Ragismic microtemperaments: Difference between revisions

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== Enneadecal ==

Enneadecal temperament tempers out the [[enneadeca]], {{monzo|-14 -19 19}}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.

Enneadecal temperament tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.

[[Subgroup]]: 2.3.5.7

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Mapping generators: ~28/27, ~3

[[POTE ~~generator~~]]: ~3/2 = 701.8804

[[Optimal tuning]] ([[POTE]]): ~3/2 = 701.8804

{{Val list|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}

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Mapping: [{{val| 19 0 14 -37 126 }}, {{val| 0 1 1 3 -2 }}]

POTE ~~generator~~: ~3/2 = 702.3603

Optimal tuning (POTE): ~3/2 = 702.3603

Optimal GPV sequence: {{Val list| 19, 133d, 152, 323e, 475de, 627de }}

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Mapping: [{{val| 19 0 14 -37 126 -20 }}, {{val| 0 1 1 3 -2 3 }}]

POTE ~~generator~~: ~3/2 = 702.2118

Optimal tuning (POTE): ~3/2 = 702.2118

Optimal GPV sequence: {{Val list| 19, 133df, 152f, 323ef }}

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Mapping generators: ~55/54, ~3

POTE ~~generator~~: ~3/2 = 701.8814

Optimal tuning (POTE): ~3/2 = 701.8814

Optimal GPV sequence: {{Val list| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}

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Mapping: [{{val| 38 0 28 -74 11 502 }}, {{val| 0 1 1 3 2 -6 }}]

POTE ~~generator~~: ~3/2 = 701.9862

Optimal tuning (POTE): ~3/2 = 701.9862

Optimal GPV sequence: {{Val list| 152, 342, 494, 1330, 1824, 2318d }}

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Mapping: [{{val| 38 0 28 -74 11 -281 }}, {{val| 0 1 1 3 2 7 }}]

POTE ~~generator~~: ~3/2 = 702.0097

Optimal tuning (POTE): ~3/2 = 702.0097

Optimal GPV sequence: {{Val list| 152f, 342f, 494 }}

Badness: 0.020782

==== Semihemienneadecal ====

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078

Mapping: [{{val| 38 1 29 -71 13 111 }}, {{val| 0 2 2 6 4 1 }}]

Mapping generators: ~55/54, ~429/250

Optimal tuning (CTE): ~429/250 = 935.1789

Optimal GPV sequence: {{Val list| 190, 304d, 494, 684, 1178, 2850, 4028ce }}

Badness: 0.014694

== Deca ==

@@ Line 550: / Line 550: @@
 == Enneadecal ==
-Enneadecal temperament tempers out the [[enneadeca]], {{monzo|-14 -19 19}}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.
+Enneadecal temperament tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 562: / Line 562: @@
 Mapping generators: ~28/27, ~3
-[[POTE generator]]: ~3/2 = 701.8804
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 701.8804
 {{Val list|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}
@@ Line 575: / Line 575: @@
 Mapping: [{{val| 19 0 14 -37 126 }}, {{val| 0 1 1 3 -2 }}]
-POTE generator: ~3/2 = 702.3603
+Optimal tuning (POTE): ~3/2 = 702.3603
 Optimal GPV sequence: {{Val list| 19, 133d, 152, 323e, 475de, 627de }}
@@ Line 588: / Line 588: @@
 Mapping: [{{val| 19 0 14 -37 126 -20 }}, {{val| 0 1 1 3 -2 3 }}]
-POTE generator: ~3/2 = 702.2118
+Optimal tuning (POTE): ~3/2 = 702.2118
 Optimal GPV sequence: {{Val list| 19, 133df, 152f, 323ef }}
@@ Line 603: / Line 603: @@
 Mapping generators: ~55/54, ~3
-POTE generator: ~3/2 = 701.8814
+Optimal tuning (POTE): ~3/2 = 701.8814
 Optimal GPV sequence: {{Val list| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}
@@ Line 616: / Line 616: @@
 Mapping: [{{val| 38 0 28 -74 11 502 }}, {{val| 0 1 1 3 2 -6 }}]
-POTE generator: ~3/2 = 701.9862
+Optimal tuning (POTE): ~3/2 = 701.9862
 Optimal GPV sequence: {{Val list| 152, 342, 494, 1330, 1824, 2318d }}
@@ Line 629: / Line 629: @@
 Mapping: [{{val| 38 0 28 -74 11 -281 }}, {{val| 0 1 1 3 2 7 }}]
-POTE generator: ~3/2 = 702.0097
+Optimal tuning (POTE): ~3/2 = 702.0097
 Optimal GPV sequence: {{Val list| 152f, 342f, 494 }}
 Badness: 0.020782
+==== Semihemienneadecal ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078
+Mapping: [{{val| 38 1 29 -71 13 111 }}, {{val| 0 2 2 6 4 1 }}]
+Mapping generators: ~55/54, ~429/250
+Optimal tuning (CTE): ~429/250 = 935.1789
+Optimal GPV sequence: {{Val list| 190, 304d, 494, 684, 1178, 2850, 4028ce }}
+Badness: 0.014694
 == Deca ==