Gammic family: Difference between revisions

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The [[Carlos Gamma]] rank ~~one~~ temperament divides 3/2 into 20 equal parts, 11 of which give a 5/4. This is closely related to the rank ~~two~~ 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|171EDO]], [[Schismatic family|schismatic]] temperament makes for a natural comparison. Schismatic, with a wedgie of {{multival|1 -8 -15}} is plainly much less complex than gammic with wedgie {{multival|20 11 -29}}, but people seeking the exotic might prefer gammic even so. The 34-note MOS is interesting, being a 1L 33s refinement of the [[34edo|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.

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|171EDO]], [[Schismatic family|schismatic]] temperament makes for a natural comparison. Schismatic, with a wedgie of {{multival|1 -8 -15}} is plainly much less complex than gammic with wedgie {{multival| 20 11 -29 }}, but people seeking the exotic might prefer gammic even so. The 34-note MOS is interesting, being a 1L 33s refinement of the [[34edo|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, giving a wedgie of {{multival|20 11 96 -29 96 192}}. 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 ==

Subgroup: 2.3.5

[[Comma]]: {{monzo| -29 -11 20 }}

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[[Badness]]: 0.087752

=== 7~~-limit ===~~

== Septimal gammic ==

Subgroup: 2.3.5.7

[[Comma list]]: 4375/4374, 6591796875/6576668672

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[[Badness]]: 0.047362

== Neptune ==

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~~, with wedgie {{multival|40 22 21 -58 -79 -13}}~~. 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|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|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. This may be described as {{multival|40 22 21 -3 ~~...~~}} or 68&103, and 171 can still be used as a tuning, with [[val]] {{val| 171 271 397 480 591 }}.

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. This may be described as {{multival|40 22 21 -3 …}} or 68&103, and 171 can still be used as a tuning, with [[val]] {{val| 171 271 397 480 591 }}.

An article on Neptune as an analog of miracle can be found [http://tech.groups.yahoo.com/group/tuning-math/message/6001 here].

Subgroup: 2.3.5.7

[[Comma list]]: 2401/2400, 48828125/48771072

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Mapping generators: 2, 7/5

[[POTE generator]]: ~7/5 = 582.452

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[[Badness]]: 0.023427

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 2465529759/2441406250

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Badness: 0.063602

[[Category:~~Theory~~]]

[[Category:Regular temperament theory]]

[[Category:Temperament family]]

[[Category:Gammic]]

[[Category:Gammic family| ]]

[[Category:Rank 2]]

~~{{todo|review}}~~

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-The [[Carlos Gamma]] rank one temperament divides 3/2 into 20 equal parts, 11 of which give a 5/4. This is closely related to the rank two 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|171EDO]], [[Schismatic family|schismatic]] temperament makes for a natural comparison. Schismatic, with a wedgie of {{multival|1 -8 -15}} is plainly much less complex than gammic with wedgie {{multival|20 11 -29}}, but people seeking the exotic might prefer gammic even so. The 34-note MOS is interesting, being a 1L 33s refinement of the [[34edo|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.
+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|171EDO]], [[Schismatic family|schismatic]] temperament makes for a natural comparison. Schismatic, with a wedgie of {{multival|1 -8 -15}} is plainly much less complex than gammic with wedgie {{multival| 20 11 -29 }}, but people seeking the exotic might prefer gammic even so. The 34-note MOS is interesting, being a 1L 33s refinement of the [[34edo|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, giving a wedgie of {{multival|20 11 96 -29 96 192}}. 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 ==
+Subgroup: 2.3.5
 [[Comma]]: {{monzo| -29 -11 20 }}
@@ Line 14: / Line 16: @@
 [[Badness]]: 0.087752
-=== 7-limit  ===
+== Septimal gammic ==
+Subgroup: 2.3.5.7
 [[Comma list]]: 4375/4374, 6591796875/6576668672
@@ Line 25: / Line 29: @@
 [[Badness]]: 0.047362
-== Neptune  ==
+== Neptune ==
-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&amp;171 temperament, with wedgie {{multival|40 22 21 -58 -79 -13}}. 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|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&amp;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|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. This may be described as {{multival|40 22 21 -3 ...}} or 68&amp;103, and 171 can still be used as a tuning, with [[val]] {{val| 171 271 397 480 591 }}.
+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. This may be described as {{multival|40 22 21 -3 …}} or 68&amp;103, and 171 can still be used as a tuning, with [[val]] {{val| 171 271 397 480 591 }}.
 An article on Neptune as an analog of miracle can be found [http://tech.groups.yahoo.com/group/tuning-math/message/6001 here].
+Subgroup: 2.3.5.7
 [[Comma list]]: 2401/2400, 48828125/48771072
@@ Line 37: / Line 43: @@
 Mapping generators: 2, 7/5
+{{Multival|legend=1| 40 22 21 -58 -79 -13 }}
 [[POTE generator]]: ~7/5 = 582.452
@@ Line 44: / Line 52: @@
 [[Badness]]: 0.023427
-=== 11-limit  ===
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
 Comma list: 385/384, 1375/1372, 2465529759/2441406250
@@ Line 57: / Line 67: @@
 Badness: 0.063602
-[[Category:Theory]]
+[[Category:Regular temperament theory]]
 [[Category:Temperament family]]
-[[Category:Gammic]]
+[[Category:Gammic family| ]] <!-- main article -->
 [[Category:Rank 2]]
-{{todo|review}}