No-threes subgroup temperaments: Difference between revisions

=== Rainy ===

In rainy, three generators make an [[8/7]]; five generators make a [[5/4]]. It is the no-3's [[restriction]] of [[tertiaseptal]] (and [[valentine]]), 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 5's and stacked 7's become easier to navigate, using the general-purpose diesis to simplify clusters.

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

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

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

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

: mapping generators: ~2, ~256/245

* [[WE]]: ~2 = 1200.0939{{c}}, ~256/245 = 77.2107{{c}}

: [[error map]]: {{val| +0.094 -0.072 -0.176 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/245 = 77.2093{{c}}

: error map: {{val| 0.000 -0.267 -0.454 }}

{{Optimal ET sequence|legend=1| 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d }}

[[Badness]] (Sintel): 0.156

=== French decimal ===

French decimal is 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 list]]: {{monzo| 372 -159 -1 }}

{{Mapping|legend=2| 1 0 372 | 0 1 -159 }}

: mapping generators: ~2, ~5

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.9901{{c}}, ~5/4 = 386.3562{{c}}

: [[error map]]: {{val| -0.010 +0.023 +0.000 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.3595{{c}}

: error map: {{val| 0.000 +0.046 +0.019 }}

{{Optimal ET sequence|legend=1| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}

[[Badness]] (Sintel): 148

==== 2.5.7.11 subgroup ====

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

Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}

Optimal tunings:  

* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}

{{Optimal ET sequence|legend=0| 264, 997e, 1261e, 1525, 1789 }}

 

 

 

Subgroup-val mapping: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}

* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}

{{Optimal ET sequence|legend=0| 261, 1261e, 1525, 1789 }}

Badness (Sintel): 10.5

=== Bastille ===

 

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

 

 

 

 

 

 

[[Badness]] (Sintel): 7.18 × 10³

 

 

Augment is related to [[augmented]].  

Comma list: 56/55, 128/125

Subgroup-val mapping: {{mapping| 3 7 0 2 | 0 0 1 1 }}

 

Gencom mapping: {{mapping| 3 0 7 0 2 | 0 0 0 1 1 }}

: mapping generators: ~5/4, ~7

* [[WE]]: ~5/4 = 398.9239{{c}}, ~7/4 = 969.1106{{c}}

* [[CWE]]: ~5/4 = 400.0000{{c}}, ~7/4 = 968.4397{{c}}

{{Optimal ET sequence|legend=0| 3, 6, 15, 21 }}

Badness (Sintel): 0.196

 

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.

 

 

 

 

 

 

 

* CTE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.003{{c}}

* CWE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.003{{c}}

Badness (Sintel): 3.415

* CTE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.005{{c}}

* CWE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.005{{c}}

Badness (Sintel): 1.985

* CTE: ~2 = 1200.000{{c}}, ~19/14 = 529.006{{c}}

* CWE: ~2 = 1200.000{{c}}, ~19/14 = 529.005{{c}}

{{Optimal ET sequence|legend=0| 93, 338, 431 }}

Badness (Sintel): 1.285

=== Tricesimoprimal miracloid ===

{{See also| Tricesimoprimal miracloid/Eliora's approach }}

Tricesimoprimal miracloid is 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

Subgroup-val mapping: {{mapping| 1 419 48 177 157 624 625 | 0 -461 -50 -192 -169 -685 -686 }}

Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~58/31 = 1084.628{{c}}

=== Huntington ===

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

[[Subgroup]]: 2.5.7.13

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

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

: mapping generators: ~2, ~16/13

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~16/13 = 357.002{{c}}

{{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 }}

[[Subgroup]]: 2.5.7.13.17

{{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 = 1200.000{{c}}, ~13/8 = 842.711{{c}}

{{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

=== Pakkanen ===

[[Comma list]]: 625/616

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

 

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

 

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

 

 

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

: mapping generators: ~2, ~5, ~100/77

* [[WE]]: ~2 = 1200.080{{c}}, ~5 = 2786.820{{c}}, ~100/77 = 454.618{{c}}

* [[CWE]]: ~2 = 1200.000{{c}}, ~5 = 2786.740{{c}}, ~100/77 = 454.590{{c}}

{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}

[[Badness]] (Sintel): 1.862

Subgroup-val mapping: {{Mapping| 1 0 2 0 1 | 0 1 1 1 1 | 0 0 -4 3 1 }}

* WE: ~2 = 1200.034{{c}}, ~5 = 2786.678{{c}}, ~13/10 = 454.569{{c}}

* CWE: ~2 = 1200.000{{c}}, ~5 = 2786.646{{c}}, ~13/10 = 454.557{{c}}

{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}

Subgroup-val mapping: {{Mapping| 1 0 2 0 1 1 | 0 1 1 1 1 1 | 0 0 -4 3 1 2 }}

* WE: ~2 = 1200.407{{c}}, ~5 = 2787.484{{c}}, ~13/10 = 455.036{{c}}

* CWE: ~2 = 1200.000{{c}}, ~5 = 2787.107{{c}}, ~13/10 = 454.906{{c}}

{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 50, 58, 66g, 87g }}

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

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

[[Optimal tuning]] ([[POTE]]): ~125/88 = 600.000{{c}}, ~5/4 = 383.768{{c}}

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

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~55/32 = 928.032{{c}}

[[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 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~5/4 = 372.236{{c}}

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

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

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~5/4 = 407.079{{c}}

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

: mapping generators, ~2, ~13/10

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~13/10 = 459.281{{c}}

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~7/4 = 968.913{{c}}

{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d }}

Badness (Sintel): 0.031

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~11/8 = 540.099{{c}}

{{Optimal ET sequence|legend=1| 5, 7, 9, 11, 20 }}

[[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=2| 1 0 1 3 | 0 8 7 2 }}

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~14/11 = 421.309{{c}}

{{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 }}

: mapping generators: ~2, ~16/13

[[Optimal tuning]]:  

* [[POTE]]: ~2 = 1200.000{{c}}, ~16/13 = 357.677{{c}}

* [[TOP tuning|POTT]]: ~2 = 1200.000{{c}}, ~16/13 = 357.794{{c}} (1200 - 300 log₂(7))

{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104 }}

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

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~17/14 = 324.446{{c}}

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~17/14 = 336.858{{c}}

[[Optimal tuning]]s ([[POTE]]): ~2 = 1200.000{{c}}, ~64/53 = 323.034{{c}}

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]]

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~13/11 = 279.318{{c}}

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

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

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

[[Optimal tuning]] ([[POTE]]): ~13/11 = 300.000{{c}}, ~11/8 = 546.660{{c}}

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

=== Bluebirds ===

{{Distinguish| Bluebird }}

{{See also| Chromatic pairs #Bluebirds }}

[[Subgroup]]: 2.11.13

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

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

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

: mapping generators, ~2, ~143/128

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~143/128 = 182.368{{c}}

[[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

{{Mapping|legend=2| 1 5 1 1 0 | 0 -4 7 8 11 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~17/13 = 462.9606{{c}}

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

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~26/19 = 539.886{{c}}

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

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