I ran across a video that played around with the equipentatonic scale [1]. Instead of dividing the octave into 12 equal parts, as is most common in Western music, you divide the octave into 5 equal parts. Each note in the equipentatonic scale has a frequency 21/5 times its predecessor.
The equipentatonic scale is used in several non-Western music systems. For example, the Javanese slendro tuning system is equipentatonic, as is Babanda music from Uganda.
In the key of C, the nearest approximants of the notes in the equipentatonic scale are C, D, F, G, A#. 
In the image above [2], the D is denoted as half sharp, 50 cents higher than D. (A cent is 1/100 of a half step.) The actual pitch is a D plus 40 cents, so the half sharp is closer, but still not exactly right.
The F should be 20 cents lower, the G should be 20 cents higher, and the A# should be 10 cents higher.
Notation
The conventional chromatic scale is denoted 12-TET, an abbreviation for 12 tone equal temperament. In general n-TET denotes the scale that results from dividing the octave into n parts. The previous discussion was looking at how 5-TET aligns with 12-TET.
A step in 5-TET corresponds to 2.4 steps in 12-TET. This is approximately 5 steps in 24-TET, the scale we’d get by adding quarter tones between all the notes in the chromatic scale.
Math
When we talk of dividing the octave evenly into n parts, we mean evenly on a logarithmic scale because we perceive music on a log scale.
The notation will be a little cleaner if we start counting from 0. Then the kth note the n-TET scale is proportional to 2k/n.
The proportionality constant is the starting pitch. So if we start on middle C, the frequencies are 262 × 2k/n Hz. The nth frequency is twice the starting frequency, i.e. an octave higher.
We can measure how well an m-TET scale can be approximated by notes from an n-TET scale with the following function:
Note that this function is asymmetric: d(m, n) does not necessarily equal d(n, m). For example, d(12, 24) = 0 because a quarter tone scale contains exact matches for every note in a semitone scale. But d(24, 12) = 1/24 because the quarter tone scale contains notes in between the notes of the semitone scale.
The equipentatonic scale lines up better with the standard chromatic scale than would a 7-note scale or an 11-note scale. That is, d(5, 12) is smaller than d(7, 12) or d(11, 12). Something similar holds if we replace 12 with 24: d(5, 24) is smaller than d(m, 24) for any m > 1 that is relatively prime to 24.
Related posts
[1] The video first presents 10-TET and then defines 5-TET as taking every other note from this scale.
[2] The image was created with the following Lilypond code.
\score {
\new Staff {
\relative c' {
c1 dih f g aih c \bar "|."
}
}
\layout {
\context {
\Staff
\remove "Bar_engraver"
\remove "Time_signature_engraver"
}
}
}
