12 is a composite number = 2^2 x 3, so only 1/11 or 5/7 can serve as generators for the cyclic group of 12 pitches.

Compare to 34 equal (which is better thanmeantone tuning in terms of harmonic accuracy) where the circle of fifths gives us...17 equal, so obviously tonal modulation and chord patterns there are different than these in 12 equal music theory.any

(12 equal has subgrous C2, C3, C4 and C6. A coset of Cn is obtained by adding to each element of Cn the same element of Cn. Example: tempered fully diminished chord is C4 in 12 equal and is represented in atonal integer notation as (0,3,6,9). The other cosets are 1,4,7,10 and 2,5,8, 11. Obviously in 34 equal we have 17 equal and tritones (of course, any tuning, divisible by 2 has this one) as subgroups. We can construct all the modes of limited transposition, various looping chord progressions and represent various other elements as generated by these cosets or intervals in them. For reference, check any abstract algebra/group theory text/wikipedia.)

From

https://en.wikipedia.org/wiki/Modulatory_space

"Toroidal modulatory spaces

If we divide the octave into n parts, where n = rs is the product of two relatively prime integers r and s, we may represent every element of the tone space as the product of a certain number of "r" generators times a certain number of "s" generators; in other words, as the direct sum of two cyclic groups of orders r and s. We may now define a graph with n vertices on which the group acts, by adding an edge between two pitch classes whenever they differ by either an "r" generator or an "s" generator (the so-called Cayley graph of Z 12 with generators r and s). The result is a graph of genus one, which is to say, a graph with a donut or torus shape. Such a graph is called a toroidal graph.

An example is equal temperament; twelve is the product of 3 and 4, and we may represent any pitch class as a combination of thirds of an octave, or major thirds, and fourths of an octave, or minor thirds, and then draw a toroidal graph by drawing an edge whenever two pitch classes differ by a major or minor third.

We may generalize immediately to any number of relatively prime factors, producing graphs can be drawn in a regular manner on an n-torus. "

Translated in more normal language we can say that every interval in 12 equal can be decomposed into major thirds and minor thirds -example: P5=M3 + m3

So we don't need chains or circles of generators to get to diatonic or 12 tone (tuned to equal, meantone, just intonation, diaschismic like 34, schismic or whatever temperament). Of course, there exist even more different methods to construct 7note diatonic/12 equal.

If we are that concerned about very good perfect fifths, the only alternative to diatonic scale actually is 17 notes and this is based on some patterns of log3 base2 - we get 8 major chords, 8 minor chords and one dissonant chord. Pythagorean/syntonic commas becomes a small step in this tuning. I guess this is similar to some kind of Indian music gamut. 41 equal or 53 equal supports it.