Ploidacot/Omega-pentacot: Difference between revisions

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== Intervals and notation ==

While there is no agreed-upon notation system for omega-pentacot, the following is based on interpreting the generator as a semitone (1/5 of a fourth), allowing for an ^ or v to stand for a 1/5 of an ''inversed'' diminished second (the difference between diatonic semitone and chromatic semitone, equivalent to the [[Pythagorean comma]]), so vvvC# and ^^Db are enharmonic.

While there is no agreed-upon notation system for omega-pentacot, the following is based on interpreting the generator as a semitone (1/5 of a fourth), allowing for an ^ or v to stand for 1/5 of an ''inversed'' diminished second (the difference between diatonic semitone and chromatic semitone, equivalent to the [[Pythagorean comma]]), so vvvC# and ^^Db are enharmonic.

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Omega-pentacot temperaments are generally interpretated as quinticular temperaments; the generator is [[18/17]], five of them gives 4/3, so the [[quinticular comma]] (1419857/1417176) is tempered out.

=== Quintilischis ===

In quintilischis, the generator is 18/17, three generators make [[19/16]], five make 4/3, and 40 make [[10/1|10th harmonic]] in the 2.3.5.17.19 subgroup, so [[4624/4617]], [[6144/6137]], and [[6885/6859]] are tempered out. This temperament is a weak extension of [[schismic]], splitting the fourth in five, and supported by {{Optimal ET sequence| 12, 253, 265, 277, 289, 301, 313, and 325 }} edos.

In the 2.3.5.7.17.19 subgroup, tempering out [[400/399]] (equating 20/19 and 21/20) leads to [[quintilipyth]] (12 & 253), and tempering out [[476/475]] (equating 19/17 with 28/25) leads to [[quintaschis]] (12 & 289).

=== Quindromeda ===

In [[quindromeda]], the generator is 18/17, three generators make [[19/16]], five make 4/3, and 28 make [[5/1|5th harmonic]] in the 2.3.5.17.19 subgroup, so [[1216/1215]], [[1445/1444]], and [[6144/6137]] are tempered out. This temperament is supported by {{Optimal ET sequence| 12, 169, 181, 193, 205, 217, 229, and 241 }} edos.

In [[quindromeda]], the generator is 18/17, three generators make 19/16, five make 4/3, and 28 make [[5/1|5th harmonic]] in the 2.3.5.17.19 subgroup, so [[1216/1215]], [[1445/1444]], and [[6144/6137]] are tempered out. This temperament is supported by {{Optimal ET sequence| 12, 169, 181, 193, 205, 217, 229, and 241 }} edos.

Equating 225/224 with 256/255 leads to [[quintakwai]] (12 & 193), which tempers out [[400/399]] (also equating 20/19 and 21/20) in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads to [[quintagar]] (12 & 217), which tempers out [[476/475]] (also equating 19/17 with 28/25) in the 2.3.5.7.17.19 subgroup.

Equating 225/224 with 256/255 leads to [[quintakwai]] (12 & 193), which tempers out 400/399 (also equating 20/19 and 21/20) in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads to [[quintagar]] (12 & 217), which tempers out 476/475 (also equating 19/17 with 28/25) in the 2.3.5.7.17.19 subgroup.

=== Quintaleap ===

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In [[quintaleap]], the generator is 18/17, three generators make 19/16, five make 4/3, and 16 make [[5/2]] in the 2.3.5.17.19 subgroup, so [[256/255]], [[361/360]], and [[4624/4617]] are tempered out. This temperament is supported by {{Optimal ET sequence| 12, 109, 121, 133, 145, and 157 }} edos.

In the 2.3.5.7.17.19 subgroup, tempering out 400/399 ~~(equating 20/19 and 21/20)~~ leads to [[quintupole]] (12 & 121), and tempering out 476/475 ~~(equating 19/17 with 28/25)~~ leads to [[quinticosiennic]] (12 & 145).

In the 2.3.5.7.17.19 subgroup, tempering out 400/399 leads to [[quintupole]] (12 & 121), and tempering out 476/475 leads to [[quinticosiennic]] (12 & 145).

