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Statement of the Theory

 

On this page we discuss a particular class of drinks called sours and propose a precise mathematical formalism that describes their structure.

 

Taxonomy of Cocktails:

Attacking the problem one family at a time

The development of a mathematical theory that can predict all great cocktail recipes from a few basic parameters describing each ingredient is an exceedingly ambitious endeavor.  Most cocktails are known to fall into one of a plethora of different families, each consisting of many different recipes sharing some primitive similarities.  This is discussed, for example, by the eminent Gary Regan.   There are substantial variations in style as one surveys the landscape of cocktails from the French-Italian family, where the spirit of choice is tempered only by the herbal, vinous essence of vermouth and seasoned with bitters, to the family of trios, where cream is used to soften and provide a lush backdrop for the spirit of choice and a liqueur is added to harmonize the two.

As a more modest undertaking, we will focus only on one family of drinks: those known as sours.  This is partly because they are my personal favorite and partly because this category includes a large number of venerable classics (e.g., Whiskey Sour, Ward 8, Margarita, Aviation, Pegu Club, Daquiri, Sidecar, Jack Rose, Pisco Sour).

Dissecting the Sour:

Thoughts on Balance

Dating back to at least 1856, the sour was one of the earliest forms of drink to arise as a simpler, smaller, and more personal beverage than the more collaboratively imbibed bowls of punch, which had previously dominated American drinking.   The shift in direction of the drinking culture from leisurely sipped bowls of punch to more efficiently consumed short drinks like the sour was perhaps a consequence of industrialization, but the true reasons are not well documented.

In any case, the simplicity of the sour lends itself particularly well to a thoughtful analysis of why it is so palatable when made correctly.  A sour is generally composed of three things.  First, there is a base spirit, typically something traditional like gin, tequila, rum, whiskey, or brandy, but occasionally something exotic like an amaro or liqueur.  Second, there is citrus juice, usually lemon or lime, sometimes grapefruit or orange, but always something with an element of sourness.  Third, there is a sweetener, which could be simple syrup, flavored syrup, sweet liqueur, or anything with an element of sweetness.  When these components come to together in the right amounts, the cocktail tastes balanced.  Much has been written on the topics of defining and achieving balance in cocktails.  For example, see  for a thorough, quantitative discussion of these topics in regard to all forms of cocktail, including sours.  Below is my humble take on the subject.

Consider, as a case study, one of the most minimal entries in the category: the whiskey sour.  An archetypal recipe looks like this:

(1)   \begin{eqnarray*} \textnormal{\bf{Brand X Whiskey}} & A_{w} \nonumber \\ \textnormal{\bf{Lemon Juice}} & A_{l} \nonumber \\ \textnormal{\bf{Simple Syrup}} & A_{s} \end{eqnarray*}

where A_{w},A_{l},A_{s} are the amounts of each ingredient.  In the limit A_{l} \rightarrow 0, we have the makings of a very ancient cocktail on our hands: the Old Fashioned (sans bitters).

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The Old Fashioned cocktail.

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The Whiskey Sour.

 

 

This drink exploits the curious phenomenon that sweetness (in this case simple syrup) can reduce the fiery burn associated with alcohol, resulting in a more balanced drink than pure whiskey.  This observation may well have been the birth of cocktails as we know them.  The earliest known definition of the word “cocktail” appeared in an 1806 edition of a Federalist newspaper, the Balance and Columbian Repository, which explained that it is “a stimulating liquor, composed of spirits of any kind, sugar, water, and bitters.”  Interestingly,  this was penned by Harry Croswell, the same man who reported that Thomas Jefferson had hired a goon to besmirch the character of George Washington to help defeat his successor in the election of 1800.  See  for an entertaining read.  Anyway, these types of cocktails consisting of just spirit, sweetener, and bitters are great ways of tasting the complex flavors of spirits without incurring as much of their unpleasant burn as when taken neat.  Nonetheless, these primordial cocktails could universally be described as fairly strong, regardless of one’s personal preference.  Suppose we wished to take this balance further and soften the strength of the alcohol in our Old Fashioned while still retaining a pleasing balance of tastes, what are the options?

