Sensamagic clan: Difference between revisions

Revision as of 05:46, 16 January 2023

The sensamagic clan tempers out the sensamagic comma, 245/243, a triprime comma with no factors of 2, ⟨0 -5 1 2] to be exact.

For full 7-limit extensions, we have sensi, bohpier, escaped, salsa, pycnic, cohemiripple, superthird, magus and leapweek discussed below, as well as

Father, {16/15, 28/27} → Father family
Sidi, {25/24, 245/243} → Dicot family
Godzilla, {49/48, 81/80} → Meantone family
Hedgehog, {50/49, 245/243} → Porcupine family
Superpyth, {64/63, 245/243} → Archytas clan
Hemiaug, {128/125, 245/243} → Augmented family
Magic, {225/224, 245/243} → Magic family
Rodan, {245/243, 1029/1024} → Gamelismic clan
Shrutar, {245/243, 2048/2025} → Diaschismic family
Octacot, {245/243, 2401/2400} → Tetracot family
Clyde, {245/243, 3136/3125} → Kleismic family
Pental, {245/243, 16807/16384} → Pental family
Bamity, {245/243, 64827/64000} → Amity family
Fourfives, {245/243, 235298/234375} → Fifive family

Tempering out 245/243 alone in the full 7-limit leads to a rank-3 temperament, sensamagic, for which 283edo is the optimal patent val.

BPS

The BPS, for Bohlen–Pierce–Stearns, is the 3.5.7 subgroup temperament tempering out 245/243. This subgroup temperament was formerly called the lambda temperament, which was named after the lambda scale.

Subgroup: 3.5.7

Comma list: 245/243

Sval mapping: [⟨1 1 2], ⟨0 -2 1]]

Sval mapping generators: ~3, ~9/7

Optimal tuning (POTE): ~3 = 1\1edt, ~9/7 = 440.4881

Optimal GPV sequence: b4, b9, b13, b56, b69, b82, b95

Sensi

Main article: Sensi

See also: Sensipent family § Sensi

Sensi tempers out 126/125, 686/675 and 4375/4374 in addition to 245/243, and can be described as the 19&27 temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. 46edo is an excellent sensi tuning, and MOS of size 11, 19 and 27 are available. The name "sensi" is a play on the words "semi-" and "sixth."

Septimal sensi

Subgroup: 2.3.5.7

Comma list: 126/125, 245/243

Mapping: [⟨1 -1 -1 -2], ⟨0 7 9 13]]

Mapping generators: ~2, ~9/7

Wedgie: ⟨⟨ 7 9 13 -2 1 5 ]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.383

Template:Val list

Badness: 0.025622

Sensation

Subgroup: 2.3.5.7.13

Comma list: 91/90, 126/125, 169/168

Sval mapping: [⟨1 -1 -1 -2 0], ⟨0 7 9 13 10]]

Gencom mapping: [⟨1 -1 -1 -2 0 0], ⟨0 7 9 13 0 10]]

Gencom: [2 9/7; 91/90 126/125 169/168]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.322

Optimal GPV sequence: Template:Val list

Sensor

Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/243, 385/384

Mapping: [⟨1 -1 -1 -2 9], ⟨0 7 9 13 -15]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.294

Optimal GPV sequence: Template:Val list

Badness: 0.037942

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168, 385/384

Mapping: [⟨1 -1 -1 -2 9 0], ⟨0 7 9 13 -15 10]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.321

Optimal GPV sequence: Template:Val list

Badness: 0.025575

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 154/153, 169/168, 256/255

Mapping: [⟨1 -1 -1 -2 9 0 10], ⟨0 7 9 13 -15 10 -16]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.365

Optimal GPV sequence: Template:Val list

Badness: 0.022908

Sensis

Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 245/243

Mapping: [⟨1 -1 -1 -2 2], ⟨0 7 9 13 4]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.962

Optimal GPV sequence: Template:Val list

Badness: 0.028680

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 91/90, 100/99

Mapping: [⟨1 -1 -1 -2 2 0], ⟨0 7 9 13 4 10]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.945

Optimal GPV sequence: Template:Val list

Badness: 0.020017

Sensus

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 245/243

Mapping: [⟨1 -1 -1 -2 -8], ⟨0 7 9 13 31]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.626

Optimal GPV sequence: Template:Val list

Badness: 0.029486

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168, 352/351

Mapping: [⟨1 -1 -1 -2 -8 0], ⟨0 7 9 13 31 10]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.559

Optimal GPV sequence: Template:Val list

Badness: 0.020789

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 136/135, 154/153, 169/168

Mapping: [⟨1 -1 -1 -2 -8 0 -7], ⟨0 7 9 13 31 10 30]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.551

