I found this on the web and thought it made pretty good sense. I’ve reposted it here without permission. It’s been modified only for format.

Sweetness, Bitterness, and Balance

Hop bitterness and sweetness from unfermentable extract are two primary factors which determine a beer’s balance. Beers which are perceived as neither bitter nor sweet are termed “in balance,” though this is largely subjective and can mean different things to different people. Nevertheless, it is helpful to quantify the relationship between sweetness and bitterness, leaving the individual to decide what’s “balanced.”

In his excellent book, Designing Great Beers, Ray Daniels suggests looking at the ratio between a beer’s International Bitterness rating (expressed in IBUs, mg of iso-alpha acid per liter of beer) to the beer’s starting gravity expressed in °Oeschle (what many people refer to as “Gravity Units”). He refers to this as the “BU:GU” (for “Bittering Units to Gravity Units”) ratio.

The BU:GU Ratio

Let’s consider three beers: a Bock, a Vienna, and a Dry Stout. Most people would say the Bock is sweeter than the others, the Dry Stout more bitter, and the Vienna somewhere in between. What do the numbers tell us? Check out the table below.

Style Bock Vienna Dry Stout IBU 22 22 40 OG, °Oe 66 50 40 BU:GU 0.33 0.44 1.00

Table 1. BU:GU Ratios for selected styles.

The bittering and gravity levels are within those specified by the 2001 BJCP guidelines.

Most people would agree that these numbers track, at least in an ordinal sense, their perception of each beer’s bitterness relative to its sweetness. The Bock tastes the sweetest and it has the lowest BU:GU ratio; the Stout is the most bitter and it has the highest ratio. Seems to work great!

Nevertheless, some beers refuse to fit in. Consider a Dubbel. It should certainly be malty, but not especially sweet. Most experienced beer tasters would place a classic Dubbel between the Bock and the Vienna in terms of sweetness/bitterness balance. Let’s see how a typical Dubbel fits in:

Style Bock Dubbel Vienna Dry Stout IBU 22 22 22 40 OG, °Oe 66 66 50 40 BU:GU 0.33 0.33 0.44 1.00

Table 1a. BU:GU Ratios for selected styles.

This says the Dubbel is as sweet as the Bock, and we know that’s not the case. Dubbels are more balanced than Bocks, and closer, in terms of balance, to something between the Bock and the Vienna. The BU:GU ratio has, in this case, led us astray. That’s surprising, as the BU:GU ratio has proven quite useful, and seems to work well for many beer styles. What went wrong? Can we do anything about it?

Attenuation and Real Terminal Extract

The three beers examined in Table 1 all had roughly the same Apparent Attenuation (AA), about 72 percent. We compute AA as:

AA = (OG – FG) / OG (1)

where OG is the Original Extract (starting gravity) and FG is the Apparent Extract (final gravity). Strictly speaking, AA is the ratio of measurements in °Plato, but we use °Oeschle as an approximation.

The BU:GU ratio works quite nicely for beers with similar apparent attenuation ratios. The Dubbel is highly attenuated, so its starting gravity belies its dryness relative to the Bock. Thus, the BU:GU ratio is less useful for beers which are relatively over- or under-attenuated. Can we take attenuation into account?

We refer to the ratio given in the formula above as apparent attenuation because it fails to account for two simple facts:

Some of the extract has been converted into alcohol; and

Alcohol is less dense than water.

We may approximate the Real Terminal Extract (RTE) with a formula by Balling (here, slightly simplified):

RTE = 0.82 x FG + 0.18 x OG (2)

We submit that RTE is a better measure of a beer’s sweetness than its Original Gravity.

A New Measure of Balance

In order to account for differing levels of attenuation, we replace the “GU,” the beer’s original gravity in °Oe, with its Real Terminal Extract, expressed in the same units. For a reason we’ll explain later, let’s multiply the ratio by 0.8. We compute:

BV = 0.8 x BU / RTE (3)

where BV stands for “Balance Value.”

Because the RTE will never be higher than the OG, we shall wind up with a larger quotient. In fact, for beers with typical levels of attenuation, the new formula will give results which will be about twice as high as those given by the old one. This shouldn’t bother us, as long as we recognize that the two formulae will produce numbers on different scales.

