Minortonic family: Difference between revisions

(4 intermediate revisions by 4 users not shown)

Line 1:

The '''minortonic family''' tempers out the minortone comma (also known as "minortonma"), {{monzo| -16 35 -17 }}. The head of this family is 5-limit minortone temperament, with generator a minor tone.

Line 15:

Line 16:

== Mitonic ==

As a 5-limit temperament, mitonic becomes minortonic, a super-accurate microtemperament tempering out the minortone comma, {{monzo| -16 35 -17 }}. ~~Flipping that gives the 5-limit wedgie {{multival| 17 35 16 }}, which tells us that~~ 10/9 can be taken as the generator, with 17 of them giving a ~6, 18 of them a ~20/3, and 35 of them giving a ~40. The generator should be tuned about 1/16 of a cent flat, with 61/17 being 0.06423 cents flat and 401/35 being 0.06234 cents flat. 171, 559 and 730 are possible equal temperament tunings.

As a 5-limit temperament, mitonic becomes minortonic, a super-accurate microtemperament tempering out the minortone comma, {{monzo| -16 35 -17 }}. 10/9 can be taken as the generator, with 17 of them giving a ~6, 18 of them a ~20/3, and 35 of them giving a ~40. The generator should be tuned about 1/16 of a cent flat, with 61/17 being 0.06423 cents flat and 401/35 being 0.06234 cents flat. 171, 559 and 730 are possible equal temperament tunings.

However, as noted before, 32/21 is only a ragisma shy of (10/9)4, and so a 7-limit interpretation, if not quite so super-accurate, is more or less inevitable. While 559 or 730 are still fine as tunings, the error of the 7-limit is lower by a whisker in [[171edo]]. 21 generators gives a ~64/7. [[Mos scale]]s of size 20, 33, 46 or 79 notes can be used for mitonic.

Line 24:

Line 25:

~~{{Multival|legend=1| 17 35 -21 16 -81 -147 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 182.458

Line 159:

Line 158:

Domain adds the [[landscape comma]], 250047/250000, to the minortone comma, giving a temperament which is perhaps most notable for its inclusion of the remarkable subgroup temperament [[~~Chromatic pairs~~ #Terrain|terrain]].

Domain adds the [[landscape comma]], 250047/250000, to the minortone comma, giving a temperament which is perhaps most notable for its inclusion of the remarkable subgroup temperament [[Subgroup temperaments #Terrain|terrain]].

[[Subgroup]]: 2.3.5.7

Line 166:

Line 165:

~~{{Multival|legend=1| 51 105 108 48 28 -44 }}~~

[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467

Line 189:

Line 186:

[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Minortonic family| ]]

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
+{{Technical data page}}
 The '''minortonic family''' tempers out the minortone comma (also known as "minortonma"), {{monzo| -16 35 -17 }}. The head of this family is 5-limit minortone temperament, with generator a minor tone.
@@ Line 15: / Line 16: @@
 == Mitonic ==
-As a 5-limit temperament, mitonic becomes minortonic, a super-accurate microtemperament tempering out the minortone comma, {{monzo| -16 35 -17 }}. Flipping that gives the 5-limit wedgie {{multival| 17 35 16 }}, which tells us that 10/9 can be taken as the generator, with 17 of them giving a ~6, 18 of them a ~20/3, and 35 of them giving a ~40. The generator should be tuned about 1/16 of a cent flat, with 6<sup>1/17</sup> being 0.06423 cents flat and 40<sup>1/35</sup> being 0.06234 cents flat. 171, 559 and 730 are possible equal temperament tunings.
+As a 5-limit temperament, mitonic becomes minortonic, a super-accurate microtemperament tempering out the minortone comma, {{monzo| -16 35 -17 }}. 10/9 can be taken as the generator, with 17 of them giving a ~6, 18 of them a ~20/3, and 35 of them giving a ~40. The generator should be tuned about 1/16 of a cent flat, with 6<sup>1/17</sup> being 0.06423 cents flat and 40<sup>1/35</sup> being 0.06234 cents flat. 171, 559 and 730 are possible equal temperament tunings.
 However, as noted before, 32/21 is only a ragisma shy of (10/9)<sup>4</sup>, and so a 7-limit interpretation, if not quite so super-accurate, is more or less inevitable. While 559 or 730 are still fine as tunings, the error of the 7-limit is lower by a whisker in [[171edo]]. 21 generators gives a ~64/7. [[Mos scale]]s of size 20, 33, 46 or 79 notes can be used for mitonic.
@@ Line 24: / Line 25: @@
 {{Mapping|legend=1| 1 -1 -3 6 | 0 17 35 -21 }}
-{{Multival|legend=1| 17 35 -21 16 -81 -147 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 182.458
@@ Line 159: / Line 158: @@
 {{See also| Landscape microtemperaments #Domain }}
-Domain adds the [[landscape comma]], 250047/250000, to the minortone comma, giving a temperament which is perhaps most notable for its inclusion of the remarkable subgroup temperament [[Chromatic pairs #Terrain|terrain]].
+Domain adds the [[landscape comma]], 250047/250000, to the minortone comma, giving a temperament which is perhaps most notable for its inclusion of the remarkable subgroup temperament [[Subgroup temperaments #Terrain|terrain]].
 [[Subgroup]]: 2.3.5.7
@@ Line 166: / Line 165: @@
 {{Mapping|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}
-{{Multival|legend=1| 51 105 108 48 28 -44 }}
 [[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467
@@ Line 189: / Line 186: @@
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Minortonic family| ]] <!-- main article -->
 [[Category:Rank 2]]