Ragismic microtemperaments: Difference between revisions

This is a collection of [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the ragisma, [[4375/4374]] = {{monzo| -1 -7 4 1 }}. The ragisma is 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.

Microtemperaments considered below are ennealimmal, supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, orga, chlorine, 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) → [[Jubilismic clan #Crepuscular|Jubilismic clan]] and [[Fifive family #Crepuscular|Fifive family]]

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

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

== Ennealimmal ==

Ennealimmal 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]], {{monzo| 1 -27 18 }}, which leads to the identification of (27/25)⁹ with the [[octave]], and gives ennealimmal a [[period]] of 1/9 octave. Its [[pergen]] is (P8/9, P5/2). While 27/25 is a 5-limit interval, a stack of two periods 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.

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 49/40~60/49, all of which have their own interesting advantages. Possible tunings are 441-, 612-, or 3600edo, 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 "[[tritave]]s" 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.

Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]], [[Tritrizo clan #Undecentic|undecentic]], [[Tritrizo clan #Schisennealimmal|schisennealimmal]], and [[Tritrizo clan #Lunennealimmal|lunennealimmal]].

7-limit ennealimmal's S-expression-based comma list is {[[4375/4374|S25/S27]], [[2401/2400|S49]]}. Interestingly, the [[landscape comma]] is equal to ([[4375/4374|S25/S27]])/([[2401/2400|S49]]) while the [[wizma]] is equal to the product of the [[4375/4374|ragisma]] and the [[2401/2400|breedsma]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 4375/4374

{{Multival|legend=1| 18 27 18 1 -22 -34 }}

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

[[Optimal tuning]] ([[POTE]]): ~27/25 = 1\9, ~5/3 = 884.3129 (~36/35 = 49.0205)

[[Tuning ranges]]:  

* 7- and 9-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 49.179]

{{Optimal ET sequence|legend=1| 27, 45, 72, 99, 171, 441, 612 }}

[[Badness]]: 0.003610

The ennealimmal temperament can be described as 99e & 171e, which tempers out [[5632/5625]] (vishdel comma) and [[19712/19683]] (symbiotic comma).

Subgroup: 2.3.5.7.11

Mapping: {{mapping| 9 1 1 12 -75 | 0 2 3 2 16 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4679 (~36/35 = 48.8654)

{{Optimal ET sequence|legend=1| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}

Badness: 0.027332

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374

Mapping: {{mapping| 9 1 1 12 -75 93 | 0 2 3 2 16 -9 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)

{{Optimal ET sequence|legend=1| 99e, 171e, 270 }}

Badness: 0.029404

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 1001/1000, 1716/1715, 4096/4095, 4375/4374

Mapping: {{mapping| 9 1 1 12 -75 93 -3 | 0 2 3 2 16 -9 6 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)

{{Optimal ET sequence|legend=1| 99e, 171e, 270 }}

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 1001/1000, 1216/1215, 1716/1715, 4096/4095, 4375/4374

Mapping: {{mapping| 9 1 1 12 -75 93 -3 -48 | 0 2 3 2 16 -9 6 13 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)

{{Optimal ET sequence|legend=1| 99e, 171e, 270 }}

==== Ennealimmalis ====

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4375/4374, 5632/5625

Mapping: {{mapping| 9 1 1 12 -75 -106 | 0 2 3 2 16 21 }}

Optimal tuning (CTE): ~27/25 = 1\9, ~5/3 = 884.4560 (~36/35 = 48.8773)

{{Optimal ET sequence|legend=1| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}

Badness: 0.022068

=== Ennealimmia ===

The ennealimmia temperament is an alternative extension and can be described as 99 & 171, which tempers out [[131072/130977]] (olympia).

Comma list: 2401/2400, 4375/4374, 131072/130977

Mapping: {{mapping| 9 1 1 12 124 | 0 2 3 2 -14 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4089 (~36/35 = 48.9244)

{{Optimal ET sequence|legend=1| 99, 171, 270, 711, 981, 1251, 2232e }}

Badness: 0.026463

==== 13-limit ====

Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374

Mapping: {{mapping| 9 1 1 12 124 93 | 0 2 3 2 -14 -9 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)

{{Optimal ET sequence|legend=1| 99, 171, 270, 711, 981, 1692e, 2673e }}

Badness: 0.016607

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935, 2080/2079, 2401/2400, 4096/4095, 4375/4374

Mapping: {{mapping| 9 1 1 12 124 93 -3 | 0 2 3 2 -14 -9 6 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)

Comma list: 936/935, 1216/1215, 2080/2079, 2401/2400, 4096/4095, 4375/4374

Mapping: {{mapping| 9 1 1 12 124 93 -3 -48 | 0 2 3 2 -14 -9 6 13 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)

