Minortonic family: Difference between revisions

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

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

== Minortone ==

Subgroup: 2.3.5

[[Comma]]: {{~~Monzo~~|-16 35 -17}}

[[Comma]]: {{monzo| -16 35 -17 }}

[[Mapping]]: [{{val|1 -1 -3}}, {{val|0 17 35}}]

[[Mapping]]: [{{val| 1 -1 -3 }}, {{val| 0 17 35 }}]

[[POTE generator]]: ~10/9 = 182.466

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== 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^(1/17) being 0.06423 cents flat and 40^(1/35) 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 }}. 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.

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~~|171 EDO~~]]. ~~The wedgie is now {{Multival|17 35 -21 16 -81 -147}}, with~~ 21 generators ~~giving~~ a 64/7. MOS of size 20, 33, 46 or 79 notes can be used for mitonic.

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 of size 20, 33, 46 or 79 notes can be used for mitonic.

Subgroup: 2.3.5.7

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[[Comma list]]: 4375/4374, 2100875/2097152

[[Mapping]]: [{{val|1 -1 -3 6}}, {{val|0 17 35 -21}}]

[[Mapping]]: [{{val| 1 -1 -3 6 }}, {{val| 0 17 35 -21 }}]

[[POTE generator]]: ~10/9 = 182.458

[[Badness]]: 0.025184

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Line 40:

Comma list: 441/440, 4375/4374, 16384/16335

Mapping: [{{val|1 -1 -3 6 10}}, {{val|0 17 35 -21 -43}}]

Mapping: [{{val| 1 -1 -3 6 10 }}, {{val| 0 17 35 -21 -43 }}]

POTE generator: ~10/9 = 182.482

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Comma list: 364/363, 441/440, 3584/3575, 4375/4374

Mapping: [{{val|1 -1 -3 6 10 11}}, {{val|0 17 35 -21 -43 -48}}]

Mapping: [{{val| 1 -1 -3 6 10 11 }}, {{val| 0 17 35 -21 -43 -48 }}]

POTE generator: ~10/9 = 182.481

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Comma list: 364/363, 441/440, 595/594, 1156/1155, 3584/3575

Mapping: [{{val|1 -1 -3 6 10 11 5}}, {{val|0 17 35 -21 -43 -48 -6}}]

Mapping: [{{val| 1 -1 -3 6 10 11 5 }}, {{val| 0 17 35 -21 -43 -48 -6 }}]

POTE generator: ~10/9 = 182.481

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Comma list: 385/384, 1331/1323, 4375/4374

Mapping: [{{val|1 -1 -3 6 3}}, {{val|0 17 35 -21 3}}]

Mapping: [{{val| 1 -1 -3 6 3 }}, {{val| 0 17 35 -21 3 }}]

POTE generator: ~10/9 = 182.449

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Comma list: 352/351, 385/384, 1331/1323, 3267/3250

Mapping: [{{val|1 -1 -3 6 3 11}}, {{val|0 17 35 -21 3 -48}}]

Mapping: [{{val| 1 -1 -3 6 3 11 }}, {{val| 0 17 35 -21 3 -48 }}]

POTE generator: ~10/9 = 182.470

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Comma list: 352/351, 385/384, 561/560, 715/714, 1452/1445

Mapping: [{{val|1 -1 -3 6 3 11 5}}, {{val|0 17 35 -21 3 -48 -6}}]

Mapping: [{{val| 1 -1 -3 6 3 11 5 }}, {{val| 0 17 35 -21 3 -48 -6 }}]

POTE generator: ~10/9 = 182.471

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Line 122:

Comma list: 169/168, 325/324, 385/384, 1331/1323

Mapping: [{{val|1 -1 -3 6 3 4}}, {{val|0 17 35 -21 3 -2}}]

Mapping: [{{val| 1 -1 -3 6 3 4 }}, {{val| 0 17 35 -21 3 -2 }}]

POTE generator: ~10/9 = 182.437

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Comma list: 169/168, 273/272, 325/324, 385/384, 1331/1323

Mapping: [{{val|1 -1 -3 6 3 4 5}}, {{val|0 17 35 -21 3 -2 -6}}]

Mapping: [{{val| 1 -1 -3 6 3 4 5 }}, {{val| 0 17 35 -21 3 -2 -6 }}]

POTE generator: ~10/9 = 182.444

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[[Comma list]]: 250047/250000, 645700815/645657712

[[Mapping]]: [{{val|3 -3 -9 -8}}, {{val|0 17 35 36}}]

[[Mapping]]: [{{val| 3 -3 -9 -8 }}, {{val| 0 17 35 36 }}]

[[POTE generator]]: ~10/9 = 182.467

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[[Badness]]: 0.013979

[[Category:~~Theory~~]]

[[Category:Regular temperament theory]]

[[Category:Temperament families]]

[[Category:Minortonic]]

[[Category:Minortonic family| ]]

