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.

=== 2.3.5.17 subgroup ===

The interval of 3 generators represents 1/3 of [[6/5]], which is very close to [[17/16]], with the comma between 6/5 and (17/16)³ being [[24576/24565]] ([[S-expression|S16/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

 

Subgroup-val mapping: {{mapping| 1 1 2 4 | 0 20 11 3 }}

 

* WE: ~2 = 1199.9899{{c}}, ~51/50 = 35.1008{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~51/50 = 35.1008{{c}}

 

{{Optimal ET sequence|legend=0| 34, 103, 137, 171, 376, 547 }}

 

Badness (Sintel): 0.320

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 [[7-odd-limit]] 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]].  

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

[[Category:Rank 2]]