No-threes subgroup temperaments
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
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
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
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¢
Estates general
Named so because it is defined as the 1789 & 3125 temperament due to 3125 providing optimal patent val for the jacobin comma, 3125 is 5 to the 5th power, and Estates General were called by Louis XVI on 5th May 1789 (05/05). Defined starting with the 2.5.11.13.19 subgroup, upwards to the 2.5.11.13.19.23.29.31 subgroup.
Subgroup: 2.5.11.13.19
Comma list: 6656/6655, 40960000000/40943078891, [-133 50 -7 18 -6⟩
Mapping: [⟨1 118 -107 -212 450], ⟨0 -266 254 496 -1025]]
Optimal tuning (CTE): ~2588443885831192576/1914932769775390625 = 521.856
2.5.11.13.19.23 subgroup
Subgroup: 2.5.11.13.19.23
Comma list: 6656/6655, 62500/62491, 190676992/190653125, [-92 23 -2 14 -10 8⟩
Mapping: [⟨1 118 -107 -212 450 579], ⟨0 -266 254 496 -1025 -1321]]
Optimal tuning (CTE): ~2592407900127232/1918105439453125 = 521.856
2.5.11.13.19.23.29 subgroup
Subgroup: 2.5.11.13.19.23.29
Comma list: 6656/6655, 62500/62491, 190676992/190653125, 7592198144/7591796875, 897740062375/897648164864
Mapping: [⟨1 118 -107 -212 450 579 251], ⟨0 -266 254 496 -1025 -1321 -566]]
Optimal tuning (CTE): ~184000/136097 = 521.856
2.5.11.13.19.23.29.31 subgroup
Subgroup: 2.5.11.13.19.23.29.31
Comma list: 6656/6655, 62500/62491, 9425/9424, 190676992/190653125, 507528125/507510784, 519411073024/519363934375
Mapping: [⟨1 118 -107 -212 450 579 251 -179], ⟨0 -266 254 496 -1025 -1321 -566 423]]
Optimal tuning (CTE): ~80275/59392 = 521.856