# Why are 1/3 stop apertures uneven numbers apart?

Why do 1/3 stop apertures go like 8, 9, 10, 11, 13, 14, 16, 18?

There's a difference of 2 between 11 and 13, it goes back to 1 between 13 and 14, and it goes back up to 2.

• Related: Why are f-stops not linear?
– scottbb
Jul 30, 2018 at 1:52
• Related: What does f-stop mean?, especially the accepted answer
– scottbb
Jul 30, 2018 at 1:53
• – scottbb
Jul 30, 2018 at 1:53
• Jul 30, 2018 at 4:21

For f/stops, there is a precise multiplied difference of 1.122462 X intervals (cube root of √2) between all third stops. The precise third stops are actually numbers like 8.98 or 10.08. My meaning of the Precise Numbers is of course the theoretical precise goal numbers that the camera designer certainly aims for. There can be no question about those (even if the physical camera mechanisms may not necessarily be precisely accurate to as many decimal places). But the nominal numbers that are marked and shown are arbitrarily rounded to numbers like 9 or 10, but the camera and lens design tries to actually compute with the actual precise values.

Precise Nominal Stop
8 8 Full
8.98 9
10.08 10
11.31 11 Full
12.7 13
14.25 14
16 16 Full

The same concept (of there being precise and nominal values) is true of f/stops, shutter speeds, and ISO. For shutter speed and ISO, then thirds are 1.259921 X intervals (∛2).

These are valid results, but not the fundamental definition, and complete detail is shown at my site at https://www.scantips.com/lights/fstop2.html

• Note the reason apertures use √2 and not 2 like shutter speed and ISO - the light admitted by the aperture is proportional to the area not the diameter, and the area is proportional to the square of the diameter. Double the diameter, and you've made a 2 stop change - as shown by your table going from F/16 to F/8. Jan 15, 2022 at 21:43

Whole f-numbers are an expression of the powers of the square root of two (√2). Every odd-numbered or fractional power of the square root of two is a non-integer with an endless number of places to the right of the decimal. Such a number is defined as an irrational number. In photography we round the actual values of many irrational numbers to a simpler number.

Note the "basic" whole stop f-number scale:

1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45, 64, 90, etc.

Every other value in the list is an irrational number based on the square root of two (√2) that has been rounded to two significant digits. Taken to twenty (20) significant digits, √2 is 1.4142135623730950488...

Eleven (11) is not exactly twice five and six-tenths (5.6), even though the actual powers of the square root of two we represent using f/5.6 and f/11 to represent them are: taken to 14 decimal places they are f/5.65685424949238 and f/11.31370849898476, respectively.

f/1.4 is a rounded version of √2 and so are all of the other f-stops that include odd-numbered powers of the √2: f/2.8, 5.6, 11, 22, etc. are actually (carried out to 16 significant digits) f/2.828427124746919, 5.65685424949238, 11.31370849898476, 22.62741699796952, 45.25483399593904, 90.50966799187808, etc.

Notice that f/5.6 actually rounds closer to f/5.7, f/22 actually rounds closer to f/23, and f/90 actually rounds closer to f/91. We use f/5.6 instead of f/5.7 because when we double 2.8 (the number we use to approximate 2.828427124746919...) we get 5.6. We use f/22 instead of f/23 because when we double 11 (the number we use to approximate 11.31370849898476) we get 22. We use f/45 instead of f/44, which would be the doubling of 22, because the 'actual' f/45 rounds closer to 45 than to 44, and even though 22 doubled is 44, 45 is a "rounder" number. These differences are totally insignificant because all but the most precise laboratory grade lenses can't control the aperture precisely enough to create that small of a difference anyway.

For non-laboratory grade cameras that allow one-third (1/3) stop settings, anything within one-sixth (1/6) stop of the actual target number is considered acceptable. Back in the film days when cameras only allowed full-stop setting of aperture and shutter times, anything within one-half (1/2) stop was considered accurate enough.

With 1/2 stop, 1/3 stop, 1/4 stop, or even more precise f-numbers all except every other whole f-number (1, 2, 4, 8, 16, 32, etc.) are irrational numbers with unending numbers of digits past the decimal. For values above eight (8), we round them to more or less the nearest whole number or integer, e.g. f/11, f/13, f/14, etc.. For values less than eight, we round them to the first significant digit to the right of the decimal, e.g f/1.4, f/6.3, f/7.2. In other words, most f-numbers that are not exact integers are rounded to two significant digits if they are not rounded even further to another number, such as f/22 for f/22.6274... and f/90 for f/90.5096... because they are twice the rounded values of f/11 and f/45.