{{Optimal ET sequence|legend=1| 99, 171, 270 }}

=== Ennealimnic ===

Ennealimnic (72 & 171) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 4375/4356

Mapping: {{mapping| 9 1 1 12 -2 | 0 2 3 2 5 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9386 (~36/35 = 49.3948)

* 11-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)

* 11-odd-limit diamond tradeoff: ~36/35 = [48.920, 52.592]

* 11-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 52.592]

{{Optimal ET sequence|legend=1| 72, 171, 243 }}

Badness: 0.020347

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 625/624

Mapping: {{mapping| 9 1 1 12 -2 -33 | 0 2 3 2 5 10 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9920 (~36/35 = 49.3414)

* 13- and 15-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)

* 13- and 15-odd-limit diamond tradeoff: ~36/35 = [48.825, 52.592]

{{Optimal ET sequence|legend=1| 72, 171, 243 }}

Badness: 0.023250

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 375/374, 441/440, 595/594

Mapping: {{mapping| 9 1 1 12 -2 -33 -3 | 0 2 3 2 5 10 6 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9981 (~36/35 = 49.3353)

Tuning ranges:  

* 17-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)

* 17-odd-limit diamond monotone and tradeoff: ~36/35 = [48.485, 50.000]

{{Optimal ET sequence|legend=1| 72, 171, 243 }}

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 243/242, 364/363, 375/374, 441/440, 513/512, 595/594

Mapping: {{mapping| 9 1 1 12 -2 -33 -3 78  | 0 2 3 2 5 10 6 -6 }}

{{Optimal ET sequence|legend=1| 72, 171, 243 }}

==== Ennealim ====

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 243/242, 325/324, 441/440

Mapping: {{mapping| 9 1 1 12 -2 20 | 0 2 3 2 5 2 }}

Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)

{{Optimal ET sequence|legend=1| 27e, 45ef, 72 }}

Badness: 0.020697

Subgroup: 2.3.5.7.11.13.17

Mapping: {{mapping| 9 1 1 12 -2 20 -3 | 0 2 3 2 5 2 6 }}

Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)

{{Optimal ET sequence|legend=1| 27eg, 45efg, 72 }}

Subgroup: 2.3.5.7.11.13.17.19

 

 

Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)

 

 

=== Ennealiminal ===

Comma list: 385/384, 1375/1372, 4375/4374

Mapping: {{mapping| 9 1 1 12 51 | 0 2 3 2 -3 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.8298 (~36/35 = 49.5036)

{{Optimal ET sequence|legend=1| 27, 45, 72, 171e, 243e, 315e }}

Badness: 0.031123

==== 13-limit ====

Comma list: 169/168, 325/324, 385/384, 1375/1372

Mapping: {{mapping| 9 1 1 12 51 20 | 0 2 3 2 -3 2 }}

Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)

{{Optimal ET sequence|legend=1| 27, 45f, 72, 171ef, 243eff }}

Badness: 0.030325

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 325/324, 385/384, 1375/1372

Mapping: {{mapping| 9 1 1 12 51 20 50 | 0 2 3 2 -3 2 -2 }}

Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)

{{Optimal ET sequence|legend=1| 27, 45f, 72 }}

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 169/168, 221/220, 325/324, 385/384, 1375/1372

 

 

 

Hemiennealimmal (72 & 198) has a period of 1/18 octave and tempers out the four smallest superparticular commas of the 11-limit JI, 2401/2400, 3025/3024, 4375/4374, and 9801/9800. Tempering out [[9801/9800]] leads an octave split into two equal parts.

Comma list: 2401/2400, 3025/3024, 4375/4374

Mapping: {{mapping| 18 0 -1 22 48 | 0 2 3 2 1 }}

: mapping generators: ~80/77, ~400/231

Optimal tuning (POTE): ~80/77 = 1\18, ~400/231 = 950.9553

* 11-odd-limit diamond monotone: ~99/98 = [13.333, 22.222] (1\90 to 1\54)

* 11-odd-limit diamond tradeoff: ~99/98 = [17.304, 17.985]

* 11-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 17.985]

 

Badness: 0.006283

Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024

Mapping: {{mapping| 18 0 -1 22 48 -19 | 0 2 3 2 1 6 }}

Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837

* 13-odd-limit diamond monotone: ~99/98 = [16.667, 22.222] (1\72 to 1\54)

* 15-odd-limit diamond monotone: ~99/98 = [16.667, 19.048] (1\72 to 2\126)

* 13-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.309]

* 15-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.926]

* 13-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.309]

* 15-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.926]

Badness: 0.012505

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 1001/1000, 1716/1715, 3025/3024

Mapping: {{mapping| 18 0 -1 22 48 -19 -12 | 0 2 3 2 1 6 6 }}

Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837

{{Optimal ET sequence|legend=1| 72, 198g, 270 }}

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 715/714, 1001/1000, 1331/1330, 1716/1715, 3025/3024

