Mintaka: Difference between revisions

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As perhaps the simplest temperament of this subgroup delivering decent accuracy - and, in particular, the simplest supported by tunings such as 17edt and 22edt - Mintaka can be considered the 3.7.11 analog of 3.5.7 [[Bohlen-Pierce-Stearns|BPS]] or 2.3.5 [[meantone]], using 7:9:11 as its fundamental consonant chord in the place of 3:5:7 or of 4:5:6.

Several extensions of this temperament are possible to incorporate additional harmonics. Off the bat, given that 1331/1323 is a [[~~https://en.xen.wiki/w/Square_superparticular~~#Sk2_.2A_S.28k_.2B_1.29_and_S.28k_-_1.29_.2A_Sk2_.28lopsided_commas.29|lopsided comma]] with S-expression S22<sup>2</sup> * S23, one can reliably choose to temper both S22 = [[484/483]] and S23 = [[529/528]] in the 3.7.11.23/4 subgroup, which equates the 11/7 generator to [[36/23]], and the interval [[11/9]] to [[28/23]]. Furthermore, the tiny comma S161 = [[25921/25920]] can be tempered to add harmonic 20 to the subgroup, finding it 8 generators down. More neatly, this can be expressed as the temperament that tempers out the commas [[253/252]], 484/483, and [[540/539]] in the 3.7.11.20.23/4 subgroup.

Several extensions of this temperament are possible to incorporate additional harmonics. Off the bat, given that 1331/1323 is a [[Square superparticular#Sk2_.2A_S.28k_.2B_1.29_and_S.28k_-_1.29_.2A_Sk2_.28lopsided_commas.29|lopsided comma]] with S-expression S22<sup>2</sup> * S23, one can reliably choose to temper both S22 = [[484/483]] and S23 = [[529/528]] in the 3.7.11.23/4 subgroup, which equates the 11/7 generator to [[36/23]], and the interval [[11/9]] to [[28/23]]. Furthermore, the tiny comma S161 = [[25921/25920]] can be tempered to add harmonic 20 to the subgroup, finding it 8 generators down. More neatly, this can be expressed as the temperament that tempers out the commas [[253/252]], 484/483, and [[540/539]] in the 3.7.11.20.23/4 subgroup.

[[Mos scale]]s of reasonable tunings have cardinalities of 5 (2L 3s), 7 (5L 2s), 12 (5L 7s), or 17 (5L 12s, 12L 5s).

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 As perhaps the simplest temperament of this subgroup delivering decent accuracy - and, in particular, the simplest supported by tunings such as 17edt and 22edt - Mintaka can be considered the 3.7.11 analog of 3.5.7 [[Bohlen-Pierce-Stearns|BPS]] or 2.3.5 [[meantone]], using 7:9:11 as its fundamental consonant chord in the place of 3:5:7 or of 4:5:6.
-Several extensions of this temperament are possible to incorporate additional harmonics. Off the bat, given that 1331/1323 is a [[https://en.xen.wiki/w/Square_superparticular#Sk2_.2A_S.28k_.2B_1.29_and_S.28k_-_1.29_.2A_Sk2_.28lopsided_commas.29|lopsided comma]] with S-expression S22<sup>2</sup> * S23, one can reliably choose to temper both S22 = [[484/483]] and S23 = [[529/528]] in the 3.7.11.23/4 subgroup, which equates the 11/7 generator to [[36/23]], and the interval [[11/9]] to [[28/23]]. Furthermore, the tiny comma S161 = [[25921/25920]] can be tempered to add harmonic 20 to the subgroup, finding it 8 generators down. More neatly, this can be expressed as the temperament that tempers out the commas [[253/252]], 484/483, and [[540/539]] in the 3.7.11.20.23/4 subgroup.
+Several extensions of this temperament are possible to incorporate additional harmonics. Off the bat, given that 1331/1323 is a [[Square superparticular#Sk2_.2A_S.28k_.2B_1.29_and_S.28k_-_1.29_.2A_Sk2_.28lopsided_commas.29|lopsided comma]] with S-expression S22<sup>2</sup> * S23, one can reliably choose to temper both S22 = [[484/483]] and S23 = [[529/528]] in the 3.7.11.23/4 subgroup, which equates the 11/7 generator to [[36/23]], and the interval [[11/9]] to [[28/23]]. Furthermore, the tiny comma S161 = [[25921/25920]] can be tempered to add harmonic 20 to the subgroup, finding it 8 generators down. More neatly, this can be expressed as the temperament that tempers out the commas [[253/252]], 484/483, and [[540/539]] in the 3.7.11.20.23/4 subgroup.
 [[Mos scale]]s of reasonable tunings have cardinalities of 5 (2L 3s), 7 (5L 2s), 12 (5L 7s), or 17 (5L 12s, 12L 5s).