Optimal GPV sequence: Template:Val list

Badness: 0.016238

Sensa

Subgroup: 2.3.5.7.11

Comma list: 55/54, 77/75, 99/98

Mapping: [⟨1 -1 -1 -2 -1], ⟨0 7 9 13 12]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.518

Optimal GPV sequence: Template:Val list

Badness: 0.036835

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 66/65, 77/75, 143/140

Mapping: [⟨1 -1 -1 -2 -1 0], ⟨0 7 9 13 12 11]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.506

Optimal GPV sequence: Template:Val list

Badness: 0.023258

Hemisensi

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 245/242

Mapping: [⟨1 -1 -1 -2 -3], ⟨0 14 18 26 35]]

Optimal tuning (POTE): ~2 = 1\1, ~25/22 = 221.605

Optimal GPV sequence: Template:Val list

Badness: 0.048714

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168, 243/242

Mapping: [⟨1 -1 -1 -2 -3 0], ⟨0 14 18 26 35 30]]

Optimal tuning (POTE): ~2 = 1\1, ~25/22 = 221.556

Optimal GPV sequence: Template:Val list

Badness: 0.033016

Bisensi

Subgroup: 2.3.5.7.11

Comma list: 121/120, 126/125, 245/243

Mapping: [⟨2 5 7 9 9], ⟨0 -7 -9 -13 -8]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 156.692

Optimal GPV sequence: Template:Val list

Badness: 0.041723

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 126/125, 169/168

Mapping: [⟨2 5 7 9 9 10], ⟨0 -7 -9 -13 -8 -10]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 156.725

Optimal GPV sequence: Template:Val list

Badness: 0.026339

Bohpier

For the 5-limit version of this temperament, see High badness temperaments #Bohpier.

Main article: Bohpier

Bohpier is named after its interesting relationship with the non-octave Bohlen-Pierce equal temperament.

Subgroup: 2.3.5.7

Comma list: 245/243, 3125/3087

Mapping: [⟨1 0 0 0], ⟨0 13 19 23]]

Wedgie: ⟨⟨ 13 19 23 0 0 0 ]]

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 146.474

Minimax tuning:

7-odd-limit: ~27/25 = [0 0 1/19⟩

Eigenmonzo basis: 2.5

9-odd-limit: ~27/25 = [0 1/13⟩

Eigenmonzo basis: 2.3

Template:Val list

Badness: 0.068237

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243, 1344/1331

Mapping: [⟨1 0 0 0 2], ⟨0 13 19 23 12]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 146.545

Minimax tuning:

11-odd-limit: ~12/11 = [1/7 1/7 0 0 -1/14⟩

Eigenmonzo basis: 2.11/9

Optimal GPV sequence: Template:Val list

Badness: 0.033949

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 196/195, 275/273

Mapping: [⟨1 0 0 0 2 2], ⟨0 13 19 23 12 14]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 146.603

Minimax tuning:

13- and 15-odd-limit: ~12/11 = [0 0 1/19⟩

Eigenmonzo basis: 2.5

Optimal GPV sequence: Template:Val list

Badness: 0.024864

Music

by Chris Vaisvil:

Triboh

Triboh is named after "Triple Bohlen-Pierce scale", which divides each step of the equal-tempered Bohlen-Pierce scale into three equal parts.

Subgroup: 2.3.5.7.11

Comma list: 245/243, 1331/1323, 3125/3087

Mapping: [⟨1 0 0 0 0], ⟨0 39 57 69 85]]

Optimal tuning (POTE): ~2 = 1\1, ~77/75 = 48.828

Optimal GPV sequence: Template:Val list

Badness: 0.162592

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 245/243, 275/273, 847/845, 1331/1323

Mapping: [⟨1 0 0 0 0 0], ⟨0 39 57 69 85 91]]

Optimal tuning (POTE): ~2 = 1\1, ~77/75 = 48.822

Optimal GPV sequence: Template:Val list

Badness: 0.082158

Escaped

See also: Escapade family § Escaped

This temperament is also known as "sensa" because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. Not to be confused with the 19e & 27 temperament (sensi extension).

Subgroup: 2.3.5.7

Comma list: 245/243, 65625/65536

Mapping: [⟨1 2 2 4], ⟨0 -9 7 -26]]

Wedgie: ⟨⟨ 9 -7 26 -32 16 80 ]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.122

Template:Val list

Badness: 0.088746

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 4000/3993

Mapping: [⟨1 2 2 4 3], ⟨0 -9 7 -26 10]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.126

Optimal GPV sequence: Template:Val list

Badness: 0.035844

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 245/243, 352/351, 385/384, 625/624

Mapping: [⟨1 2 2 4 3 2], ⟨0 -9 7 -26 10 37]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.138

