No-threes subgroup temperaments: Difference between revisions

 

* For no-fives, see [[#No-threes-or-fives subgroup temperaments]].

== 2.5.7 temperaments ==

 

[[Comma list]]: 78125/76832

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

: Sval mapping generators: ~2, ~28/25

[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389

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

Subgroup: 2.5.7.11

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

: Sval mapping generators: ~2, ~28/25

Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745

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

{{See also| Chromatic pairs #Mabilic }}

Given below is the no-three version of [[Mavila family#Armodue|armodue]], [[Mabila family#Semabila|semabila]], and [[Magic family#Trismegistus|trismegistus]]. It is the 7 & 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma.

[[Comma list]]: 1071875/1048576

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

: [[gencom]]: [2 175/128; 1071875/1048576]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 527.236

{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305bc }}

[[Tp tuning #T2 tuning|RMS error]]: 0.7729 cents

Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]] (and [[valentine]]). Rainy is notable theoretically as it equates ([[2/1]])/([[5/4]])³ (128/125, the lesser diesis) with ([[2/1]])/([[8/7]])⁵ (the 2.7-subgroup [[cloudy comma]], which is similar to the 2.5-subgroup lesser diesis in that tempering it out tunes the 8/7 about 8.8{{cent}} sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7{{cent}}). By tempering out their difference, stacked 5s and stacked 7s become easier to navigate, using the general-purpose diesis to simplify clusters. (Note that this analysis assumes a [[lattice]]-based conceptualization of [[JI]] which is often called "stacking-based"; see [[taxonomies of xen approaches]].)

A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.

[[Comma list]]: [[2100875/2097152]]

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

[[Gencom]]: [2 256/245; 2100875/2097152]

[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]

Optimal tuning ([[POTE]]): ~256/245 = 77.205

{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents

=== French decimal ===

Conceived upon the fact that 1789edo has an excellent 5/4, and uses it as the generator. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, a 1525 & 1789 temperament is obtained.

Subgroup: 2.5.7

Comma basis: {{monzo|372 -159 -1}}

Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}]

Optimal tuning (CTE): ~5/4 = 386.360

{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...

==== 2.5.7.11 subgroup ====

Subgroup: 2.5.7.11

Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}

Sval mapping:  [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]

Optimal tuning (CTE): ~5/4 = 386.361

{{Optimal ET sequence|legend=1|264, 733}}, ...

Subgroup: 2.5.7.11.13

Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]

Optimal tuning (CTE): ~5/4 = 386.361

{{Optimal ET sequence|legend=1|1525, 1789}}, ...

Described as the 2.5.7 subgroup 1407 & 1789 temperament, and named after an eponymous historical event which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Double bastille|double bastille]].

Subgroup: 2.5.7

Comma list: {{Monzo|1426 -596 -15}}

Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]

Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243

{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...

[[Subgroup]]: 2.5.7.11

[[Comma list]]: 56/55, 128/125

{{Mapping|legend=3| 3 0 7 9 11| 0 0 0 -1 -1 }}

: [[gencom]]: [5/4 8/7; 56/55 128/125]

[[Optimal tuning]] ([[POTE]]): ~5/4 = 1\3, ~8/7 = 228.275

{{Optimal ET sequence|legend=1| 3, 6, 9, 15, 21 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.422 cents

'''Ostara''' is a temperament that is derived from 93 & 524 solar calendar leap rule scale. It was initially defined by taking the 2.7.13.17.19 subgroup, but it can also be intepreted in general no-threes 19-limit.  

Ostara can also refer to a collection of temperaments which temper out 16807/16796.

[[Subgroup]]: 2.5.7.11

[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200

[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]

[[Optimal tuning]] ([[POTE]]): ~5120/3773 = 529.003¢

Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]

Optimal tuning (POTE): ~1664/1225 = 529.003¢

{{Optimal ET sequence|legend=1| 93, 245e, 338, 431, 1386c }}

Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]

Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251

Optimal tuning (POTE): ~1664/1225 = 529.003¢

{{Optimal ET sequence|legend=1| 93, 338, 431, 955c, 1386cg }}

Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]

Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875

Optimal tuning (POTE): ~19/14 = 529.003¢

{{See also|Tricesimoprimal miracloid/Eliora's approach|l1=Eliora's approach to tricesimoprimal miracloid}}

Described as the 52 & 1789 temperament in the 2.5.7.11.19.29.31 subgroup, with harmonics specifically selected for 52edo and 1789edo. Its generator is [[31/29]], which is also close to the secor. Since it is conceived as the temperament in the above specific subgroup, it makes no sense to name it for smaller subgroups. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable [[well temperament]] for 52edo.

Subgroup: 2.5.7.11.19.29.31

Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688

Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]

Optimal tuning (CTE): ~58/31 = 1084.628

{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...

{{See also| Chromatic pairs #Huntington }}

Huntington may be described as the 10 & 27 temperament in the 2.5.7.13 subgroup.  

[[Comma list]]: [[640/637]], [[10985/10976]]

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

: [[gencom]]: [2 16/13; 640/637 10985/10976]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/13 = 357.002

{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3452 cents

==== Silver ====

{{See also| Chromatic pairs #Silver }}

Silver can be described as the 10 & 27 temperament in the 2.5.7.13.17 subgroup.  

