Colin Wright pointed out a pattern in my previous post that I hadn’t seen before. He wrote about it here a couple years ago.

Start with the powers of 2 from the top of the post:

2¹ = 2
2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32
2⁶ = 64
2⁷ = 128
2⁸ = 256
2⁹ = 512

and sort the numbers on the right hand side in lexical order, i.e. sort them as you’d sort words. This is almost never what you want to do, but here it is [1].

128
16
2
256
32
4
512
64
8

Next, add a decimal point after the first digit in each number.

1.28
1.6
2
2.56
3.2
4
5.12
6.4
8

Now compare the decibel values listed at the bottom of the previous post.

1.25
1.6
2
2.5
3.2
4
5
6.3
8

Five of the numbers are the same and the remaining 4 are close. And where the numbers differ, the exact decibel value is between the the two approximate values.

Here’s a table to compare the exact value rounded to 3 decimals, the approximation given before, and the value obtained by sorting powers of 2 and moving the decimal.

    |---+-----------+--------+-------|
    | n | 10^(n/10) | approx | power |
    |---+-----------+--------+-------|
    | 1 |     1.259 |   1.25 |  1.28 |
    | 2 |     1.585 |   1.60 |  1.60 |
    | 3 |     1.995 |   2.00 |  2.00 |
    | 4 |     2.512 |   2.50 |  2.56 |
    | 5 |     3.162 |   3.20 |  3.20 |
    | 6 |     3.981 |   4.00 |  4.00 |
    | 7 |     5.012 |   5.00 |  5.12 |
    | 8 |     6.310 |   6.30 |  6.40 |
    | 9 |     7.943 |   8.00 |  8.00 |
    |---+-----------+--------+-------|

So the numbers at the top and bottom of my list are practically the same, but in a different order.

Related post: 2s, 5s, and decibels.

[1] One reason I use ISO dates (YYYY-MM-DD) in my personal work is that that lexical order equals chronological order. Otherwise you can get weird things like December coming before March because D comes before M or because 1 (as in 12) comes before 3. Using year-month-day and padding days and months with zeros as needed eliminates this problem.

When you see the word “almost” in a mathematical context, it might be used informally, but often it has a precise meaning. I wrote about this before in the post Common words that have a technical meaning.

Often the technical meaning of “almost” is “within any finite tolerance.” That’s how it is used in the context of almost periodic functions.

A function f is periodic with period T if for all x,

f(x + T) = f(x)

For example, the sine function is periodic with period 2π.

So what does almost periodic mean? It means that for any ε > 0, there exists a T > 0 such that

|f(x + T) − f(x)| < ε

for all x. Note that the value of T depends on the value of ε. You tell me your tolerance for “almost” and in theory I could hand you back T that meets your tolerance.

That’s in theory. Can we actually do it in practice? Let’s consider

f(t) = sin(2παt) + sin(2πβt)

where α/β is irrational. In the previous post we had α = 1/log 2 and β = 1/log 5.

The Hurwitz approximation theorem says that because α/β is irrational, there are infinitely many integers p and q such that

|α/β – p/q| < 1/(√5q²).

To put it another way, we can find p and q such that

|qα – pβ| < β/√5q,

i.e. we can find p and q such that qα and pβ are close together as we wish by looking for a large enough q, and Hurwitz promises us that we can find a q as large as we want.

If we let T = q/α then the first component of f is exactly periodic, i.e.

sin(2πα(t + q/α) = sin(2παt).

With a little effort we can show that

sin(2πβ(t + q/α)) = sin(2πβt + 2π(qβ − pα)t/α)

and we said above we can make

qβ − pα

as small as we like by choosing a large enough q in Hurwitz’ theorem. And so we can choose q large enough that

sin(2πβ(t + q/α)) − sin(2πβt)

is uniformly as small as we’d like.

Last night I ran across a delightful Twitter thread by Nathan Mishra-Linger that starts out by saying

It’s nice that 2^7 ≈ 5^3, because that lets you approximate any integer valued decibel quantity as a product of powers of 2 and 5.

The observation that

2⁷ ≈ 5³,

is more familiar if you multiply both sides by 2³. That is,

2¹⁰ ≈ 10³

This is the basis of using “kilo” to mean both 1000 and 1024 in different contexts. More on that here.

Decibel approximations

Nathan points out that this happy coincidence means that small integer decibels can be written as small powers of 2 and 5. Here’s a table based on one of the tweets in Nathan’s thread.

0 dB = 10^0.0 = 2⁰5⁰
1 dB = 10^0.1 ≈ 2⁻²5¹
2 dB = 10^0.2 ≈ 2³5⁻¹
3 dB = 10^0.3 ≈ 2¹5⁰
4 dB = 10^0.4 ≈ 2⁻¹5¹
5 dB = 10^0.5 ≈ 2⁻³5²
6 dB = 10^0.6 ≈ 2²5⁰
7 dB = 10^0.7 ≈ 2⁰5¹
8 dB = 10^0.8 ≈ 2⁵5⁻¹
9 dB = 10^0.9 ≈ 2³5⁰

You can show that the maximum relative error above is 1.4% with the following Python code.

    approx = [
        2** 0 * 5** 0,
        2**-2 * 5** 1,
        2** 3 * 5**-1,
        2** 1 * 5** 0,
        2**-1 * 5** 1,
        2**-3 * 5** 2,
        2** 2 * 5** 0,
        2** 0 * 5** 1,
        2** 5 * 5**-1,
        2** 3 * 5** 0,
    ]
    
    err = [abs(1 - approx[n]/10**(n/10)) for n in range(10)]
    print(max(err))

Almost in step

The last tweet in Nathan’s thread is a diagram showing how powers of 2 and 5 sort of cycle. Here’s another to look at that.

Taking logs of both sides of

2⁷ ≈ 5³,

shows that

7 log 2 ≈ 3 log 5.

Suppose two people start out walking together, one taking lots of small steps and the other taking fewer larger steps. Specifically, one takes 7 steps of size log 2 in the time that it takes the other to make 3 step of size log 5. Then periodically they’ll almost be in step.

This reminded me of my recent thread about how Earth and Mars line up approximately every 17 years.

Suppose you have two planets orbiting a star. One takes time log 2 to orbit, and another one, further from the star, takes time log 5 to orbit. Then after the inner planet takes 7 orbits and the outer planet takes 3, the two planets will be approximately in a straight line with their star.

Plots

One last observation. Above I said “periodically they’ll almost be in step.” There’s a rigorous way to use the words “almost” and “periodic.” The sum of a sine wave of period log 2 and a sine wave of period log 5 is an “almost periodic” function. See my next post for the rigorous definition of an almost periodic function, but for now I’ll just include a few plots that the function described in this paragraph is “almost periodic” in the colloquial sense.

Here’s a plot of two sine waves, one with 7 periods of length log 2 and the other with 3 periods of length 5.

    Plot[{Sin[2 Pi t/Log[2]], Sin[2 Pi t/Log[5]]}, {t, 0, 7 Log[2] }]

Here’s their sum.

    Plot[Sin[2 Pi t/Log[2]] + Sin[2 Pi t/Log[5]], {t, 0, 7 Log[2] }]

And here’s two approximate periods of the sum.

    Plot[Sin[2 Pi t/Log[2]] + Sin[2 Pi t/Log[5]], {t, 0, 14 Log[2] }]

Related posts

• Decibels are logs base 10^0.1
• Circle of fifths and roots of 2