No-threes subgroup temperaments: Difference between revisions
documenting french decimal |
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Subgroup: 2.5.7 | Subgroup: 2.5.7 | ||
Comma basis: 372 -159 -1 | Comma basis: {{monzo|372 -159 -1}} | ||
Sval mapping: [{{val| 1 2 54 | Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}] | ||
Optimal tuning (CTE): ~5/4 = 386.360 | Optimal tuning (CTE): ~5/4 = 386.360 | ||
Vals: 205, 264, 469, 733, 997, 1261, 1525, 1789, ... | Vals: {{EDOs|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ... | ||
=== 2.5.7.11 subgroup === | === 2.5.7.11 subgroup === | ||
Subgroup: 2.5.7.11 | Subgroup: 2.5.7.11 | ||
Comma basis: -49 8 17 -5, 45 -27 10 -3 | Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}} | ||
Sval mapping: [{{val| 1 2 54 -177 | Sval mapping: [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}] | ||
Optimal tuning (CTE): ~5/4 = 386.361 | Optimal tuning (CTE): ~5/4 = 386.361 | ||
Revision as of 14:26, 8 December 2022
This is a collection of subgroup temperaments which omit the prime harmonic of 3.
Llywelyn aka shoe
Subgroup: 2.5.7
Comma list: 4194304/4117715
Sval mapping: [⟨1 1 3], ⟨0 7 -1]]
Mapping generators: 2, ~8/7
Gencom mapping: [⟨1 0 1 3], ⟨0 0 7 -1]]
Gencom: [2 8/7; 4194304/4117715]
Optimal tuning (POTE): ~8/7 = 226.910
2.5.7.11 subgroup
Subgroup: 2.5.7.11
Comma list: 176/175, 1310720/1294139
Sval mapping: [⟨1 1 3 1], ⟨0 7 -1 13]]
Gencom: [2 8/7; 176/175 1310720/1294139]
Gencom mapping: [⟨1 0 1 3 1], ⟨0 0 7 -1 13]]
Optimal tuning (POTE): ~8/7 = 227.114
Optimal GPV sequence: Template:Val list
2.5.7.11.13 subgroup
Subgroup: 2.5.7.11.13
Comma list: 176/175, 640/637, 847/845
Sval mapping: [⟨1 1 3 1 2], ⟨0 7 -1 13 9]]
Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]
Gencom mapping: [⟨1 0 1 3 1 2], ⟨0 0 7 -1 13 9]]
Optimal tuning (POTE): ~8/7 = 227.108
Optimal GPV sequence: Template:Val list
2.5.7.11.13.17 subgroup
Subgroup: 2.5.7.11.13.17
Comma list: 176/175, 221/200, 640/637, 833/832
Sval mapping: [⟨1 1 3 1 2 2], ⟨0 7 -1 13 9 11]]
Gencom: [2 8/7; 176/175 221/200, 640/637, 833/832]
Gencom mapping: [⟨1 0 1 3 1 2 2], ⟨0 0 7 -1 13 9 11]]
Optimal tuning (POTE): ~8/7 = 227.242
Optimal GPV sequence: Template:Val list
Didacus
Related temperaments: roulette, hemithirds
Subgroup: 2.5.7
Comma list: 3136/3125
Sval mapping: [⟨1 2 2], ⟨0 2 5]]
Gencom: [2 28/25; 3136/3125]
Gencom mapping: [⟨1 0 2 2], ⟨0 0 2 5]]
Optimal tuning (POTE): ~28/25 = 93.772
RMS error: 0.2138 cents
Rainy
Three generators make an 8/7; five generators make a 5/4. This is the no-threes version of tertiaseptal.
Subgroup: 2.5.7
Sval mapping: [⟨1 2 3], ⟨0 5 -3]]
Gencom: [2 256/245; 2100875/2097152]
Gencom mapping: [⟨1 0 2 3], ⟨0 0 5 -3]]
Optimal tuning (POTE): ~256/245 = 77.205
RMS error: 0.0586 cents
Mercy
Two generators make an 8/7; seven generators make an 8/5. Mercy can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of 31edo, and is the no-threes or elevens version of miracle.
