The subject of Pentatonic Scales reminded my of a problem I have with the Equal Temperment scales used in musical instuments and in musical notation. Everytime I look at musical notation (which I can't read, btw) I can not help but to be reminded that these notes are not the true notes, but imposters.

My entire argument is based on an error function I derive below. You may take offense to this function specifically, but you will see that other weighting functions will have little effect on the final conclusion: There *is* error in the equal temperament scales.

First we will assume the notes are equi-spaced up the scale. We make this assumption so that any song may begin with any note, and all other notes can be played relative to that. This assumption is wrong for situations where the instrument is tuned for a specific song; and is capable of providing a pure harmonic match with irregular spacing.

Now the question is, What is the best spacing?

We will only consider notes relative to a fundamental note, and we will call them the "Just Notes".

Please let the Just Notes be represented by

A just note is only considered "good" if it is in a simple ratio from fundamental. For example, a perfect fifth:

Reads as: "One 3/2 step from fundamental is 3/2 of fundamental". The next note up the harmonic pentatonic scale is

... and so on. We can also see "Zero 3/2 steps from fundamental is simply the fundamental

We will want to play the Just Notes, but the notes we have come from the equal temperament scale. How close can we get to the just note?

We will denote rounding to the nearest integer with the "round" function

BEST is the note on from the equal temperament scale "closest" to the actual note we wish to create. This BEST note will probably not be perfect, so we track how much they differ

Furthermore, we can not simply add the error terms for all possible `"ratio"` and `"power"` because there is an infinity of them, and some ratios are simply too big to be considered musical. Therefore we will give weight to the DIFF, more weight for the simple ratios, and more weight to notes close to fundamental:

WLOG, Let

We can define WEIGHT as

Some examples:

We can now assign an error term

We square the error to

1. make all error positive
2. weight the simpler ratios heavily
3. ensure the infinite series converges
4. ensure the infinite series converges fast (so we can approximate)

`"ERROR" = sum_("over all Just Notes") (( "note" ^("round"(ln( "note" )/ln( "spacing" ))) - "note") / "note" /( "max"^|"power"\|))^2`

We can graph this for all spacings that fit evenly into an octave { `"spacing"^n = 2` where `n in NN}`}

Notes per Octave	ERROR (x1,000,000)
1	38,740.54
2	6,097.04
3	2,247.67
4	1,598.79
5	415.10
6	1,336.95
7	235.05
8	479.01
9	252.64
10	125.29
11	368.07
12	37.23

Notes per Octave	ERROR (x1,000,000)
13	219.32
14	114.13
15	69.73
16	113.63
17	60.61
18	137.35
19	18.46
20	78.21
21	61.81
22	14.48
23	83.63
24	18.59

The pentatonic scale (5) is not perfect, and there is error. The heptatonic (7) scale is closer to true harmonic notes, and the chromatic (12) scale is the best unless we accept 19 notes per octave, which may be to difficult to play.

But it is the assumption that notes must fit perfectly within an octave that prevents us from finding an even better equal temperament scale. The error function above can be used to explore all note spacings. But please note, the weighting function gives a relatively heavy weighting (WEIGHT = 1/4) to matching octaves compared to matching 2/3 ratio (WEIGHT = 1/9). This heavy preference for octave matching means this error function will still prefer something close to `"n"` notes per octave

And we can graph this error term for all known spacings.

We can zoom in a little

The matches are a little off. Specifically, the equi-spaced pentatonic scale should have a note ratio of 1.1483 (about 5.01 notes per octave) , not the 1.1487 (5 notes per octave). This is a small difference only because of the weight function: If we give no weight to matching octaves, then the best note spacing is 1.1452 (5.11 notes per octave). Do not dismiss the small size of these differences, they are perfectly audible (more below).

Most pentatonic scales are played on a chromatic scale (12 notes per octave), so they are more true to their simple ratios, but still not perfect.

I will leave you with words from a better writer than me: