Keenan Pepper left a comment on my previous post saying that the DTMF tones used by touch tone phones “are actually quite close to 14 equal divisions of the octave (rather than the usual 12).”

Let’s show that this is right using a little Python.

    import numpy as np

    freq = np.array([697, 770, 852, 941, 1209, 1336, 1477])
    lg = np.log2(freq)
    spacing = lg[1:] - lg[:-1]
    print(spacing)
    print([2/14, 2/14, 2/14, 5/14, 2/14, 2/14])

If the tones are evenly spaced on a 14-note scale, we’d expect the logarithms base 2 of the notes to differ by multiples of 1/14.

This produces the following:

    [0.144 0.146 0.143 0.362 0.144 0.145]
    [0.143 0.143 0.143 0.357 0.143 0.143]

By dividing the octave into 14 points the DFMT system largely avoids overlap between the set of tones and the set of their harmonics. However, the first overtone of the first tone (1394 Hz) is kinda close to the fundamental of the 6th tone (1336 Hz).

Related posts

• Circle of fifths and roots of 2
• 53-note scale
• 2s, 5s, and decibels
• What key has 30 sharps?

In music, a perfect fifth is the interval between two notes whose frequencies are in 3:2 ratio. For example, the interval from an A at 440 Hz and an E at 660 Hz is a perfect fifth.

Going up by 12 perfect fifths is very nearly the same as going up 7 octaves. That is,

(3/2)¹² ≈ 2⁷

or in other words,

2^7/12 ≈ 3/2.

This is why equal temperament tuning works. More on that here.

53-note scale

Going back to our first approximation, we can say that (12, 7) is an approximate solution to the equation

(3/2)^x = 2^y.

One could naturally ask whether there are better approximate solutions. One such solution, dating back to 40 BC, came from the Chinese scholar King Fang [1]. He discovered that (53, 31) is a substantially more accurate solution than (12, 7), between 6 and 7 times better.

    >>> 1.5**12/2**7
    1.0136432647705078
    >>> 1.5**53/2**31
    1.0020903140410862

In other words, going up 53 perfect fifths is nearly the same as going up 31 octaves. You could divide the octave into 53 parts using perfect fifths; we will demonstrate this visually below. However, we’re moving more the realm of number theory than practical music theory at this point.

Powers of 2 and 3

We could change our problem slightly, making it mathematically simpler though less obviously related to music, by putting all our powers of 2 on one side and looking for powers of 3 that approximately equal powers of two. That is, we could look at approximate solutions to

3^x = 2^y.

where now x and y have different meanings; our new y is our old y plus our old x. No power of 3 will ever equal a power of 2, but you can find powers of 3 close to powers of 2, making the ratio as small as you’d like.

Ergodic map

Taking the logarithm of both sides in base 2, we recast the problem as looking for integers n such that

log₂(3) n

is approximately an integer.

If we define k = log₂(3), we are looking for n such that kn mod 1, the integer part of kn, is near 0.

The map

n → kn mod 1

is ergodic because k is irrational, and so its image in the unit interval is dense as n goes to infinity. This means we can find solutions as close we’d like to 0. It also means we can find solutions as close as we’d like to any other number in the interval.

Here’s a plot of our map for n up to 12.

Note that the range of the map falls very nearly on the 12 evenly-spaced horizontal lines. This corresponds to the circle of fifths filling in the 12 notes of the chromatic scale.

Now let’s keep going for n up to 53.

Look closely at the bottom of the plot. The graph gets close to 0 at 12, and it gets even closer to 0 at 53. The value of kn mod 1 doesn’t get smaller than the value at n = 53 until n = 359 where the value is about twice as small.

When we go up by 12 perfect fifths, we don’t end up exactly on the note we started on; 12 fifths is a little more than 7 octaves. If we keep going up in fifths we fill in notes a little higher than the original chromatic scale. Here’s a plot for n up to 24, with the values sorted so we can focus on the range rather than the ups and downs of filling in the range.

Every note in the original chromatic scale now has a counterpart that’s somewhere around 25 cents sharp.

If we keep going to n = 53, we fill in the octave more evenly. Here’s a plot of the sorted values.

Related posts

• What key has 30 sharps?
• Listening to golden angles
• What does ergodic mean?
• The mathematics of Deep Note

[1] A. L. Leigh Silver. Some Musico-Mathematical Curiosities. The Mathematical Gazette , Feb., 1964, Vol. 48, No. 363 (Feb., 1964), pp. 1-17.

The Hamming distance between two sequences of symbols is the number of places in which they differ. For example, the Hamming distance between the words “hamming” and “farming” is 2, because the two worlds differ in their first and third letters.

Hamming distance is natural when comparing sequences of bits because bits are either the same or different. But when the sequence of symbols comes from a larger alphabet, Hamming distance may not be the most appropriate metric.

Here “alphabet” is usually used figuratively to mean the set of available symbols, but it could be a literal alphabet. As English words, “hamming” seems closer to “hanning” than to “farming” because m is closer to n, both in the alphabet and phonetically, than it is to f or r. [1]

The Lee distance between two sequences x₁x₂…x_n and y₁y₂…y_n of symbols from an alphabet of size q is defined as

So if we use distance in the English alphabet, the words “hamming” and “hanning” are a Lee distance of 1 + 1 = 2 apart, while “hamming” and “farming” are a Lee distance of 2 + 5 = 7 apart.

Coding theory uses both Hamming distance and Lee distance. In some contexts, it only matters whether symbols are different, and in other contexts it matters how different they are. If q = 2 or 3, Hamming distance and Lee distance coincide. If you’re working over an alphabet of size q > 3 and symbols are more likely to be corrupted into nearby symbols, Lee distance is the appropriate metric. If all corruptions are equally likely, then Hamming distance is more appropriate.

Application to music

Lee distance is natural in music since notes are like integers mod 12. Hence the circle of fifths.

My wife and I were discussing recently which of two songs was in a higher key. My wife is an alto and I’m a baritone, so we prefer lower keys. But if you transpose a song up so much that it’s comfortable to sing an octave lower, that’s good too.

If you’re comfortable singing in the key of C, then the key of D is two half-steps higher. But what about they key of A? You could think of it as 9 half-steps higher, or 3 half-steps lower. In the definition of Lee distance, measured in half-steps, the distance from C to D is

min{2, 12 – 2} = 2,

i.e. you could either go up two half-steps or down 10. Similarly the distance between C and A is

min{9, 12-9} = 3.

So you could think of the left side of the minimum in the definition of Lee distance as going up from x to y and the right side as going down from x to y.

Using Lee distance, the largest interval is the tritone, the interval from C to F#. It’s called the tritone because it is three whole steps. If C is your most comfortable key, F# would be your least comfortable key: the notes are as far away from your range as possible. Any higher up and they’d be closer because you could drop down an octave.

The tritone is like the hands of a clock at 6:00. The hour and minute hands are as far apart as possible. Just before 6:00 the hands are closer together on the left side of the clock and just after they are closer on the right side of the clock.

Related posts

• Circle of fifths and number theory
• Morse code separation
• Introduction to Hamming codes

[1] I bring up “Hanning” because Hamming and Hanning are often confused. In signal processing there is both a Hamming window and a Hanning window. The former is named after Richard Hamming and the latter after Julius von Hann. The name “Hanning window” rather than “Hann window” probably comes from the similarity with the Hamming window.