The 77% of the pH drop that is 31% of the acid

Dark navy briefing slide, masthead “Process control briefing · spontaneous sauerkraut fermentation · 15 °C · 42 days” on the left and “pH · 13 sampling points · measured in triplicate” on the right, headlined “The kraut was made in week one.” A standfirst says there are thirteen sampling points, one line and every reading on the page; that the figures are the laboratory’s own, from a spontaneous sauerkraut fermentation held at 15 °C for 42 days with pH measured in triplicate at each point; that nothing is smoothed, interpolated, indexed or rebased, no point dropped and none averaged into a bucket; that the days are plotted where they fall so a week of calendar occupies a week of chart, and the axis is linear, evenly ticked and unbroken. It then reads off the chart: pH falls from 5.87 to 3.93 in the first seven days — 1.94 of the 2.51-point drop, 77.3% of it — and then goes flat, moving 0.57 of a point over the remaining five weeks, of which four — days 14 to 42 — are worth 0.38 of a point between them. The main panel, headed “The fall in pH, day 0 to day 42”, subtitled “All 13 sampling points · linear time axis, days plotted where they fall · linear pH axis, 3.0 to 6.0 in even half-point steps · each point the mean of three replicates”, is a gold line chart with a marked point at each of the thirteen samples: 5.87 at day 0, 5.76, 5.71 and 5.70 through day 2, then a steep fall through 5.15, 4.69 and 4.36 to 3.93 at day 7, after which the line runs almost horizontally through 3.74, 3.67, 3.54 and 3.49 to end at 3.36 on day 42. A dashed vertical marks day 7. A gold dashed rule at 5.87 runs from day 0 to day 7 under the label “Week one: −1.94 points · 77.3% of the whole drop”, and a pale dashed rule at 3.93 runs from day 7 to day 42 under “Days 7–42: −0.57 of a point”. Six cards run down the right. “The drop, in one figure” prints 1.94 in large type over a note that this is the points of pH lost in the first seven days out of a 2.51-point fall across the whole 42 — 77.3% of the total — and that the remaining five weeks are worth 0.57 of a point between them. “Read-out for the production committee” says the curve is essentially flat from day 7, that every week after it buys a fraction of what the first week bought while keeping a tank, a headspace and a cold room out of service, and that days 14 to 42 are worth 0.38 of a point or 15.1% of the drop — recommending the standard cycle be cut from 42 days to 14 and the capacity re-used. “Schedule options, in the order the chart gives them” tabulates six pull points with their pH and cumulative share of the drop: day 7 at 3.93 and 77.3%, day 14 at 3.74 and 84.9%, day 21 at 3.67 and 87.6%, day 28 at 3.54 and 92.8%, day 35 at 3.49 and 94.8%, day 42 at 3.36 and 100.0%. “Why the axis starts at 3.0” explains that pH has no zero to start from — pH 0 is one mole of hydrogen ion per litre and the scale runs negative below it — so the axis is drawn across the range the readings occupy, in even half-point steps, with no break and no cap. “The 13 sampling points, as measured” lists every reading in two columns. “How to read the chart” says to follow the line left to right, that steeper means the reading moved faster, and that the gap between two points read off the axis is the change in pH between them. A final card gives the basis of measurement: Tlais, Lemos Junior, Filannino, Campanaro, Gobbetti and Di Cagno, Microbiology Spectrum 10(4):e00168-22 (2022), Table 1, with pH values the means of three replicates. A method line runs along the foot: bench pH meter, three replicates per sampling point, means plotted; 13 points across 42 days; linear time axis; linear pH axis in even half-point steps; no smoothing, no interpolation, no exclusions. A footnote credits the source, notes all 13 pH readings are as published, and says the slide is a demonstration drawn by us in the manner of a process control briefing — the committee, the read-out and the schedule options are invented, and every figure behind them is real. — Showing the misleading chart

