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:

The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2¹⁵)/3, 46 give (2¹⁹)/5, and 75 give (2³⁰)/7. 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 is not enough notes for you, there is always the 171-note mos.

[[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

[[Badness]]: 0.010836

=== Semisupermajor ===

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 ==

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 }}

[[Badness]]: 0.010954

=== 11-limit ===

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 }}

Badness: 0.043734

==== 13-limit ====

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 }}

Badness: 0.033545

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

Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256

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)

{{Optimal ET sequence|legend=1| 152f, 342f, 494 }}

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

Subgroup: 2.3.5.7.11.13

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)

{{Optimal ET sequence|legend=1| 190, 304d, 494, 684, 1178, 2850, 4028ce }}

Badness: 0.014694

=== Kalium ===

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.

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}

[[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

Subgroup: 2.3.5.7.11

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

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 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