3.2 Pythagorean Intervals

As discussed in the previous section, Pythagoras was interested in understanding the notes and scales used in Greek music. In particular, he studied the Greek stringed instruments, called the lyre. His first observation was that if you have two strings with the same length, tension and thickness, they sound the same when you pluck them. This means they have the same pitch and sound good (or consonant) when played together. The second observation was that if the strings have different lengths (keeping the tension and thickness the same) the strings have different pitches and generally sound bad (or dissonant) when played together. Finally, he noticed that for certain lengths, the two strings still had different pitches, but now sounded consonant rather than dissonant. Pythagoras called the relationship between two notes an interval.

For example, as mentioned above, when two strings have the same length, they have the same pitch, and the relationship, or interval, between the notes is called a unison. If one string is exactly one-half of the length of the other string, its pitch is much higher, but they still sound consonant when played together. This interval is called an octave. Finally, if one string has a length that is two-thirds the length of the other, the strings again sound consonant when played together and this interval is called a Perfect Fifth.

Already, we have a very important distinction. These intervals are defined by the lengths of the strings being in a certain ratio. One does not form an octave by reducing the length of on string by a fixed amount, like 10 cm. Rather, one forms an octave by dividing the length of a string by a factor of 2. In other words, if the lengths of the strings are in a ratio of 2 to 1, the pitches of the strings will form an interval of an octave.

Musically, there are many different intervals, each with its own characteristic sound and each with a special role in music harmony. We will encounter quite a few intervals during the course. Each interval will have a name, like an octave, and it will be defined by a ratio, like 2:1 for the octave.

So far, our list of intervals includes:

 Name Ratio Unison 1:1 Octave 2:1 Perfect Fifth 3:2

As mentioned above, each interval has a particular characteristic. The unison may not seem so interesting, as the two notes are the same. This may be true, but it is the simplest and most fundamental interval. However, it can be used to great effect. Imagine a singer singing a simple melody. Now imagine a chorus of a hundred voices, all singing the same melody in unison, i.e. everyone singing exactly the same pitch. It is actually very hard to have that many people sing perfectly enough, but when they do the effect is quite dramatic. So, great composers can use even the simplest interval effectively.

The octave is the next most important interval. As discussed in the previous section, it defines the range of the music scale. Two notes an octave apart sound so similar that they are always given the same name. For example, elementary piano pieces often start on middle C. However, if you go up an octave from there, the note is still called a C. In fact, it is sometime hard to tell whether two notes are one or two octaves apart. If someone has a deep voice and cannot sing a particular melody because the notes are too high, it is always acceptable to sing the melody an octave lower. Musically, the melody is the same.

Finally, we come to the Perfect Fifth. In this case, there is no question that two notes a Perfect Fifth apart are really two different notes. On the piano, the note a Perfect Fifth above Middle C is called a G. Thus, the Perfect Fifth is the first interval to get us into harmony, the question of how different pitches interact with each other. Once again, the Perfect Fifth has a special quality to it – anyone who has heard Gregorian chant will recognize the open haunting quality of the Perfect Fifth.

So far, we have only talked about intervals in terms of going up in pitch. However, we can go down, as well. If you start with a string and you double its length, the pitch will go down by an octave.

 String 2 1/2 2 2/3 3/2 String 1 1 1 1 1 Ratio 1/2 2/1 2/3 3/2 Interval Up an octave Down an octave Up a Fifth Down a Fifth

It is also very important to notice that intervals do not depend on the initial frequency, length, tension or type of string. Intervals only give a relationship between two notes, but so far, like Pythagoras, we only defined them in terms of lengths of strings. It turns out that for a string, if you divide the length of the string by a factor of two, the frequency goes up by a factor of two. So, whether we are considering the lengths of the strings or their pitches, the ratios for the intervals are the same.

Musically, intervals are very important, but why are we discussing them so carefully in a physics course? From a scientific standpoint, by far the most important characteristic of these simple consonant intervals is that they are defined by ratios of small integers. This was very exciting to Pythagoras. Consider the fact that Pythagoras had just invented mathematics and integers, in particular. And, he had just invented arithmetic. But, this was just abstract thought. Pythagoras had no way of knowing whether mathematics would actually be useful for anything. It then turns out that integers and ratios could describe the consonant musical intervals.

Actually, this also highlights a deep connection between the human experience and abstract mathematics. What appears to be a subjective judgment, this interval sounds good (consonant), that interval sounds bad (dissonant) can actually be predicted using abstract mathematics. Apparently, our emotional response to the world sometimes even follows mathematical laws!

Returning now to intervals, we can define a new operation, namely, combining intervals. For example, we could go up in pitch by an interval of an octave and down by an interval of a Perfect Fifth. In this case, going up by an octave means multiplying the frequency by a factor of 2. Going down by a Fifth means dividing by 3/2. All together we have 2/(3/2) = 4/3. Thus, by combining intervals, we have actually produced a new interval, called the Perfect Fourth. The Perfect Fourth is defined by a ratio of 4/3.

To summarize:

Ratios of 1/2 and 2/1 give octaves

Ratios of 2/3, 3/2 give fifths

Ratios of 3/4, 4/3 give fourths

Notice that the ratios above only involve the integers 1, 2, 3, and 4. As an exercise it is helpful to write out ALL ratios involving these integers:

 1/1 1/2 1/3 1/4 2/1 2/2 2/3 2/4 3/1 3/2 3/3 3/4 4/1 4/2 4/3 4/4

Now, try to determine what interval each of these corresponds to. For example, 2/3 corresponds to going down by a Perfect Fifth. 4/1 corresponds to going up by an octave TWICE. In other words, since going up an octave means multiplying by 2, going up two octaves means multiplying by 2 twice, or 2x2 = 4.

At this point, we can ask, why stop at 4? Why not use 5, 6, 7, and 8 as ratios for intervals. The answer is that we can use larger integers. However, Pythagoras stopped 4 because he realized that he had all the intervals that he needed to construct the musical scale, and that was his original goal. Moreover, Pythagoras was a great believer in simplicity – he wanted the simplest explanation for the musical as possible. He believed that simplest explanation must be the correct one. To this day, this continues to be a central conviction in physics and science.

In fact, Pythagoras was so happy with these intervals that he called them "Perfect", i.e. the Perfect Fifth and the Perfect Fourth. (The unison and octave are also perfect, but as discussed above, they are so perfect they are not really considered to be a new note.)

In a later section, we will discuss how to create a musical scale from these intervals, but first we will organize what we have learned in a slightly different way, which will make constructing the scale a bit easier.

The following table shows the relationship between notes on the scale, intervals and frequencies, starting with the note D:

 Name of note Name of interval Ratio of interval Decimal equivalent Frequency of note. D Octave 2/1 2.000 587 Hz A Perfect Fifth 3/2 1.500 440 Hz G Perfect Fourth 4/3 1.333 391 Hz D Unison 1/1 1.000 293 Hz

Although we don’t yet have enough notes for a complete musical scale, these were, in fact, the notes to which the strings on the Greek lyre were tuned. Also, Gregorian chant was mostly based on fourths and fifths and, so, used these notes primarily.

Of course, we can start on any note we want, D = 293 Hz is just an example.