No-twos subgroup temperaments: Difference between revisions

For extensions to this temperament that include the octave, see [[sensamagic clan]]. Non-octave extensions will be documented below.

This temperament uses the canonical extension for prime 13 described at [[No-twos subgroup temperaments #Tridecimal mintaka|Tridecimal mintaka]].

For extensions to this temperament that include the prime 2, see [[Canopic clan]]. No-twos extensions will be documented below.

This temperament is the no-twos restriction of [[squares]], and as such is named after a star that belonged to the obsolete constellation of Quadrans Muralis, whose name has to do with squares. However, seeing the sheer complexity and size of the commas, nekkar is much more naturally thought of as 3.5.7.11 than 3.5.7, whereupon it becomes a strong extension of [[mintaka]].

Polaris tempers out the comma 177147/171875, and thus equates 7 5/3's with 15/11, or equivalently 7 9/5's with 11/9.  

[[Gencom]]: [3/1 5/3; 177147/171875]

[[Sval]] [[mapping]]: [{{val|1 2 1}}, {{val|0 1 -6}}]

[[POTE generator]]: ~5/3 = 892.6

 

[[EDT]]s: 17, 15, 32, 49, 13[+11], 47, 19, 11[+11], 81, 66, 79[+11], 62[+11], 28[+11], 21[-11]

{{main|Deneb}}

[[Gencom]]: [3/1 11/9; 6655/6561]

 

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

[[POTE generator]]: ~[[11/9]] = 340.242

[[EDT]]s: 28, 11, 17, 6, 39, 5, 67, 45, 50, 16, 23, 73, 61, 62

Fomalhaut is an extension of Deneb to higher limits that splits the interval of [[11/3]] in three.  

The 23-limit version of Fomalhaut was created first, as an attempt to approximate the no-2s, no-7s 23-limit as accurately as possible using 25 to 35 notes per equave, defined as the b28 & b33 temperament in this limit. Then the lower limit versions were created by simply extrapolating the temperament downwards.

[[Subgroup]]: 3.5.11.13

 

[[Gencom]]: [3/1 99/65; 6655/6561 274625/264627]

[[Sval]] [[mapping]]: [{{val|1 5 1 -2}}, {{val|0 -9 3 11}}]

[[POTE generator]]: ~[[99/65]] = 748.0156

[[EDT]]s: {{EDs|b28, b5, b33, b23f, b61, b56f, b38c, b10cf, b66c, b51ff|equave=t}}

* Complexity: 1.561892

* Adjusted Error: 6.495941 cents

* TE Error: 1.755451 cents/octave

[[Subgroup]]: 3.5.11.13.17

[[Comma list]]: 1105/1089, 4225/4131, 6655/6561

[[Gencom]]: [3/1 99/65; 1105/1089 4225/4131 6655/6561]

[[Sval]] [[mapping]]: [{{val|1 5 1 -2 1}}, {{val|0 -9 3 11 4}}]

[[POTE generator]]: ~[[17/11]] = 748.0236

[[EDT]]s: {{EDs|b28, b5, b33, b23f, b61, b56f, b38c, b10cf, b66c, b51ffg|equave=t}}

* Complexity: 1.418914

* Adjusted Error: 6.431616 cents

* TE Error: 1.573498 cents/octave

==== 3.5.11.13.17.19 ====

[[Subgroup]]: 3.5.11.13.17.19

[[Comma list]]: 247/243, 325/323, 1105/1089, 4675/4617

[[Gencom]]: [3/1 99/65; 247/243 325/323 1105/1089 4675/4617]

[[Sval]] [[mapping]]: [{{val|1 5 1 -2 1 7}}, {{val|0 -9 3 11 4 -11}}]

[[POTE generator]]: ~[[17/11]] = 747.9960

[[EDT]]s: {{EDs|b28, b33, b5, b61, b56f, b23f, b38ch, b66ch, b89fgh, b10cfh|equave=t}}

 

 

 

 

 

 

[[EDT]]s: {{EDs|b28, b5, b33, b23f, b61, b56f, b38ch, b10cfhi, b66ch, b51ffg|equave=t}}

 

Effectively a [[microtemperament]], Alnilam takes a generator of an 81/55 flat fifth and equates 9 of them with [[11/9]]. The name was given by [[User:CompactStar|CompactStar]] to continue with the theme of naming no-twos temperaments after proper star names, but also to indirectly reference [[mavila]].

[[Comma list]]: {{monzo|0 -35 9 0 10}}

 

[[Sval]] [[mapping]]: [{{val|1 5 -1}}, {{val|0 -10 9}}]

[[CTE tuning|CTE generator]]: ~81/55 = 672.410

[[EDT]]s: 99, 17, 82, 116, 181, 65, 14[-5], 280, 48, 215, 31, 133, 314, 263

{{main|Mintaka}}

Extensions to prime 5 are covered at [[No-twos subgroup temperaments#Mintra|Mintra]] and [[No-twos subgroup temperaments#3.5.7.11 subgroup|Nekkar]].

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

 

Sval mapping generators: ~3, ~21/11

* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[11/7]] = 778.961

* [[CWE]]: ~3 = 1\1ed3, ~[[11/7]] = 778.803

[[Support]]ing [[ET]]s: {{EDs|b22, b5, b17, b39, b12, b61, b27, b7, b83, b49, b56, b32, b29, b100|equave=t}}

=== Tridecimal Mintaka ===

This extension to prime 13 works in the sharper half of the Mintaka tuning range, where the most important pental extension is [[No-twos subgroup temperaments#Mintra|Mintra]].