  • One possibility is to simply increase the level of simple syrup in the drink, since we know this reduces the strength of the alcohol.  The problem with this approach is that the drink eventually becomes too sweet; a well made Old Fashioned will already have the level of sweetness below, but probably close to, this threshold so that it can reduce the alcoholic strength as much as possible without making the drink sweet.
  • The other option is to add something new to the cocktail.  Based on our goals, we can place certain requirements on any potential addition: (1) it must have the ability to subdue the strength of alcohol and (2) it must not possess sweetness.  While it may not be the only thing that fits these criteria, sourness certainly works.  Mixing a spirit only with pure lemon or lime juice will not generally be very pleasant, but the alcoholic strength is reduced.  Repeating this with the more mildly sour orange juice is, in fact, the basis for some very popular, albeit somewhat boring, drinks (e.g., Screwdriver, Tequila Sunrise).

This justifies the whiskey sour prototype given in 1, which indeed consists of an alcoholic spirit (Whiskey), a sweetener (simple syrup), and an element of sourness (lemon juice).

The Fundamental Theorem of Sours

With this specific form, sours are endowed with the distinctive feature that each basic taste element suppresses all of the others, which actually allows us to derive some very precise consequences.

  • We’ve already said that alcoholic bite can be reduced by both sourness and sweetness.  Therefore, to keep the total alcoholic bite (T_{bite}) in check, it must be precisely balanced by the sum of the total sourness (T_{sour}) and total sweetness (T_{sweet}) in the drink:

    (2)   \begin{equation*} \frac{T_{bite}}{T_{sour}+T_{sweet}}= c_{1}=constant. \end{equation*}

  • Similarly, sweetness is lessened by alcoholic bite and obviously by sourness (hence the success of lemonade).  So to keep the drink from being too sweet we need an additional balance of flavors:

    (3)   \begin{equation*} \frac{T_{sweet}}{T_{bite}+T_{sour}}= c_{2}=constant. \end{equation*}

  • Lastly, by the same arguments, sourness is moderated by alcoholic bite and sweetness.  Thus, we can control the level of sourness in the drink:

    (4)   \begin{equation*} \frac{T_{sour}}{T_{bite}+T_{sweet}}=c_{3}=constant. \end{equation*}

To the extent that personal tastes differ, c_{1},c_{2}, and c_{3} are only constants for fixed drinking tastes; however, the spread in personal tastes is small enough to allow certain bars to achieve widespread acclaim and others to garner general disdain (just check out yelp), despite the fluctuations in tastes of their patrons.

We can describe any particular drink by its total flavor vector \vec{T}, defined as:

(5)   \begin{equation*} \vec{T} \equiv \left[ \begin{array}{c} T_{bite} \\ T_{sweet} \\ T_{sour} \end{array} \right]. \end{equation*}

We consider the drink to be balanced if \vec{T} is a solution to Eq.s 2 – 4.  Since c_{1},c_{2}, and c_{3} must be positive, it turns out that there are two possible scenarios, depending on the precise values of those constants.  One is that there is no \vec{T} that solves those equations; this could be the case for a person who hates sours.  The other more interesting scenario is that solutions exist, which must have a specific form.  If \vec{T} is a solution, then so is c \vec{T} for any number c.  This scale invariance is not surprising, since after all we know that we can double a recipe or half it or multiply it by whatever we want and it tastes the same.

So apart from the fact that we can make the drink whatever size we want, the solution is unique, and at least in the case of the whiskey sour, there is one best recipe!

Suppose for the sake of argument that c_{1}=1, c_{2}=\frac{1}{3}, c_{3}=\frac{1}{3}.  Then a balanced drink in this case must be of the form:

(6)   \begin{equation*} \vec{T} \propto \left[ \begin{array}{c} 2} \\ 1 \\ 1 \end{array} \right] \end{equation*}

for this particular taste preference in some system of units.  We could decide that we will measure alcoholic bite in units that are twice as big and we would still have the same result as long as we set c_{1}=c_{2}=c_{3}=\frac{1}{2} so that balanced drinks would be of the form:

(7)   \begin{equation*} \vec{T} \propto \left[ \begin{array}{c} 1} \\ 1 \\ 1 \end{array} \right] \end{equation*}

in our new system of units.

