No-threes subgroup temperaments: Difference between revisions
Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
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Optimal tuning (CTE): ~58/31 = 1084.628 | Optimal tuning (CTE): ~58/31 = 1084.628 | ||
{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ... | |||
== French decimal == | == French decimal == | ||
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Optimal tuning (CTE): ~5/4 = 386.360 | 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 === | === 2.5.7.11 subgroup === | ||
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Optimal tuning (CTE): ~5/4 = 386.361 | Optimal tuning (CTE): ~5/4 = 386.361 | ||
{{Optimal ET sequence|legend=1|264, 733}}, ... | |||
=== 2.5.7.11.13 subgroup === | === 2.5.7.11.13 subgroup === | ||
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Optimal tuning (CTE): ~5/4 = 386.361 | Optimal tuning (CTE): ~5/4 = 386.361 | ||
{{Optimal ET sequence|legend=1|1525, 1789}}, ... | |||
== Mabon == | == Mabon == | ||
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Optimal tuning (CTE): ~729/448 = 870.792 | Optimal tuning (CTE): ~729/448 = 870.792 | ||
{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ... | |||
=== 2.9.7.11 subgroup === | === 2.9.7.11 subgroup === | ||
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Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243 | Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243 | ||
{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ... | |||
== Shipwreck == | == Shipwreck == | ||
Revision as of 20:07, 7 May 2023
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
Optimal ET sequence: 5, 11c, 16, 21, 37
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 ET sequence: 16, 21, 37
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 ET sequence: 16, 21, 37
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 ET sequence: 16, 21, 37
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
Optimal ET sequence: 6, 19, 25, 31, 37, 99, 130, 161, 353
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
Optimal ET sequence: 31, 47, 78, 109, 140, 171, 202, 233
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
Optimal ET sequence: 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd
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
Optimal ET sequence: 10, 21, 31
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
Optimal ET sequence: 10, 21, 31
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
Optimal ET sequence: 10, 21, 31, 52f
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
Optimal ET sequence: 13, 16, 22, 28, 35, 41, 47, 57, 63, 98c
Frostburn
Subgroup: 2.5.7.11
Comma list: 245/242, 625/616
Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
Optimal ET sequence: 29, 35, 41, 47, 88e
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
Optimal ET sequence: 13, 24, 33, 37, 46, 57, 70, 127
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
Optimal ET sequence: 13, 44, 57, 70
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¢
Optimal ET sequence: 93, 431, 338, 524
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
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
Optimal ET sequence: 52, 1737, 1789, ...
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
Optimal ET sequence: 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
Optimal ET sequence: 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
Optimal ET sequence: 1525, 1789, ...
Mabon
Derived from a calendar leap cycle built for the autumn equinox, hence the name. Defined as the 11 & 62 temperament.
Subgroup: 2.9.7
Comma basis: 44957696/43046721
Sval mapping: [⟨1 1 -3], ⟨0 3 8]]
Optimal tuning (CTE): ~729/448 = 870.792
Optimal ET sequence: 7d, 11, 18d, 29, 40, 62, ...
2.9.7.11 subgroup
Subgroup: 2.9.7.11
Comma basis: 896/891, 1331/1296
Sval mapping: [⟨1 1 -3 2], ⟨0 3 8 2]]
Optimal tuning (CTE): ~16/11 = 870.966
Bastille
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 pure bastille.
Subgroup: 2.5.7
Comma list: [1426 -596 -15⟩
Sval mapping: [⟨1 -4 254], ⟨0 -15 596]]
Optimal tuning (CTE): ~[381 0 -159 -4⟩ = 694.243
Optimal ET sequence: 382, 1025, 1407, 1789, 3196, ...
Shipwreck
Subgroup: 2.7.53
Comma list: 1048576/1042139
Gencom: [2 64/53; 1048576/1042139]
Mapping: [⟨1 0 6], ⟨0 3 -1]]]
POTE generator: ~64/53 = 323.034