How do the beers we’ve already examined rate in terms of their Balance Values? Let’s compute them for our familiar examples as well as for a couple of extra styles (Dortmunder and Pilsener):

Style Bock Dubbel Vienna Dortmunder Pilsener Dry Stout IBU 22 22 22 26 35 40 OG, °Oe 66 66 50 52 50 40 FG, °Oe 18 14 14 14 14 10 RTE, °Oe 26.6 23.4 20.5 20.8 20.5 15.4 BV 0.66 0.75 0.86 1.00 1.37 2.08

Table 2. Balance Values for selected styles.

The bittering and gravity levels are within those specified by the 2001 BJCP guidelines.

Here we see the proper ordinal relationship between the beers, particularly the first three. The Dubbel is essentially midway between the Bock and the Vienna in terms of its balance. This seems quite reasonable.

Why the factor of 0.8? With it, the Dortmunder, a beer which belongs to a style which the Beer Judge Certification Program (BJCP) Style Guidelines claim is balanced, has a Balance Value of one. Sweet styles will have lower Balance Values, and more bitter styles will have higher BVs. Here, we see a BV of 0.86 for the Vienna, which is slightly sweet, and a BV of 1.37 for the Pilsener, which is bitter. Not only does the 0.8 factor make the new values about double those produced by the old formula for beers with typical attenuation levels, but it places what many consider to be the line of demarcation between sweet and bitter very close to unity. Handy!

Using the Balance Value

There are a number of ways in which the Balance Value may be applied. One way is helpful in beer appreciation. Compare the Balance Values for several styles of beer, using the midpoints of the ranges given for OG, FG, and IBUs for styles in the BJCP style guidelines. (These values appear in Table 3 below for most styles.) Compute the Balance Values for commercial examples when the necessary parameters are given, and taste the beers. See how the Balance Value tracks what you perceive as the balance between sweetness and bitterness.

Style Mid

BV Style Mid

BV 1. AMERICAN LAGER 11. ENGLISH AND SCOTTISH STRONG ALE 1A. Light/Standard/Premium 1.02 11A. Old Ale 1.26 1B. Dark 0.70 11B. Strong Scotch Ale (Wee Heavy) 0.74 1C. Classic American Pilsner 1.33 12. BARLEYWINE AND IMPERIAL STOUT 2. EUROPEAN PALE LAGER 12A. English-style Barleywine 1.56 2A. Bohemian Pilsner 1.50 12B. American-Style Barleywine 1.56 2B. Northern German Pilsner 1.64 12C. Russian Imperial Stout 1.60 2C. Dortmunder Export 1.06 13. EUROPEAN DARK LAGER 2D. Münchner Helles 0.83 13A. Munich Dunkel 0.90 3. LIGHT ALE 13B. Schwarzbier (Black Beer) 1.23 3A. Blond Ale 1.02 14. BOCK 3B. American Wheat 0.91 14A. Traditional Bock 0.85 3C. Cream Ale 0.81 14B. Helles Bock/Maibock 0.88 4. BITTER AND ENGLISH PALE ALE 14C. Doppelbock 0.65 4A. Ordinary Bitter 1.63 15. PORTER 4B. Special or Best Bitter 1.53 15A. Robust Porter 1.28 4C. Strong Bitter/English Pale Ale 1.67 15B. Brown Porter 1.17 5. SCOTTISH ALES 16. STOUT 5A. Light 60/- 0.63 16A. Dry Stout 2.13 5B. Heavy 70/- 0.81 16B. Sweet Stout 1.08 5C. Export 80/- 1.00 16C. Oatmeal Stout 1.40 6. AMERICAN PALE ALES 16D. Foreign Extra Stout 1.88 6A. American Pale Ale 1.24 17. WHEAT BEER 6B. American Amber Ale 1.24 17A. Bavarian Weizen 0.65 6C. California Common Beer 1.67 17B. Bavarian Dunkelweizen 0.65 7. India Pale Ale 1.76 17D. Weizenbock 0.64 8. KÖLSCH AND ALTBIER 18. STRONG BELGIAN ALE 8A. Kölsch-Style Ale 1.11 18A. Dubbel 0.95 8B. Düsseldorf Altbier 1.88 18B. Tripel 0.79 8C. Northern German Altbier 1.22 18C. Belgian Strong Golden Ale 0.89 9. GERMAN AMBER LAGER 18D. Belgian Strong Dark Ale 0.86 9A. Oktoberfest/Märzen 0.92 19. BELGIAN AND FRENCH ALE 9B. Vienna Lager 1.03 19A. Belgian Pale Ale 1.28 10. BROWN ALE 19B. Witbier 0.87 10A. Mild 0.81 19C. Biere de Garde 0.80 10B. Northern English Brown Ale 1.03 19D. Saison 1.16 10C. Southern English Brown 0.85 19E. Belgian Specialty Ale 1.22 10D. American Brown Ale 1.69 23A. Classic Rauchbier 0.92

