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I'm struggling with moving from a quarterly calibration of a baseline NK model to the corresponding monthly calibration, so that IRFs give the same results with the appropriate timing.

What are the steps to do to construct appropriate variables and change parameters?

The Dynare code for the (quarterly) log-linearized model is attached:

[name = 'Production function']              y = n;
[name = 'Marginal utility of consumption']  uc = - c;
[name = 'Resource constraint']              c = y;

// NK block
[name = 'Euler equation']                   -uc = -uc(+1) - (r - pi(+1)); 
[name = 'Phillips curve']                   pi = beta*pi(+1) - lambda*mu;
[name = 'Taylor rule']                      r = phi_pi*pi + mm;
[name = 'Inflation']                        pi = p - p(-1);  

% Production inputs
[name = 'Cost minimization labor']          y - n = mu + w_real;

// Wage block
[name = 'Nominal wage rigidity']            w = (1-gamma)*(w_reset + p) + gamma*w(-1);
[name = 'Labor supply']                     w_reset = phi*n - uc;  
[name = 'Real Wage']                        w_real = w - p;        

// Dynamics of shocks
[name = 'Dynamics money shock']             mm = rho_m*mm(-1) + sigma_m*e_m;

Quarterly calibration:

beta                = (1/1.04)^(1 / 4);                   %Discount factor
theta               = 1 - 1/(1/(1-.75));                  %Price stickiness (implies 6 quarters of stickiness)
gamma               = 0;                                  %Wage rigidity
rho_m               = .9^3;                               %AR(1) money shock
sigma_m             = 1;                                  %Money shock 
lambda              = (1-theta)*(1-beta*theta)/theta;
phi                 = 1;                                  %Inverse  Frisch
phi_pi              = 1.5;                                %Taylor rule

Attempt to monthly calibration:

beta                = (1/1.04)^(1 / 12);                   %Discount factor
theta               = 1 - 1/(1/(1-.75)*3);                 %Price stickiness (implies 6 quarters of stickiness)
gamma               = 0;                                   %Wage rigidity
rho_m               = .9;                                  %AR(1) money shock
sigma_m             = 1/3 (?);                             %Money shock 
lambda              = (1-theta)*(1-beta*theta)/theta;
phi                 = 1;                                  %Inverse  Frisch
phi_pi              = 1.5 (?);                            %Taylor rule

Also, what happens with gamma $\neq$ 0?

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