  A notation system was previously presented which can notate any rational
frequency in free Just Intonation. Transposition of music is carried out by
multiplying each member of a set of frequencies by a single frequency.
Transposition of JI notations up by a fixed amount requires multiplication to
be defined for any two notations. Transposition down requires inversion to be
defined for any notation, which allows division to also be defined for any two
notations. Each notation splits into four components which in decreasing size
order are octave, diatonic scale note, sharps or flats, rational comma
adjustment. Multiplication can be defined for each of the four notation
components. Since rational number multiplication is commutative, this leads to
a definition of multiplication for frequencies and thus notations. Examples of
notation inversion and multiplication are given. Examples of transposing
melodies are given. These are checked for accuracy using the rational numbers
which each notation represents. Calculation shortcuts are considered which make
notation operations quicker to carry out by hand. A question regarding whether
rational commas should be extended from 5-rough rational numbers to all
rational numbers is considered which would greatly simplify notation
multiplication. This approach is rejected since it leads to confusion about
octave number. The four component notation system is recommended instead.
Extensions to computer notation systems and stave representations are briefly
mentioned.