Optimal GPV sequence: Template:Val list

Badness: 0.031366

Salsa

See also: Schismatic family

Subgroup: 2.3.5.7

Comma list: 245/243, 32805/32768

Mapping: [⟨1 1 7 -1], ⟨0 2 -16 13]]

Wedgie: ⟨⟨ 2 -16 13 -30 15 75 ]]

Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 351.049

Template:Val list

Badness: 0.080152

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 245/242, 385/384

Mapping: [⟨1 1 7 -1 2], ⟨0 2 -16 13 5]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.014

Optimal GPV sequence: Template:Val list

Badness: 0.039444

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 243/242, 245/242

Mapping: [⟨1 1 7 -1 2 4], ⟨0 2 -16 13 5 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.025

Optimal GPV sequence: Template:Val list

Badness: 0.030793

Pycnic

See also: High badness temperaments § Stump

The fifth of pycnic in size is a meantone fifth, but four of them are not used to reach 5. This has the effect of making the Pythagorean major third, nominally 81/64, very close to 5/4 in tuning, being a cent sharp of it in the POTE tuning for instance. Pycnic has mos of size 9, 11, 13, 15, 17… which contain these alternative thirds, leading to two kinds of major triads, an official one and a nominally Pythagorean one which is actually in better tune.

Subgroup: 2.3.5.7

Comma list: 245/243, 525/512

Mapping: [⟨1 3 -1 8], ⟨0 -3 7 -11]]

Wedgie: ⟨⟨ 3 -7 11 -18 9 45 ]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 567.720

Template:Val list

Badness: 0.073735

Cohemiripple

See also: Ripple family

Subgroup: 2.3.5.7

Comma list: 245/243, 1323/1250

Mapping: [⟨1 -3 -5 -5], ⟨0 10 16 17]]

Wedgie: ⟨⟨ 10 16 17 2 -1 -5 ]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 549.944

Template:Val list

Badness: 0.190208

11-limit

Subgroup: 2.3.5.7.11

Comma list: 77/75, 243/242, 245/242

Mapping: [⟨1 -3 -5 -5 -8], ⟨0 10 16 17 25]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 549.945

Optimal GPV sequence: Template:Val list

Badness: 0.082716

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 77/75, 147/143, 243/242

Mapping: [⟨1 -3 -5 -5 -8 -5], ⟨0 -10 -16 -17 -25 -19]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 549.958

Optimal GPV sequence: Template:Val list

Badness: 0.049933

Superthird

See also: Shibboleth family

Subgroup: 2.3.5.7

Comma list: 245/243, 78125/76832

Mapping: [⟨1 -5 -5 -10], ⟨0 18 20 35]]

Wedgie: ⟨⟨ 18 20 35 -10 5 25 ]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.076

Template:Val list

Badness: 0.139379

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243, 78125/76832

Mapping: [⟨1 -5 -5 -10 2], ⟨0 18 20 35 4]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.152

Optimal GPV sequence: Template:Val list

Badness: 0.070917

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 196/195, 1375/1352

Mapping: [⟨1 -5 -5 -10 2 -8], ⟨0 18 20 35 4 32]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.119

Optimal GPV sequence: Template:Val list

Badness: 0.052835

Superenneadecal

Superenneadecal is a cousin of enneadecal but sharper fifth is used to temper 245/243.

Subgroup: 2.3.5.7

Comma list: 245/243, 395136/390625

Mapping: [⟨19 0 14 -7], ⟨0 1 1 2]]

Template:Val list

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 2560/2541, 3773/3750

Mapping: [⟨19 0 14 -7 96], ⟨0 1 1 2 -1]]

Template:Val list

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 832/825, 1001/1000

Mapping: [⟨19 0 14 -7 96 10], ⟨0 1 1 2 -1 2]]

Template:Val list

Magus

For the 5-limit version of this temperament, see High badness temperaments #Magus.

Magus temperament tempers out 50331648/48828125 (salegu) in the 5-limit. This temperament can be described as 46 & 49 temperament, which tempers out the sensamagic and 28672/28125 (sazoquingu). The alternative extension amigo (43 & 46) tempers out the same 5-limit comma as the magus, but with the starling comma (126/125) rather than the sensamagic tempered out.

Subgroup: 2.3.5.7

Comma list: 245/243, 28672/28125

Mapping: [⟨1 -2 2 -6], ⟨0 11 1 27]]

Wedgie: ⟨⟨ 11 1 27 -24 12 60 ]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.465

Template:Val list

Badness: 0.108417

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 245/243, 1331/1323

Mapping: [⟨1 -2 2 -6 -6], ⟨0 11 1 27 29]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.503

Optimal GPV sequence: Template:Val list

Badness: 0.045108

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 176/175, 245/243, 1331/1323

Mapping: [⟨1 -2 2 -6 -6 5], ⟨0 11 1 27 29 -4]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.366

Optimal GPV sequence: Template:Val list

Badness: 0.043024

Leapweek

Not to be confused with scales produced by leap week calendars such as Symmetry454.

