Introduction[edit]

By giving a transversal for a MOS for a particular rank two temperament, we can define the MOS as the tempering, in that temperament, of the scale. This is not very interesting in itself; what is more interesting is that since only three primes--2 and two odd primes--can be used for the transversal, we can put the result into Scala, and use its "Lattice and player" command under the Analyze pull-down menu to depict the MOS in terms of a lattice diagram. This can be used to better understand the chord relationships within the MOS. It should be noted that while only chords with two odd primes are depicted, larger chords are associated to them. When 3 and 5 are relatively complex as in miracle, for instance, 7 is likely to be brought along with them, and hence the lattice picture of the 5-limit triads can be used to understand relations between tetrads in miracle and other temperaments of a similar kind. Choosing two primes which bound a chord of interest often works: for instance 3 and 5 bound 7 in miracle, so that the 5-limit transversal shows 7-limit tetrads. On the other hand, in the 11-limit, 5 and 11 bound both 7 and 9, so that the complete 11-limit sextads are represented by the triads of the 2.5.11 transveral.

A difficulty with the method is that the planar arrangement does not correspond precisely with the linear arrangement of MOS; the single comma between the three primes rolls up the plane into a cylinder, hence there may be chords not represented by the transveral lattice. Rather than attempting to roll any cylinders, one may take advantage of the fact that with only three generators there is just one comma on the subgroup temperament; multiplying by it, dividing by it, and taking the original scale one gets an extended transversal which reduces to the same MOS but which shows the chords within a larger lattice. These are the scales with "ex" postfixing the name of the scale.