Subgroup temperaments: Difference between revisions

A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.  

For temperaments that omit various prime harmonics, see:

* [[No-twos subgroup temperaments]] (additionally, [[Catalog of 3.5.7 subgroup rank two temperaments]]).

= Composite subgroup temperaments =

=== Darian calendar ===

Darian calendar is described as 24 & 668 temperament in the 2.3.11.19 [[subgroup]] and is named after a certain calendar layout by the same name. The generator is close to the [[36/35]] quartertone, and this allows an extension to the 2.3.35.11.19 subgroup. 5 of them make [[11/8]], 8 of them make [[3/2]], and 6 of them make [[32/19]].

==== 2.3.11.19 subgroup ====

The temperament is simplest in this subgroup, although there is a tradeoff of breaking up the simplicity of the 36/35 quartertone.

[[Subgroup]]: 2.3.11.19

{{Mapping|legend=2| 4 5 13 18 | 0 8 5 -6 }}

: sval mapping generators: ~6291456/5285401, ~25289/24576

[[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257

[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ...

==== 2.3.35.11.19 subgroup ====

668edo does not map 36/35 consistently, with its own [[direct approximation]] being 27 steps while the direct approximations of its constituent odd harmonics do not sum to that same amount: 3/2, 8/5, and 8/7 are 391, 453, and 129 steps, respectively, and 391 + 391 + 453 + 129 - 668 - 668 = 28, ≠ 27.

Subgroup: 2.3.35.11.19

Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }}

: sval mapping generators: ~2240/1881, ~36/35

Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288

[[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...

See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].

Commatose is a [[Dual-fifth temperaments|dual-fifth temperament]] which uses the Pythagorean comma as a generator. It was developed by [[Eliora]] to highlight the near-perfect expression of 9/8 by [[1789edo]], while at the same time the fact that it completely misses 3/2. It is described as the 460 & 1329 temperament. In the 13-limit extension 24 generators are equal to [[~]][[13/9]].

[[Subgroup]]: 2.9.5.7

[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}

{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765

{{Optimal ET sequence|legend=1| 460, 869, 1329 }}

[[Badness]]: 0.611

Subgroup: 2.9.5.7.11

Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}

Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}

Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

==== 2.9.5.7.11.13 ====

Subgroup: 2.9.5.7.11.13

Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156

Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}

Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.0564

Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a [[7L 4s|daemotonic 7L 4s]] scale of reasonable hardness (hence the name ― daemo- + tertiaschis), which is notoriously difficult to approximate with simple JI or RTT methods.

Subgroup: 2.9.5.7.33.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}

Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982

[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}

Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.

[[Subgroup]]: 2.9.5.7

[[Comma list]]: 225/224, 3125/3087

{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170

{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}

Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]

Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 & 35 temperament.

[[Subgroup]]: 2.9.5.7.13

[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]

{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}

{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090

{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents

Related temperament: [[Schismatic family #Garibaldi|Cassandra]]

Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.

Subgroup: 2.9.5.7.11

Comma list: 100/99, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 | 0 1 -4 -7 -9 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743

{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}

Related temperament: [[Schismatic family #Garibaldi|Andromeda]]

Subgroup: 2.9.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414

{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}

== 2.3.25 subgroup ==

This is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a [[shrubmajor]] third (~425c) in quarter-comma tuning.

Subgroup: 2.3.25

Edo join: 17 & 12

Comma list: [[2048/2025]]

{{Mapping|legend=2| 1 1 7| 0 1 -4}}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136

''See also: [[Reversed meantone]]''

Edo join: 17 & 12

Comma list: 2048/2025, 576/575, 82/81

{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264

This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.

Subgroup: 2.3.7.23.25.41.59

Edo join: 17 & 12

Comma list: 64/63, 225/224, 162/161, 82/81, 177/175

{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387

{{See also| Chromatic pairs #Glacial }}

[[Comma list]]: 45/44, 65/64, 81/80

{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}

: [[gencom]]: [2 9/8; 45/44 65/64 81/80]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151

{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}

[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents

* ''[[Thundersnow]]'' - [[Sevish]] (2021)

== 2.9.7 subgroup ==

=== Mabon ===

Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.

Subgroup: 2.9.7

Comma basis: 44957696/43046721

Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]

Optimal tuning (CTE): ~729/448 = 870.792

{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...

 

 

Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]

 

Optimal tuning (CTE): ~16/11 = 870.966

 

{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}

 

 

[[Comma list]]: 41503/41472, 322102/321489

 

 

: mapping generators: ~2, ~77/72

 

{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}

 

 

 

{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}

 

[[Badness]]: 0.00263

 

 

 

 

 

 

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents

{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}

 

=== Machine ===

 

[[Subgroup]]: 2.9.7.11

{{Mapping|legend=2| 1 0 6 13 | 0 1 -1 -3 }}

: sval mapping generators: ~2, ~9

{{Mapping|legend=3| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}

: [[gencom]]: [2 8/7; 64/63 99/98]

* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128

* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843

{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}

[[Badness]]: 0.00233

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 896/891, 26411/26244

 

{{Mapping|legend=2| 1 0 -1 6 | 0 5 6 -4 }}

 

: sval mapping generators: ~2, ~14/9

 

{{Mapping|legend=3| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}

 

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782

 

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}

 

[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents

 

Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]

== 2.9.7.13.17 subgroup ==

=== Novisept ===

Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4. It can be formed by doubling both generator and period of [[gizzard]].

