Appendix C29B- The Ramanujan Octave and Discriminants -4m

The calculation of the g invariant for discriminants -4m requires an adjustment to the Ramanujan octave invariant for odd discriminants.  The new invariant is shown to be useful for finding solvable polynomials if the q octic continued fraction is divided by harmonics of the 12-tone scale. The 1/12 and 1/8 powers of 12 are significant in finding monic polynomials of degree 4 or less.  The numbers 8 and 12 also appear in the elements of the octahedron.  This platonic solid has 8 faces, 12 edges and 6 vertices.  The solid may have a natural relation to the 12-tone scale only based on the number 12 but it also extends into the mathematics of modular functions.  The powers 1/12 and 1/24 are found in many modular equations such as the j-invariant where  is a factor used to impose independence of the invariant for elliptic curves in any coordinate system.  The powers 12, 24 and 8 appear in the Weber function relations to the j-invariant.  Although other variants of the Ramanujan octave have appeared in the literature, the results discussed in this paper do not have any significance beyond these mathematical observations but still serves as an intriguing connection of mathematics to architectural geometry and music harmony.

The Ramanujan Octave and Discriminants -4m

Appendix C29- The Ramanujan Octave

An analogy is made between the equations derived Chapter 29 with the musical equal tempered scale.  I call this the Ramanujan octave based on his octic q continued fraction and its relation to class invariants.  The existence theorem of solvability is demonstrated with two examples representing the product and quotient of the modulus of the octic continued fraction.  These examples show a universality to finding radical forms of the class invariants for any class number.  An infinite number of semitones are shown to create the Ramanujan octave!

The Ramanujan Octave and Examples of the Existence Theorem