Why does our musical scale have twelve notes (counting both the white and black keys on the piano)? Why not ten or fifteen or twenty?

To answer this question, we first need some background information. A note's pitch or frequency is measured in cycles per second; for example, A' is 440 cycles per second. The distance between two notes, measured as the ratio of their pitches, is called an interval. If the interval between two notes is a ratio of small integers (such as 2/1, 3/2, or 4/3), they sound good together - they are consonant rather than dissonant.

The seven pure intervals smaller than or equal to an octave (and larger than unison) that are commonly considered to be consonant are:

• 2/1 - the octave
• 3/2 - the perfect fifth
• 4/3 - the perfect fourth (the harmonic inverse of 3/2)
• 5/4 - the major third
• 6/5 - the minor third
• 5/3 - the major sixth (the harmonic inverse of 6/5)
• 8/5 - the minor sixth (the harmonic inverse of 5/4)

(Two intervals are harmonic inverses if they combine to make an octave; in other words, if the ratios multiplied together make two: for example, 3/2 x 4/3 = 2.)

In the past, people constructed scales based on these pure or natural ratios. For example, the just intonation system uses the exact ratios shown in the table below. However, this method runs into serious problems. Although some of the intervals are perfect, other combinations of notes sound very bad. After the middle ages, music became more complex, with more key changes, and these bad intervals became more common.

The modern equal temperament system was invented (in the 1500s) to solve this problem. The octave is divided into twelve exactly equal intervals. In this system, the smallest interval, the semitone, is not a simple integer ratio, but is the twelfth root of two (2^1/12) or approximately 1.059. Larger intervals are multiples of the twelfth root of two, as shown in the table below. Although no interval (except the octave) is perfect in this system, the error is "spread around" evenly so there are no very bad intervals.

The table below compares just intonation with equal temperament. The intervals in both systems are never exactly the same (except the octave), but they are very close - always within about one percent or better. For example, the fifth, obtained by multiplying the twelfth root of two by itself seven times, is 1.498, which is very nearly a perfect 1.500.

Number of Semitones	Interval Name	Notes	Consonant?	Just Intonation*	Equal Temperament	Difference
0	unison	C-C	Yes	1/1=1.000	20/12=1.000	0.0%
1	semitone	C-C#	No	16/15=1.067	21/12=1.059	0.7%
2	whole tone	C-D	No	9/8=1.125	22/12=1.122	0.2%
3	minor third	C-Eb	Yes	6/5=1.200	23/12=1.189	0.9%
4	major third	C-E	Yes	5/4=1.250	24/12=1.260	0.8%
5	perfect fourth	C-F	Yes	4/3=1.333	25/12=1.335	0.1%
6	tritone	C-F#	No	7/5=1.400	26/12=1.414	1.0%
7	perfect fifth	C-G	Yes	3/2=1.500	27/12=1.498	0.1%
8	minor sixth	C-Ab	Yes	8/5=1.600	28/12=1.587	0.8%
9	major sixth	C-A	Yes	5/3=1.667	29/12=1.682	0.9%
10	minor seventh	C-Bb	No	9/5=1.800	210/12=1.782	1.0%
11	major seventh	C-B	No	15/8=1.875	211/12=1.888	0.7%
12	octave	C-C'	Yes	2/1=2.000	212/12=2.000	0.0%

* This table shows one variation of just intonation.

So, back to the original question: Why does our scale have twelve notes? We have explained that an equal-tempered scale works better in practice than a scale based on pure intervals, but we have not yet explained why we prefer the twelve-tone equal-tempered scale. Why do we not use a ten-tone or twenty-tone equal-tempered scale? Is there something special about twelve?

The answer is: Yes, the twelve-tone equal-tempered scale is remarkable. The nearly perfect intervals seen in the table above are not typical of other equal-tempered scales. Consider the seven basic consonant intervals including the octave (described above): 2/1, 3/2, 4/3, 5/4, 6/5, 5/3, 8/5. We observe:

The twelve-tone equal-tempered scale is the smallest equal-tempered scale that contains all seven of the basic consonant intervals to a good approximation - within one percent.

Let's compare the twelve-tone equal-tempered scale to some other scales.

• All equal-tempered scales with 14 notes or fewer, except the twelve-tone equal-tempered scale, contain at most only three of the seven basic intervals (including the octave) within one percent.
• Several equal-tempered scales with between 15 and 30 notes (notably the 19-tone and 24-tone scales) contain all seven basic intervals, but in none of these scales are the intervals more nearly pure than in the twelve-tone equal-tempered scale.
• The 31-tone equal-tempered scale has all seven basic intervals to a good approximation, some with better accuracy than the twelve-tone scale, but the most important fifth (3/2) interval is less accurate than in the twelve-tone scale (2^18/31=1.495). Some Indonesian music actually uses a 31-tone equal-tempered scale.
• The 41-tone equal-tempered scale is the first with a better fifth (3/2) interval than the twelve-tone scale (2^24/41=1.5004).
• The 53-tone equal-tempered scale has all seven basic intervals with a better accuracy than the twelve-tone scale (the fifth is 2^31/53=1.49994).

But bigger is not necessarily better. Although scales with many tones have many nearly pure intervals that are consonant (ratios of small integers), they have even more intervals that are dissonant (not ratios of small integers). In contrast, the small twelve-tone equal-tempered scale has more consonant intervals (seven) than dissonant intervals (five). We observe:

The twelve-tone equal-tempered scale is the only equal-tempered scale that contains all seven of the basic consonant intervals to a good approximation - within one percent - and contains more consonant intervals than dissonant intevals.

Also, scales with many tones are too large to be really practical: a keyboard with the same range as a piano would be huge.

In summary, the twelve-tone equal-tempered scale is probably the best compromise of all possible scales, and that is why it is now standard almost everywhere.

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The Equal Temperament Musical Scales Worksheet (MS Excel spreadsheet or PDF document) shows all the ET scales (up to 100 tones) and shows how well they match the "ideal" intervals. If you don't agree with my ideal intervals, the spreadsheet allows you to enter your own ideal intervals. If you don't agree with my scoring, you can change the score function, if you know basic programming.

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