The same readings on a cream background, kicked “The same 13 readings · four measures · four zero-based axes · nothing added, nothing dropped” and headlined “The first week is 77% of the pH drop and 31% of the acid.” A standfirst says nothing on the briefing was cropped, truncated, smoothed or rescaled and every reading printed on it is correct; that what sets the picture is that pH is not a quantity but minus the base-ten logarithm of one, so a perfectly ordinary linear axis with evenly spaced ticks is already spacing equal ratios equally and every gap the eye measures is a multiplication rather than an amount; that un-logged, the same 42 days run the other way, with 10 to the minus pH climbing from 1.35 to 436.5 µmol/L, a 324-fold rise, of which the µmol/L added in the dramatic first week is 26.7% and in the flat tail 73.3%; that this is the same thirteen numbers un-logged rather than a second opinion, so what settles it is the pair the paper measured independently, in units that are ordinary numbers, both of which land far from the pH figure; and that it is not a truncation either, since widening the axis changes the steepness and nothing else that matters — on any pH axis there is, from 3.0 or from zero, week one is still 77.3% of the fall. The main panel, headed “One ferment, four rulers” and subtitled “Same 13 sampling points, same 42 days, same samples · every panel zero-based · dashed line marks day 7 · x-axis in days”, holds four small line charts side by side, each with a dashed vertical at day 7. Panel 1, “pH, as the briefing drew it”, is drawn from zero to 6.2 in grey-brown: the line drops from 5.87 to 3.93 and then runs flat to 3.36, with a note that it is drawn from zero rather than from 3.0, that widening the axis flattens the plunge and changes nothing else, and that the first 7 days are 77.3% of the fall on every pH axis there is. Panel 2, “The pH, un-logged”, labelled 10 to the minus pH as µmol/L on a zero-based axis to 450 and drawn in gold, is not a second measurement but the same 13 numbers with the logarithm taken off: barely off the floor through day 5, 117.5 at day 7, then climbing steadily through 182.0, 213.8, 288.4 and 323.6 to 436.5 at day 42 — a 324-fold rise, of which the first 7 days are 26.7% of the µmol/L added. Panel 3, “Titratable acid, measured”, in mL 0.1 M NaOH per 10 g dry weight on a zero-based axis to 18 and in blue, is near-flat to day 3 and then climbs almost straight from 0.91 to 16.9 — 0.091 to 1.69 mmol — with the first 7 days 31.2% of it. Panel 4, “Lactic acid, measured”, in mg/g DW on a zero-based axis to 220 and in green, runs from 0.36 to 199.4 with the day-28 dip left unsmoothed, the first 7 days 41.2%. Three panels sit below. “The same acid, two different drops” takes two steps from the paper’s own table that added almost the same amount of acid and tabulates them: day 3 to day 4, one day, +1.30 mL of titratable acid, 0.46 of a point on the pH chart; day 28 to day 35, seven days, +1.25 mL, 0.05 of a point; ratio 1.04× of acid against 9.2× of drama. It notes that measured as free hydrogen ion the later step is the bigger of the two — 35.2 µmol/L against 13.3 — while moving the line one ninth as far, that the spans differ and the point is the vertical distance rather than the rate, and that the eye reads the steep part as the part where things happened when the steep part is only where the ratio moved fastest, which early in a ferment is when there was almost no acid to multiply. “What the read-out got wrong” says the briefing proposed cutting the cycle from 42 days to 14 on the grounds that the last four weeks are worth 0.38 of a point or 15.1% of the drop, that the figure is right, and that it is the only one of the four measures that calls the tail small: days 14 to 42 are 15.1% of the change in pH points, 51.0% of the titratable acid, 40.6% of the lactic acid and 58.5% of the hydrogen-ion activity — so half the acid in the jar arrives after the day the read-out picks and better than two-thirds of it after day 7, where the line goes flat, and kraut pulled at day 14 is a different food from kraut pulled at day 42. “What one point of pH costs” draws four evenly spaced pH ticks as gold bars on one linear µmol/L scale, each labelled ×10 above the next: pH 6.0 at 1 µmol/L, pH 5.0 at 10, pH 4.0 at 100 and pH 3.0 at 1 000, with a note that the pH 4.0 bar is a tenth of the one below it while the two above are drawn at a 3.5-pixel minimum, since at that scale they come to half a pixel and a twentieth of one. Six cards run down the right. “Four measures of one ferment” tabulates the share of each measure’s total change falling in days 0–7 and days 7–42: pH 77.3% and 22.7%, the same pH un-logged 26.7% and 73.3%, titratable acid measured 31.2% and 68.8%, lactic acid measured 41.2% and 58.8% — with a note that the second row is the first with the logarithm taken off rather than a fresh reading, and that the two independent measurements under it both land nearer to it than to the pH figure. “A mean pH is not a mean acidity” notes the 13 readings average 4.54, while averaging the same readings un-logged and converting back gives 3.90, or 4.34 times as much acid, because adding logs multiplies the quantities — so a mean of pH readings is the geometric mean of the acid and a gap in points is a ratio rather than an amount; it adds that this is an illustration of the gap rather than a time-average, the 13 points being unevenly spaced, and that averaging pH is right where the paper does it, over three replicate readings of one sample. “The minus sign, while we are here” notes that pH is minus the log so the line falls as the acid rises, and that stellar magnitude does the same. “Other units with the log already inside” tabulates what one step buys: pH ×10 in hydrogen-ion activity, a decibel ×1.259 in power with 10 dB being ×10, moment magnitude ×31.6 in radiated energy, stellar magnitude ×2.512 in brightness and backwards, an f-stop ×2 in light gathered, a semitone ×1.0595 in frequency. “What pH is still the right unit for” says thresholds, which is a real job: a meter reads pH directly and 4.6 is where the rule for acidified foods is drawn, at the growth limit of proteolytic C. botulinum — but even there inhibition tracks undissociated acid as well as pH, acetic biting harder than lactic at the same reading, so a linear acid measure belongs beside it for how much. A “Basis of measurement” card credits Table 1 of the paper for pH, total titratable acidity and lactic acid at 13 points across 42 days at 15 °C, says panel 2 is 10 to the minus pH computed by us, and notes that pH is an activity so 10 to the minus pH is that activity written as an equivalent concentration, the free-hydrogen-ion concentration being somewhat higher and shifting with the brine’s ionic strength — one more reason to decide on the titrated figure. Five notes run along the foot: “Four measures, four zero-based axes”, “Days plotted where they fall”, “Every reading printed”, “Nothing smoothed or interpolated” and “The log named, not hidden”. — Showing the misleading chart

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A process briefing plots the pH of a sauerkraut fermentation across 42 days: 13 sampling points, days where they fall, no smoothing and no break in the axis. The line plunges 1.94 points in the first week — 77.3% of the whole fall — and then goes flat, so the committee proposes emptying the tanks at day 14. But pH is not a quantity, it is minus the base-ten logarithm of one, so an ordinary linear axis is already spacing equal ratios equally. Un-log the same thirteen numbers and the first week is 26.7% of the rise; the two acid measurements the paper took on the same samples say 31.2% and 41.2%. Better than two-thirds of the acid arrives after day 7, where the line goes flat.

Tlais, Lemos Junior, Filannino, Campanaro, Gobbetti & Di Cagno, “How Microbiome Composition Correlates with Biochemical Changes during Sauerkraut Fermentation: a Focus on Neglected Bacterial Players and Functionalities,” Microbiology Spectrum 10(4):e00168-22 (2022), Table 1. Spontaneous sauerkraut fermentation at 15 °C for 42 days, sampled at 13 points; pH 5.87 at day 0, 3.93 at day 7 and 3.36 at day 42; total titratable acidity 0.91, 5.90 and 16.9 mL 0.1 M NaOH per 10 g dry weight; lactic acid 0.36, 82.4 and 199.4 mg/g DW. Hydrogen-ion activity is 10 to the minus pH, computed by us, 2022Added September 5, 2026

01The claim

Thirteen sampling points, one line, and every reading on the page: the pH of a spontaneous sauerkraut fermentation held at 15 °C for 42 days, measured in triplicate. Nothing is smoothed, interpolated, indexed or rebased. No point has been dropped and none averaged into a bucket. The days are plotted where they fall, so a week of calendar occupies a week of chart. The axis is linear, evenly ticked and unbroken, and it starts at 3.0 for the reason the slide gives plainly — pH has no zero to start from, since pH 0 is one mole of hydrogen ion per litre and the scale runs negative below it. Read off the chart: pH falls from 5.87 to 3.93 in the first seven days, which is 1.94 of the 2.51-point drop, or 77.3% of it, and then goes flat, moving 0.57 of a point across the remaining five weeks. Read-out for the production committee: the fermentation is essentially over by day 7. Days 14 to 42 — four weeks of the schedule — are worth 0.38 of a point, or 15.1% of the drop, and each of those weeks keeps a tank, a headspace and a cold room out of service. Recommendation: cut the standard cycle from 42 days to 14 and re-use the capacity.

02The trick

Everything on that slide is true, and the read-out it produces is wrong, because pH is not a quantity. It is minus the base-ten logarithm of one — of hydrogen-ion activity — so a perfectly ordinary linear axis with evenly spaced ticks is already doing what a log axis does: equal distances on it are equal ratios, and every gap the eye measures is a multiplication rather than an amount. A log axis at least announces itself, with 1, 10 and 100 printed on the ticks. Here the ticks read 6.0, 5.5, 5.0, and there is nothing on the page to give it away. Un-log the same thirteen numbers and the 42 days run the other way. Ten to the minus pH climbs from 1.35 µmol/L at day 0 to 117.5 at day 7 and 436.5 at day 42, a 324-fold rise; of the µmol/L it added, the dramatic first week is 26.7% and the flat tail is 73.3%. That is the same thirteen numbers with the logarithm taken off rather than a second reading, so it settles nothing by itself — but the same paper measured the acid twice more on the same samples, in units that are ordinary numbers, and both of those land far from the pH figure. Total titratable acidity rises from 0.91 to 16.9 mL of 0.1 M NaOH per 10 g dry weight — 0.091 to 1.69 mmol of acid — with 31.2% of it arriving in the first week and 68.8% after. Lactic acid, quantified by HPLC, rises from 0.36 to 199.4 mg/g, 41.2% then and 58.8% after. Two independent measurements, both saying the tail is where the acid is; one logarithm, saying the opposite. The clearest way to see it is a pair of steps from the paper’s own table. Between day 3 and day 4 the ferment gained 1.30 mL of titratable acid and the line fell 0.46 of a point. Between day 28 and day 35 it gained 1.25 mL — 4% less acid — and the line fell 0.05 of a point, one ninth as far; measured as free hydrogen ion the later step is actually the larger of the two, 35.2 µmol/L against 13.3. Same acid, ninefold difference in drama, and the difference is the logarithm. Two smaller consequences ride along, and both are worth naming because they are what makes a logged unit different from a bent axis. The first is what happens when these values go through arithmetic. Adding logs multiplies the quantities, so a mean of pH readings is the geometric mean of the acid rather than the arithmetic one: the thirteen readings average pH 4.54, but average the same readings un-logged and convert back and the answer is 3.90 — 4.34 times as much acid. A gap quoted in “points” is a ratio, not an amount. (Averaging pH is exactly right where this paper does it, over three replicate readings of one sample; what it will not give you is a mean acidity. And the 13 points here are unevenly spaced in time, so the figure illustrates the gap rather than averaging anything over the 42 days.) The second is that widening the axis does not help. A truncated axis is cured by putting the zero back, and here putting it back does flatten the plunge — the week-one drop takes 64.7% of the plot height on a 3.0–6.0 axis, 31.3% on a zero-based one and 13.9% across the full 0–14 — while changing nothing that matters, because on every one of those axes week one is still 77.3% of the fall. The compression lives in the unit rather than in the frame, which is why no axis setting reaches it. And the minus sign at the front means the line falls while the acid rises, an inversion nobody had to draw. What the committee actually decided turns on all of this. Days 14 to 42 are 15.1% of the change in pH points — and 51.0% of the titratable acid, 40.6% of the lactic acid and 58.5% of the hydrogen-ion activity. Half the acid in the jar arrives after the day the read-out picks, and better than two-thirds of it after day 7, where the line goes flat. (Both drawings are ours; the committee, the read-out and the schedule options are invented, and every figure behind them is real.)

03The fix

Plot the quantity the decision is about. The question in front of the committee was how much sourer the kraut gets after day 14, which is a question about an amount, and the amount is not on a pH axis at any range. Put titratable acid on a zero-based linear axis and the answer is on the page in one glance: the line climbs almost straight from day 3 to day 42, and the 51% of it that arrives after day 14 sits entirely inside the stretch the pH chart draws as a horizontal line. The same four panels side by side, sharing an x-axis and each zero-based, are the whole lesson — the change in shape between panel 1 and panels 2 to 4 is not a distortion being introduced, it is the one the logged unit was applying all along. Where a series genuinely spans orders of magnitude and the linear chart collapses into a line along the bottom, that is the honest case for a log axis, which at least prints its decades on the ticks: label it loudly and give it dots rather than bars. Keep pH itself where it earns its keep, because it earns it often. A meter reads it directly, and 4.6 is where the rule for acidified foods is drawn, at the growth limit of proteolytic C. botulinum — a jar either is or is not on the safe side of that line, and a logged unit answers a threshold question perfectly. Worth knowing even there that inhibition tracks undissociated acid as well as pH, which is why acetic bites harder than lactic at the same reading, and one more reason a linear acid measure belongs beside the meter. The distinction to hold on to is between a threshold and an amount: the logged unit says which side of the line, and only the un-logged quantity says how far. So treat the logarithm the way you would treat any other unit and print it — say in the caption what the unit is a logarithm of and what one step of it multiplies, since “one pH point is ten times the acid” is a sentence that costs nothing and changes how every gap on the chart is read. And keep logged values out of a mean, a sum or a growth rate on the way to a slide unless you mean the multiplicative version of it; convert, do the arithmetic, convert back. And where a decision turns on an amount, show both charts and take the number off the linear one.