Xenharmonic Wiki

Gammic family: Difference between revisions

← Older edit
VisualWikitext
No edit summary
m added brief explanation of the extension to 17
 
(10 intermediate revisions by 4 users not shown)
Line 1: Line 1:
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. 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]] 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.
{{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 }}, 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, 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.
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 ==
Line 17: Line 18:


[[Badness]]: 0.087752
[[Badness]]: 0.087752
=== 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
[[Comma list]]: 15625/15606, 24576/24565
{{Mapping|legend=1| 1 1 2 4 | 0 20 11 3 }}
: mapping generators: ~2, ~51/50
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~51/50 = 35.1011
{{Optimal ET sequence|legend=1| 34, 103, 137, 171, 376, 547, 2564g, 3111cg, 3658cgg }}
[[Badness]] (Sintel): 0.320


== Septimal gammic ==
== Septimal gammic ==
Line 24: Line 42:


{{Mapping|legend=1| 1 1 2 0 | 0 20 11 96 }}
{{Mapping|legend=1| 1 1 2 0 | 0 20 11 96 }}
Wedgie: {{multival| 20 11 96 -29 96 192 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~234375/229376 = 35.0904
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~234375/229376 = 35.0904
Line 78: Line 94:


[[Mapping]]: {{mapping| 1 1 2 1 | 0 20 11 62 }}
[[Mapping]]: {{mapping| 1 1 2 1 | 0 20 11 62 }}
{{Multival|legend=1|20 11 62 -29 42 113}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1990656/1953125 = 34.984
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1990656/1953125 = 34.984
Line 129: Line 143:
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]] 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]] 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.  


[[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].  


=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 140: Line 155:


: mapping generators: 2, ~7/5
: mapping generators: 2, ~7/5
{{Multival|legend=1| 40 22 21 -58 -79 -13 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 582.452
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 582.452
Line 148: Line 161:


[[Badness]]: 0.023427
[[Badness]]: 0.023427
==== 2.3.5.7.17 subgroup ====
[[Subgroup]]: 2.3.5.7.17
[[Comma list]]: 1225/1224, 2401/2400, 24576/24565
{{Mapping|legend=1| 1 21 13 13 7 | 0 -40 -22 -21 -6 }}
: mapping generators: ~2, ~7/5
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, 7/5 = 582.450
{{Optimal ET sequence|legend=1| 35, 68, 103, 171, 581, 752, 923, 1094 }}
[[Badness]] (Sintel): 0.404


=== 11-limit ===
=== 11-limit ===
Line 243: Line 271:


[[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]]
Retrieved from "https://en.xen.wiki/w/Gammic_family"