Breedsmic temperaments: Difference between revisions

This page discusses miscellaneous rank-2 temperaments tempering out the [[breedsma]], {{monzo|-5 -1 -2 4}} = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12EDO, for example) which does not possess a neutral third cannot be tempering out the breedsma.

The breedsma is also the amount by which four stacked [[10/7]] intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, [[25/24]]. We might note also that 49/40 × 10/7 = 7/4 and 49/40 × (10/7)² = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.

* ''[[Decimal]]'', {25/24, 49/48} → [[Dicot family #Decimal]]

* ''[[Beatles]]'', {64/63, 686/675} → [[Archytas clan #Beatles]]

* [[Squares]], {81/80, 2401/2400} → [[Meantone family #Squares]]

* [[Myna]], {126/125, 1728/1715} → [[Starling temperaments #Myna]]

* [[Miracle]], {225/224, 1029/1024} → [[Gamelismic clan #Miracle]]

* ''[[Octacot]]'', {245/243, 2401/2400} → [[Tetracot family #Octacot]]

* ''[[Greenwood]]'', {405/392, 1323/1280} → [[Greenwoodmic temperaments #Greenwood]]

* ''[[Quasitemp]]'', {875/864, 2401/2400} → [[Keemic temperaments #Quasitemp]]

* ''[[Quadrimage]]'', {2401/2400, 3125/3072} → [[Magic family #Quadrimage]]

* ''[[Hemiwürschmidt]]'', {2401/2400, 3136/3125} → [[Würschmidt family #Hemiwürschmidt]]

* [[Ennealimmal]], {2401/2400, 4375/4374} → [[Ragismic microtemperaments #Ennealimmal]]

* ''[[Quadritikleismic]]'', {2401/2400, 15625/15552} → [[Kleismic family #Quadritikleismic]]

* ''[[Sesquiquartififths]]'', {2401/2400, 32805/32768} → [[Schismatic family #Sesquiquartififths]]

* ''[[Neptune]]'', {2401/2400, 48828125/48771072} → [[Gammic family #Neptune]]

* ''[[Eagle]]'', {2401/2400, 10485760000/10460353203} → [[Vulture family #Eagle]]

Hemififths tempers out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator, with [[99edo|99EDO]] and [[140edo|140EDO]] providing good tunings, and [[239edo|239EDO]] an even better one; and other possible tunings are 160^(1/25), giving just 5s, the 7- and 9-odd-limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&58 temperament. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS{{clarify}}.

By adding [[243/242]] (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99EDO is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice.

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 1 -5 -1 }}, {{val| 0 2 25 13 }}]

{{Multival|legend=1| 2 25 13 35 15 -40 }}

[[POTE generator]]: ~49/40 = 351.477

* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo|1/5 0 1/25}}

: [{{monzo|1 0 0 0}}, {{monzo|7/5 0 2/25 0}}, {{monzo|0 0 1 0}}, {{monzo|8/5 0 13/25 0}}]

: Eigenmonzos: 2, 5

{{Val list|legend=1| 41, 58, 99, 239, 338, 1253bbc, 1591bbc }}

[[Badness]]: 0.022243

Mapping: [{{val| 1 1 -5 -1 2 }}, {{val| 0 2 25 13 5 }}]

POTE generator: ~11/9 = 351.521

Vals: {{Val list| 17c, 41, 58, 99e }}

Badness: 0.023498

Mapping: [{{val| 1 1 -5 -1 2 4 }}, {{val| 0 2 25 13 5 -1 }}]

POTE generator: ~11/9 = 351.573

Vals: {{Val list| 17c, 41, 58, 99ef }}

Badness: 0.019090

Comma list: 2401/2400, 3388/3375, 9801/9800

Mapping: [{{val| 2 0 -35 -15 -47 }}, {{val| 0 2 25 13 34 }}]

