Xenharmonic Wiki

Fractional-octave temperaments

Revision as of 22:28, 11 January 2023 by Eliora (talk | contribs)

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 simple 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. When tuned in 742edo, it is generated by a 53edo fifth intermingled with 14edo periods.

Subgroup: 2.3.5.7

Comma list: 14348907/14336000, 56358560858112/56296884765625

Mapping: [⟨14 0 -145 239], ⟨0 1 8 -9]]

Mapping generators: ~6125/5832, ~3

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

Template:Val list

Badness: 0.196

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨14 0 -145 239 115], ⟨0 1 8 -9 -3]]

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

Optimal GPV sequence: Template:Val list

Badness: 0.0450

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 6656/6655, 9801/9800, 24192/24167

Mapping: [⟨14 0 -145 239 115 74], ⟨0 1 8 -9 -3 -1]]

Optimal tuning (CTE): ~3/2 = 701.8733

Optimal GPV sequence: Template:Val list

Badness: 0.0269

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]]

Mapping generators: ~50/49, ~3

Optimal tuning (POTE): ~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]]

Optimal tuning (POTE): ~3/2 = 703.0355

Optimal GPV sequence: 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]]

Optimal tuning (POTE): ~3/2 = 703.0520

Optimal GPV sequence: Template:Val list

Badness: 0.048732

Triacontaheptoid

Subgroup: 2.3.5.7

Comma list: 244140625/242121642, 283115520/282475249

Mapping: [⟨37 2 67 85], ⟨0 3 1 1]]

Mapping generator: ~50/49, ~24000/16807

Optimal tuning (CTE): ~24000/16807 = 612.4003

Template:Val list

Badness: 0.784746

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨37 2 67 85 128], ⟨0 3 1 1 0]]

Optimal tuning (CTE): ~768/359 = 612.4003

Optimal GPV sequence: 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 2 67 85 128 118], ⟨0 3 1 1 0 1]]

Optimal tuning (CTE): ~462/325 = 612.4206

Optimal GPV sequence: 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 2 67 85 128 118 189], ⟨0 3 1 1 0 1 -2]]

Optimal tuning (CTE): ~121/85 = 612.4187

Optimal GPV sequence: Template:Val list

Badness: 0.052475

44th-octave temperaments

One step of 44edo is very close to the septimal comma, 64/63. The relationship is preserved even up thousands of edos.

Ruthenium

Ruthenium is named after the 44th element, and can be expressed as the 1848 & 2684 temperament.

Subgroup: 2.3.5.7

Comma list: [-8  23 -5 -6⟩, [51 -13 -1 -10⟩

Mapping: [⟨44 0 -386 263], ⟨0 1 7 -2]]

Mapping generators: ~64/63, ~3

Optimal tuning (CTE): ~3/2 = 701.9420

Template:Val list

Badness: 0.111

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 1771561/1771470, 67110351/67108864

Mapping: [⟨44 0 -386 263 -57], ⟨0 1 7 -2 3]]

Optimal tuning (CTE): ~3/2 = 701.9429

Optiml GPV sequence: Template:Val list

Badness: 0.0209

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 9801/9800, 196625/196608, 823680/823543, 1771561/1771470

Mapping: [⟨44 0 -386 263 -57 1976], ⟨0 1 7 -2 3 -26]]

Optimal tuning (CTE): ~3/2 = 701.939

Optiml GPV sequence: Template:Val list

Badness: 0.0396

56th-octave temperaments

Barium

One step of 56edo is close to a syntonic comma. Named after the 56th element, barium tempers out the [-225 224 -56⟩ comma, which sets 56 syntonic commas equal to the octave. It can be expressed as the 224 & 2072 temperament.

Subgroup: 2.3.5

Comma list: [-225 24 -56⟩

Mapping: [⟨56 0 -225], ⟨0 1 4]]

Mapping generators: ~81/80, ~3

Optimal tuning (CTE): ~3/2 = 701.9379

Template:Val list

Badness: 4.70

7-limit

Subgroup: 2.3.5.7

Comma list: [-12 29 -11 -3⟩, [47 -7 -7 -7⟩

Mapping: [⟨56 0 -225 601], ⟨0 1 4 -5]]

Optimal tuning (CTE): ~3/2 = 701.9433

Template:Val list

Badness: 0.227

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 1019215872/1019046875, 14765025303/14763950080

Mapping: [⟨56 0 -225 601 460], ⟨0 1 4 5 -3]]

Optimal tuning (CTE): ~3/2 = 701.9431

Optimal GPV sequence: Template:Val list

Badness: 0.0345

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]]

Mapping generators: ~81/80, ~7

Optimal tuning (POTE): ~7/4 = 969.1359

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]]

Optimal tuning (POTE): ~7/4 = 969.5715

Optimal GPV sequence: 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]]

Optimal tuning (POTE): ~7/4 = 969.9612

Optimal GPV sequence: Template:Val list

Badness: 0.036267

91st-octave temperaments

Protactinium

Protactinium is described as the 364 & 1547 temperament and named after the 91st element.

