Gammic family: Difference between revisions
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{{Technical data page}} | {{Technical data page}} | ||
The [[Carlos Gamma]] rank-1 temperament divides 3/2 into 20 equal parts, 11 of which give a 5/4. This is closely related to the rank-2 microtemperament tempering out {{monzo| -29 -11 20 }}. This temperament, '''gammic''', takes 11 [[generator]] steps to reach 5/4, and 20 to reach 3/2. The generator in question is 1990656/1953125 = {{monzo| 13 5 -9 }}, which when suitably tempered is very close to 5/171 octaves, which makes for an ideal gammic tuning. As a 5-limit temperament supported by [[171edo]], [[Schismatic family|schismatic]] temperament makes for a natural comparison. | The [[Carlos Gamma]] rank-1 temperament divides 3/2 into 20 equal parts, 11 of which give a 5/4. This is closely related to the rank-2 microtemperament tempering out {{monzo| -29 -11 20 }}, the [[gammic comma]]. This temperament, '''gammic''', takes 11 [[generator]] steps to reach 5/4, and 20 to reach 3/2. The generator in question is 1990656/1953125 = {{monzo| 13 5 -9 }}, which when suitably tempered is very close to 5/171 octaves, which makes for an ideal gammic tuning. As a 5-limit temperament supported by [[171edo]], [[Schismatic family|schismatic]] temperament makes for a natural comparison. Schismic, tempering out {{monzo| -15 8 1 }}, the [[schisma]], is plainly much less complex than gammic, but people seeking the exotic might prefer gammic even so. The 34-note mos is interesting, being a 1L 33s refinement of the [[34edo]] tuning. Of course gammic can be tuned to 34, which makes the two equivalent, and would rather remove the point of Carlos Gamma if used for it. | ||
Because 171 is such a strong [[7-limit]] system, it is natural to extend gammic to the 7-limit. This we may do by adding [[4375/4374]] to the comma list | Because 171 is such a strong [[7-limit]] system, it is natural to extend gammic to the 7-limit. This we may do by adding [[4375/4374]] to the comma list. 96 gammic generators finally reach 7, which is a long way to go compared to the 39 generator steps of pontiac. If someone wants to make the trip, a 103-note mos is possible. | ||
== Gammic == | == Gammic == | ||
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=== 2.3.5.17 subgroup === | === 2.3.5.17 subgroup === | ||
The interval of 3 generators represents one-third of [[6/5]], which is very close to [[17/16]], with the comma between 6/5 and (17/16)<sup>3</sup> being [[24576/24565]] = {{S|16/S17}}. This then naturally interprets the generator as [[51/50]] with two generators representing [[25/24]], tempering out [[15625/15606]] = S49×S50<sup>2</sup>. | |||
[[Subgroup]]: 2.3.5.17 | [[Subgroup]]: 2.3.5.17 | ||
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{{Mapping|legend=1| 1 1 2 0 | 0 20 11 96 }} | {{Mapping|legend=1| 1 1 2 0 | 0 20 11 96 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~234375/229376 = 35.0904 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~234375/229376 = 35.0904 | ||
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[[Mapping]]: {{mapping| 1 1 2 1 | 0 20 11 62 }} | [[Mapping]]: {{mapping| 1 1 2 1 | 0 20 11 62 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1990656/1953125 = 34.984 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1990656/1953125 = 34.984 | ||
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A more interesting extension is to neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds [[2401/2400]] to the gammic comma, and may be described as the 68&171 temperament. The generator chain goes merrily on, stacking one 10/7 over another, until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of septimal consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending [[Carlos Gamma]]. | A more interesting extension is to neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds [[2401/2400]] to the gammic comma, and may be described as the 68&171 temperament. The generator chain goes merrily on, stacking one 10/7 over another, until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of septimal consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending [[Carlos Gamma]]. | ||
Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)<sup>3</sup> equates to 11/4 | Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)<sup>3</sup> equates to 11/4. | ||
[[Gene Ward Smith]] once described [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6001.html neptune as an analog of miracle]. | [[Gene Ward Smith]] once described [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6001.html neptune as an analog of miracle]. | ||
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: mapping generators: 2, ~7/5 | : mapping generators: 2, ~7/5 | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 582.452 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 582.452 | ||
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[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Pages with mostly numerical content]] | |||
[[Category:Gammic family| ]] <!-- main article --> | [[Category:Gammic family| ]] <!-- main article --> | ||
[[Category:Gammic| ]] <!-- key article --> | [[Category:Gammic| ]] <!-- key article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||