Mapping: {{mapping| 18 0 -1 22 48 -19 -12 48 105 | 0 2 3 2 1 6 6 -2 }}

Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837

{{Optimal ET sequence|legend=1| 72, 198g, 270 }}

==== Semihemiennealimmal ====

Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374

Mapping: {{mapping| 18 0 -1 22 48 88 | 0 4 6 4 2 -3 }}

: mapping generators: ~80/77, ~1053/800

Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727

{{Optimal ET sequence|legend=1| 126, 144, 270, 684, 954 }}

Badness: 0.013104

Subgroup: 2.3.5.7.11.13.17

Comma list: 2401/2400, 2431/2430, 3025/3024, 4225/4224, 4375/4374

Mapping: {{mapping| 18 0 -1 22 48 88 -119 | 0 4 6 4 2 -3 27 }}

: mapping generators: ~80/77, ~1053/800

Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727

{{Optimal ET sequence|legend=1| 270, 684, 954 }}

Badness: 0.013104

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2401/2400, 2431/2430, 2926/2925, 3025/3024, 4225/4224, 4375/4374

Mapping: {{mapping| 18 0 -1 22 48 88 -119 -2 | 0 4 6 4 2 -3 27 11 }}

: mapping generators: ~80/77, ~1053/800

Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727

{{Optimal ET sequence|legend=1| 270, 684h, 954h, 1224 }}

Badness: 0.013104

Semiennealimmal tempers out [[4000/3993]], and uses a ~140/121 semifourth generator. Notably, however, two generator steps do not reach ~4/3, despite that the name may suggest so. In fact, it splits the generator of ennealimmal into three.  

Comma list: 2401/2400, 4000/3993, 4375/4374

Mapping: {{mapping| 9 3 4 14 18 | 0 6 9 6 7 }}

: mapping generators: ~27/25, ~140/121

Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3367

{{Optimal ET sequence|legend=1| 72, 369, 441 }}

Badness: 0.034196

==== 13-limit ====

Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374

Mapping: {{mapping| 9 3 4 14 18 -8 | 0 6 9 6 7 22 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3375

{{Optimal ET sequence|legend=1| 72, 297ef, 369f, 441 }}

Badness: 0.026122

=== Quadraennealimmal ===

Comma list: 2401/2400, 4375/4374, 234375/234256

Mapping: {{mapping| 9 1 1 12 -7 | 0 8 12 8 23 }}

: mapping generators: ~27/25, ~25/22

Optimal tuning (POTE): ~27/25 = 1\9, ~25/22 = 221.0717

{{Optimal ET sequence|legend=1| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374, 2097152/2096325

Mapping: {{mapping| 27 1 0 34 177 | 0 2 3 2 -4 }}

: mapping generators: ~2744/2673, ~2352/1375

Optimal tuning (POTE): ~2744/2673 = 1\27, ~2352/1375 = 928.8000

{{Optimal ET sequence|legend=1| 27, 243, 270, 783, 1053, 1323 }}

Rhodium splits the ennealimmal period in five parts and thereby features a period of 9 × 5 = 45, thus the name is given after the 45th element.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374, 117440512/117406179

Mapping: {{mapping| 45 1 -1 56 226 | 0 2 3 2 -2 }}

: mapping generators: ~3072/3025, ~55/32

Optimal tuning (CTE): ~3072/3025 = 1\45, ~55/32 = 937.6658

Badness: 0.0381

Subgroup: 2.3.5.7.11.13

Comma list: 2401/2400, 4225/4224, 4375/4374, 6656/6655

Mapping: {{mapping| 45 1 -1 56 226 272 | 0 2 3 2 -2 -3 }}

Optimal tuning (CTE): ~66/65 = 1\45, ~55/32 = 937.657

Badness: 0.0226

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, leading to a wedgie of {{multival| 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 is not enough notes for you, there is always the 171-note mos.

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

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

{{Multival|legend=1| 37 46 75 -13 15 45 }}

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

 

=== Semisupermajor ===

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.

[[Subgroup]]: 2.3.5.7

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

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

{{Multival|legend=1| 19 19 57 -14 37 79 }}

: 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

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)

Badness: 0.043734

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)

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

==== Hemienneadecalis ====

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)

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)

 

==== 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.

[[Subgroup]]: 2.3.5.7

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

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

{{Multival|legend=1| 55 73 93 -12 -7 11 }}

[[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.

[[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

{{Multival|legend=1| 21 56 -77 40 -181 -336 }}

[[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

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

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

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>

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

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

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

{{Multival|legend=1| 22 48 -38 25 -122 -223 }}

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

[[Badness]]: 0.037000

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

Badness: 0.012860

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

[[Subgroup]]: 2.3.5.7

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

: mapping generators: ~2, ~8/7

{{Multival|legend=1| 23 40 1 10 -63 -110 }}