Hemimean clan: Difference between revisions

Line 67:

Badness (Dirichlet): 0.195

==== ~~Roulette~~ ====

==== Tridecimal didacus ====

~~Roulette~~ is equivalent to [[hemiwur]] or [[grosstone]] with no mapping for prime 3. The mapping of prime 13 is somewhat strange, because it is the only mapping that requires a negative amount of generators (and a large amount of them), but it can be rationalized in a variety of ways, such as that because [[~]][[8/7]] is already tuned almost 3{{cent}} flat, it makes sense to equate two of it with [[~]][[13/10]] (tempering out the 8{{cent}} [[huntma]]). This mapping of 13 increases the [[badness]] of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension to didacus if prime 3 is not included.

Tridecimal didacus (formerly ''roulette''; that name has now been reassigned to the no-threes 19-limit extension 37 & 68) is equivalent to [[hemiwur]] or [[grosstone]] with no mapping for prime 3. The mapping of prime 13 is somewhat strange, because it is the only mapping that requires a negative amount of generators (and a large amount of them), but it can be rationalized in a variety of ways, such as that because [[~]][[8/7]] is already tuned almost 3{{cent}} flat, it makes sense to equate two of it with [[~]][[13/10]] (tempering out the 8{{cent}} [[huntma]]). This mapping of 13 increases the [[badness]] of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension to didacus if prime 3 is not included.

Subgroup: 2.5.7.11.13

Line 88:

Badness (Dirichlet): 0.324

==== Mediantone ====

===== Mediantone =====

Mediantone is named after its whole tone generator serving as the [[mediant]] of [[9/8]] and [[10/9]], namely [[19/17]], in addition to [[28/25]], as well as by the observation that this temperament seems to have been repeatedly rediscovered in parts in a variety of contexts, so that it seems to exist as a "median" of all of these temperaments' logics. It is also an intentional play on "[[meantone]]", as the context one is most likely to first discover this logic is when the tone also represents [[~]][[10/9]][[~]][[9/8]].

Line 107:

Badness (Dirichlet): 0.612

===== 2.5.7.11.13.17.19 subgroup =====

====== 2.5.7.11.13.17.19 subgroup ======

Subgroup: 2.5.7.11.13.17.19

Line 121:

Badness (Dirichlet): 0.618

===== Roulette =====

Roulette is an alternative no-twos 19-limit extension of tridecimal didacus to mediantone (the two mappings converging at [[37edo]]), equating (8/7)<sup>2</sup> to [[17/13]] in addition to 13/10, tempering out [[170/169]] and [[833/832]]; in doing so, it also tempers out the micro-comma [[2000033/2000000]] so that ([[50/49]])<sup>3</sup> is equated to [[17/16]]. The generator is then equated to 19/17 in the same way as in mediantone.

Subgroup: 2.5.7.11.13.17

Comma list: [[170/169]], [[176/175]], [[640/637]], [[1375/1372]]

Sval mapping: {{mapping| 1 2 2 2 5 7 | 0 2 5 9 -8 -18 }}

: sval mapping generators: ~2, ~28/25

Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.285

Optimal ET sequence: {{Optimal ET sequence| 6g, 31, 37, 68, 105 }}

Badness (Dirichlet): 0.685

====== 2.5.7.11.13.17.19 subgroup ======

Subgroup: 2.5.7.11.13.17.19

Comma list: [[170/169]], [[176/175]], [[476/475]], [[640/637]], [[1375/1372]]

