Quindromeda family: Difference between revisions

The '''quindromeda family''' tempers out {{monzo| 56 -28 -5 }}, equating the [[Pythagorean comma]] with a stack of five [[schisma]]s, making it a member of the [[schismic-Pythagorean equivalence continuum]].  

[[Comma list]]: {{monzo| 56 -28 -5 }}

[[POTE generator]]: ~4428675/4194304 = 99.526

{{Val list|legend=1| 12, 169, 181, 193, 205, 217, 422 }}

[[Badness]]: 0.399849

{{See also| 28ed5 #Regular temperaments }}

The ''quintagar'' temperament (12&217) tempers out the hemimean comma (3136/3125) and the [[garischisma]] (33554432/33480783) in the 7-limit. In the 2.3.5.7.17.19 subgroup, 256/255 (the difference between 16/15 and 17/16), 289/288 (between 17/16 and 18/17), 324/323 (between 18/17 and 19/18), 361/360 (between 19/18 and 20/19), and 400/399 (between 20/19 and 21/20) are equated together, and 476/475 (between 28/25 and 19/17) is tempered out. Immediate 2.3.5.7.11.17.19 extensions include ''quintoneum'' (12&217, tempering out 441/440), ''quintasandra'' (217&229, equating 385/384 with 400/399), and ''quintasandroid'' (12&229, equating 400/399 with 441/440). Full 19-limit extensions include ''quintoneum'' (12f&217), ''quintoneoid'' (12&217), ''quintasandra'' (217&229), ''quintasandroid'' (12f&229), and ''quintasand'' (12&229). The name ''quintagar'' is so named because the generator is 1/5 of the [[Schismatic family #Garibaldi|garibaldi]] fourth (~4/3, about 497.8 cents) aside from tempering out the garischisma.

[[Mapping]]: [{{val|1 2 0 -3}}, {{val|0 -5 28 70}}]

 

[[POTE generator]]: ~200/189 = 99.555

{{Val list|legend=1| 12, 217, 229, 446, 675c }}

[[Badness]]: 0.142897

The name ''quintoneum'' is a play on the words "quintans" (Latin for "one fifth") and "[[Hemimage temperaments #Cotoneum|cotoneum]]".

Mapping: [{{val|1 2 0 -3 -5}}, {{val|0 -5 28 70 102}}]

POTE generator: ~35/33 = 99.539

Vals: {{Val list| 12, 205d, 217 }}

Badness: 0.087157

Mapping: [{{val|1 2 0 -3 -5 -7}}, {{val|0 -5 28 70 102 129}}]

POTE generator: ~35/33 = 99.541

Vals: {{Val list| 12f, 205df, 217 }}

Badness: 0.052361

Mapping: [{{val|1 2 0 -3 -5 -7 5}}, {{val|0 -5 28 70 102 129 -11}}]

POTE generator: ~18/17 = 99.540

Vals: {{Val list| 12f, 205df, 217 }}

Badness: 0.035653

Mapping: [{{val|1 2 0 -3 -5 -7 5 4}}, {{val|0 -5 28 70 102 129 -11 3}}]

POTE generator: ~18/17 = 99.541

Vals: {{Val list| 12f, 205df, 217 }}

Badness: 0.025782

Mapping: [{{val|1 2 0 -3 -5 11}}, {{val|0 -5 28 70 102 -88}}]

POTE generator: ~35/33 = 99.537

Vals: {{Val list| 12, 205d, 217 }}

Badness: 0.072826

Mapping: [{{val|1 2 0 -3 -5 11 5}}, {{val|0 -5 28 70 102 -88 -11}}]

POTE generator: ~18/17 = 99.537

Vals: {{Val list| 12, 205d, 217 }}

Badness: 0.042339

Mapping: [{{val|1 2 0 -3 -5 11 5 4}}, {{val|0 -5 28 70 102 -88 -11 3}}]

POTE generator: ~18/17 = 99.537

Vals: {{Val list| 12, 205d, 217 }}

Badness: 0.028983

=== Quintasandra ===

The name ''quintasandra'' is a play on the words "quintans" and "[[Schismatic family #Garibaldi|cassandra]]". This temperament tempers out 19712/19683 and 41503/41472 in the 2.3.7.11 subgroup as the cassandra temperament, but with the hemimean comma rather than the [[schisma]] tempered out.

