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.

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

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

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

Subgroup-val mapping: {{mapping| 1 3 4 5 | 0 -4 -7 -9 }}

=== Mabilic ===

{{Main| Mabilic and trismegistus }}

{{See also| Chromatic pairs #Mabilic }}

Mabilic is the no-3 [[restriction]] 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 }}

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

: mapping generators: ~2, ~175/128

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1201.2543{{c}}, ~175/128 = 527.7872{{c}}

: [[error map]]: {{val| +1.254 -1.698 -1.491 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~175/128 = 527.2041{{c}}

: error map: {{val| 0.000 -4.701 -4.846 }}

{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd }}

[[Badness]] (Sintel): 1.70

=== Huntington ===

{{See also| Chromatic pairs #Huntington }}

[[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 ====

Subgroup: 2.5.7.13

Comma list: 640/637, 10985/10976

Subgroup-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}

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

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

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

{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}

==== Silver ====

Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.

Subgroup: 2.5.7.13.17

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

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

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

* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}

Badness (Sintel): 0.504

=== Ostara ===

Ostara is a temperament that is derived from 93 & 524 solar calendar leap rule scale, interpreted in general no-3's 19-limit. It is a weak extension of the unnamed 2.5.7-subgroup 28 & 31 temperament, which tempers out 8589934592/8544921875.

Subgroup: 2.5.7.11

Comma list: 8589934592/8544921875, 30691800524/30517578125

Subgroup-val mapping: {{mapping| 1 1 20 -49 | 0 3 -39 119 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~5120/3773 = 529.0037{{c}}

{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}

Badness (Sintel): 11.7

Subgroup: 2.5.7.11.13

Subgroup-val mapping: {{mapping| 1 1 20 -49 35 | 0 3 -39 119 -71 }}

* WE: ~2 = 1199.9194{{c}}, ~1664/1225 = 528.9681{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0036{{c}}

{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}

Badness (Sintel): 3.42

Subgroup: 2.5.7.11.13.17

Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}

* WE: ~2 = 1199.9054{{c}}, ~1664/1225 = 528.9628{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0046{{c}}

{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}

Badness (Sintel): 1.99

Subgroup: 2.5.7.11.13.17.19

Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}

* WE: ~2 = 1199.9081{{c}}, ~19/14 = 528.9639{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~19/14 = 529.0045{{c}}

{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}

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.

[[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}}

: [[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

Subgroup: 2.5.7.11

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

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

* 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: 2.5.7.11.13

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

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

=== Tricesimoprimal miracloid ===

{{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.

 

 

 

 

 

 

 

 

 

 

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

 

 

 

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

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

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

=== Movila ===

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]].

[[Comma list]]: 1331/1280

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

[[Optimal tuning]] (CTE): ~2 = 1200.000{{c}}, ~11/8 = 529.846{{c}}

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

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

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

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

[[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

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

[[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 }}

: mapping generators: ~2, ~5/4

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

 

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

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

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

: mapping generators, ~2, ~5/4

[[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

=== Score ===

[[Subgroup]]: 2.7.11.13

[[Comma list]]: 343/338, 847/832

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

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

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

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

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

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

[[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]]

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

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

: mapping generators, ~2, ~13/11

[[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]]

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

{{Mapping|legend=3| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}

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

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

[[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

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

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

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

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

[[Category:Temperament collections]]

[[Category:Subgroup temperaments]]