POTE generator: ~49/40 = 351.505

Vals: {{Val list| 58, 140, 198, 734bc, 932bcd, 1130bcd }}

Badness: 0.042487

Mapping: [{{val| 2 0 -35 -15 -47 -37 }}, {{val| 0 2 25 13 34 28 }}]

POTE generator: ~49/40 = 351.502

Vals: {{Val list| 58, 140, 198, 536f, 734bcf, 932bcdf }}

Badness: 0.021188

Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the 31&171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo|171EDO]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 3 2 3 }}, {{val| 0 -22 5 -3 }}]

[[POTE generator]]: ~256/245 = 77.191

{{Val list|legend=1| 31, 109, 140, 171 }}

Mapping: [{{val| 1 3 2 3 7 }}, {{val| 0 -22 5 -3 -55 }}]

POTE generator: ~256/245 = 77.227

Vals: {{Val list| 31, 109e, 140e, 171, 202 }}

Mapping: [{{val| 1 3 2 3 7 1 }}, {{val| 0 -22 5 -3 -55 42 }}]

POTE generator: ~117/112 = 77.203

Vals: {{Val list| 31, 109e, 140e, 171 }}

Mapping: [{{val| 1 3 2 3 7 1 1 }}, {{val| 0 -22 5 -3 -55 42 48 }}]

POTE generator: ~68/65 = 77.201

Vals: {{Val list| 31, 109eg, 140e, 171 }}

Mapping: [{{val| 1 3 2 3 5 }}, {{val| 0 -22 5 -3 -24 }}]

POTE generator: ~22/21 = 77.173

Vals: {{Val list| 31, 109, 140, 171e, 311e }}

Mapping: [{{val| 1 3 2 3 5 1 }}, {{val| 0 -22 5 -3 -24 42 }}]

POTE generator: ~22/21 = 77.158

Vals: {{Val list| 31, 109, 140, 311e, 451ee }}

Mapping: [{{val| 1 3 2 3 5 1 1 }}, {{val| 0 -22 5 -3 -24 42 48 }}]

POTE generator: ~22/21 = 77.162

Vals: {{Val list| 31, 109g, 140, 311e, 451ee }}

=== Hemitert ===

Comma list: 2401/2400, 3025/3024, 65625/65536

Mapping: [{{val| 1 3 2 3 6 }}, {{val| 0 -44 10 -6 -79 }}]

POTE generator: ~45/44 = 38.596

Vals: {{Val list| 31, 280, 311, 342 }}

Badness: 0.015633

Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095

Mapping: [{{val| 1 3 2 3 6 1 }}, {{val| 0 -44 10 -6 -79 84 }}]

POTE generator: ~45/44 = 38.588

Vals: {{Val list| 31, 280, 311, 964f, 1275f, 1586cff }}

Badness: 0.033573

Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095

Mapping: [{{val| 1 3 2 3 6 1 1 }}, {{val| 0 -44 10 -6 -79 84 96 }}]

POTE generator: ~45/44 = 38.589

Vals: {{Val list| 31, 280, 311, 653f, 964f }}

Badness: 0.025298

== Harry ==

Harry adds [[cataharry comma|cataharry]], 19683/19600, to the set of commas. It may be described as the 58&72 temperament. The period is half an octave, and the generator 21/20, with generator tunings of 9\130 or 14\202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices.

Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is {{multival| 12 34 20 30 …}}.

Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with {{multival| 12 34 20 30 52 …}} as the octave wedgie. [[130edo|130EDO]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.