Table 3. Midpoint Balance Values for Most Styles in the 2001 BJCP Style Guidelines.

Styles which are usually soured and/or unhopped do not appear.

Another application is in recipe formulation. When formulating a recipe, one usually starts with a target Original Gravity. If you know what apparent attenuation to expect, you can predict the Real Terminal Extract:

RTE = OG x [1 – 0.82 x AA] (4)

where AA is the Apparent Attenuation, expressed as a decimal fraction. Typical values are 0.70-0.76, depending on the yeast strain and the fermentability of your wort, and other factors. If you have a Balance Value in mind (based, perhaps, on a commercial example you like, or computed from the mid-range of the parameters of the BJCP style guideline for the style you’re brewing), you can compute a target Bitterness Level, in IBUs:

BU = 1.25 x BV x RTE (5)

Equations (4) and (5) may be combined:

BU = 1.25 x BV x OG x [1 – 0.82 x AA] (6)

You can then compute hopping rates according to your favorite method.

An Example

Let’s use Equation (6) to determine the bittering level of a Northern Brown Ale. We want an OG of 1.046 (46 °Oe), and an apparent attenuation of 0.74. The BJCP Style Guidelines say that a Northern Brown should possess a “Gentle to moderate sweetness . . . . Balance is nearly even.” We want something that is leaning just a tiny bit towards sweet. Let’s use a Balance Value of 0.97, which should be “nearly even” in balance. Plugging these figures into Equation (6) yields:

BU = 1.25 x 0.97 x 46 x [1 – 0.82 x 0.74] = 1.25 x 0.97 x 46 x 0.3932 = 21.9

We would round this to 22 IBU, which we see is near the middle of the range of 15 to 30 IBU.

Where the Balance Value Can Fall Short

There are many factors which influence how a person will perceive the balance of a beer. Some are specific to each person, and the new formula can’t address them. Nor can any other relatively simple formula, so let’s not let that bother us.

Some beer-dependent factors which can influence the perceived balance, besides Real Terminal Extract and the Bittering Level, include:

Ester Profile: Fruity esters can enhance apparent sweetness

Sourness: Beers which are soured may taste less sweet to some people

Roastiness: Roast grains lend a bitterness of their own

Phenol Concentration: Phenols can be perceived as bitter

Terminal Extract Profile: Not all terminal extact is sweet. Proteins and long dextrins are essentially flavorless; simpler non-fermentables may be quite sweet

The old formula didn’t take these factors into consideration either, so we’re no worse off than we were before. However, Michael Lewis, in Stout, argues that experienced tasters can distinguish between various flavors in beers, even when they are similar. That’s a point in favor of a simple approach like that outlined here (as well as the earlier BU:GU ratio).

Weber and Fechner Were Here (and Stevens, too!)

Weber was a psychologist who stated that a Just Noticible Difference (JND) in the level of a stimulus (i.e., the amount of change necessary for a person to tend to notice the change) was proportional to the level of that stimulus. For example, if an audio stimulus is presented to a listener at a level of 10 milliwatts per square meter, and they do not notice a change in level until the stimulus is increased to 11 milliwatts per square meter (the JND is here 1 milliwatt per square meter, or 10 percent of the original stimulus), Weber’s Law would posit that the same listener would require an increase from 40 to 44 milliwatts per square meter (the JND is here 4 milliwatts per square meter; again, 10 percent) or a decrease from 20 to 18 milliwatts per square meter (a JND of, yet again, 10 percent) before noticing a difference in level. Presumably, this would also apply to stimuli like sweetness and bitterness.

