Fractional-octave temperaments

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All temperaments on this page have a fractional-octave period.

Temperaments discussed elsewhere include:


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

Optimal GPV sequence: 176, 660, 836, 1848, 2684, 4532, 19976, 24508, 29040, 33572

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: 176, 660, 836, 1848, 2684, 4532, 15444, 19976e

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: 836, 1848, 2684, 7216, 9900, 12584

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: [-225 224 -56⟩

Mapping: [⟨56 0 -225], ⟨0 1 4]]

Mapping generators: ~81/80, ~3

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

Optimal GPV sequence: 224, 1176, 1400, 1624, 1848, 2072, 5992, 8064, 26264, 34328b, 42392b

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

Optimal GPV sequence: 224, 1176, 1400, 1624, 1848, 2072, 5768, 7616, 17080, 24696cd

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: 224, 1176, 1400, 1624, 1848, 3920, 5768, 7616, 21000cd, 28616cd

Badness: 0.0345

13-limit

Subgroup: 2.3.5.7.11

Comma list: 4225/4224, 9801/9800, 67392/67375, 26802913280/26795786661

Mapping: [⟨56 0 -225 601 460 651], ⟨0 1 4 -5 -3 -5]]

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

Vals: 224, 1848, 2072, ...

61st-octave temperaments

Promethium

Promethium tempers out the dipromethia and can be described as the 183 & 2684 temperament. By tempering out 4100625/4100096 promethium identifies the diaschisma with 2025/2002 in the 13-limit and also in the 17-limit.

Subgroup: 2.3.5.7.11.13

Comma list: 10648/10647, 196625/196608, 4100625/4100096, 204800000/204788493

Mapping: [⟨61 0 335 703 66 -161], ⟨0 2 -4 -11 3 8]]

Mapping generators: ~2025/2002 = 1\61, ~6875/3969 = 950.970

Optimal tuning (CTE): ~6875/3969 = 950.970

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 14400/14399, 37180/37179, 121875/121856, 140800/140777, 3536379/3536000

Mapping: [⟨61 0 335 703 66 -161 201], ⟨0 2 -4 -11 3 8 1]]

Mapping generators: ~2025/2002 = 1\61, ~11907/6875 = 950.970

Optimal tuning (CTE): ~11907/6875 = 950.970

Vals: 183, 2684, ...

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

Optimal GPV sequence: 65, 130

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: 65d, 130

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: 65d, 130

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: 364, 819e, 1183, 1547

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: 364, 1183, 1547, 1911

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

Optimal GPV sequence: 118, 236, 354, 472, 2242, 2714b, 3186b, 3658b

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: 118, 354, 472

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: 354, 944e, 1298

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: 354, 944e, 1298

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

Optimal GPV sequence: 354, 2360, 2714, 3068, 3442, 3776, 7198cd, 10974bccdd

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: 354, 3068e, 3442, 3776, 11682ccdde

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: 354, 3068e, 3422, 3776

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: 354, 3068e, 3422, 3776

Badness: 0.105

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