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Fractional-octave temperaments

Revision as of 13:01, 15 June 2023 by Eliora (talk | contribs) (i'll finish up the page after lunch)

Fractional-octave temperaments, when viewed from a regular temperament theory perspective, are temperaments which have a period which corresponds to a just interval mapped to a fraction of the octave, that is one step of an EDO.

Theory

Fractional-octave temperaments are valuable with regards to polysystemicism and polychromatics. They are acoustically significant with regards to containing modes of limited transposition, as well as their ability to expand on the harmony of the equal division they are a superset of. Such temperaments are also a way of introducing less common and harmonically less performing equal divisions into music that prefers consonance and is based on regular temperament theory.

Terminology

The terminology was developed by Eliora. The equal division containing the mos scale of such a temperament, starting from the tonic, is referred to as a wireframe, and individual notes of that equal division are called hinges. Thus in this context, the wireframe is the tuning consisting of only stacks of the period and no stacks of the generator. Temperament-agnostically, this can be used to refer to any structure embedded in an (x,y)-ET which repeats y times within that period, its "wireframe" is y-ET.

The most common way to produce a fractional-octave temperament is through an excellent approximation of an interval relative to the size of the wireframe edo. For example, compton family tempers out the Pythagorean comma and maps 7 steps of 12edo to 3/2. Likewise, a lot of 10th-octave temperaments have a 13/8 as 7\10, and 26th-octave temperaments often have a 7/4 for 21\26.

Temperament collections

Temperaments discussed as a part of a commatic family, or otherwise in temperament lists unrelated to fractional-octave theory 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 ET 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 ET 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 ET 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 ET sequence: 224, 1176, 1400, 1624, 1848, 3920, 5768, 7616, 21000cd, 28616cd

Badness: 0.0345

13-limit

Subgroup: 2.3.5.7.11.13

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

Optimal ET sequence: 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

Optimal ET sequence: 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 ET 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 ET 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 ET 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

Comma list: [47 -7 -7 -7⟩, [-2 -25 1 14⟩

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

mapping generators: ~1728/1715, ~3

Optimal tuning (CTE): ~3/2 = 701.991

Supporting ETs: 364, 819, 1183, 1547, 1911, 2730, 3094, 3913, 4277

11-limit

Subgroup: 2.3.5.7.11

Comma list: 234375/234256, 26214400/26198073, 514714375/514434888

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

mapping generators: ~1728/1715, ~3

Optimal tuning (CTE): ~3/2 = 702.015

Supporting ETs: 364, 819e, 1183, 1547, 1911, 2275, 2730e, 3458

13-limit

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 ET 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 ET 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...

Optimal ET sequence: 3219c, 4884, 8103, 12987, ...

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