Mapping: [{{val|1 2 0 -3 13}}, {{val|0 -5 28 70 -115}}]

POTE generator: ~200/189 = 99.551

Vals: {{Val list| 12e, 217, 446 }}

Badness: 0.109908

Mapping: [{{val|1 2 0 -3 13 11}}, {{val|0 -5 28 70 -115 -88}}]

POTE generator: ~55/52 = 99.548

Vals: {{Val list| 12e, 217, 446, 663c }}

Badness: 0.067730

Mapping: [{{val|1 2 0 -3 13 11 5}}, {{val|0 -5 28 70 -115 -88 -11}}]

POTE generator: ~18/17 = 99.548

Vals: {{Val list| 12e, 217, 446, 663c }}

Badness: 0.038153

Mapping: [{{val|1 2 0 -3 13 11 5 4}}, {{val|0 -5 28 70 -115 -88 -11 3}}]

POTE generator: ~18/17 = 99.547

Vals: {{Val list| 12e, 217, 446, 663ch }}

Badness: 0.026654

=== Quintasandroid ===

Mapping: [{{val|1 2 0 -3 -6}}, {{val|0 -5 28 70 114}}]

POTE generator: ~200/189 = 99.570

Vals: {{Val list| 12, 217e, 229, 470cd, 699cd }}

Badness: 0.093971

Mapping: [{{val|1 2 0 -3 -6 -8}}, {{val|0 -5 28 70 114 141}}]

POTE generator: ~55/52 = 99.578

Vals: {{Val list| 12f, 217ef, 229, 241, 470cd, 711ccd }}

Badness: 0.065701

Mapping: [{{val|1 2 0 -3 -6 -8 5}}, {{val|0 -5 28 70 114 141 -11}}]

POTE generator: ~18/17 = 99.574

Vals: {{Val list| 12f, 217ef, 229, 241, 470cd }}

Badness: 0.046624

Mapping: [{{val|1 2 0 -3 -6 -8 5 4}}, {{val|0 -5 28 70 114 141 -11 3}}]

POTE generator: ~18/17 = 99.575

Vals: {{Val list| 12f, 217ef, 229, 241, 470cd }}

Badness: 0.033145

==== Quintasand ====

Mapping: [{{val|1 2 0 -3 -6 11}}, {{val|0 -5 28 70 114 -88}}]

Vals: {{Val list| 12, 217e, 229, 446e, 675ceef }}

Badness: 0.100195

Mapping: [{{val|1 2 0 -3 -6 11 5}}, {{val|0 -5 28 70 114 -88 -11}}]

POTE generator: ~18/17 = 99.556

Vals: {{Val list| 12, 217e, 229, 446e, 675ceef }}

Badness: 0.057851

Mapping: [{{val|1 2 0 -3 -6 11 5 4}}, {{val|0 -5 28 70 114 -88 -11 3}}]

POTE generator: ~18/17 = 99.557

Vals: {{Val list| 12, 217e, 229, 446e, 675ceefh }}

Badness: 0.040410

The ''quintakwai'' temperament (12&193) tempers out the hemifamity comma (5120/5103) and the compass comma (9765625/9680832, quinruyoyo) in the 7-limit; 1375/1372 and 4375/4356 in the 11-limit. In the 2.3.5.7.17.19 subgroup, 225/224 (the difference between 15/14 and 16/15), 256/255 (between 16/15 and 17/16), 289/288 (between 17/16 and 18/17), 324/323 (between 18/17 and 19/18), and 361/360 (between 19/18 and 20/19) are equated together, and 400/399 (between 20/19 and 21/20) is tempered out. The name ''quintakwai'' is so named because the generator is 1/5 of the [[Mirkwai clan #Kwai|kwai]] fourth (~4/3, about 497.4 cents).