[[Subgroup]]: 3.7.11.13

[[Comma list]]: 1331/1323, 218491/216513

[[Sval]] [[mapping]]: [{{val| 1 0 1 10}}, {{val| 0 3 2 -13}}]

 

Sval mapping generators: ~3, ~21/11

[[Optimal tuning]]s:

* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[11/7]] = 780.155

* [[CWE]]: ~3 = 1\1ed3, ~[[11/7]] = 780.183

[[Support]]ing [[ET]]s: {{EDs|b39, b22, b17, b5f, b61, b56, b100, b139f, b95, b178ef, b83f, b134, b73f, b217ef|equave=t}}

This extension to prime 13 works in the flatter half of the Mintaka tuning range, where the most important pental extension is [[No-twos subgroup temperaments#3.5.7.11 subgroup|Nekkar]].

[[Subgroup]]: 3.7.11.13

[[Comma list]]: 1331/1323, 351/343

[[Sval]] [[mapping]]: [{{val| 1 0 1 -3}}, {{val| 0 3 2 9}}]

Sval mapping generators: ~3, ~21/11

[[Optimal tuning]]s:

* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[11/7]] = 774.432

* [[CWE]]: ~3 = 1\1ed3, ~[[11/7]] = 774.782

[[Support]]ing [[ET]]s: {{EDs|b5, b27, b22, b32, b17f, b37f, b12ff, b49, b59, b42df, b76, b39ff, b86d, b71f|equave=t}}

Mebsuta is a microtemperament in the 3.7.11 subgroup that sets the relative sizes of [[9/7]] and [[11/9]] to be in the ratio of 5:4; its generator is identifiable as the ratio between these intervals, 81/77. It produces a 21L 1s [[MOS scale]] against the tritave, which serves as a well-temperament of [[22edt]]; that scale's chroma is identified with [[1331/1323]].

[[Sval]] [[mapping]]: [{{val| 1 2 2}}, {{val| 0 -5 4 }}]

 

Sval mapping generators: ~3, ~81/77

* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[81/77]] = 86.957

* [[CWE]]: ~3 = 1\1ed3, ~[[81/77]] = 86.957

[[Support]]ing [[ET]]s: {{EDs|b22, b175, b153, b197, b131, b328, b109, b21, b219, b87, b43, b372, b65, b23|equave=t}}

[[Subgroup]]: 3.7.11.19

 

[[Sval]] [[mapping]]: [{{val| 1 2 2 3}}, {{val| 0 -5 4 -7}}]

Sval mapping generators: ~3, ~81/77

[[Optimal tuning]]s:

* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[81/77]] = 86.929

* [[CWE]]: ~3 = 1\1ed3, ~[[81/77]] = 86.932

[[Support]]ing [[ET]]s: {{EDs|b22, b175, b197, b153, b131, b219, b372, b109, b328, b241, b87, b21, b65, b43|equave=t}}

==== 3.5.7.11.19 subgroup ====

Tempering out [[12005/11979]], the unisquary comma, sets the chroma 1331/1323 equal to [[245/243]], producing an accurate if complex mapping for prime 5 at 32 generators up; it is notable that this sets eight [[11/9]]s equal to [[5/1]], which is the 3.5.11 restriction of [[mohaha]].

 

[[Subgroup]]: 3.5.7.11.19

 

 

[[Sval]] [[mapping]]: [{{val| 1 0 2 2 3}}, {{val| 0 32 -5 4 -7}}]

 

[[Optimal tuning]]s:

* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[81/77]] = 87.065

* [[CWE]]: ~3 = 1\1ed3, ~[[81/77]] = 87.066

 

Adhara cleaves the step of Mebsuta in three to produce a remarkable Don Page temperament for the chord 7:9:11:13:17 (that is, setting [[13/11]] to two-thirds of 9/7, and [[17/13]] to four-thirds of 11/9). It can be extended to even higher subgroups fairly naturally, and encompasses several prominent tunings within its structure (such as [[65edt]]~[[41edo]], [[131edt]], and [[197edt]]).

[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2}}, {{val| 0 -15 12 22 38}}]

 

Sval mapping generators: ~3, ~119/117

* PETE (Pure-Equaves TE): ~3 = 1\1ed3, ~[[119/117]] = 28.979

* [[CTE]]: ~3 = 1\1ed3, ~[[119/117]] = 28.970

This includes the natural extension of Mebsuta to prime 19.

 

 

 

 

 

 

 

 

[[Comma list]]: 513/512, 729/728, 833/832, 969/968, 3971/3969

[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2 3}}, {{val| 0 -15 -7 12 22 38 -21}}]

Sval mapping generators: ~3, ~64/63

[[Optimal tuning]]s:

* PETE (Pure-Equaves TE): ~3 = 1\1ed3, ~[[64/63]] = 28.978

* [[CTE]]: ~3 = 1\1ed3, ~[[64/63]] = 28.975

[[Optimal ET sequence]]: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[328edt|b328]], [[525edt|b525]], [[722edt|b722]], [[1247edt|b1247âf]], [[1969edt|b1969ââf]]

 

 

[[Subgroup]]: 3.5.7.8.11.13.17.19.23

 

[[Comma list]]: 361/360, 441/440, 513/512, 729/728, 833/832, 969/968, 1127/1125

 

[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2 2 3 4}}, {{val| 0 -35 -15 -7 12 22 38 -21 -75}}]

 

 

 

[[Optimal ET sequence]]: [[65edt|b65i]], [[66edt|b66i]], [[131edt|b131]]