Sval mapping: {{mapping| 1 2 2 2 5 7 7 | 0 2 5 9 -8 -18 -17 }}

: sval mapping generators: ~2, ~28/25

Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.259

Optimal ET sequence: {{Optimal ET sequence| 6g, 31, 37, 68, 105 }}

Badness (Dirichlet): 0.676

=== Rectified hebrew ===

@@ Line 67: / Line 67: @@
 Badness (Dirichlet): 0.195
-==== Roulette ====
+==== Tridecimal didacus ====
-Roulette is equivalent to [[hemiwur]] or [[grosstone]] with no mapping for prime 3. The mapping of prime 13 is somewhat strange, because it is the only mapping that requires a negative amount of generators (and a large amount of them), but it can be rationalized in a variety of ways, such as that because [[~]][[8/7]] is already tuned almost 3{{cent}} flat, it makes sense to equate two of it with [[~]][[13/10]] (tempering out the 8{{cent}} [[huntma]]). This mapping of 13 increases the [[badness]] of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension to didacus if prime 3 is not included.
+Tridecimal didacus (formerly ''roulette''; that name has now been reassigned to the no-threes 19-limit extension 37 & 68) is equivalent to [[hemiwur]] or [[grosstone]] with no mapping for prime 3. The mapping of prime 13 is somewhat strange, because it is the only mapping that requires a negative amount of generators (and a large amount of them), but it can be rationalized in a variety of ways, such as that because [[~]][[8/7]] is already tuned almost 3{{cent}} flat, it makes sense to equate two of it with [[~]][[13/10]] (tempering out the 8{{cent}} [[huntma]]). This mapping of 13 increases the [[badness]] of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension to didacus if prime 3 is not included.
 Subgroup: 2.5.7.11.13
@@ Line 88: / Line 88: @@
 Badness (Dirichlet): 0.324
-==== Mediantone ====
+===== Mediantone =====
 Mediantone is named after its whole tone generator serving as the [[mediant]] of [[9/8]] and [[10/9]], namely [[19/17]], in addition to [[28/25]], as well as by the observation that this temperament seems to have been repeatedly rediscovered in parts in a variety of contexts, so that it seems to exist as a "median" of all of these temperaments' logics. It is also an intentional play on "[[meantone]]", as the context one is most likely to first discover this logic is when the tone also represents [[~]][[10/9]][[~]][[9/8]].
@@ Line 107: / Line 107: @@
 Badness (Dirichlet): 0.612
-===== 2.5.7.11.13.17.19 subgroup =====
+====== 2.5.7.11.13.17.19 subgroup ======
 Subgroup: 2.5.7.11.13.17.19
@@ Line 121: / Line 121: @@
 Badness (Dirichlet): 0.618
+===== Roulette =====
+Roulette is an alternative no-twos 19-limit extension of tridecimal didacus to mediantone (the two mappings converging at [[37edo]]), equating (8/7)<sup>2</sup> to [[17/13]] in addition to 13/10, tempering out [[170/169]] and [[833/832]]; in doing so, it also tempers out the micro-comma [[2000033/2000000]] so that ([[50/49]])<sup>3</sup> is equated to [[17/16]]. The generator is then equated to 19/17 in the same way as in mediantone.
+Subgroup: 2.5.7.11.13.17
+Comma list: [[170/169]], [[176/175]], [[640/637]], [[1375/1372]]
+Sval mapping: {{mapping| 1 2 2 2 5 7 | 0 2 5 9 -8 -18 }}
+: sval mapping generators: ~2, ~28/25
+Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.285
+Optimal ET sequence: {{Optimal ET sequence| 6g, 31, 37, 68, 105 }}
+Badness (Dirichlet): 0.685
+====== 2.5.7.11.13.17.19 subgroup ======
+Subgroup: 2.5.7.11.13.17.19
+Comma list: [[170/169]], [[176/175]], [[476/475]], [[640/637]], [[1375/1372]]
+Sval mapping: {{mapping| 1 2 2 2 5 7 7 | 0 2 5 9 -8 -18 -17 }}
+: sval mapping generators: ~2, ~28/25
+Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.259
+Optimal ET sequence: {{Optimal ET sequence| 6g, 31, 37, 68, 105 }}
+Badness (Dirichlet): 0.676
 === Rectified hebrew ===