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).

 

 

Comma list: 15625/15606, 24576/24565

 

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

 

 

 

Badness (Sintel): 0.320