Subgroup: 2.5.7
Comma list: 823543/819200
Sval mapping: [⟨1 3 3], ⟨0 -7 -2]]
Gencom: [2 2744/2560; 823543/819200]
Gencom mapping: [⟨1 0 3 3], ⟨0 0 -7 -2]]
Optimal tuning (POTE): ~343/320 = 116.291
2.5.7.13
Subgroup: 2.5.7.13
Comma list: 343/338, 640/637
Sval mapping: [⟨1 3 3 4], ⟨0 -7 -2 -3]]
Gencom: [2 14/13; 343/338 640/637]
Gencom mapping: [⟨1 0 3 3 4], ⟨0 0 -7 -2 -3]]
Optimal tuning (POTE): ~14/13 = 116.094
2.5.7.13.17
Subgroup: 2.5.7.13.17
Comma list: 170/169, 224/221, 640/637
Sval mapping: [⟨1 3 3 4 4], ⟨0 -7 -2 -3 1]]
Gencom: [2 14/13; 170/169 224/221 640/637]
Gencom mapping: [⟨1 0 3 3 4 4], ⟨0 0 -7 -2 -3 1]]
Optimal tuning (POTE): ~14/13 = 115.769
2.5.7.13.17.19
Subgroup: 2.5.7.13.17.19
Comma list: 170/169, 343/338, 640/637, 16384/16055
Sval mapping: [⟨1 3 3 4 4 3], ⟨0 -7 -2 -3 1 13]]
Gencom mapping: [⟨1 0 3 3 4 4 3], ⟨0 0 -7 -2 -3 1 13]]
Gencom: [2 14/13; 170/169 343/338 640/637 16384/16055]
Optimal tuning (POTE): ~14/13 = 115.716
Pakkanen (rank 3)
Subgroup: 2.5.7.11
Comma list: 625/616
Optimal tuning (TE): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
Frostburn
Subgroup: 2.5.7.11
Comma list: 245/242, 625/616
Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
Yer (rank 3)
Subgroup: 2.11.13.17.19
Comma list: 209/208, 2057/2048
Sval mapping: [⟨1 0 0 11 4], ⟨0 1 0 -2 -1], ⟨0 0 1 0 1]]
Optimal tuning (TE): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
Yamablu
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 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: [⟨1 5 1 1 0], ⟨0 -4 7 8 11]]
Optimal tuning (POTE): ~17/13 = 462.9606
RMS error: 0.4898 cents
Ostara
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: [⟨1 1 20 -49], ⟨0 3 -39 119]]
Optimal tuning (POTE): ~5120/3773 = 529.003¢
2.5.7.11.13 subgroup
Subgroup: 2.5.7.11.13
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
Sval Mapping: [⟨1 1 20 -49 35], ⟨0 3 -39 119 -71]]
Optimal tuning (POTE): ~1664/1225 = 529.003¢
2.5.7.11.13.17 subgroup
Subgroup: 2.5.7.11.13.17
Sval Mapping: [⟨1 1 20 -49 35 42], ⟨0 3 -39 119 -71 -86]]
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
Optimal tuning (POTE): ~1664/1225 = 529.003¢
2.5.7.11.13.17.19 subgroup
Subgroup: 2.5.7.11.13.17.19
Sval Mapping: [⟨1 1 20 -49 35 42], ⟨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¢
Pure onzonic
The 2.5.11.13 subgroup primarily contains temperaments developed for 1789edo, since it tempers out the jacobin comma 6656/6655, for which 2.5.11.13 is the subgroup, and the year 1789 is hallmark for the French revolution.
Subgroup: 2.5.11.13
Comma list: 6656/6655, [-119 -46 15 47⟩
Mapping: [⟨1 74 3 74], ⟨0 -156 1 -153]]
Optimal tuning (POTE): ~11/8 = 551.370
Optimal GPV sequence: 37, 1789
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: [⟨1 419 48 177 157 624 625], ⟨0 -461 -50 -192 -169 -685 -686]]
Optimal tuning (CTE): ~58/31 = 1084.628
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: [372 -159 -1⟩
Sval mapping: [⟨1 2 54], ⟨0 1 -159]]
Optimal tuning (CTE): ~5/4 = 386.360
Vals: 205, 264, 469, 733, 997, 1261, 1525, 1789, ...
2.5.7.11 subgroup
Subgroup: 2.5.7.11
Comma basis: [-49 8 17 -5⟩, [45 -27 10 -3⟩
Sval mapping: [⟨1 2 54 -177], ⟨0 1 -159 -539]]
Optimal tuning (CTE): ~5/4 = 386.361
Vals: 264, 733, ...
2.5.7.11.13 subgroup
Subgroup: 2.5.7.11.13
Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
Sval mapping: [⟨1 2 54 -177 52], ⟨0 1 -159 -539 173]]
Optimal tuning (CTE): ~5/4 = 386.361
Vals: 1525, 1789, ...