=== Passion ===

@@ Line 3: / Line 3: @@
 == Intervals and notation ==
-While there is no agreed-upon notation system for omega-pentacot, the following is based on interpreting the generator as a semitone (1/5 of a fourth), allowing for an ^ or v to stand for a 1/5 of an ''inversed'' diminished second (the difference between diatonic semitone and chromatic semitone, equivalent to the [[Pythagorean comma]]), so vvvC# and ^^Db are enharmonic.
+While there is no agreed-upon notation system for omega-pentacot, the following is based on interpreting the generator as a semitone (1/5 of a fourth), allowing for an ^ or v to stand for 1/5 of an ''inversed'' diminished second (the difference between diatonic semitone and chromatic semitone, equivalent to the [[Pythagorean comma]]), so vvvC# and ^^Db are enharmonic.
 {| class="wikitable"
@@ Line 326: / Line 326: @@
 Omega-pentacot temperaments are generally interpretated as quinticular temperaments; the generator is [[18/17]], five of them gives 4/3, so the [[quinticular comma]] (1419857/1417176) is tempered out.
+=== Quintilischis ===
+{{See also| Schismatic family }}
+In quintilischis, the generator is 18/17, three generators make [[19/16]], five make 4/3, and 40 make [[10/1|10th harmonic]] in the 2.3.5.17.19 subgroup, so [[4624/4617]], [[6144/6137]], and [[6885/6859]] are tempered out. This temperament is a weak extension of [[schismic]], splitting the fourth in five, and supported by {{Optimal ET sequence| 12, 253, 265, 277, 289, 301, 313, and 325 }} edos.
+In the 2.3.5.7.17.19 subgroup, tempering out [[400/399]] (equating 20/19 and 21/20) leads to [[quintilipyth]] (12 &amp; 253), and tempering out [[476/475]] (equating 19/17 with 28/25) leads to [[quintaschis]] (12 &amp; 289).
 === Quindromeda ===
 {{Main| Quindromeda family }}
-In [[quindromeda]], the generator is 18/17, three generators make [[19/16]], five make 4/3, and 28 make [[5/1|5th harmonic]] in the 2.3.5.17.19 subgroup, so [[1216/1215]], [[1445/1444]], and [[6144/6137]] are tempered out. This temperament is supported by {{Optimal ET sequence| 12, 169, 181, 193, 205, 217, 229, and 241 }} edos.
+In [[quindromeda]], the generator is 18/17, three generators make 19/16, five make 4/3, and 28 make [[5/1|5th harmonic]] in the 2.3.5.17.19 subgroup, so [[1216/1215]], [[1445/1444]], and [[6144/6137]] are tempered out. This temperament is supported by {{Optimal ET sequence| 12, 169, 181, 193, 205, 217, 229, and 241 }} edos.
-Equating 225/224 with 256/255 leads to [[quintakwai]] (12 &amp; 193), which tempers out [[400/399]] (also equating 20/19 and 21/20) in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads to [[quintagar]] (12 &amp; 217), which tempers out [[476/475]] (also equating 19/17 with 28/25) in the 2.3.5.7.17.19 subgroup.
+Equating 225/224 with 256/255 leads to [[quintakwai]] (12 &amp; 193), which tempers out 400/399 (also equating 20/19 and 21/20) in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads to [[quintagar]] (12 &amp; 217), which tempers out 476/475 (also equating 19/17 with 28/25) in the 2.3.5.7.17.19 subgroup.
 === Quintaleap ===
@@ Line 339: / Line 346: @@
 In [[quintaleap]], the generator is 18/17, three generators make 19/16, five make 4/3, and 16 make [[5/2]] in the 2.3.5.17.19 subgroup, so [[256/255]], [[361/360]], and [[4624/4617]] are tempered out. This temperament is supported by {{Optimal ET sequence| 12, 109, 121, 133, 145, and 157 }} edos.
-In the 2.3.5.7.17.19 subgroup, tempering out 400/399 (equating 20/19 and 21/20) leads to [[quintupole]] (12 &amp; 121), and tempering out 476/475 (equating 19/17 with 28/25) leads to [[quinticosiennic]] (12 &amp; 145).
+In the 2.3.5.7.17.19 subgroup, tempering out 400/399 leads to [[quintupole]] (12 &amp; 121), and tempering out 476/475 leads to [[quinticosiennic]] (12 &amp; 145).
 === Passion ===