Subgroup: 2.3.5.7

Comma list: 245/243, 2097152/2066715

Mapping: [⟨1 1 17 -6], ⟨0 1 -25 15]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.536

Template:Val list

Badness: 0.140577

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 1331/1323

Mapping: [⟨1 1 17 -6 -3], ⟨0 1 -25 15 11]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.554

Optimal GPV sequence: Template:Val list

Badness: 0.050679

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 245/243, 352/351, 364/363

Mapping: [⟨1 1 17 -6 -3 -1], ⟨0 1 -25 15 11 8]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.571

Optimal GPV sequence: Template:Val list

Badness: 0.032727

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 154/153, 169/168, 245/243, 256/255, 273/272

Mapping: [⟨1 1 17 -6 -3 -1 -10], ⟨0 1 -25 15 11 8 24]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.540

Optimal GPV sequence: Template:Val list

Badness: 0.026243

Leapweeker

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 169/168, 221/220, 245/243, 364/363

Mapping: [⟨1 1 17 -6 -3 -1 17], ⟨0 1 -25 15 11 8 -22]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.537

Optimal GPV sequence: Template:Val list

Badness: 0.026774

Semiwolf

Subgroup: 3/2.7/4.5/2

Comma list: 245/243

Mapping: [⟨1 1 3], ⟨0 1 -2]]

POL2 generator: ~7/6 = 262.1728

Optimal GPV sequence: 3edf, 5edf, 8edf

Semilupine

Subgroup: 3/2.7/4.5/2.11/4

Comma list: 100/99, 245/243

Mapping: [⟨1 1 3 4], ⟨0 1 -2 -4]]

POL2 generator: ~7/6 = 264.3771

Optimal GPV sequence: 8edf, 13edf

Hemilycan

Subgroup: 3/2.7/4.5/2.11/4

Comma list: 245/243, 441/440

Mapping: [⟨1 1 3 1], ⟨0 1 -2 4]]