[[Subgroup]]: 2.9.7.13.17

[[Comma list]]: 729/728, 442/441, 833/832

[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836

Badness (Dirichlet): 0.142

Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[1331/1296]]

{{Mapping|legend=2|1 1 2|0 3 2}}

[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060

{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}

Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[131769/131072]]

{{Mapping|legend=2|1 1 4|0 4 -1}}

[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863

{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]

Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].

[[Subgroup]]: 2.9.15.7

[[Comma list]]: 225/224, 245/243

{{Mapping|legend=2| 1 0 2 -1 | 0 5 3 6 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Comma list]]: 1029/1024

 

 

 

 

 

 

{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}

==== 2.9.5.21 ====

''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''

 

 

 

 

 

 

 

===== 2.9.5.21.11 =====

Subgroup: 2.9.5.21.11

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

{{Mapping|legend=2| 1 2 5 | 0 3 1 }}

 

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9692{{c}}, ~85/64 = 491.5853{{c}}

 

 

== 2.75.9/7.85 subgroup ==

 

 

[[Comma list]]: 2025/2023 ({{monzo| 2 -2 1 0 }}), 24576/24565 ({{monzo| 13 1 0 -3 }})

: mapping generators: ~2, ~85/32

[[Optimal tuning]]s:  

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0241{{c}}, ~85/64 = 491.3358{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.3290{{c}}

{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441* }}

== 4.3.5 subgroup ==

[[Comma list]]: 15625/15552

{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}

: Mapping generators: ~4, ~5/3

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941

[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}

{{Main| Tetrameantone }}

[[Comma list]]: 81/80

{{Mapping|legend=2| 1 1 2 | 0 -1 -4 }}

: Mapping generators: ~4, ~4/3

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761

[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}

[[Subgroup]]: 4.3.5

[[Comma list]]: 3125/3072

{{Mapping|legend=2| 1 0 1 | 0 5 1 }}

: Mapping generators: ~4, ~5/4

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059

[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}

[[Subgroup]]: 4.3.5

[[Comma list]]: 256/243

{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}

: Mapping generators: ~4, ~16/15

[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062

[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}

[[Subgroup]]: 4.6.5

[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}

{{Mapping|legend=2| 1 0 -4| 0 1 4 }}

: mapping generators: ~4, ~6

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214

[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69

[[Subgroup]]: 4.6.5.7

[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}

{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622

[[Support]]ing [[ET]]s: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]

The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.

Fourwar is named after the closely related [[hemiwar]] temperament.

[ ⟨ 1 0 1 ]

⟨25.21211, 47.81337]

⟨2399.397, 3101.829, 2787.126]

[8, 1, -8⟩ (393216:390625)

q99, q62, q37, q161, q136, q198, q25, q124, q74, q235

[ ⟨ 1 0 1 1 ]

⟨2399.420, 3101.846, 2787.150, 3368.747]

⟨-0.580, -0.109, 0.837, -0.079]

[-2, -1, -2, 4⟩ (2401:2400)

[3, 0, -5, 2⟩ (3136:3125)

[5, 1, -3, -2⟩ (6144:6125)

</pre>

==== 4.6.5.7.11 ====

[1, 0, 1, -4, 2⟩ (2420:2401)

==== 4.6.5.7.11.13 ====

[ ⟨ 1 0 1 1 1 0 ]

⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]

⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]

[-1, 0, 3, -3, 1, 0⟩ (1375:1372)

</pre>

[ ⟨ 1 0 1 1 1 0 1 ]

⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]

⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]

[0, 1, -1, 0, 1, -1, 0⟩ (66:65)

[1, 1, 1, -1, 0, 0, -1⟩ (120:119)

[1, 2, 0, 0, -1, -1, 0⟩ (144:143)

[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)

q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg

[ ⟨ 1 0 1 1 1 0 1 1 ]

⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]

⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]

[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)

q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh

[ ⟨ 1 0 1 1 1 0 1 1 0 ]

⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]

⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]

Complexity 1.115920

TE Error 2.100561 cents/octave

[[Subgroup]]: 4.9.25

[[Comma list]]: [[6561/6400]]

{{Mapping|legend=2| 1 3 4 | 0 1 4 }}

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429

[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]

[[Subgroup]]: 4.9.49

[[Comma list]]: 4096/3969

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190

[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49

Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."

[[Comma list]]: 64/63

{{Mapping|legend=2| 1 0 2 | 0 1 -1 }}

: sval mapping generators: ~8, ~9

: [[gencom]]: [8 9/8; 64/63]

[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898

[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}

[[Badness]]: 0.0215 × 10^-3

== 2.5/3.… subgroups ==

=== Magicaltet ===

Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.

[[Subgroup]]: 2.5/3.7.11

[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})

: mapping generators: ~2, ~5/3

: [[gencom]]: [2 6/5; 100/99 385/384]

[[Optimal tuning]]s:

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351

{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}

: <nowiki/>* wart for 5/3

[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents

{{See also | Chromatic pairs #Starlingtet }}

Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.  

[[Subgroup]]: 2.5/3.7/3

[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})

{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}