Subgroup temperaments: Difference between revisions

[[Comma list]]: 29282/29241 ({{monzo| 1 -4 0 4 -2 }}), 42875/42768 ({{monzo| -4 -5 3 -1 0 }}), 885115/884736 ({{monzo| -15 -3 1  3 1 }})

* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1199.4965{{c}}, ~3/2 = 703.1192{{c}}

* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}

Related temperaments include [[#Bridgetown|bridgetown]], [[aberschismic family #Namaka|namaka]], [[schismatic family #Hemigari|hemigari]], [[#Barbados|barbados]], and [[the Archipelago #Parizekmic|parizekmic]].  

* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1199.7072{{c}}, ~26/15 = 951.2727{{c}}

* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5016{{c}}

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

* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}}

* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}}

[[Comma list]]: 2401/2400

* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1200.0206{{c}}, ~21/20 = 350.9724{{c}}

* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 350.9743{{c}}

==== 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)