[[Subgroup]]: 2.5.7.13.17

[[Comma list]]: [[170/169]], [[640/637]], [[5525/5488]]

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

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/8 = 842.711

{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents

[[Comma list]]: 625/616

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

: mapping generators: ~2, ~5, ~11

[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653

{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}

Temperaments discussed elsewhere include:

* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]

This temperament has a structure very similar to [[mavila]] but is somewhat more accurate because the generator is a flat [[11/8]] rather than a sharp [[4/3]]. The major third is still ~[[5/4]], but the minor third is now ~[[64/55]] instead of ~[[6/5]].

[[Subgroup]]: 2.5.11

[[Comma list]]: 1331/1280

[[Mapping]]: [{{val|1 1 3}}, {{val|0 3 1}}]

[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[11/8]] = 529.846

{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}

Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, is related to [[wizard]].  

[[Comma list]]: [[15625/15488]]

: [[gencom]]: [125/88 5/4; 15625/15488]

 

[[Optimal tuning]] ([[POTE]]): ~125/88 = 1\2, ~5/4 = 383.768

{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3997

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

: Mapping generators, ~2, ~[[55/32]]

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[55/32]] = 928.032

Sephiroth is the no-7 restriction of [[muggles]].  

[[Subgroup]]: 2.5.11.13.17

[[Comma list]]: 65/64, 170/169, 221/220

{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}

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

: [[gencom]]: [2 5/4; 65/64 170/169 221/220]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 372.236

{{Optimal ET sequence|legend=1| 10, 13, 16, 29 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.774 cents

=== Trader ===

[[Subgroup]]: 2.5.13

[[Comma list]]: [[26/25]]

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

: Mapping generators, ~2, ~[[5/4]]

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[5/4]] = 407.079

=== Superquintal ===

[[Comma list]]: 64000000/62748517

{{Mapping|legend=2|1 5 6|0 -7 -6}}

: Mapping generators, ~2, ~13/10

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281

=== Vengeance ===

Another lower-error replica of mavila, with the fifth being ~[[25/17]] instead of ~[[3/2]].

 

[[Comma list]]: 78608/78125

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[34/25]] = 529.095

{{Optimal ET sequence|legend=1|7g, 9, 25, 34, 93, 127, 288, 415}}

== No-threes-or-fives subgroup temperaments ==

Amaranthine is the high-accuracy 2.7.11 subgroup strong restriction of [[Gamelismic clan#11-limit 3|undecimal mothra]].

{{Mapping|legend=2| 1 2 -3 | 0 1 8 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~7/4 = 968.913

Badness (Dirichlet): 0.031

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

: [[gencom]]: [2 11/8; 343/338 847/832]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 540.099

 

[[Tp tuning #T2 tuning|RMS error]]: 1.282 cents

Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup.  

[[Subgroup]]: 2.7.11.13

[[Comma list]]: [[1573/1568]], [[15488/15379]]

{{Mapping|legend=3| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}

: [[gencom]]: [2 14/11; 1573/1568 15488/15379]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/11 = 421.309

{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 225, 319cd, 413bcd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.4043 cents

Voltage is the 3 & 7 temperament in the 2.7.13 subgroup.  

[[Comma list]]: [[28672/28561]]

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

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

: [[gencom]]: [2, 16/13; 28672/28561]

[[Optimal tuning]]:

* [[POTE]]: ~2 = 1\1, ~16/13 = 357.677

* [[TOP tuning|POTT]]: ~2 = 1\1, ~16/13 = 357.794 (1200 - 300 log₂(7))

 

[[Tp tuning #T2 tuning|RMS error]]: 0.1423 cents

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

: Mapping generators, ~2, ~[[17/14]]

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 324.446

{{Mapping|legend=2|1 0 1|0 10 11}}

: Mapping generators, ~2, ~[[17/14]]

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 336.858

[[Gencom]]: [2 64/53; 1048576/1042139]

[[Mapping]]: [{{val|1 0 6}}, {{val|0 3 -1}}]]

[[POTE generator]]: ~64/53 = 323.034

Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, is generated by ~13/11. Two generator steps give ~11/8 and three generator steps give ~13/8.  

[[Comma list]]: [[1352/1331]]

: [[gencom]]: [2 13/11; 1352/1331]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/11 = 279.318

{{Optimal ET sequence|legend=1| 13, 30, 43, 73, 116 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.8449 cents

[[Comma list]]: [[29282/28561]]

: [[gencom]]: [13/11 11/8; 29282/28561]

[[Optimal tuning]] ([[POTE]]): ~13/11 = 1\4, ~11/8 = 546.660

{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 44, 68, 112c, 180bc }}

[[Tp tuning #T2 tuning|RMS error]]: 0.8685 cents

[[Subgroup]]: 2.11.13

[[Comma list]]: [[265837/262144]]

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

: [[gencom]]: [2 143/128; 265837/262144]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~143/128 = 182.368

[[Tp tuning #T2 tuning|RMS error]]: 0.4444 cents

Yamablu, with a generator of ~17/13, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The [[Kite's Method of Naming Rank-2 Scales using Mode Numbers|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].

[[Subgroup]]: 2.11.13.17.19

[[Comma list]]: 209/208, 2057/2048, 83521/83486

[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]

Optimal tuning ([[POTE]]): ~17/13 = 462.9606

{{Optimal ET sequence|legend=1| 13, 44, 57, 70}}

[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents

[[Mapping]]: [{{val|1 1 2}}, {{val|0 6 5}}]

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/19 = 539.886

{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}

[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}

[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638