Gammic family: Difference between revisions

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. 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.

[[Subgroup]]: 2.3.5

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

: mapping generators: ~2, ~1990656/1953125

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1990656/1953125 = 35.0964

{{Optimal ET sequence|legend=1| 34, 103, 137, 171, 547, 718, 889, 1607 }}

[[Badness]]: 0.087752

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)³ 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×S50².

[[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 ==

[[Subgroup]]: 2.3.5.7

 

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

 

{{Mapping|legend=1| 1 1 2 0 | 0 20 11 96 }}

 

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

 

{{Optimal ET sequence|legend=1| 34d, 171, 205, 1402, 1573, 1744, 1915 }}

 

[[Badness]]: 0.047362

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

Comma list: 243/242, 4375/4356, 100352/99825

 

 

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 35.089

 

{{Optimal ET sequence|legend=1| 34d, 137d, 171 }}

 

Badness: 0.097061

 

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

 

Comma list: 243/242, 364/363, 625/624, 2200/2197

 

 

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 35.091

 

 

 

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

 

Comma list: 243/242, 364/363, 375/374, 595/594, 2200/2197

 

Mapping: {{mapping| 1 1 2 0 2 3 4 | 0 20 11 96 50 24 3 }}

 

 

{{Optimal ET sequence|legend=1| 34d, 137d, 171 }}

 

Badness: 0.031466

 

[[Subgroup]]: 2.3.5.7

 

[[Comma list]]: 225/224, 94143178827/91913281250

 

[[Mapping]]: {{mapping| 1 1 2 1 | 0 20 11 62 }}

 

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

 

{{Optimal ET sequence|legend=1| 34d, 69d, 103, 240, 343b }}

 

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

Comma list: 225/224, 243/242, 215622/214375

 

 

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 34.985

 

 

Badness: 0.065326

 

Subgroup: 2.3.5.7.11.13

 

Comma list: 225/224, 243/242, 351/350, 1188/1183

 

Mapping: {{mapping| 1 1 2 1 2 3 | 0 20 11 62 50 24 }}

 

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 34.988

 

{{Optimal ET sequence|legend=1| 34d, 69de, 103, 240, 343be }}

 

Badness: 0.033418

 

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

 

Comma list: 225/224, 243/242, 351/350, 375/374, 1188/1183

 

Mapping: {{mapping| 1 1 2 1 2 3 4 | 0 20 11 62 50 24 3 }}

 

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 34.997

 

 

Badness: 0.025030

 

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)³ equates to 11/4.  

 

[[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

 

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

 

{{Mapping|legend=1| 1 21 13 13 | 0 -40 -22 -21 }}

 

: mapping generators: 2, ~7/5

 

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

 

{{Optimal ET sequence|legend=1| 35, 68, 103, 171, 1094, 1265, 1436, 1607, 1778 }}

 

[[Badness]]: 0.023427

 

[[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

 

Subgroup: 2.3.5.7.11

 

Comma list: 385/384, 1375/1372, 78408/78125

 

Mapping: {{mapping| 1 21 13 13 2 | 0 -40 -22 -21 3 }}

 

 

{{Optimal ET sequence|legend=1| 35, 68, 103, 171e, 274e, 445ee }}

 

 

 

Comma list: 385/384, 625/624, 1188/1183, 1375/1372

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Category:Gammic| ]] <!-- key article -->