Pythagorean Comma As Twelve Justly Tuned Perfect Fifths

Equal Temperament

By GLENN DAVIS DOCTOR G

A ‘pure’ or ‘perfect’ interval is one in which a harmonic of one pitch beats very slowly with a different harmonic of the other pitch. After the octave (in which the second harmonic of the lower tone is consonant with the fundamental of the higher tone) comes the fifth, in which the third harmonic of the lower tone beats with the second harmonic of the higher tone. The ratio of the fundamentals of two tones a perfect fifth apart is 3:2 – as you can see, the numbers in the ratio are the same as the numbers of the harmonics that form the consonance.

This phenomenon suggests a method of tuning a keyboard instrument to pure intervals. We can start with one reference pitch (such as Middle C), then tune the pitch a perfect fifth above that, then add another perfect fifth. This will take us up more than an octave from where we started, so we drop back an octave, tuning the D directly above Middle C so its second harmonic corresponds to the fundamental of the higher D. Repeating this process twelve times and dropping down an octave whenever necessary, we will find ourselves almost a perfect octave above the starting pitch. Going through the Circle of Fifths like this ought to be a good way to get a keyboard instrument in tune. The only trouble is that the last, almost-a-perfect-octave note is slightly out of tune with the first: it’s sharp. The ‘circle’ doesn’t close. This tuning error is called the Pythagorean Comma, and it’s about a quarter of a semitone. So piano tuners actually tune all of the fifths just a bit flat, so that the tuning circle closes and all of the fifths sound the same.

Let’s review what we’ve done. We started with the notion that a perfect fifth (3:2 frequency ratio) is one of the most important intervals in music, second in importance only to the octave. We constructed a scale from twelve perfect fifths. We then noticed that the last note of the scale is a little sharp compared to the first, so we flattened all the fifths an equal amount so that the last note is exactly an octave above the first. This process of stretching or compressing perfect intervals to make a scale come out ‘even’ (so that the ending note is an exact integral number of perfect octaves from the first) is called ‘tempering.’ Furthermore, when you temper the tuning of a scale so that all intervals between adjacent notes are the same, then you have ‘equal temperament.’ An equally tempered scale with twelve notes to the octave is called the chromatic scale, which – of course – is what your ivories produce.

All of the keyboard music we hear is based on the chromatic scale, so we are inclined to think of it as a heaven-sent standard. Actually, it is a set of compromises, in which all of the intervals are out of tune to one extent or another. The table below shows the frequency ratios between the various notes of the chromatic scale and the root note of the scale.

NOTE              RATIO TO ROOT
Root                    1.0000 to 1
2                          1.0595 to 1
3                          1.1225 to 1
4                          1.1892 to 1
5                          1.2599 to 1
6                          1.3348 to 1
7                          1.4142 to 1
8                          1.4983 to 1
9                          1.5874 to 1
10                        1.6818 to 1
11                        1.7818 to 1
12                        1.8877 to 1
Octave                2.0000 to 1

Don’t worry, you don’t have to memorize all those numbers. In fact they are important to us only in that they enable us to compare the intervals of the chromatic scale to perfect (just) intervals.

To compare intervals, we use a unit of frequency measurement called the ‘cent.’ There are 100 cents in a chromatic (equally tempered) semitone, and therefore 1200 cents in an octave. One cent is a very small interval. Two notes a semitone apart have about a 6% frequency difference, so a cent is a frequency difference of 0.06%. If you were to play the G which is 2-1/2 octaves above Middle C, then a note one cent above it would beat with it once per second. Most people cannot tell two notes a cent apart if they are played one after the other. However, many musicians can tell if two notes are two cents apart, and nearly everybody can hear a pitch difference if two notes are five cents apart. If the notes of a scale are more than ten cents from where they should be, the resulting music will sound out of tune.

This should give you an idea of how big an interval a cent is. Now let’s see how out-of-tune the intervals of our beloved chromatic scale are when we compare them to perfect intervals.

First, let’s look at the perfect fifth. Its frequency ratio is 3:2, or 1.5 to 1. The ratio of the chromatic fifth (note #8 from the above table) is 1.4983 to 1, which is mighty close to 1.5. They are about 0.1% apart. In terms of cents, the chromatic fifth is about two cents flat – close enough to be considered ‘perfect’ for most musical applications. The same is true of the fourth: 1.3333 to 1 for the perfect fourth versus 1.3348 to 1 for the chromatic fourth, which makes the chromatic fourth about two cents sharp.

How about the major third? The perfect major third has a frequency ratio of 1.25 to 1, whereas the ratio of the chromatic major third (note #5 from the table) is 1.2599 to 1. The error here is 0.8%, or about thirteen cents. The minor third (note #4) error is 0.9%, or about fifteen cents. The minor sixth has the same error as the major third, because  the minor sixth is an octave minus a major third, and the octave is ‘perfect’ in the chromatic scale. Similarly, the major sixth has the same error as the minor third.

The most important mistuning error of the chromatic scale is the minor seventh. The perfect minor seventh has a frequency ratio of 7:4, or 1.75 to 1. The chromatic minor seventh (note #11 in the table) has a frequency ratio of 1.7818 to 1, which gives an error of 1.8%, or 30 cents! This is why a chromatic minor seventh always sounds discordant. The fact is that most of us keyboard players don’t really know what a consonant minor seventh sounds like. Vocalists and orchestral musicians, on the other hand, produce consonant seconds, thirds, sixths, and sevenths, as well as fourths and fifths, because they instinctively bend toward consonance as they play