Several types of metabolai can be considered: the first, the simplest one, consists of only a turn of the first lattice; it is the same, but with a different reference or direction 0. An example can be two concentric 8-lattices with a circular shift of half basic angle. See in Fig.3 above this metabolê and its beautiful effect, in a RIGE that can be found in the Alhambra, Granada (Spain). The lattice could be a unique 16-one, but also two concentric 8's. The choice would depend on the simplicity of computer implementation (see below).
A second type of metabolê, also simple, can appear: a change which keeps reference angles but jumps to a multiple or submultiple order. In those cases, both lattices share the main directions, but one of them introduces another set of main directions, in a symmetrical way, as the new directions will be inserted between the old ones. For example we can change from 8 to 16, or from 24 to 12. As before, a global or universal lattice can include both if desired.
A somewhat more complex metabolê, similar to the former, is a change to an order which share some divisor, their integers have a maximum common divisor, as in the cases 8 to 12 (m.c.d.=4) or 9 to 24 (m.c.d.=3). In this case, a global lattice can include both of them, too.
The most complex metabolê consists of a change between two prime orders, like 12-13 (never found by the author!). However, more simple changes between numbers which share only the factor '2' are used sometimes, as 10-12, in a wood door in the National Museum in Baghdad. Another rare example can be found in the Alcazaba museum of Malaga, where a Nazari relief (about XV century) shows a rare metabolê from a central part on 8-lattice, and 4 parts on a 7-lattice. This example is, perhaps, a "tour de force" from an Art in decline. But an even more complex example found by the author is the one in Konya, Turkey, which uses three different lattices, 16, 12 and 10, that coexist harmoniously here. See below, in Further Examples, the graphic analysis of both forms.
The metabolê has an exemplary value for us: it changes references, and/or circle divisions, in the same way that musical metabolê change the reference, in tonal or modal tonic, or the division of octave or tetrachord: genus Rast on note rast can change to Rast on naua, reference change, or to Nahauand on note rast, division change (see below, Musical Relationship, and [Barker,1989. Sanchez,1993-a]), The RIGE metabolê, like its musical counterpart, allows keeping simplicity while introducing variations: the psychoaesthetic effect is to allow easy recognition and appreciation, which becomes denser after some study, as with some masterpieces.
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