Subgroup temperaments: Difference between revisions

=== 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.

{{Mapping|legend=3| 4 5 13 18 | 0 8 5 -6 }}

[[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}}, ...

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

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

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

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

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

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

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

===== Sburb =====

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

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

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

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

Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]

[[Comma list]]: 169/168, 540/539, 729/728

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518

{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}

A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.

[[Comma list]]: [[50/49]]

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

[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962

[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}

Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].  

[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830

Badness (Sintel): 0.119

Edson is related to [[pele]] and [[andromeda]].  

[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}

{{Mapping|legend=5| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414

Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].  

[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]

{{Mapping|legend=5| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}

: [[gencom]]: [2 15/13; 352/351 676/675 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491

{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}

{{Distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.

Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.

[[Comma list]]: 364/363, 441/440, 1001/1000

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016

{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}

Related temperament: [[hemithirds]], [[newt]]

[[Comma list]]: 2401/2400

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966

{{Optimal ET sequence|legend=1|7, 10, 17}}

==== Bridgetown ====

{{See also| Chromatic pairs #Bridgetown }}

Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].  

[[Subgroup]]: 2.3.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]]

{{Mapping|legend=3| 1 0 -6 -1 | 0 2 9 3 }}

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

: [[gencom]]: [2 15/13; 352/351 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents

=== Blackweed ===

Blackweed is a [[restriction]] of undecimal [[blackwood]] as it tempers out 256/243 alike but in the 2.3.11/7 subgroup. 20edo is close to the optimum, which has 4\20 as the period and 420{{c}} as the generator.

[[Subgroup]]: 2.3.11/7

[[Comma list]]: {{monzo| 8 -5 }} (256/243)

{{Mapping|legend=3| 5 8 0 | 0 0 1 }}

: mapping generators: ~9/8, ~11/7

[[Optimal tuning]]s:

* [[Tp tuning|subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}}

* [[Tp tuning|subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}

: error map: {{val| 0.000 +18.045 +2.475 }}

=== Pepperoni ===

{{See also| Chromatic pairs #Pepperoni }}

Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.

[[Subgroup]]: 2.3.11/7.13/7

[[Comma list]]: 352/351, 364/363

{{Mapping|legend=3| 1 0 7 12 | 0 1 -4 -7 }}

{{Mapping|legend=5| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}

: [[gencom]]: [2 3/2; 352/351 364/363]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856

{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*^† }}

The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

[[Subgroup]]: 2.3.13/5

[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}

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

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}

* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]

==== Tobago ====

{{See also| Chromatic pairs #Tobago }}

Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].

[[Subgroup]]: 2.3.11.13/5

[[Comma list]]: [[243/242]], [[676/675]]

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

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

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312

{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents

[[Subgroup]]: 2.3.11.13/5.17

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

* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)

* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)

{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}

Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]

[[Subgroup]]: 2.3.7.13/5

[[Comma list]]: 64/63, 91/90

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910

{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents

Scales: [[Oceanfront scales]]

=== Seventeen-cot ===

Seventeen-cot is a rank-2 temperament in the 2.3.13/5 and 2.3.11/5.13/5 subgroups. It tempers out the [[Tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~2250/2197 or ~169/165 generator which is a 17th of a ~3/2 perfect fifth. It can be described as the 29 & 146 temperament in these subgroups.

====2.3.13/5 subgroup====

Comma basis: {{monzo| -6 -11 17 }} (2.3.13/5)

edo join: 29 & 146

{{Mapping|legend=3| 1 1 1 | 0 17 11 }}

* WE: ~2 = 1200.0001354{{c}}, ~2250/2197 = 41.2914993{{c}}

: error map: {{val| +0.0001354 +0.0006234 -0.0073197}}

* CWE: ~2 = 2/1, ~2250/2197 = 41.2915011{{c}}

edos: 29, 465, 494, 436, 523, 407, 378, 349, 30[-3], 28[+3], 320, 291, 59[-3], 262

====2.3.11/5.13/5 subgroup====

Comma basis: 225000/224939, 43940/43923

 

edo join: 29 & 146

 

{{Mapping|legend=3| 1 1 1 1 | 0 17 4 11 }}

: mapping generators: ~2, ~169/165

 

* WE: ~2 = 1199.9934923{{c}}, ~169/165 = 41.2918271{{c}}

: error map: {{val| -0.0065077 -0.0004485 +0.1565720 -0.0103579}}

* CWE: ~2 = 2/1, ~169/165 = 41.2917463{{c}}

: error map: {{val| 0.0000000 +0.0046870 +0.1627569 -0.0043781}}

 

edos: 29, 465, 494, 436, 523, 407, 378, 349, 320, 291, 30[-3], 262, 28[+3], 233

 

[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})

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]].

{{Mapping|legend=3| 1 9 6 13 | 0 -298 -188 -521 }}

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

[[Badness]]: 0.611

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 }}

Badness: 0.165

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

{{See also|Schismatic family#Tertiaschis}}

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.

{{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}}

{{See also|Schismatic family #Garibaldi}}

{{See also|No-threes subgroup temperaments #Frostburn}}

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.

{{Mapping|legend=3| 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]]

==== 2.9.5.7.13 ====

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=3| 1 0 15 25 -28 | 0 1 -4 -7 10 }}

 

{{Mapping|legend=5| 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.

{{Mapping|legend=3| 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]]

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

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

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