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨91 0 644 -33 1036 481], ⟨0 1 -3 -2 -5 -1]]

Mapping generators: ~1728/1715, ~3

Optimal tuning (CTE): ~3/2 = 702.0195

Optimal GPV sequence: Template:Val list

Badness: 0.0777

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨91 0 644 -33 1036 481 -205], ⟨0 1 -3 -2 -5 -1 4]]

Optimal tuning (CTE): ~3/2 = 702.0269

Optimal GPV sequence: Template:Val list

Badness: 0.0582

111th-octave temperaments

Roentgenium

Roentgenium is defined as 4884 & 8103 in the 19-limit and is named after the 111th element. 111 is 37 x 3, and what's particularly remarkable about this temperament is that it still preserves the relationship of 11/8 to 37edo in EDOs the size of thousands. Developed for a musical composition in 8103edo by Eliora.

Subgroup: 2.3.5.7.11

Comma list: [-25 -12 -3 12  5⟩, [-27  27  0  3 -7⟩, [26  -8 -2  8 -9⟩

Mapping: [⟨111 111 2855 896 384], ⟨0 1 -40 -9 0]]

Optimal tuning (CTE): ~3/2 = 701.964

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 31213/31212, 486400/486387, 633556/633555, 653429/653400, 1037232/1037153, 9714446/9713275, 24764600/24762387

Mapping: [⟨111 111 2855 896 384 410 452 472], ⟨0 1 -40 -9 0 -11 -25 7]]

Optimal tuning (CTE): ~3/2 = 701.9...

Vals: 3219c, 4884, 8103, 12987, ...

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]]

Mapping generators: ~15625/15552, ~7

Optimal tuning (POTE): ~7/4 = 968.7235

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]]

Optimal tuning (POTE): ~7/4 = 968.5117

Optimal GPV sequence: Template:Val list

Badness: 0.049316

Centenniamajor

Named after the fact that 18 is the age of majority in most countries, and 100 (centennial) + 18 (major) = 118.

Subgroup: 2.3.5.7.11

Comma list: 32805/32768, 151263/151250, 1224440064/1220703125

Mapping: [⟨118 187 274 0 -420], ⟨0 0 0 2 5]]

Mapping generators: ~15625/15552, ~405504/153125

Optimal tuning (CTE): ~202752/153125 = 484.4837

Optimal GPV sequence: Template:Val list

Badness: 0.357

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨118 187 274 0 -420 271], ⟨0 0 0 2 5 1]]

Optimal tuning (CTE): ~8125/6144 = 484.4867

Optimal GPV sequence: Template:Val list

Badness: 0.122

Oganesson

Named after the 118th element. In the 13-limit, the period corresponds to 169/168, and in the 17-limit, it corresponds also to 170/169, meaning that 28561/28560 is tempered out. As opposed to being an extension of parakleischis, this has the generator that splits the third harmonic into three equal parts.

In the 7-limit and 11-limit, the period corresponds to bronzisma.

Subgroup: 2.3.5.7

Comma list: [30 10 -27 6⟩, [77 -20 -5 -12⟩

Mapping: [⟨118 0 274 643], ⟨0 3 0 -5]]

Mapping generators: ~2097152/2083725, ~1953125/1354752

Optimal tuning (CTE): ~1953125/1354752 = 634.0068

Template:Val list

Badness: 2.66

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, [13 -1 4 -16 7⟩, [55 -7 -15 -2 -1⟩

Mapping: [⟨118 0 274 643 1094], ⟨0 3 0 -5 -11]]

Optimal tuning (CTE): ~1953125/1354752 = 634.0085

Optimal GPV sequence: Template:Val list

Badness: 0.568

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 9801/9800, 537403776/537109375, 453874312332/453857421875

Mapping: [⟨118 0 274 643 1094 499], ⟨0 3 0 -5 -11 -1]]

Mapping generators: ~169/168, ~1124864/779625

Optimal tuning (CTE): ~1124864/779625 = 634.0087

Optimal GPV sequence: Template:Val list

Badness: 0.172

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 4096/4095, 9801/9800, 34391/34375, 361250/361179, 562432/562275

Mapping: [⟨118 0 274 643 1094 499 607], ⟨0 3 0 -5 -11 -1 2]]

Mapping generators: ~170/169, ~238/165

Optimal tuning (CTE): ~238/165 = 634.0080

Optimal GPV sequence: Template:Val list

Badness: 0.105

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