Why seven notes in octave

The diatonic scale is the least populated Pythagorean scale to include every interval found in the chromatic scale.

And he proceeded to divide after this manner: —First of all, he took away one part of the whole, and then he separated a second part which was double the first, and then he took away a third part which was half as much again as the second and three times as much as the first, and then he took a fourth part which was twice as much as the second, and a fifth part which was three times the third, and a sixth part which was eight times the first, and a seventh part which was twenty-seven times the first. After this he filled up the double intervals and the triple cutting off yet other portions from the mixture and placing them in the intervals, so that in each interval there were two kinds of means, the one exceeding and exceeded by equal parts of its extremes, the other being that kind of mean which exceeds and is exceeded by an equal number. Where there were intervals of ³⁄₂ and of ⁴⁄₃ and of ⁹⁄₈, made by the connecting terms in the former intervals, he filled up all the intervals of ⁴⁄₃ with the interval of ⁹⁄₈, leaving a fraction over; and the interval which this fraction expressed was in the ratio of 256 to 243. And thus the whole mixture out of which he cut these portions was all exhausted by him.

Plato. Timæus (transl. by Benjamin Jowett)

Why are there seven different notes in an octave?—My answer (not necessarily a novelty) is, in brief: because the diatonic scale is the least populated Pythagorean scale to include every interval found in the chromatic scale.

For most of us such an answer indeed entails no end of further questions: what is a Pythagorean scale? what is the chromatic scale? the diatonic? an interval? and what makes the chromatic scale so special?

Pythagorean scale(s)

Mathematically, we can model any tone pitch as a single number, its fundamental frequency. A ratio of the fundamental frequencies of two tones expresses what musicians call an interval.

There are three intervals, their consonance aurally obvious. Moreover, they are æsthetically (mathematically) beautiful and are called the unison, octave and fifth, corresponding to ¹⁄₁, ²⁄₁ and ³⁄₂ ratios, respectively.

Pythagoreans were among the first philosophical schools of Greece, back in 5th through 4th centuries BC, who investigated various aspects of the universe to discover how it was built from rational numbers, preferably “simpler” ones involving integers from the first ten in both numerator and in denominator. Once a nasty custom persisted to describe the guys as some mystic numerologists dwelling in ivory towers and ignoring the nature; this surely is utter nonsense, Pythagoreans being perhaps the earliest school to have turned to actual measurements and a kind of scientific experiments. And it was music to which they applied experimental methods first and foremost. (They would have never killed an elephant to build themselves a house anyway.)

The Pythagorean scale is a way to build all other intervals applicable in music from the three, or, to be more exact, from the octave and the fifth. The method employed is thus simple: Let an arbitrary number (frequency) ν be the first member of a sequence. Then let us check if ³⁄₂ · previous number (corresponding to the tone up a fifth from the previous tone) is less than (i. e. lower than the tone an octave up from the first tone). If it proves so, the subsequent member is just ³⁄₂ · previous number. If not, \frac{\frac{3}{2} \cdot previous number}{\frac{2}{1}} (up a fifth then down an octave from the previous tone). Thus the interval from the first tone to its double (the octave) is gradually populated with intermediate tones.

Then, starting from 349.23 Hz (and why not?) we will bump into a Pythagorean sequence Π:
Π = {349.23, 523.85, 392.88, 589.33, 441.99 …}

Let us call a sorted list of n first members of the Pythagorean sequence—n-tonic Pythagorean scale Πn:
Π1 = {349.23}
Π2 = {349.23, 523.85}
Π3 = {349.23, 392.88, 523.85}

Chromatic scale

Let us investigate intervals between adjacent tones in various Pythagorean scales for patterns (Fig. 1).

Fig. 1. n-tonic Pythagorean scales

Fig. 1. n-tonic Pythagorean scales

One peculiarity discovered in the 12-tonic P-scale is that the intervals between any pair of adjacent tones, unlike any P-scale for n in [3..11], are roughly equal. The 13th tone seems to spoil the pattern again, but if we are not to divide the frequency by two we may discover that it is close to the octave up from our first tone, and is positioned accordingly, its interval from the 12th approximately equal to the remaining intervals.

