No-threes subgroup temperaments

This is a collection of subgroup temperaments which omit the prime harmonic of 3.

Llywelyn

Subgroup: 2.5.7

Comma list: 4194304/4117715

Sval mapping: [⟨1 1 3], ⟨0 7 -1]]

Gencom: [2 8/7; 4194304/4117715]

Gencom mapping: [⟨1 0 1 3], ⟨0 0 7 -1]]

Optimal tuning (POTE): ~8/7 = 226.910

RMS error: 0.5391 cents

2.5.7.11

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

2.5.7.11.13

Subgroup: 2.5.7.11.13

Comma list: 176/175, 640/637, 1304576/1294139

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

2.5.7.11.13.17

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

Didacus

See also: Hemimean clan § 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

Comma list: 2100875/2097152

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

See also: Quince clan § 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.

2.5.7.11

Subgroup: 2.5.7.11

Mapping: [⟨1 1 20 -49], ⟨0 3 -39 119]]

Comma list: 8589934592/8544921875, 53710650917/53687091200

POTE Generator: ~5120/3773 = 529.003¢

Vals: 93, 431, 338, 524

2.5.7.11.13

Subgroup: 2.5.7.11.13

Mapping: [⟨1 1 20 -49 35], ⟨0 3 -39 119 -71]]

Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125

POTE Generator: ~1664/1225 = 529.003¢

2.5.7.11.13.17

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

POTE Generator: ~1664/1225 = 529.003¢

2.5.7.11.13.17.19

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

POTE Generator: ~1664/1225 = 529.003¢

2.7.13.17.19

Subgroup: 2.7.13.17.19

Comma list: 16807/16796, 157339/157216, 47071232/47045881

POTE Generator: 529.009¢ ~19/14

Vals: 93, 338, 431, 524, 617, 710