Ragismic microtemperaments: Difference between revisions

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)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo|19EDO]] up to just ones. [[171edo|171EDO]] is a good tuning for either the 5 or 7 limits, and [[494edo|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|665EDO]] for a tuning.

Subgroup: 2.3.5.7

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

{{Multival|legend=1|19 19 57 -14 37 79}}

[[POTE generator]]: ~3/2 = 701.880

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

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

POTE generator: ~3/2 = 702.360

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

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

POTE generator: ~3/2 = 702.212

Optimal GPV sequence: {{Val list| 19, 152f, 323e }}

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

POTE generator: ~3/2 = 701.881

Optimal GPV sequence: {{Val list| 152, 342, 494, 836, 1178, 2014 }}

==== 13-limit ====

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

POTE generator: ~3/2 = 701.986

Optimal GPV sequence: {{Val list| 152, 342, 494, 836 }}