Subgroup: 2.3.5.7

[[Comma list]]: 2401/2400, 19683/19600

[[Mapping]]: [{{val| 2 4 7 7 }}, {{val| 0 -6 -17 -10 }}]

[[POTE generator]]: ~21/20 = 83.156

{{Val list|legend=1| 14c, 58, 72, 130, 202, 534, 736b, 938b }}

[[Badness]]: 0.034077

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 4000/3993

Mapping: [{{val| 2 4 7 7 9 }}, {{val| 0 -6 -17 -10 -15 }}]

POTE generator: ~21/20 = 83.167

Vals: {{Val list| 14c, 58, 72, 130, 202 }}

Badness: 0.015867

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 364/363, 441/440

Mapping: [{{val| 2 4 7 7 9 11 }}, {{val| 0 -6 -17 -10 -15 -26 }}]

POTE generator: ~21/20 = 83.116

Vals: {{Val list| 14cf, 58, 72, 130, 332f, 462ef }}

Badness: 0.013046

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 243/242, 289/288, 351/350, 441/440

Mapping: [{{val| 2 4 7 7 9 11 9 }}, {{val| 0 -6 -17 -10 -15 -26 -6 }}]

POTE generator: ~21/20 = 83.168

Vals: {{Val list| 14cf, 58, 72, 130, 202g }}

Badness: 0.012657

In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = {{monzo|22 -1 -10 1}}. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)^1/8, giving just 7s, or 384^1/38, giving pure fifths.

Adding 3025/3024 extends to the 11-limit and gives {{multival| 38 -3 8 64 …}} for the initial wedgie, and as expected, 270 remains an excellent tuning.

Subgroup: 2.3.5.7

[[Comma list]]: 2401/2400, 29360128/29296875

[[Mapping]]: [{{val|1 31 0 9}}, {{val|0 -38 3 -8}}]

{{Multival|legend=1| 38 -3 8 -93 -94 27 }}

[[POTE generator]]: ~1024/875 = 271.107

{{Val list|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}

[[Badness]]: 0.035832

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 5632/5625

Mapping: [{{val|1 31 0 9 53}}, {{val|0 -38 3 -8 -64}}]

POTE generator: ~90/77 = 271.111

Vals: {{Val list| 31, 208, 239, 270 }}

 

=== 13-limit ===

Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095

Mapping: [{{val|1 31 0 9 53 -59}}, {{val|0 -38 3 -8 -64 81}}]

POTE generator: ~90/77 = 271.107

Vals: {{Val list| 31, 239, 270, 571, 841, 1111 }}

Badness: 0.017921

== Decoid ==

Decoid tempers out 2401/2400 and 67108864/66976875, as well as the [[15/14ths equal temperament|linus comma]], {{monzo| 11 -10 -10 10 }}. Either 8/7 or 16/15 can be used its generator. It may be described as the 130&270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the [[Qintosec family|qintosec temperament]].

Subgroup: 2.3.5.7

[[Comma list]]: 2401/2400, 67108864/66976875

[[Mapping]]: [{{val| 10 0 47 36 }}, {{val| 0 2 -3 -1 }}]

Mapping generators: ~15/14, ~8192/4725

 

 

[[POTE generator]]: ~16/15 = 111.099

{{Val list|legend=1| 10, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}

 

 

=== 11-limit ===

Comma list: 2401/2400, 5632/5625, 9801/9800

Mapping: [{{val| 10 0 47 36 98 }}, {{val| 0 2 -3 -1 -8 }}]

POTE generator: ~16/15 = 111.070

Vals: {{Val list| 10e, 130, 270, 670, 940, 1210, 2150c }}

Badness: 0.018735

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095

Mapping: [{{val| 10 0 47 36 98 37 }}, {{val| 0 2 -3 -1 -8 0 }}]

POTE generator: ~16/15 = 111.083

Vals: {{Val list| 10e, 130, 270, 940, 1210f, 1480cf }}

 

== Neominor ==

The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''.

Subgroup: 2.3.5.7

[[Comma list]]: 2401/2400, 177147/175616

[[Mapping]]: [{{val|1 3 12 8}}, {{val|0 -6 -41 -22}}]

 

 

 

[[Badness]]: 0.088221

Comma list: 243/242, 441/440, 35937/35840

Mapping: [{{val|1 3 12 8 7}}, {{val|0 -6 -41 -22 -15}}]

POTE generator: ~33/28 = 283.276

Vals: {{Val list| 72, 161, 233, 305 }}

Badness: 0.027959

Comma list: 169/168, 243/242, 364/363, 441/440

Mapping: [{{val|1 3 12 8 7 7}}, {{val|0 -6 -41 -22 -15 -14}}]

POTE generator: ~13/11 = 283.294

Vals: {{Val list| 72, 161f, 233f }}

Badness: 0.026942

== Emmthird ==

The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.