POL2 generator: ~7/6 = 261.5939

Optimal GPV sequence: 8edf, 11edf

@@ Line 17: / Line 17: @@
 * ''[[Fourfives]]'', {245/243, 235298/234375} → [[Fifive family #Fourfives|Fifive family]]
-Tempering out 245/243 alone in the full 7-limit leads to a [[Planar temperament|rank-3 temperament]], [[sensamagic]], for which [[283edo|283EDO]] is the [[optimal patent val]].
+Tempering out 245/243 alone in the full 7-limit leads to a [[Planar temperament|rank-3 temperament]], [[sensamagic]], for which [[283edo]] is the [[optimal patent val]].
 == BPS ==
-The ''BPS'', for ''Bohlen–Pierce–Stearns'', is the 3.5.7 subgroup temperament tempering out 245/243. This subgroup temperament was formerly called as ''lambda temperament'', which was named after [[4L 5s (tritave-equivalent)|lambda scale]].
+The ''BPS'', for ''Bohlen–Pierce–Stearns'', is the 3.5.7 subgroup temperament tempering out 245/243. This subgroup temperament was formerly called the ''lambda'' temperament, which was named after the [[4L 5s (tritave-equivalent)|lambda scale]].
-Subgroup: 3.5.7
+[[Subgroup]]: 3.5.7
 [[Comma list]]: 245/243
@@ Line 30: / Line 30: @@
 Sval mapping generators: ~3, ~9/7
-[[POTE generator]]: ~9/7 = 440.4881
+[[Optimal tuning]] ([[POTE]]): ~3 = 1\1edt, ~9/7 = 440.4881
 [[Optimal GPV sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
@@ Line 51: / Line 51: @@
 {{Multival|legend=1| 7 9 13 -2 1 5 }}
-[[POTE generator]]: ~9/7 = 443.383
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 443.383
 {{Val list|legend=1| 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd }}
@@ Line 68: / Line 68: @@
 Gencom: [2 9/7; 91/90 126/125 169/168]
-POTE generator: ~9/7 = 443.322
+Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.322
 Optimal GPV sequence: {{Val list| 19, 27, 46, 111de, 157de }}
@@ Line 79: / Line 79: @@
 Mapping: [{{val| 1 -1 -1 -2 9 }}, {{val| 0 7 9 13 -15 }}]
-POTE generator: ~9/7 = 443.294
+Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.294
 Optimal GPV sequence: {{Val list| 19, 27, 46, 111d, 157d, 268cdd }}
@@ Line 92: / Line 92: @@
 Mapping: [{{val| 1 -1 -1 -2 9 0 }}, {{val| 0 7 9 13 -15 10 }}]
-POTE generator: ~9/7 = 443.321
+Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.321
 Optimal GPV sequence: {{Val list| 19, 27, 46, 111df, 157df }}
@@ Line 105: / Line 105: @@
 Mapping: [{{val| 1 -1 -1 -2 9 0 10 }}, {{val| 0 7 9 13 -15 10 -16 }}]
-POTE generator: ~9/7 = 443.365
+Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.365
 Optimal GPV sequence: {{Val list| 19, 27, 46, 157df, 203cdff, 249cddff }}
@@ Line 118: / Line 118: @@
 Mapping: [{{val| 1 -1 -1 -2 2 }}, {{val| 0 7 9 13 4 }}]
-POTE generator: ~9/7 = 443.962
+Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.962
 Optimal GPV sequence: {{Val list| 8d, 19, 27e, 73ee }}
@@ Line 131: / Line 131: @@
 Mapping: [{{val| 1 -1 -1 -2 2 0 }}, {{val| 0 7 9 13 4 10 }}]
-POTE generator: ~9/7 = 443.945
+Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.945
 Optimal GPV sequence: {{Val list| 19, 27e, 46e, 73ee }}
@@ Line 144: / Line 144: @@
 Mapping: [{{val| 1 -1 -1 -2 -8 }}, {{val| 0 7 9 13 31 }}]
-POTE generator: ~9/7 = 443.626
+Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.626
 Optimal GPV sequence: {{Val list| 19e, 27e, 46, 119c, 165c }}
@@ Line 157: / Line 157: @@
 Mapping: [{{val| 1 -1 -1 -2 -8 0 }}, {{val| 0 7 9 13 31 10 }}]
-POTE generator: ~9/7 = 443.559
+Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.559
 Optimal GPV sequence: {{Val list| 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff }}
@@ Line 170: / Line 170: @@
 Mapping: [{{val| 1 -1 -1 -2 -8 0 -7 }}, {{val| 0 7 9 13 31 10 30 }}]
-POTE generator: ~9/7 = 443.551
+Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.551
 Optimal GPV sequence: {{Val list| 19eg, 27eg, 46 }}
@@ Line 183: / Line 183: @@
 Mapping: [{{val| 1 -1 -1 -2 -1 }}, {{val| 0 7 9 13 12 }}]
-POTE generator: ~9/7 = 443.518
+Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.518
 Optimal GPV sequence: {{Val list| 19e, 27, 46ee }}
@@ Line 196: / Line 196: @@
 Mapping: [{{val| 1 -1 -1 -2 -1 0 }}, {{val| 0 7 9 13 12 11 }}]
-POTE generator: ~9/7 = 443.506
+Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.506
 Optimal GPV sequence: {{Val list| 19e, 27, 46ee }}
@@ Line 209: / Line 209: @@
 Mapping: [{{val| 1 -1 -1 -2 -3 }}, {{val| 0 14 18 26 35 }}]
-POTE generator: ~25/22 = 221.605