There's a difference of 2 between 11 and 13, it goes back to 1 between 13 and 14, and it goes back up to 2!

In the specific case of the one-third (1/3) stop f-numbers between f/11 and f/16 the disparity you have observed is due to the vagarities of the rounding being used.

f/11 is ≈ f/11.313708...
f/13 is ≈ f/12.697741...
f/14 is ≈ f/14.254544...
f/16 is actually f/16

It is also the case that sometimes the same rounded numbers are used for slightly different target values when one is a 1/3 stop value and the other is a half-stop or quarter-stop value. For example, both the quarter-stop above f/2 and the third-stop above f/2 are both notated as f/2.2, even though the two target numbers are different (f/2.1818 and f/2.2449, respectively), or the one-third stop above f/11 and the one-half stop above f/11 are both notated as f/13, even though the two target numbers (f/12.6977 and f/13.4543, respectively) are different.

• One expects the % change for 1 full stop to be 100% when you open up and 50% when you stop down. This you get but -- 1/2 f-stop change is only 41% up 29% down instead of expected 50% / 25%. 1/3 f-stop 26% change per increment when you open up and 21% when you close down. Strange but true! Jul 30, 2018 at 15:30
• @AlanMarcus It's not strange at all. The scale is logarithmic, not linear. Jul 30, 2018 at 15:31
• Let's say "counterintuitive", then. Most people aren't used to thinking that way. Jul 31, 2018 at 6:54
• Most people have trouble understanding why one must increase 100 by 50% to get 150, but one must then decrease 150 by 33% to get back to 100. It is because they have no conceptual grasp of fractions and the relationship between the reciprocals 3/2 and 2/3. That doesn't mean it is strange. It just means we have to learn multiplication and division as well as addition and subtraction. Exponential/logarithmic functions are the next evolutionary steps of mathematics past multiplication/division. That doesn't make logarithmic sequences strange. They're still a basic part of number theory. Jul 31, 2018 at 7:07
• If one looks at the 'C' and 'D' scale on a slide rule, understanding why '5' isn't exactly halfway between '1' and '10' should be easily intuitive. Jul 31, 2018 at 7:17

No question, the f-number sequence seems weird!. The 1/3 f-stop number set might not appear so weird if you were dealing with money. Suppose you have one dollar to invest at the bank and the they promise that after three compounding periods your money will double. Further, if you keep the principal and interest in the bank, the money will continue to double after each third period. In other words the 1/3 f-number sequence progresses identically as such a compound money number set.

$1.00$1.26 $1.59$2.00 $2.52$3.17 $4.00$5.04 $6.35$8.00 $10.08$12.70 $16.00$20.16 $25.40$32.00 $40.32$50.79 $64.00 A tip of the hat to WayneF I used 1/2 f-stop set not 1/3 f-stop set: Let's use the sixth root of 2 -- note the f-number doubles each third period. I always said I am full of gobbledygook!$1.00 $1.12$1.26 $1.41$1.59 $1.78$2.00 $2.24$2.52 $2.83$3.17 $3.56$4.00 $4.49$5.04 $5.66$6.35 $7.13$8.00 $8.98$10.08 $11.31$12.70 $14.25$16.00 $17.96$20.16 $22.63$25.40 $28.51$32.00 $35.92$40.32 $45.25$50.80 $57.02$64.00

• FWIW, f/1 to f/2 or f/2 to f/4 etc is two stops, not one. And you said f-number, but are using shutter speed steps of ∛2 instead of ∛1.414. Edit would be more clear. Jul 30, 2018 at 0:24

Factors for computing fractional f-stop numbers

Full stop: Square root(2) = 1.4142135623731

Half stop: square root(square root(2)) = fourth root(2) = 1.18920711500272

Third stop: third root(square root(2)) = sixth root(2) = 1.12246204830937

Tenth stop: tenth root(square Root(2)) = twentiest root(2) = 1.03526492384138

• This doesn't answer the question. It states what the multiplicative factors are, but the question is why are the factors the way they are.
– scottbb
Nov 27 at 1:40