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 }}. 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.

Subgroup: 2.3.5

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

[[Mapping]]: [{{val|1 1 2}}, {{val|0 20 11}}]

[[POTE generator]]: ~1990656/1953125 = 35.0964

[[Badness]]: 0.087752

Subgroup: 2.3.5.7

[[Mapping]]: [{{val|1 1 2 0}}, {{val|0 20 11 96}}]

[[POTE generator]]: ~234375/229376 = 35.0904

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

[[Badness]]: 0.047362

Mapping: [{{val|1 1 2 0 2}}, {{val|0 20 11 96 50}}]

POTE generator: ~45/44 = 35.089

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

Badness: 0.097061

Mapping: [{{val|1 1 2 0 2 3}}, {{val|0 20 11 96 50 24}}]

POTE generator: ~45/44 = 35.091

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

Badness: 0.047822

Mapping: [{{val|1 1 2 0 2 3 4}}, {{val|0 20 11 96 50 24 3}}]

POTE generator: ~45/44 = 35.090

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

Badness: 0.031466

Subgroup: 2.3.5.7

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

 

{{Multival|legend=1|20 11 62 -29 42 113}}

[[POTE generator]]: ~1990656/1953125 = 34.984

[[Badness]]: 0.230839

Mapping: [{{val|1 1 2 1 2}}, {{val|0 20 11 62 50}}]

POTE generator: ~45/44 = 34.985

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

Badness: 0.065326

Mapping: [{{val|1 1 2 1 2 3}}, {{val|0 20 11 62 50 24}}]

POTE generator: ~45/44 = 34.988

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

Badness: 0.033418

Mapping: [{{val|1 1 2 1 2 3 4}}, {{val|0 20 11 62 50 24 3}}]

POTE generator: ~45/44 = 34.997

{{Optimal ET sequence|legend=1| 34d, 69de, 103, 137, 240 }}

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|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. 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 }}.

Subgroup: 2.3.5.7

[[Mapping]]: [{{val|1 21 13 13}}, {{val|0 -40 -22 -21}}]

Mapping generators: 2, ~7/5

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

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

[[Badness]]: 0.023427

Mapping: [{{val|1 21 13 13 2}}, {{val|0 -40 -22 -21 3}}]

POTE generator: ~7/5 = 582.475

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

Badness: 0.063602

Mapping: [{{val|1 21 13 13 2 27}}, {{val|0 -40 -22 -21 3 -48}}]

POTE generator: ~7/5 = 582.480

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

Badness: 0.037156

Mapping: [{{val|1 21 13 13 2 27 7}}, {{val|0 -40 -22 -21 3 -48 -6}}]

POTE generator: ~7/5 = 582.475

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

Badness: 0.025909

Mapping: [{{val|1 21 13 13 52}}, {{val|0 -40 -22 -21 -100}}]

POTE generator: ~7/5 = 582.478

{{Optimal ET sequence|legend=1| 68e, 103, 171, 274, 719be, 993bcde, 1267bbcde }}

Badness: 0.069721

Mapping: [{{val|1 21 13 13 52 27}}, {{val|0 -40 -22 -21 -100 -48}}]

POTE generator: ~7/5 = 582.477

{{Optimal ET sequence|legend=1| 68e, 103, 171, 274, 719be, 993bcde }}

Badness: 0.034977

Mapping: [{{val|1 21 13 13 52 27 7}}, {{val|0 -40 -22 -21 -100 -48 -6}}]

POTE generator: ~7/5 = 582.475

{{Optimal ET sequence|legend=1| 68e, 103, 171, 274, 445e, 719be, 1164bcdeef }}

Badness: 0.024577

Mapping: [{{val|2 2 4 5 8}}, {{val|0 40 22 21 -37}}]

Mapping generators: ~99/70, ~99/98

POTE generator: ~99/98 = 17.545

{{Optimal ET sequence|legend=1| 68, 206b, 274, 342 }}

Badness: 0.041727

[[Category:Gammic family| ]] <!-- main article -->

[[Category:Rank 2]]