+Optimal tuning (POTE): ~2 = 1\1, ~25/22 = 221.605
 Optimal GPV sequence: {{Val list| 27e, 38d, 65, 157de, 222cde }}
@@ Line 222: / Line 222: @@
 Mapping: [{{val| 1 -1 -1 -2 -3 0 }}, {{val| 0 14 18 26 35 30 }}]
-POTE generator: ~25/22 = 221.556
+Optimal tuning (POTE): ~2 = 1\1, ~25/22 = 221.556
 Optimal GPV sequence: {{Val list| 27e, 38df, 65f }}
@@ Line 235: / Line 235: @@
 Mapping: [{{val| 2 5 7 9 9 }}, {{val| 0 -7 -9 -13 -8 }}]
-POTE generator: ~11/10 = 156.692
+Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 156.692
 Optimal GPV sequence: {{Val list| 8d, …, 38d, 46, 176dde, 222cdde, 268cddee }}
@@ Line 248: / Line 248: @@
 Mapping: [{{val| 2 5 7 9 9 10 }}, {{val| 0 -7 -9 -13 -8 -10 }}]
-POTE generator: ~11/10 = 156.725
+Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 156.725
 Optimal GPV sequence: {{Val list| 8d, …, 38df, 46 }}
@@ Line 256: / Line 256: @@
 == Bohpier ==
 : ''For the 5-limit version of this temperament, see [[High badness temperaments #Bohpier]].''
-{{main|Bohpier}}
+{{Main| Bohpier }}
 '''[[Bohpier]]''' is named after its [[Relationship between Bohlen-Pierce and octave-ful temperaments|interesting relationship with the non-octave Bohlen-Pierce equal temperament]].
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 245/243, 3125/3087
@@ Line 268: / Line 268: @@
 {{Multival|legend=1| 13 19 23 0 0 0 }}
-[[POTE generator]]: ~27/25 = 146.474
+Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 146.474
 [[Minimax tuning]]:
 * 7-odd-limit: ~27/25 = {{monzo| 0 0 1/19 }}
-: Eigenmonzos (unchanged intervals): 2, 5/4
+: [[Eigenmonzo basis]]: 2.5
 * 9-odd-limit: ~27/25 = {{monzo| 0 1/13 }}
-: Eigenmonzos (unchanged intervals): 2, 4/3
+: [[Eigenmonzo basis]]: 2.3
 {{Val list|legend=1| 41, 131, 172, 213c }}
@@ Line 287: / Line 287: @@
 Mapping: [{{val| 1 0 0 0 2 }}, {{val| 0 13 19 23 12 }}]
-POTE generator: ~12/11 = 146.545
+Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 146.545
 Minimax tuning:
 * 11-odd-limit: ~12/11 = {{monzo| 1/7 1/7 0 0 -1/14 }}
-: Eigenmonzos (unchanged intervals): 2, 11/9
+: Eigenmonzo basis: 2.11/9
 Optimal GPV sequence: {{Val list| 41, 90e, 131e }}
@@ Line 304: / Line 304: @@
 Mapping: [{{val| 1 0 0 0 2 2 }}, {{val| 0 13 19 23 12 14 }}]
-POTE generator: ~12/11 = 146.603
+Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 146.603
 Minimax tuning:
 * 13- and 15-odd-limit: ~12/11 = {{monzo| 0 0 1/19 }}
-: Eigenmonzos (unchanged intervals): 2, 5/4
+: Eigenmonzo basis: 2.5
 Optimal GPV sequence: {{Val list| 41, 90ef, 131ef, 221bdeff }}
@@ Line 328: / Line 328: @@
 Mapping: [{{val| 1 0 0 0 0 }}, {{val| 0 39 57 69 85 }}]
-POTE generator: ~77/75 = 48.828
+Optimal tuning (POTE): ~2 = 1\1, ~77/75 = 48.828
 Optimal GPV sequence: {{Val list| 49, 123ce, 172 }}
@@ Line 341: / Line 341: @@
 Mapping: [{{val| 1 0 0 0 0 0 }}, {{val| 0 39 57 69 85 91 }}]
-POTE generator: ~77/75 = 48.822
+Optimal tuning (POTE): ~2 = 1\1, ~77/75 = 48.822
 Optimal GPV sequence: {{Val list| 49f, 123ce, 172f, 295ce, 467bccef }}
@@ Line 348: / Line 348: @@
 == Escaped ==
-{{see also| Escapade family #Escaped }}
+{{See also| Escapade family #Escaped }}
-This temperament is also called as "sensa" because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. ''Not to be confused with 19e&amp;27 temperament (sensi extension).''
+This temperament is also known as "sensa" because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. ''Not to be confused with the 19e &amp; 27 temperament (sensi extension).''
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 245/243, 65625/65536
@@ Line 360: / Line 360: @@
 {{Multival|legend=1| 9 -7 26 -32 16 80 }}
-[[POTE generator]]: ~28/27 = 55.122
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 55.122
 {{Val list|legend=1| 22, 65, 87, 196, 283 }}
@@ Line 373: / Line 373: @@
 Mapping: [{{val| 1 2 2 4 3 }}, {{val| 0 -9 7 -26 10 }}]
-POTE generator: ~28/27 = 55.126
+Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.126
 Optimal GPV sequence: {{Val list| 22, 65, 87, 196, 283 }}
@@ Line 386: / Line 386: @@
 Mapping: [{{val| 1 2 2 4 3 2 }}, {{val| 0 -9 7 -26 10 37 }}]
-POTE generator: ~28/27 = 55.138
+Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.138
 Optimal GPV sequence: {{Val list| 22, 65, 87, 283 }}
@@ Line 393: / Line 393: @@
 == Salsa ==
-{{see also| Schismatic family }}
+{{See also| Schismatic family }}
