No-threes subgroup temperaments: Difference between revisions

* French decimal and trader have a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.

=== 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

 

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

 

 

 

 

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

 

* 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 }}

Badness (Sintel): 52.2

==== 2.5.7.11.13 subgroup ====

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

Optimal tunings:  

* 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

 

{{Main| Bastille }}

 

 

 

 

{{Mapping|legend=2| 1 -4 254 | 0 15 -596 }}

 

: [[error map]]: {{val| -0.009 +0.020 +0.001 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -380 159 4 }} = 505.7570{{c}}

: error map: {{val| 0.000 +0.041 +0.018 }}

 

 

[[Badness]] (Sintel): 7.18 × 10³

 

{{See also| Chromatic pairs #Augment }}

 

 

 

 

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

 

 

* [[WE]]: ~5/4 = 399.0128{{c}}, ~7/4 = 974.7085{{c}}

: [[error map]]: {{val| -2.962 +6.776 -0.040 }}

 

{{Optimal ET sequence|legend=1| 3, 6, 15, 21, 27, 102ccd, 129ccd }}

 

 

 

 

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 }}

* 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

=== Tricesimoprimal miracloid ===

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

{{Todo|inline=1| complete section | comment = Add missing data from 2.5.7 to 2.5.7.11.19.29. }}

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. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable [[well temperament]] for 52edo.

 

 

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

 

 

 

{{Optimal ET sequence|legend=0| 52, 1737, 1789 }}

 

Badness (Sintel): 4.51

 

{{See also| Chromatic pairs #Huntington }}

 

Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.  

[[Comma list]]: 40960000/40353607

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

 

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

: mapping generators: ~2, ~80/49

* [[WE]]: ~2 = 1199.5781{{c}}, ~80/49 = 842.6730{{c}}

: [[error map]]: {{val| -0.422 -0.569 +1.866 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9136{{c}}

: error map: {{val| 0.000 -0.091 +2.828 }}

{{Optimal ET sequence|legend=0| 7c, 10, 27, 37, 84, 121 }}

Badness (Sintel): 1.87

 

==== 2.5.7.13 subgroup ====

 

Comma list: 640/637, 10985/10976

 

 

* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}

 

 

Badness (Sintel): 0.319

 

 

 

 

 

 

Gencom mapping: {{mapping| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}

 

 

 

Badness (Sintel): 0.504

 

=== Pakkanen ===

 

 

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

 

* [[WE]]: ~2 = 1200.6499{{c}}, ~5/4 = 380.2990{{c}}, ~7/4 = 969.2326{{c}}

: error map: {{val| 0.000 -5.689 +0.826 +1.530 }}

 

 

 

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}}

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 }}

[[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}}

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