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Fractional-octave temperaments

Revision as of 14:02, 15 September 2022 by Eliora (talk | contribs) (→118th-octave temperaments)

All temperaments on this page have a fractional-octave period, such as 1\26, 1\31, or 1\41.

Temperaments discussed elsewhere include:

14th-octave temperaments

While 14edo is poor in LCJI harmonics, some of its multiples (such as 224edo and 742edo) are members of zeta edo list.

Silicon

The name of silicon temperament comes from the 14th element. Defined upwards to the 13-limit. In 742edo, what's also unique is that it is generated by a 53edo fifth intermingled with 14edo periods.

5-limit

Subgroup: 2.3.5

Comma list: [-145 112 -14⟩

Mapping generators: ~282429536481/268435456000, ~3/2

Mapping: [⟨14 14 -33], ⟨0 1 8]]

Optimal tuning (CTE): ~3/2 = 701.864

7-limit

Subgroup: 2.3.5.7

Comma list: 14348907/14336000, 56358560858112/56296884765625

Mapping generators: ~6125/5832, ~3/2

Mapping: [⟨14 14 -33 113], ⟨0 1 8 -9]]

Optimal tuning (CTE): ~3/2 = 701.870

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 1240029/1239040, 2359296/2358125

Mapping generators: ~605/576, ~3/2

Mapping: [⟨14 14 -33 113 73], ⟨0 1 8 -9 -3]]

Optimal tuning (CTE): ~3/2 = 701.872

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 9801/9800, 67392/67375, 59535/59488

Mapping generators: ~104/99, ~3/2

Mapping: [⟨14 14 -33 113 73 60], ⟨0 1 8 -9 -3 -1]]

Optimal tuning (CTE): ~3/2 = 701.873

Vals: 70d, 224, 294, 448, 518, 672, 742, 966, 1190, 1260

37th-octave temperaments

37EDO is accurate for harmonics 5, 7, 11, and 13, so various 37th-octave temperaments actually make sense.

Rubidium

The name of rubidium temperament comes from Rubidium, the 37th element.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 4194304/4117715

Mapping: [⟨37 0 86 104], ⟨0 1 0 0]]

POTE generator: ~3/2 = 703.3903

Template:Val list

Badness: 0.312105

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1375/1372, 65536/65219

Mapping: [⟨37 0 86 104 128], ⟨0 1 0 0 0]]

POTE generator: ~3/2 = 703.0355

Vals: Template:Val list

Badness: 0.101001

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 640/637, 847/845, 1375/1372

Mapping: [⟨37 0 86 104 128 137], ⟨0 1 0 0 0 0]]

POTE generator: ~3/2 = 703.0520

Vals: Template:Val list

Badness: 0.048732

Triacontaheptoid

Subgroup: 2.3.5.7

Comma list: 244140625/242121642, 283115520/282475249

Mapping: [⟨37 23 74 92], ⟨0 3 1 1]]

POTE generator: ~5/4 = 385.3041

Template:Val list

Badness: 0.784746

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4000/3993, 226492416/226474325

Mapping: [⟨37 23 74 92 128], ⟨0 3 1 1 0]]

POTE generator: ~5/4 = 385.3281

Vals: Template:Val list

Badness: 0.167327

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1375/1372, 4000/3993, 15379/15360

Mapping: [⟨37 23 74 92 128 125], ⟨0 3 1 1 0 1]]

POTE generator: ~5/4 = 385.3067

Vals: Template:Val list

Badness: 0.076183

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 715/714, 1225/1224, 4000/3993, 11271/11264

Mapping: [⟨37 23 74 92 128 125 175], ⟨0 3 1 1 0 1 -2]]

POTE generator: ~5/4 = 385.3427

Vals: Template:Val list

Badness: 0.052475

65th-octave temperaments

65EDO is accurate for harmonics 3, 5, and 11, so various 65th-octave temperaments actually make sense.

Terbium

The name of terbium temperament comes from Terbium, the 65th element.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 78732/78125

Mapping: [⟨65 103 151 0], ⟨0 0 0 1]]

POTE generator: ~8/7 = 230.8641

Template:Val list

Badness: 0.169778

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 4000/3993, 5632/5625

Mapping: [⟨65 103 151 0 225], ⟨0 0 0 1 0]]

POTE generator: ~8/7 = 230.4285

Vals: Template:Val list

Badness: 0.059966

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 2080/2079, 3584/3575

Mapping: [⟨65 103 151 0 225 58], ⟨0 0 0 1 0 1]]

POTE generator: ~8/7 = 230.0388

Vals: Template:Val list

Badness: 0.036267

91st-octave temperaments

Protactinium

Defined as the 364 & 1547 temperament and named after the 91st element.

Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 91125/91091, 2912000/2910897, 369754/369603

Mapping: [⟨91 91 371 149 581 390], ⟨0 1 -3 -2 -5 -1]]

Mapping generators: ~1728/1715, ~3/2

Optimal tuning (CTE): ~3/2 = 702.020c

Optimal GPV sequence: 364, 1183, 1547

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 4096/4095, 14400/14399, 42500/42471, 75735/75712, 2100875/2100384

Mapping: [Template:91 91 371 149 581 390 159, ⟨0 1 -3 -2 -5 -1 4]]

Mapping generators: ~3773/3744, ~3/2

Optimal tuning (CTE): ~3/2 = 702.027c

Optimal GPV sequence: 364, 1183, 1547

118th-octave temperaments

118EDO is accurate for harmonics 3 and 5, so various 118th-octave temperaments actually make sense.

Parakleischis

118EDO and its multiples are members of both parakleismic and schismic, and from this it derives its name.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 1224440064/1220703125

Mapping: [⟨118 187 274 0], ⟨0 0 0 1]]

POTE generator: ~8/7 = 231.2765

Template:Val list

Badness: 0.145166

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 32805/32768, 137781/137500

Mapping: [⟨118 187 274 0 77], ⟨0 0 0 1 1]]

POTE generator: ~8/7 = 231.4883

Vals: Template:Val list

Badness: 0.049316

Oganesson

Named after the 118th element, since a simpler temperament was already named. 82 periods plus a generator correspond to 13/8.

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 34398/34375, 32805/32768, 384912/384475

Mapping: 118 187 274 332 410 473 0 0 0 -2 -5 1

Mapping generators: ~3185/3168, ~8125/6144

Optimal tuning (CTE): ~8125/6144 = 484.487

Vals: 354, 590ee, 708e, 944e, 1298, 1652, 2006

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