Subgroup: 2.3.5.7

[[Comma list]]: 2401/2400, 14348907/14336000

[[Mapping]]: [{{val|1 -3 -17 -8}}, {{val|0 14 59 33}}]

[[POTE generator]]: ~2744/2187 = 392.988

{{Val list|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}

[[Badness]]: 0.016736

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Comma list: 2401/2400, 3025/3024, 4000/3993

Map: [{{val|1 -13 -14 -9 -8}}, {{val|0 42 47 34 33}}]

POTE generator: ~14/11 = 416.718

Vals: {{Val list| 72, 167, 239, 311, 1316c }}

Badness: 0.022926

Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400

Mapping: [{{val|1 -13 -14 -9 -9 -47}}, {{val|0 42 47 34 33 146}}]

POTE generator: ~14/11 = 416.716

Vals: {{Val list| 72, 311, 694, 1005c, 1699cd }}

Badness: 0.020888

== Newt ==

This temperament has a generator of neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]].

Subgroup: 2.3.5.7

[[Comma list]]: 2401/2400, 33554432/33480783

[[Mapping]]: [{{val|1 1 19 11}}, {{val|0 2 -57 -28}}]

{{Multival|legend=1|2 -57 -28 -95 -50 95}}

[[POTE generator]]: ~49/40 = 351.113

{{Val list|legend=1| 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bbcc }}

[[Badness]]: 0.041878

Comma list: 2401/2400, 3025/3024, 19712/19683

Mapping: [{{val|1 1 19 11 -10}}, {{val|0 2 -57 -28 46}}]

POTE generator: ~49/40 = 351.115

Vals: {{Val list| 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b }}

Badness: 0.019461

Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095

 

POTE generator: ~49/40 = 351.117

Vals: {{Val list| 41, 229, 270, 581, 851, 2283b, 3134b }}

Badness: 0.013830

The amicable temperament tempers out the [[amity comma]] and the [[canousma]] in addition to the breedsma, and is closely associated with the canou temperament.  

While it extends well into 2.3.5.7.13/11, there are multiple reasonable places for the prime 11 and 13 in the interval chain. Amical (311 & 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 & 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 & 113) and pseudoamorous (14cf & 99ef) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semioctave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two.

Subgroup: 2.3.5.7

[[Comma list]]: 2401/2400, 1600000/1594323

[[Mapping]]: [{{val|1 3 6 5}}, {{val|0 -20 -52 -31}}]

[[POTE generator]]: ~21/20 = 84.880

[[Badness]]: 0.045473

=== Amical ===

Comma list: 2401/2400, 131072/130977, 1600000/1594323

Mapping: [{{val| 1 3 6 5 -8 }}, {{val| 0 -20 -52 -31 162 }}]

Vals: {{val list| 99, 212e, 311, 410, 721, 1032, 1343 }}

Badness: 0.101

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206

Mapping: [{{val| 1 3 6 5 -8 -5 }}, {{val| 0 -20 -52 -31 162 123 }}]

Vals: {{val list| 99, 212ef, 311, 410, 721, 1032 }}

Badness: 0.0499

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 6250/6237, 19712/19683

Mapping: [{{val| 1 3 6 5 14 }}, {{val| 0 -20 -52 -31 -149 }}]

POTE generator: ~21/20 = 84.8896

 

 

==== 13-limit ====

 

 

 

 

 

 

=== Pseudoamical ===

Comma list: 385/384, 1375/1372, 1600000/1594323

Mapping: [{{val| 1 3 6 5 -1 }}, {{val| 0 -20 -52 -31 63 }}]

