No-threes subgroup temperaments

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

Llywelyn

Subgroup: 2.5.7

Comma: 4194304/4117715

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

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

Sval mapping: [1 1 3], 0 7 -1]]

POL2 generator: ~8/7 = 226.910

Optimal GPV sequence16, 37

RMS error: 0.5391 cents

2.5.7.11

Subgroup: 2.5.7.11

Comma: 176/175, 1310720/1294139

Gencom: [2 8/7; 176/175 1310720/1294139]

Gencom mapping: [1 0 1 3 1], 0 0 7 -1 13]]

Sval mapping: [1 1 3 1], 0 7 -1 13]]

POL2 generator: ~8/7 = 227.114

Optimal GPV sequence16, 21, 37

2.5.7.11.13

Subgroup: 2.5.7.11

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

Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]

Gencom mapping: [1 0 1 3 1 2], 0 0 7 -1 13 9]]

Sval mapping: [1 1 3 1 2], 0 7 -1 13 9]]

POL2 generator: ~8/7 = 227.108

Optimal GPV sequence16, 21, 37

2.5.7.11.13.17

Subgroup: 2.5.7.11

Comma: 176/175, 221/200, 640/637, 833/832

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]]

Sval mapping: [1 1 3 1 2 2], 0 7 -1 13 9 11]]

POL2 generator: ~8/7 = 227.242

Optimal GPV sequence16, 21, 37

Didacus

See also: Hemimean clan #Didacus

Related temperaments: roulette, hemithirds

Subgroup: 2.5.7

Comma: 3136/3125

Gencom: [2 28/25; 3136/3125]

Gencom mapping: [1 0 2 2], 0 0 2 5]]

Sval mapping: [1 2 2], 0 2 5]]

POL2 generator: ~28/25 = 93.772

Optimal GPV sequence6, 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

Comma: 2100875/2097152

Gencom: [2 256/245; 2100875/2097152]

Gencom mapping: [1 0 2 3], 0 0 5 -3]]

Sval mapping: [1 2 3], 0 5 -3]]

POL2 generator: ~256/245 = 77.205

Optimal GPV sequence31, 47, 78, 109, 140, 171, 202, 233

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

Gencom: [2 2744/2560; 823543/819200]

Gencom mapping: [1 0 3 3], 0 0 -7 -2]]

Sval mapping: [1 3 3], 0 -7 -2]]

POL2 generator: ~343/320 = 116.291

Optimal GPV sequence10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd

2.5.7.13

Subgroup: 2.5.7.13

Comma list: 343/338, 640/637

Gencom: [2 14/13; 343/338 640/637]

Gencom mapping: [1 0 3 3 4], 0 0 -7 -2 -3]]

Sval mapping: [1 3 3 4], 0 -7 -2 -3]]

POL2 generator: ~14/13 = 116.094

Optimal GPV sequence10, 21, 31

2.5.7.13.17

Subgroup: 2.5.7.13.17

Comma list: 170/169, 224/221, 640/637

Gencom: [2 14/13; 170/169 224/221 640/637]

Gencom mapping: [1 0 3 3 4 4], 0 0 -7 -2 -3 1]]

Sval mapping: [1 3 3 4 4], 0 -7 -2 -3 1]]

POL2 generator: ~14/13 = 115.769

Optimal GPV sequence10, 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

Gencom: [2 14/13; 170/169 343/338 640/637 16384/16055]

Gencom mapping: [1 0 3 3 4 4 3], 0 0 -7 -2 -3 1 13]]

Sval mapping: [1 3 3 4 4 3], 0 -7 -2 -3 1 13]]

POL2 generator: ~14/13 = 115.716

Optimal GPV sequence10, 21, 31, 52f

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]]

TE tuned generators: ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638

Optimal GPV sequence13, 24, 33, 37, 46, 57, 70, 127

Yamablu

Yamablu, with a generator of ~17/13, is named for it's 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]]

POL2 generator: ~17/13 = 462.9606

Optimal GPV sequence13, 44, 57, 70

RMS error: 0.4898 cents