Fechner was another psychologist who argued that a scale on which the unit is the JND would track human perception. Based on Weber’s Law, this implies a logarithmic relationship between the physical level of the stimulus and the perceived level. Indeed, the decibel scale uses just such a relationship for sound level/loudness. Psychologists refer to the combined law as the Weber-Fechner Law.

What does that mean for the Balance Value? If we take the logarithm of the numerator (log [0.8 x BU]), we obtain something we can expect to perceptually track bitterness. If we take the logarithm of the RTE, which appears in the denominator, we would expect that to track perceived sweetness. If we subtract one from the other, we would obtain something we would expect to track the perception on Balance.

Of course, the logarithm of the quotient of two numbers is simply the difference of their logarithms. So, one would expect the logarithm of the Balance Value to be a perceptual scale of beer balance. “Balanced” beers would tend to have log Balance Values close to zero. Sweet beers would tend to have negative log Balance Values, and bitter beers would have positive log Balance Values.

Did Weber and Fechner have the last say regarding the relationship between physical quantities and how those quantities are perceived? Not nearly. A later (a century later, in fact) psychologist, Stevens, posited a power-law relationship between physical stimulus level and perception. Under Stevens’s Law, one would determine the correct exponents for the bitterness and sweetness components (they may be different), apply them to each part individually, and subtract the two parts. Not as neat as using Weber and Fechner, but Stevens knew what he was talking about, too. Still, I think I’ll stick with the simpler Weber-Fechner law here, until I see some actual data which contradicts it.

In Conclusion

A new method of computing a beer’s balance between sweetness and bitterness was presented. It accounts for attenuation, as well as the beer’s original gravity and bittering level. The new value is referred to as the Balance Value, and assumes values close to unity for beers which many regard as “balanced.” It is computed from the beer’s original and final gravities, expressed in °Oeschle (“Gravity Units”), and the beer’s bittering level, expressed in International Bittering Units (IBUs).

Practical applications of the new quantity were briefly discussed. The metric’s use in determining a bittering level in recipe formulation was illustated through an example.

The new value does not take into consideration complex factors, such as ester levels and types, roastiness, phenol concentration, and how the terminal extract is distributed between sweet and flavorless components. Therefore, it cannot be expected to perform perfectly. Rather, it is offered as an improvement over an existing formula which has proven quite useful. Hopefully, the new value will prove even more useful.

Finally, it was suggested that the logarithm of the Bitterness Value would better track the perceived balance. This has the advantage of placing the balance point at or near zero, with sweet beers having negative log Balance Value, and bitter beers having positive log Balance Values.

A Final Remark

We think that the BU:GU ratio is a useful tool. Like nearly any simple formula not derived from first principles, it has numerous limitations. Yet, in spite of these, it has served the craft- and homebrewing communities well. Our aim here was to provide an alternative which remains almost as simple, but is more robust (and, hence, hopefully even more useful), not to rip apart someone else’s suggestion. We’re building on someone else’s suggestion. And that’s how progress is made.

References

Ray Daniels, Designing Great Beers. Boulder, Co: Brewers Publications, 1996, p. 126. BJCP Beer Style Committee: Bruce Brode, Steve Casselman, Tim Dawson, Peter Garofalo, Bryan Gros, Bob Hall, David Houseman, Al Korzonas, Martin Lodahl, Craig Pepin, Bob Rogers, Beer Judge Certification Program (BJCP) Guide to Beer Styles For Home Brew Beer Competitions. Available at: http://www.bjcp.org/style-index.html, visited 2003-12-02. Michael Lewis, Stout. Boulder, CO: Brewers Publications, 1995, p. 71-73. Gustav T Fechner, Elemente der Psychophysik. Leipzig: Breitkopf und Härtel, 1860. (Available translated as: Elements of Psychophysics New York: Holt, Rinehart and Winston, 1966.) S S Stevens, To honor Fechner and repeal his law, Science, 1961. 133, p. 80-86.

Copyright ©2003-2004, J A S Viggiano. All Rights Reserved.