POTE generator: ~21/20 = 84.9091

Vals: {{val list| 99, 113, 212, 961ccdeee }}

Badness: 0.0858

==== 13-limit ====

Comma list: 325/324, 385/384, 1375/1372, 19773/19712

Mapping: [{{val| 1 3 6 5 -1 2 }}, {{val| 0 -20 -52 -31 63 24 }}]

POTE generator: ~21/20 = 84.9127

Vals: {{val list| 99, 113, 212, 537cdeff, 749ccdeefff }}

Badness: 0.0470

=== Pseudoamorous ===

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 980000/970299

Mapping: [{{val| 1 3 6 5 7 }}, {{val| 0 -20 -52 -31 -50 }}]

POTE generator: ~21/20 = 84.9164

Vals: {{val list| 99e, 212e }}

Badness: 0.0566

==== 13-limit ====

Comma list: 243/242, 364/363, 441/440, 1875/1859

 

POTE generator: ~21/20 = 84.9164

 

 

 

=== Floral ===

Comma list: 2401/2400, 9801/9800, 14641/14580

Mapping: [{{val| 2 6 12 10 13 }}, {{val| 0 -20 -52 -31 -43 }}]

POTE generator: ~21/20 = 84.8788

Vals: {{val list| 198, 212, 410 }}

Badness: 0.0651

==== 13-limit ====

Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580

Mapping: [{{val| 2 6 12 10 13 19 }}, {{val| 0 -20 -52 -31 -43 -82 }}]

POTE generator: ~21/20 = 84.8750

Vals: {{val list| 198, 410 }}

Badness: 0.0370

 

 

 

 

 

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024

Mapping: [{{val| 1 3 6 5 3 6 }}, {{val| 0 -40 -104 -62 13 -65 }}]

POTE generator: ~40/39 = 42.4391

Vals: {{val list| 85c, 113, 198, 311, 509, 820f }}

 

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 -1 6 4 }}, {{val| 0 26 -37 -12 }}]

[[POTE generator]]: ~15/14 = 119.297

{{Val list|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}

Mapping: [{{val| 1 -1 6 4 -3 }}, {{val| 0 26 -37 -12 65 }}]

POTE generator: ~15/14 = 119.279

Vals: {{Val list| 10, 151, 161, 171, 332 }}

Mapping: [{{val| 1 -1 6 4 -3 4 }}, {{val| 0 26 -37 -12 65 -3 }}]

POTE generator: ~15/14 = 119.281

Vals: {{Val list| 10, 151, 161, 171, 332, 835eeff }}

Mapping: [{{val| 1 -1 6 4 -3 4 2 }}, {{val| 0 26 -37 -12 65 -3 21 }}]

POTE generator: ~15/14 = 119.281

Vals: {{Val list| 10, 151, 161, 171, 332, 503ef, 835eeff }}

{{see also| Ragismic microtemperaments #Parakleismic }}

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 31 34 26 }}, {{val| 0 -52 -56 -41 }}]

[[POTE generator]]: ~1296/875 = 678.810

{{Val list|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 19 0 6 }}, {{val| 0 -60 8 -11 }}]

[[POTE generator]]: ~57344/46875 = 348.301

{{Val list|legend=1| 31, 348, 379, 410, 441, 1354, 1795, 2236 }}

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 13 33 21 }}, {{val| 0 -32 -86 -51 }}]

[[POTE generator]]: ~2800/2187 = 428.066

{{Val list|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696, 6955dd }}

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 5 1 3 }}, {{val| 0 -18 7 -1 }}]

{{Multival|legend=1|18 -7 1 -53 -49 22}}

[[POTE generator]]: ~8/7 = 227.512

{{Val list|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}

Mapping: [{{val| 1 5 1 3 1 }}, {{val| 0 -18 7 -1 13 }}]

POTE generator: ~8/7 = 227.500

Vals: {{Val list| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}