadecal ====

Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078

Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}

: mapping generators: ~55/54 = 1\38, ~55/54, ~429/250

Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)

Badness: 0.014694

Named after the 19th element, potassium, and after an archaic variant of the element's name to resolve a name conflict. [[19/16]] can be used as a generator. Since it is enfactored in the 17-limit and lower, it makes no sense to name it for the lower subgroups.

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344

Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}

Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244

{{Optimal ET sequence|legend=1| 855, 988, 1843 }}

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].''

The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374.

[[Comma list]]: 4375/4374, 3955078125/3954653486

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270

{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}

[[Badness]]: 0.015075

The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.  

Early in the design of the [[Sagittal]] notation system, Secor and Keenan found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4 ¢ many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of Brahmagupta temperament that has pure octaves and pure fifths, which can also be described as a 17-limit extension having 1/7th octave period (171.4286 ¢) and 1/21st apotome generator (5.4136 ¢).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 70368744177664/70338939985125

{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}

: mapping generators: ~1157625/1048576, ~27/20

[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716

{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }}

[[Badness]]: 0.029122

=== 11-limit ===

Comma list: 4000/3993, 4375/4374, 131072/130977

Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}

Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704

{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }}

Badness: 0.052190

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374

Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}

Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.706

{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771eef }}

Badness: 0.023132

== Abigail ==

Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930]: "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things."</ref>

''For the 5-limit temperament, see [[Very high accuracy temperaments#Abigail]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 2147483648/2144153025

{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }}

: mapping generators: ~46305/32768, ~27/20

[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899

{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}

[[Badness]]: 0.037000

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 131072/130977

Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901

{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }}

Badness: 0.012860

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095

Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903

{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }}

Badness: 0.008856

== Gamera ==

''For the 5-limit temperament, see [[High badness temperaments#Gamera]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 589824/588245

: mapping generators: ~2, ~8/7

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336

{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}

[[Badness]]: 0.037648

=== Hemigamera ===

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 589824/588245

Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}

: mapping generators: ~99/70, ~8/7

Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3370

{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646, 1068d }}

Badness: 0.040955

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024

Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373

{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646f, 1068df }}

Comma list: 4375/4374, 14641/14580, 15488/15435

Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }}

: mapping generators: ~2, ~77/72

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1642

{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }}

Badness: 0.078

Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580

Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }}

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1628

{{Optimal ET sequence|legend=1| 73f, 125f, 198, 323, 521 }}

: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''

Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament. [[1106edo]] is an strong tuning.  

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}

{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}

: mapping generators: ~332150625/234881024, ~1125/1024

* [[CTE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7475

* [[error map]]: {{val| 0.0000 +0.0253 -0.0514 -0.0133 }}

* [[CWE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7474

* error map: {{val| 0.0000 +0.0244 -0.0508 -0.0218 }}

{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}

[[Badness]] (Smith): 0.0394

Comma list: 3025/3024, 4375/4374, 2791309312/2790703125

Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}

Optimal tunings:

* CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481

{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}

Badness (Smith): 0.0170

Badness (Sintel): 2.041

Badness (Sintel): 0.804

Badness (Sintel): 0.695

Badness (Sintel): 0.556

: ''For the 5-limit version of this temperament, see [[Schismic–Mercator equivalence continuum #Countritonic]].''

{{See also| Syntonic–kleismic equivalence continuum }}

In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, 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-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, and 11-limit adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.