We will exploit this freedom in what follows and choose to use units such that balanced drinks always have the form of Eq. 7.  In this view, c_{1}=c_{2}=c_{3}=\frac{1}{2} and Eq. 7 always hold, and the only thing that changes from person to person with different taste preferences is their system of units so that \vec{T} may be different for two people for the same drink.

How exactly does one calculate \vec{T} for some particular drink?  Recall that the components in \vec{T} represent the total alcoholic bite, sweetness, and sourness in the cocktail.  It then stands to reason that we need to add up the alcoholic bite, sweetness, and sourness inherent in each ingredient.  Toward that end, let us introduce the concept of the flavor density vector.  Any potential cocktail ingredient can be described by its flavor density vector:

(8)   \begin{equation*} \vec{\rho} \equiv \left[ \begin{array}{c} \rho_{bite} \\ \rho_{sweet} \\ \rho_{sour} \end{array} \right]. \end{equation*}

Here \rho_{bite} is the alcoholic bite per unit volume of the ingredient, \rho_{sweet} is the sweetness per unit volume of the ingredient, and \rho_{sour} is the sourness per unit volume of the ingredient.  Note that \vec{\rho} is a property of the ingredient that is independent of the quantity of the ingredient and independent of the cocktail that we want to make with it.

Consider a cocktail composed of N ingredients.  The i’th ingredient has a flavor density vector of \vec{\rho_{i}} and the amount used in the cocktail is A_{i}.  The drink therefore has the total flavor vector:

(9)   \begin{equation*} \vec{T} = \sum\limits_{i=1}^{N} A_{i} \vec{\rho_{i}} = \begin{bmatrix} \rho_{bite, 1} & \rho_{bite, 2} & \ldots & \rho_{bite, N} \\ \rho_{sweet, 1} & \rho_{sweet, 2} & \ldots & \rho_{sweet, N} \\ \rho_{sour, 1} & \rho_{sour, 2} & \ldots & \rho_{sour, N} \end{bmatrix} \begin{bmatrix} A_{1} \\ A_{2} \\ \vdots \\ A_{N} \end{bmatrix}. \end{equation*}

 

Finally, we can equate the RHS of Eq. 9 with the RHS of Eq. 7 to yield the seminal result of this project, which we will call the Fundametal Theorem of Sours:

(10)   \begin{equation*} \begin{bmatrix} \rho_{bite, 1} & \rho_{bite, 2} & \ldots & \rho_{bite, N} \\ \rho_{sweet, 1} & \rho_{sweet, 2} & \ldots & \rho_{sweet, N} \\ \rho_{sour, 1} & \rho_{sour, 2} & \ldots & \rho_{sour, N} \end{bmatrix} \begin{bmatrix} A_{1} \\ A_{2} \\ \vdots \\ A_{N} \end{bmatrix} =c \left[ \begin{array}{c} 1} \\ 1 \\ 1 \end{array} \right]. \end{equation*}

c is just an arbitrary constant that controls the size of the drink, we can set c=1 if we like.  If we would like to make a drink out of some set of ingredients, and we know what the flavor density vectors (\vec{\rho_{i}}) are for each ingredient, then all we have to do is solve Eq. 10 for the amounts of each ingredient (A_{i}) to use for a balanced tasting recipe.  In principle, this is easy, it’s just math that we can let the computer do for us (…well it’s not entirely straightforward as discussed in The Algorithm).  The only difficult part is figuring out what the flavor density vectors are for the ingredients we want to use.  This is discussed in Ingredient Calibration.

Lastly, Eq. 10 can be written in a different form that will also prove useful:

(11)   \begin{equation*} \sum\limits_{i=1}^{N} A_{i}\rho_{bite, i}=\sum\limits_{i=1}^{N} A_{i}\rho_{sweet, i}=\sum\limits_{i=1}^{N} A_{i}\rho_{sour, i} \end{equation*}

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Wondrich D. Imbibe!: from absinthe cocktail to whiskey smash, a salute in stories and drinks to “Professor” Jerry Thomas, pioneer of the American bar. New York, N.Y.: Perigee Book/Penguin Group; 2007.

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