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 245/243, 32805/32768
@@ Line 403: / Line 403: @@
 {{Multival|legend=1| 2 -16 13 -30 15 75 }}
-[[POTE generator]]: ~128/105 = 351.049
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 351.049
 {{Val list|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
@@ Line 416: / Line 416: @@
 Mapping: [{{val| 1 1 7 -1 2 }}, {{val| 0 2 -16 13 5 }}]
-POTE generator: ~11/9 = 351.014
+Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.014
 Optimal GPV sequence: {{Val list| 17, 24, 41, 106d, 147d }}
@@ Line 429: / Line 429: @@
 Mapping: [{{val| 1 1 7 -1 2 4 }}, {{val| 0 2 -16 13 5 -1 }}]
-POTE generator: ~11/9 = 351.025
+Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.025
 Optimal GPV sequence: {{Val list| 17, 24, 41, 106df, 147df }}
@@ Line 436: / Line 436: @@
 == Pycnic ==
-{{see also| High badness temperaments #Stump }}
+{{See also| High badness temperaments #Stump }}
-The fifth of pycnic in size is a meantone fifth, but four of them are not used to reach 5. This has the effect of making the Pythagorean major third, nominally 81/64, very close to 5/4 in tuning, being a cent sharp of it in the POTE tuning for instance. Pycnic has MOS of size 9, 11, 13, 15, 17... which contain these alternative thirds, leading to two kinds of major triads, an official one and a nominally Pythagorean one which is actually in better tune.
+The fifth of pycnic in size is a meantone fifth, but four of them are not used to reach 5. This has the effect of making the Pythagorean major third, nominally 81/64, very close to 5/4 in tuning, being a cent sharp of it in the POTE tuning for instance. Pycnic has [[mos]] of size 9, 11, 13, 15, 17… which contain these alternative thirds, leading to two kinds of major triads, an official one and a nominally Pythagorean one which is actually in better tune.
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 245/243, 525/512
@@ Line 448: / Line 448: @@
 {{Multival|legend=1| 3 -7 11 -18 9 45 }}
-[[POTE generator]]: ~45/32 = 567.720
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~45/32 = 567.720
 {{Val list|legend=1| 17, 19, 55c, 74cd, 93cdd }}
@@ Line 455: / Line 455: @@
 == Cohemiripple ==
-{{see also| Ripple family }}
+{{See also| Ripple family }}
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 245/243, 1323/1250
@@ Line 465: / Line 465: @@
 {{Multival|legend=1| 10 16 17 2 -1 -5 }}
-[[POTE generator]]: ~7/5 = 549.944
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 549.944
 {{Val list|legend=1| 11cd, 13cd, 24 }}
@@ Line 478: / Line 478: @@
 Mapping: [{{val| 1 -3 -5 -5 -8 }}, {{val| 0 10 16 17 25 }}]
-POTE generator: ~7/5 = 549.945
+Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 549.945
 Optimal GPV sequence: {{Val list| 11cdee, 13cdee, 24 }}
@@ Line 491: / Line 491: @@
 Mapping: [{{val| 1 -3 -5 -5 -8 -5 }}, {{val| 0 -10 -16 -17 -25 -19 }}]
-POTE generator: ~7/5 = 549.958
+Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 549.958
 Optimal GPV sequence: {{Val list| 11cdeef, 13cdeef, 24 }}
@@ Line 498: / Line 498: @@
 == Superthird ==
-{{see also| Shibboleth family }}
+{{See also| Shibboleth family }}
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 245/243, 78125/76832
@@ Line 508: / Line 508: @@
 {{Multival|legend=1| 18 20 35 -10 5 25 }}
-[[POTE generator]]: ~9/7 = 439.076
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 439.076
 {{Val list|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
@@ Line 521: / Line 521: @@
 Mapping: [{{val| 1 -5 -5 -10 2 }}, {{val| 0 18 20 35 4 }}]
-POTE generator: ~9/7 = 439.152
+Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.152
 Optimal GPV sequence: {{Val list| 11cd, 30d, 41, 153be, 194be, 235bcee }}
@@ Line 534: / Line 534: @@
 Mapping: [{{val| 1 -5 -5 -10 2 -8 }}, {{val| 0 18 20 35 4 32 }}]
-POTE generator: ~9/7 = 439.119
+Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.119
 Optimal GPV sequence: {{Val list| 11cdf, 30df, 41 }}
@@ Line 541: / Line 541: @@
 == Superenneadecal ==
 Superenneadecal is a cousin of [[enneadecal]] but sharper fifth is used to temper 245/243.
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 245/243, 395136/390625
-Mapping: [⟨19 0 14 -7], ⟨0 1 1 2]]
+[[Mapping]]: [{{val| 19 0 14 -7 }}, {{val| 0 1 1 2 }}]
 {{Val list|legend=1| 19, 76bcd, 95, 114, 133, 247b, 380bcd }}
@@ Line 555: / Line 554: @@
 Subgroup: 2.3.5.7.11