Fig. 2. Chromatic Scale

Fig. 2. Chromatic Scale

Thus, 12 tones positioned by fifths fill the octave with approximately (logarithmically) equal intervals (the 13th being basically of the same pitch as the first one, an octave up). Fig. 2 depicts the resulting scale, indexing individual tones as frequences (Hz), common notation note blobs, and common English letter names of the notes. The numbers between the frequences indicate their ratios. Indeed, we can extend the scale up and down, covering more than just one octave: the above procedure does not depend on any particular first value of ν.

If we are willing to tolerate the roughness of the equality—as many cultures of the world do, most traditions in our own European culture included—we can use the scale to play or sing any music. If we are not, we are on the way along the Pythagorean sequence to more-than-12-tone music, like that found in certain oriental traditions or in some ultra-super-modern-avant-gard academic music in the West, but this falls beyond our current study.

Both the fact of equality of all the intervals between adjacent tones in the 12-tone Pythagorean scale and the fact of non-exactness of the equality are of utter importance for our music. The first means that we can sing or play any tune starting from any tone, while the second raises an important issue of scale temperaments which is also beyond the scope of this essay. Musicians call the 12-tone scale—either Pythagorean or tempered in some way—chromatic.

Diatonic scale

Ok, our musical culture is essentially chromatic but how it comes that so many simpler tunes subsist on mere seven tones, and why do we only have seven names for the notes?

The elementary interval, the interval equal to the interval between any two adjacent tones in the chromatic scale, is called the semitone in music. Thus we can measure all the intervals between any two tones in semitones. Chromatic scale features 12 quantitively different intervals of 1, 2 … 12 semitones not exceeding the octave (or 13, if we are to include the 0-semitone unison).

Fig 3. Diatonic scale and its intervals

Fig 3. Diatonic scale and its intervals

Let us return to our n-tonic Pythagorean scales and find which of the intervals appear when we add to n considering not only the first tone but also the upper-bound tone of the octave and the previously introduced tones (Fig. 3).
With 1-tonic scale we have only the 12-semitone interval (the perfect octave as a musician would call it), 2-tonic scale adds 7-semitone (the perfect fifth) and 5-semitone (the perfect fourth, the inversion of perfect fifth, that is, the octave minus the perfect fifth). Three-tonic scale brings in the 2-semitone major second and its inversion, the 10-semitone minor seventh. Four-tonic—the 9-semitone major sixth and the 3-semitone minor third. Five-tonic—the 4-semitone major third and the 8-semitone minor sixth. Six-tonic involves the 11-semitone major seventh and the 1-semitone minor second. And finally, 7-tonic completes the repertoire with the 6-semitone, or 3-tone, tritone, aka diminished fifth, aka augmented fourth.

So, seven tones comprising diatonic scale suffice to expose an interval quantitively (i. e. by the number of semitones) equal to any interval found in the chromatic scale.

Free bonus: why “fifth” and why “octave”?

If we enumerate all the steps, or degrees, in the diatonic scale, we’ll see that the fifth from the first step is... er... the fifth degree, and the octave is the eighth (“octava” in Latin).

Appendix: some code to play with

#!python3
octave = 2
fifth = 3/2
first_tone = 347.654

def next_tone(tone):
  fifth_up = tone * fifth
  return fifth_up if fifth_up < first_tone * octave else fifth_up / octave

def n_tonic_scale(n):
# return [] if n <= 0 else [first_tone] if n == 1 else n_tonic_scale(n - 1) + [next_tone(n_tonic_scale(n - 1)[-1])]
  if n <= 0 :
    return []
  elif n == 1 :
    return [first_tone]
  else :
    temp = n_tonic_scale(n - 1)
  return temp + [next_tone(temp[-1])]

def print_scale(n, scale):
  print(n, "-tonic Pythagorean scale:", sep="")
  if n > 1:
    for i in range(len(scale)-1):
      print (" {0:6.2f}\t".format(scale[i]), "\n  ↕ {0:10.8f}\t".format(scale[i+1]/scale[i]))
  if n > 0:
    print (" {0:6.2f}\t".format(scale[-1]))

def print_n_tonic_scale(n):
  print_scale(n, sorted(n_tonic_scale(n)))
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