 

{{Multival|legend=1| 5 -28 -58 -56 -106 -56 }}

 

[[POTE generator]]: ~625/588 = 99.483

{{Val list|legend=1| 12, 169, 181, 193 }}

[[Badness]]: 0.155536

POTE generator: ~35/33 = 99.472

Vals: {{Val list| 12, 181, 193, 374, 567ce }}

Badness: 0.073158

POTE generator: ~35/33 = 99.468

Vals: {{Val list| 12, 181, 193, 374, 567ce, 941bce }}

Badness: 0.062737

POTE generator: ~18/17 = 99.469

Vals: {{Val list| 12, 181, 193, 374, 567ce, 941bceg }}

Badness: 0.037855

POTE generator: ~18/17 = 99.469

Vals: {{Val list| 12, 181, 193, 374, 567ce, 941bcegh, 1508bccdeegghh }}

Badness: 0.025861

POTE generator: ~35/33 = 99.456

Vals: {{Val list| 12f, 169e, 181 }}

Badness: 0.061873

POTE generator: ~18/17 = 99.458

Vals: {{Val list| 12f, 169e, 181, 374ff, 555cff }}

Badness: 0.043506

POTE generator: ~18/17 = 99.459

Vals: {{Val list| 12f, 169e, 181, 374ff, 555cff }}

Badness: 0.031313

POTE generator: ~35/33 = 99.484

Vals: {{Val list| 12f, 181f, 193 }}

Badness: 0.057357

POTE generator: ~18/17 = 99.484

Vals: {{Val list| 12f, 181f, 193 }}

Badness: 0.039289

POTE generator: ~18/17 = 99.485

Vals: {{Val list| 12f, 181f, 193 }}

Badness: 0.028101

The ''quindro'' temperament (205&217) is an extension of the [[28ed5 #Regular temperaments|''quindromeda'' temperament]] which tempers out the ragisma (4375/4374), the vishdel comma (5632/5625), and the ibnsinma (2080/2079).  

[[Mapping]]: [{{val|1 2 0 15}}, {{val|0 -5 28 -147}}]

 

[[POTE generator]]: ~4428675/4194304 = 99.529

{{Val list|legend=1| 205, 217, 422, 639, 1061 }}

[[Badness]]: 0.316850

Mapping: [{{val|1 2 0 15 -5}}, {{val|0 -5 28 -147 102}}]

POTE generator: ~17325/16384 = 99.529

Vals: {{Val list| 205, 217, 422, 639, 1061e }}

Badness: 0.085219

Mapping: [{{val|1 2 0 15 -5 11}}, {{val|0 -5 28 -147 102 -88}}]

POTE generator: ~4096/3861 = 99.529

Vals: {{Val list| 205, 217, 422, 639, 1061ef }}

Badness: 0.039203

Mapping: [{{val|1 2 0 15 -5 11 5}}, {{val|0 -5 28 -147 102 -88 -11}}]

POTE generator: ~18/17 = 99.529

Vals: {{Val list| 205, 217, 422, 639, 1061ef }}

Badness: 0.024110

Mapping: [{{val|1 2 0 15 -5 11 5 4}}, {{val|0 -5 28 -147 102 -88 -11 3}}]

POTE generator: ~18/17 = 99.529

Vals: {{Val list| 205, 217, 422, 639h, 1061efh }}

Badness: 0.016764

The ''semiquindromeda'' temperament (410&422) is an extension of the [[28ed5 #Regular temperaments|''quindromeda'' temperament]] with a half-octave period.

[[Mapping]]: [{{val|2 4 0 -5}}, {{val|0 -5 28 64}}]

[[POTE generator]]: ~1323/1250 = 99.521

{{Val list|legend=1| 12, 398, 410, 422, 832, 1254d, 2086bd }}

[[Badness]]: 0.233140

Mapping: [{{val|2 4 0 -5 -10}}, {{val|0 -5 28 64 102}}]

POTE generator: ~1323/1250 = 99.525

Vals: {{Val list| 12, 410, 422 }}

Badness: 0.093926

Mapping: [{{val|2 4 0 -5 -10 -13}}, {{val|0 -5 28 64 102 123}}]

POTE generator: ~1323/1250 = 99.523

Vals: {{Val list| 12f, 410, 422, 1254df, 1676bdff, 2098bcddff }}

Badness: 0.053361

Mapping: [{{val|2 4 0 -5 -10 -13 10}}, {{val|0 -5 28 64 102 123 -11}}]

POTE generator: ~18/17 = 99.522

Vals: {{Val list| 12f, 410, 422, 832, 1254df, 1676bdff }}

Badness: 0.034659

Mapping: [{{val|2 4 0 -5 -10 -13 10 8}}, {{val|0 -5 28 64 102 123 -11 3}}]

POTE generator: ~18/17 = 99.523

Vals: {{Val list| 12f, 410, 422, 1254dfhh, 1676bdffhh }}

Badness: 0.025439