-[[Comma list]]: 245/243, 2560/2541, 3773/3750
+Comma list: 245/243, 2560/2541, 3773/3750
-Mapping: [⟨19 0 14 -7 96], ⟨0 1 1 2 -1]]
+Mapping: [{{val| 19 0 14 -7 96 }}, {{val| 0 1 1 2 -1 }}]
 {{Val list|legend=1| 19, 76bcd, 95, 114e }}
@@ Line 564: / Line 563: @@
 Subgroup: 2.3.5.7.11.13
-[[Comma list]]: 196/195, 245/243, 832/825, 1001/1000
+Comma list: 196/195, 245/243, 832/825, 1001/1000
-Mapping: [⟨19 0 14 -7 96 10], ⟨0 1 1 2 -1 2]]
+Mapping: [{{val| 19 0 14 -7 96 10 }}, {{val| 0 1 1 2 -1 2 }}]
 {{Val list|legend=1| 19, 76bcdf, 95, 114e }}
@@ Line 573: / Line 572: @@
 : ''For the 5-limit version of this temperament, see [[High badness temperaments #Magus]].''
-Magus temperament tempers out 50331648/48828125 (salegu) in the 5-limit. This temperament can be described as 46&amp;49 temperament, which tempers out the sensamagic and 28672/28125 (sazoquingu). Alternative extension [[Starling temperaments #Amigo|amigo]] (43&amp;46) tempers out the same 5-limit comma as the magus, but with the [[126/125|starling comma]] (126/125) rather than the sensamagic tempered out.
+Magus temperament tempers out 50331648/48828125 (salegu) in the 5-limit. This temperament can be described as 46 &amp; 49 temperament, which tempers out the sensamagic and 28672/28125 (sazoquingu). The alternative extension [[Starling temperaments #Amigo|amigo]] (43 &amp; 46) tempers out the same 5-limit comma as the magus, but with the [[126/125|starling comma]] (126/125) rather than the sensamagic tempered out.
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 245/243, 28672/28125
@@ Line 583: / Line 582: @@
 {{Multival|legend=1| 11 1 27 -24 12 60 }}
-[[POTE generator]]: ~5/4 = 391.465
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 391.465
 {{Val list|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}
@@ Line 596: / Line 595: @@
 Mapping: [{{val| 1 -2 2 -6 -6 }}, {{val| 0 11 1 27 29 }}]
-POTE generator: ~5/4 = 391.503
+Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.503
 Optimal GPV sequence: {{Val list| 46, 95, 141bc }}
@@ Line 609: / Line 608: @@
 Mapping: [{{val| 1 -2 2 -6 -6 5 }}, {{val| 0 11 1 27 29 -4 }}]
-POTE generator: ~5/4 = 391.366
+Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.366
 Optimal GPV sequence: {{Val list| 46, 233bcff, 279bccff }}
@@ Line 618: / Line 617: @@
 ''Not to be confused with scales produced by leap week calendars such as [[Symmetry454]].''
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 245/243, 2097152/2066715
@@ Line 624: / Line 623: @@
 [[Mapping]]: [{{val| 1 1 17 -6 }}, {{val| 0 1 -25 15 }}]
-[[POTE generator]]: ~3/2 = 704.536
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.536
 {{Val list|legend=1| 17, 29c, 46, 109, 155, 264b, 419b }}
@@ Line 637: / Line 636: @@
 Mapping: [{{val| 1 1 17 -6 -3 }}, {{val| 0 1 -25 15 11 }}]
-POTE generator: ~3/2 = 704.554
+Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.554
 Optimal GPV sequence: {{Val list| 17, 29c, 46, 109, 264b, 373b, 637bbe }}
@@ Line 650: / Line 649: @@
 Mapping: [{{val| 1 1 17 -6 -3 -1 }}, {{val| 0 1 -25 15 11 8 }}]
-POTE generator: ~3/2 = 704.571
+Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.571
 Optimal GPV sequence: {{Val list| 17, 29c, 46, 63, 109 }}
@@ Line 663: / Line 662: @@
 Mapping: [{{val| 1 1 17 -6 -3 -1 -10 }}, {{val| 0 1 -25 15 11 8 24 }}]
-POTE generator: ~3/2 = 704.540
+Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.540
 Optimal GPV sequence: {{Val list| 17g, 29cg, 46, 109, 155f, 264bfg }}
@@ Line 676: / Line 675: @@
 Mapping: [{{val| 1 1 17 -6 -3 -1 17 }}, {{val| 0 1 -25 15 11 8 -22 }}]
-POTE generator: ~3/2 = 704.537
+Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.537
 Optimal GPV sequence: {{Val list| 17, 29c, 46, 109g, 155fg, 264bfgg }}

Sensamagic clan: Difference between revisions

Revision as of 05:46, 16 January 2023

BPS

Sensi

Septimal sensi

Sensation

Sensor

13-limit

17-limit

Sensis

13-limit

Sensus

13-limit

17-limit

Sensa

13-limit

Hemisensi

13-limit

Bisensi

13-limit

Bohpier

11-limit

13-limit

Triboh

13-limit

Escaped

11-limit

13-limit

Salsa

11-limit

13-limit

Pycnic

Cohemiripple

11-limit

13-limit

Superthird

11-limit

13-limit

Superenneadecal

11-limit

13-limit

Magus

11-limit

13-limit

Leapweek

11-limit

13-limit

17-limit

Leapweeker

Semiwolf

Semilupine

Hemilycan

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