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Ragismic microtemperaments: Difference between revisions

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→Enneadecal: +semihemienneadecal, a very obvious extension
→Enneadecal: POTE -> CTE, +reduced generators
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== Enneadecal ==
== 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
[[Subgroup]]: 2.3.5.7
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Mapping generators: ~28/27, ~3
Mapping generators: ~28/27, ~3


[[Optimal tuning]] ([[POTE]]): ~3/2 = 701.8804
[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.9275 (~225/224 = 7.1907)


{{Val list|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}
{{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 }}]
Mapping: [{{val| 19 0 14 -37 126 }}, {{val| 0 1 1 3 -2 }}]


Optimal tuning (POTE): ~3/2 = 702.3603
Optimal tuning (CTE): ~3/2 = 702.1483 (~225/224 = 7.4115)


Optimal GPV sequence: {{Val list| 19, 133d, 152, 323e, 475de, 627de }}
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 }}]
Mapping: [{{val| 19 0 14 -37 126 -20 }}, {{val| 0 1 1 3 -2 3 }}]


Optimal tuning (POTE): ~3/2 = 702.2118
Optimal tuning (CTE): ~3/2 = 701.9258 (~225/224 = 7.1890)


Optimal GPV sequence: {{Val list| 19, 133df, 152f, 323ef }}
Optimal GPV sequence: {{Val list| 19, 133df, 152f, 323ef }}
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Mapping generators: ~55/54, ~3
Mapping generators: ~55/54, ~3


Optimal tuning (POTE): ~3/2 = 701.8814
Optimal tuning (CTE): ~3/2 = 701.9351 (~225/224 = 7.1983)


Optimal GPV sequence: {{Val list| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}
Optimal GPV sequence: {{Val list| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}
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Badness: 0.009985
Badness: 0.009985


==== Hemienneadec ====
==== Hemienneadecalis ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256


Mapping: [{{val| 38 0 28 -74 11 502 }}, {{val| 0 1 1 3 2 -6 }}]
Mapping: [{{val| 38 0 28 -74 11 -281 }}, {{val| 0 1 1 3 2 7 }}]


Optimal tuning (POTE): ~3/2 = 701.9862
Optimal tuning (CTE): ~3/2 = 701.9955 (~225/224 = 7.2587)


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


Badness: 0.030391
Badness: 0.020782


==== Hemienneadecalis ====
==== Hemienneadec ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213


Mapping: [{{val| 38 0 28 -74 11 -281 }}, {{val| 0 1 1 3 2 7 }}]
Mapping: [{{val| 38 0 28 -74 11 502 }}, {{val| 0 1 1 3 2 -6 }}]


Optimal tuning (POTE): ~3/2 = 702.0097
Optimal tuning (CTE): ~3/2 = 701.9812 (~225/224 = 7.2444)


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


Badness: 0.020782
Badness: 0.030391


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


Optimal tuning (CTE): ~429/250 = 935.1789
Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)


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