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Why are there 7 notes in music

To answer this childish question, let us first consider the usual well-known circle of fifths. Usually it begins with G# and ends with F#, in order to have as few flats and sharps as possible. However, let us start it with C and write down all the enharmonics, that is, notes which are of the same pitch but are written differently, in the 12-grade equally tempered scale:

B#/C - G - D - A - E - B/Cb - F#/Gb - C#/Db - G#/Ab - D#/Eb - A#/Bb - E#/F - B#/C

It is clear why it is called a circle of fifths (or sometimes a circle of fifths and fourths, since all notes are considered within one octave): it is made of fifths. But what are those fifths that make this circle of fifths? Before the 12-grade equally tempered scale was discovered, there was just one fifth known, a natural one, which exists due to acoustic dependencies [1]. If we consider natural fifths [2], rather than tempered ones, then there will be no circle as there is an unlimited series of natural fifths as in the following:

In this table, the lower row shows the same circle of fifths, while the upper row shows the ratio between the respective sounds from the natural fifth series and the left C (in cents [3]). The tempered fifth amounts to 700 cents, while the natural fifth is 702 cents [4]. To make the picture more visual, from sounds which are equal in octave terms we take the one nearest to C, so if the sound is more than half an octave (triton, or 600 cents) from C, then we take the same sound one octave down (702 - 1200 = -498). You can see that the right C (which is highlighted in bold, like the left one) is 23 cents (or more exactly, 23.46 cents), or almost 1/8 of a tone) higher than the left C. Why then do we close this segment of the series of fifths into a circle? The usual explanation is that we get the same sound. However, it is the same in the 12-grade equally tempered scale only.

I would like to suggest a different explanation concerning the creation of this circle of fifths. The fifth ratio series is closed into the circle due to the fact that the ratio between its circle’s extreme sounds is less than any the ratio between any of its sounds [5]. Let us call this "the closing principle".

In the quoted part of the fifth ratio series, there are also other segments, which correspond to that principle. In the segment limited by C (left) and B (five steps to the right in the series), the ratio between those two is also less than any other ratio within this segment. Let us write down the notes of this segment according to their pitches (scale wise), but replacing B with C:

C - D - E - G - A - C

We now have got the pentatonic scale. However, how could we replace one sound with another? The reason here is the same as when making the 12-tone fifth ratio circle (which would be logically called dodecatonic). To be exact, in that dodecatonic we equate B# and C. For us, that is the same sound (which we shall call "the closing tone"). In the same way, B and C make the same sound in pentatonic, although there is a 90 c distance between them. In the fifth ratio series, there is another segment which satisfies the closing principle as follows:

Although there is a very large ratio between the extreme sounds (a major second in our terms), that is still less than other ratios within the segment. It does not matter whether we change the notes to signify sounds for which ratios are shown in the upper low of the above table. Of course, notes reflect an established system of ratios between sounds, but they reflect that system only. Other systems will require other notes.

Parts of the sound space (the fifth ratio series is a one-dimensional sound space, if, of course, we don’t take octaves into consideration), selected on the basis of the closing principle, are tone systems [6] (in the broadest meaning of that term). But, what about 7 notes? The fifth ratio series does not provide an answer as to where they derive from (for your reference, the next segment of the fifth ratio series which is a tone system is 41 sound long). So let us try and use the next overtone, which produces new sounds, i.e. the 5^th overtone.

If we use natural major thirds, a new dimension appears in the sound space: here we get a fifth and third ratio plane. On this plane, the ratio between neighboring sounds in one dimension (let us say, a horizontal one) is the same as before, the natural fifth (702 c); in the other (vertical) dimension, it is equal to the natural major third (386 c). And it is here, in this fifth and third ratio plane, that we find a tone system, which corresponds to major and minor system as follows:

Here, the closing tone, Bb, is shown in bold, as above. To show that the modern notation corresponds to this particular system, we started with Bb, and you can see that notes are similar to an alphabetical list [7], as well as the fact that the B note has a special position. This is the answer to the question on 7 notes, and there is no real need to write the whole scale.

Here I am leaving out many very questions of great interest (for example, the embedded tone systems, neighbouring tone systems (modulations), a tone system and its "environment", formation of chords and their movement, assigning of different tones as a center of tone system (modes), a scales regulations, etc.). However, I cannot miss one question being of particular importance to me: the one on tone systems and 12-grade equally tempered scale. All the above tone systems can exist within that scale, despite the pitch differences between pure and tempered tones. Let us consider, however, yet another tone system within the fifth and third ratio plane which is as follows:

This tone system has 17 tones. It should be pointed out that sounds which sound the same in the 12-grade equally tempered scale (not enharmonically, but the same), and noted the same in our notation - E, B, F#, C#, G# - from upper and lower rows - are now different and independent tones in this system. This tone system cannot exist anymore within the 12-grade equally tempered scale.

Nowadays, we can avoid temperament, and this is due to the use of computers. I am not against the 12-grade equally tempered scale, or any other scale for that matter, but only to the extent to which that scale turns from a swaddling band into a cerement.

Footnotes

[1] A sounding body is oscillating not only as a whole, but also with its every part: every half (producing an octave), every third part (a fifth), etc. The entire number of such oscillations is called the overtone series.

[2] Please bear in mind that, hereinafter, the word tempered means “belonging to the 12-grade equally tempered scale”.

[3] Cent is 1/100 of semitone in the 12-grade equally tempered scale. Cents are convenient to use, because they provide a visual representation. Let us abbreviate cents as c. The octave is then 1200 c.

[4] According to some data, that 2 cents is the smallest ratio between sounds the human ear can distinguish in terms of physiology.

[5] It is also clear even without calculations that the next minimal ratio between sounds within the fifth ratio circle will be semitone, or more exactly, 90 cents.

[6] If we call the proximity of overtones as kindred (and this is how it is often called), then it is possible to create a more playful definition: a tone system appears in such segment of the sound space where the least kindred sounds turn out to be the closest ones.

[7] In the Italian notation where B is designated as H, and Bb as B it is clearer:

Of course, there are also other notations, right up to Indian “sa-ri-ga-ma-pa-dha-ni”.

© Constantine Shushpanov, 2001.

Revised 04.07.01.