# Technical Studies Reference

### Murrey Math

This study calculates and displays Murrey Math Lines, which were developed by Thomas Henning Murrey.

The calculation is performed over a Square of fixed width. We denote the base value of the Square Width as $$n$$, and this base value is modified by the Multiplier Input, which we denote as $$v$$. The width of the Square is $$[nv]$$, where $$[\cdot]$$ denotes the Rounding Function. The value of $$n$$ is determined as follows.

• If the Custom Square Width Input is set to zero, then $$n$$ is determined by the setting of the Square Width Input. The values of this Input are restricted to powers of $$2$$, from $$4$$ through $$512$$.
• If the Custom Square Width Input is set to a nonzero value, then $$n$$ is determined by this value. These values are not restricted to powers of $$2$$.

The Square over which the calculations are performed begins at Index $$t_{start}$$ and ends at Index $$t_{end}$$. These are related via the equation $$t_{start} = t_{end} - [nv] + 1$$, and $$t_{end}$$ is determined by the Inputs as follows.

• If the Use Last Bar as End Date-Time is set to Yes, then $$t_{end}$$ is the Index of the last bar in the chart.
• If the Use Last Bar as End Date-Time is set to No, then $$t_{end}$$ is the Index of the bar determined by the setting of the End Date-Time Input.

Let $$H$$ and $$L$$ be random variables denoting the High and Low Prices, respectively. We begin by computing the Highest High, Lowest Low, and Range over the Square. These are denoted as $$\max_{t_{end}}\left(H,[nv]\right)$$, $$\min_{t_{end}}\left(L,[nv]\right)$$, and $$\textrm{Range}_{t_{end}}(H,L,[nv])$$. Note that these are not moving calculations; the calculations are performed only at Index $$t_{end}$$.

Next we define a function $$SR_t(n,v)$$, which we calculate at Index $$t_{end}$$ as follows.

• If $$\max_{t_{end}}\left(H,[nv]\right) > 25$$ and $$\displaystyle{\frac{\ln(0.4)\max_{t_{end}}\left(H,[nv]\right))}{\ln(10)} - \left\lfloor \frac{\ln(0.4)\max_{t_{end}}\left(H,[nv]\right))}{\ln(10)} \right\rfloor > 0}$$, then $$\displaystyle{SR_{t_{end}}(n,v) = \exp\left(\ln(10)\left\lfloor \frac{\ln(0.4\max_{t_{end}}\left(H,[nv]\right))}{\ln(10)} \right\rfloor + 1\right)}$$.
• If $$\max_{t_{end}}\left(H,[nv]\right) > 25$$ and $$\displaystyle{\frac{\ln(0.4)\max_{t_{end}}\left(H,[nv]\right))}{\ln(10)} - \left\lfloor \frac{\ln(0.4)\max_{t_{end}}\left(H,[nv]\right))}{\ln(10)} \right\rfloor \leq 0}$$, then $$\displaystyle{SR_{t_{end}}(n,v) = \exp\left(\ln(10)\left\lfloor \frac{\ln(0.4\max_{t_{end}}\left(H,[nv]\right))}{\ln(10)} \right\rfloor\right)}$$.
• If $$\max_{t_{end}}\left(H,[nv]\right) \leq 25$$, then $$\displaystyle{SR_{t_{end}}(n,v) = 100\exp\left(\ln(8)\left\lfloor \frac{\ln(0.005)\max_{t_{end}}\left(H,[nv]\right))}{\ln(8)} \right\rfloor\right)}$$.

For an explanation of the logarithmic function $$\ln$$, refer to our description here.

The Murrey Math Interval Range function is denoted as $$\textrm{Range}^{(MMI)}_{t_{end}}(n,v)$$, and we calculate it at Index $$t_{end}$$ as follows.

$$\displaystyle{\textrm{Range}^{(MMI)}_{t_{end}}(n,v) = \left. \ln\left(\frac{SR_{t_{end}}(n,v)}{\textrm{Range}_{t_{end}}(H,L,[nv])}\right) \middle/ \ln(8) \right.}$$

Next we define a function called the Octave Count, dentoted as $$OC_t(n,v)$$. We calculate this at Index $$t_{end}$$ as follows.

• If $$\textrm{Range}^{(MMI)}_{t_{end}}(n,v) \leq 0$$, then $$OC_{t_{end}}(n,v) = 0$$.
• If $$\textrm{Range}^{(MMI)}_{t_{end}}(n,v) > 0$$ and $$\displaystyle{\textrm{Range}^{(MMI)}_{t_{end}}(n,v) - \ln\left(\frac{SR_{t_{end}}(n,v)}{\textrm{Range}_{t_{end}}(H,L,[nv])}\right) + \ln(8)\left\lfloor \textrm{Range}^{(MMI)}_{t_{end}}(n,v) \right\rfloor = 0}$$, then $$OC_{t_{end}}(n,v) = \left\lfloor \textrm{Range}^{(MMI)}_{t_{end}}(n,v) \right\rfloor$$.
• If $$\textrm{Range}^{(MMI)}_{t_{end}}(n,v) > 0$$ and $$\displaystyle{\textrm{Range}^{(MMI)}_{t_{end}}(n,v) - \ln\left(\frac{SR_{t_{end}}(n,v)}{\textrm{Range}_{t_{end}}(H,L,[nv])}\right) + \ln(8)\left\lfloor \textrm{Range}^{(MMI)}_{t_{end}}(n,v) \right\rfloor \neq 0}$$, then $$OC_{t_{end}}(n,v) = \left\lfloor \textrm{Range}^{(MMI)}_{t_{end}}(n,v) \right\rfloor + 1$$.

Next we define functions $$Si_t(n,v)$$, $$M_t(n,v)$$, $$I_t(n,v)$$, $$B^{(+)}_t(n,v)$$, and $$B^{(-)}_t(n,v)$$, which we calculate at Index $$t_{end}$$ as follows.

$$\displaystyle{Si_{t_{end}}(n,v) = SR_{t_{end}}(n,v)\exp(-\ln(8)OC_{t_{end}}(n,v))}$$

For an explanation of the exponential function $$\exp$$, refer to our description here.

$$\displaystyle{M_{t_{end}}(n,v) = \left\{ \begin{matrix} \left\lfloor \left. \ln\left|\frac{\textrm{Range}^{(MMI)}_{t_{end}}(n,v)}{Si_{t_{end}}(n,v)}\right| \middle/ \ln(2) \right. + 0.00001 \right\rfloor & \textrm{Range}^{(MMI)}_{t_{end}}(n,v) \neq 0 \\ 0 & \textrm{Range}^{(MMI)}_{t_{end}}(n,v) = 0 \end{matrix}\right .}$$

$$\displaystyle{I_{t_{end}}(n,v) = \frac{\frac{1}{2}=(\max_{t_{end}}(H,[nv]) + \min_{t_{end}}(L,[nv]))}{Si_{t_{end}}(n,v)\exp((M_{t_{end}}(n,v) - 1)\ln(2))}}$$

$$\displaystyle{B^{(+)}_{t_{end}}(n,v)} = (I_{t_{end}}(n,v) + 1)Si_{t_{end}}(n,v)\exp((M_{t_{end}}(n,v) - 1)\ln(2))$$

$$\displaystyle{B^{(-)}_{t_{end}}(n,v)} = (I_{t_{end}}(n,v) - 1)Si_{t_{end}}(n,v)\exp((M_{t_{end}}(n,v) - 1)\ln(2))$$

Finally, we are ready to calculate the Murrey Math Lines. These are enumerated using a Line Index $$\ell$$, which takes on integer values from $$0$$ to $$24$$. The function used for calculating the lines is denoted as $$MML_t(n,v,\ell)$$. To obtain the values of the Murrey Math Lines, we evaluate this function at $$t_{end}$$ as follows.

$$\displaystyle{MML_t(n,v,\ell) = B^{(-)}_{t_{end}}(n,v) + \left(\left(\frac{\ell - 8}{8}\right)\left(B^{(+)}_{t_{end}}(n,v) - B^{(-)}_{t_{end}}(n,v)\right)\right)}$$

The Murrey Math Lines are displayed as horizontal line segments between $$t_{start}$$ and the current Index $$t$$. The values of these Lines are rounded to the nearest Tick Size if the Round to Tick Size Input is set to Yes. See the Inputs section below.

The values of $$\ell$$ correspond to the Octaves as follows.

$$\ell$$ Octave
$$0$$ $$-8/8$$
$$1$$ $$-7/8$$
$$2$$ $$-6/8$$
$$3$$ $$-5/8$$
$$4$$ $$-4/8$$
$$5$$ $$-3/8$$
$$6$$ $$-2/8$$
$$7$$ $$-1/8$$
$$8$$ $$0/8$$
$$9$$ $$1/8$$
$$10$$ $$2/8$$
$$11$$ $$3/8$$
$$12$$ $$4/8$$
$$13$$ $$5/8$$
$$14$$ $$6/8$$
$$15$$ $$7/8$$
$$16$$ $$8/8$$
$$17$$ $$+1/8$$
$$18$$ $$+2/8$$
$$19$$ $$+3/8$$
$$20$$ $$+4/8$$
$$21$$ $$+5/8$$
$$22$$ $$+6/8$$
$$23$$ $$+7/8$$
$$24$$ $$+8/8$$

#### Inputs

• Square Width: The width of the Square. In other words, the number of bars to use in the calculation. There is a list of preset choices. If you want to use a different width, then set this through the Custom Square Width Input setting instead. When Custom Square Width is set to a nonzero number, it overrides this Square Width Input setting.
• Square Width Multiplier: The default for this Input is 1.5. This value is multiplied with Square Width to increase the Square Width value.
• Custom Square Width: When this Input is set to a nonzero value, this overrides the Square Width Input and specifies the number of bars to use for the width of the Square. Normally this should be set to 0 unless you want to override Square Width.
• Use Last Bar as End Date-Time:If this is set to Yes, the Murray Math study will be calculated using the bars at the end of the chart. The Murray Math lines will be redrawn for every new bar added to the chart. If this Input is set to No, the Murray Math will be calculated using the bars beginning with the bar at the End Date-Time Input, and going back the number of bars specified with the Square Width Input. This Input is set to Yes by default.
• End Date-Time: The Date-Time which will be the last bar used in the calculation. The calculation will go back from this Date-Time the number of bars specified with the Square Width Input. By default, the End Date-Time Input is not set and will not be used unless the Use Last Bar as End Date-Time Input is set to No.
• Round to Tick Size: When this Input is set to Yes, then the Murray Math lines are rounded to the nearest Tick Size value which is set through Chart >> Chart Settings.