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
-'''Minortonic family''' tempers out the minortone comma (also known as "minortonma"), {{Monzo|-16 35 -17}}. The head of this family is five-limit minortone temperament, with generator a minor tone.
+The '''minortonic family''' tempers out the minortone comma (also known as "minortonma"), {{monzo| -16 35 -17 }}. The head of this family is five-limit minortone temperament, with generator a minor tone.
 == Minortone ==
 Subgroup: 2.3.5
-[[Comma]]: {{Monzo|-16 35 -17}}
+[[Comma]]: {{monzo| -16 35 -17 }}
-[[Mapping]]: [{{val|1 -1 -3}}, {{val|0 17 35}}]
+[[Mapping]]: [{{val| 1 -1 -3 }}, {{val| 0 17 35 }}]
 [[POTE generator]]: ~10/9 = 182.466
@@ Line 15: / Line 15: @@
 == 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^(1/17) being 0.06423 cents flat and 40^(1/35) 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 }}. 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.
-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|171 EDO]]. The wedgie is now {{Multival|17 35 -21 16 -81 -147}}, with 21 generators giving a 64/7. MOS of size 20, 33, 46 or 79 notes can be used for mitonic.
+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 of size 20, 33, 46 or 79 notes can be used for mitonic.
 Subgroup: 2.3.5.7
@@ Line 23: / Line 23: @@
 [[Comma list]]: 4375/4374, 2100875/2097152
-[[Mapping]]: [{{val|1 -1 -3 6}}, {{val|0 17 35 -21}}]
+[[Mapping]]: [{{val| 1 -1 -3 6 }}, {{val| 0 17 35 -21 }}]
+{{Multival|legend=1| 17 35 -21 16 -81 -147 }}
 [[POTE generator]]: ~10/9 = 182.458
-{{Val list|legend=1| 46, 125, 171 }}
+{{Val list|legend=1| 46, 125, 171, 1927d, 2098d, …, 3637bcdd }}
 [[Badness]]: 0.025184
@@ Line 38: / Line 40: @@
 Comma list: 441/440, 4375/4374, 16384/16335
-Mapping: [{{val|1 -1 -3 6 10}}, {{val|0 17 35 -21 -43}}]
+Mapping: [{{val| 1 -1 -3 6 10 }}, {{val| 0 17 35 -21 -43 }}]
 POTE generator: ~10/9 = 182.482
@@ Line 51: / Line 53: @@
 Comma list: 364/363, 441/440, 3584/3575, 4375/4374
-Mapping: [{{val|1 -1 -3 6 10 11}}, {{val|0 17 35 -21 -43 -48}}]
+Mapping: [{{val| 1 -1 -3 6 10 11 }}, {{val| 0 17 35 -21 -43 -48 }}]
 POTE generator: ~10/9 = 182.481
@@ Line 64: / Line 66: @@
 Comma list: 364/363, 441/440, 595/594, 1156/1155, 3584/3575
-Mapping: [{{val|1 -1 -3 6 10 11 5}}, {{val|0 17 35 -21 -43 -48 -6}}]
+Mapping: [{{val| 1 -1 -3 6 10 11 5 }}, {{val| 0 17 35 -21 -43 -48 -6 }}]
 POTE generator: ~10/9 = 182.481
@@ Line 79: / Line 81: @@
 Comma list: 385/384, 1331/1323, 4375/4374
-Mapping: [{{val|1 -1 -3 6 3}}, {{val|0 17 35 -21 3}}]
+Mapping: [{{val| 1 -1 -3 6 3 }}, {{val| 0 17 35 -21 3 }}]
 POTE generator: ~10/9 = 182.449
@@ Line 92: / Line 94: @@
 Comma list: 352/351, 385/384, 1331/1323, 3267/3250
-Mapping: [{{val|1 -1 -3 6 3 11}}, {{val|0 17 35 -21 3 -48}}]
+Mapping: [{{val| 1 -1 -3 6 3 11 }}, {{val| 0 17 35 -21 3 -48 }}]
 POTE generator: ~10/9 = 182.470
@@ Line 105: / Line 107: @@
 Comma list: 352/351, 385/384, 561/560, 715/714, 1452/1445
-Mapping: [{{val|1 -1 -3 6 3 11 5}}, {{val|0 17 35 -21 3 -48 -6}}]
+Mapping: [{{val| 1 -1 -3 6 3 11 5 }}, {{val| 0 17 35 -21 3 -48 -6 }}]
 POTE generator: ~10/9 = 182.471
@@ Line 120: / Line 122: @@
 Comma list: 169/168, 325/324, 385/384, 1331/1323
-Mapping: [{{val|1 -1 -3 6 3 4}}, {{val|0 17 35 -21 3 -2}}]
+Mapping: [{{val| 1 -1 -3 6 3 4 }}, {{val| 0 17 35 -21 3 -2 }}]
 POTE generator: ~10/9 = 182.437
@@ Line 133: / Line 135: @@
 Comma list: 169/168, 273/272, 325/324, 385/384, 1331/1323
-Mapping: [{{val|1 -1 -3 6 3 4 5}}, {{val|0 17 35 -21 3 -2 -6}}]
+Mapping: [{{val| 1 -1 -3 6 3 4 5 }}, {{val| 0 17 35 -21 3 -2 -6 }}]
 POTE generator: ~10/9 = 182.444
@@ Line 148: / Line 150: @@
 [[Comma list]]: 250047/250000, 645700815/645657712
-[[Mapping]]: [{{val|3 -3 -9 -8}}, {{val|0 17 35 36}}]
+[[Mapping]]: [{{val| 3 -3 -9 -8 }}, {{val| 0 17 35 36 }}]
 [[POTE generator]]: ~10/9 = 182.467
@@ Line 156: / Line 158: @@
 [[Badness]]: 0.013979
-[[Category:Theory]]
+[[Category:Regular temperament theory]]
 [[Category:Temperament families]]
-[[Category:Minortonic]]
+[[Category:Minortonic family| ]] <!-- main article -->
+[[Category:Rank 2]]