The current music scale system that we know of is credited to Pythagoras, a Greek philosopher and mathematician who lived around 550 BC. Legend has it that Pythagoras listened to the blacksmith’s hammer striking the anvil and discovered the tones produced by the hammering was related to the weight of hammer, and to his great surprise, the weights came in simple whole number ratios to each other. With further investigation, it did not take him very long to realize that consonant sounds, tones that sound “pleasing to the ear” when played together, are all in simple ratios to each other (Fauvel, Flood, &Wilson, 2003; Parker, 2009).
There are two presumptions when we try to make the Pythagorian scale (Giordano, 2010).
1. Our scale will consist of a series of notes. The first note can be any note of frequency f, but the last one should be an octave higher, which has a frequency 2f.
2. Our scale should contain notes that make a "pleasing sound" when played together, which means the frequencies of the notes should be in simple ratios to each other.
We will follow the example given by Fauvel, Flood and Wilson (2003). Let us start with a tone of frequency f. We call this note C. Next we consider its second harmonic with frequency 2f. We name this note C', which is by definition an octave higher. The interval between notes C' and C in terms of the ratio of the frequencies is 2:1. If we choose a note with frequency 3f, the intervals between the notes are 3:2:1. Since we want to confine our scale within one octave, we get our third note by moving 3f one octave down, which is 3f/2 = 1.5f. We name this note G.
Now our scale looks like this:
Therefore we now have two new intervals. The interval between C and G is 3/2:1 = 3/2, which is also called a perfect fifth. And the interval between G and C' is 2:3/2 = 4/3, which is a perfect fourth in the western music convention.
With the two new intervals, we will generate more notes between C and C'. If we lower C' by a perfect fifth, which means 2f divided by 3/2, or 4f/3, we get note F. In the same way, if we lower G by a perfect fourth, we get D, which is (3f/2) / (4/3) = 9f/8. Now in an ascending order our scale looks like the following:
There are two presumptions when we try to make the Pythagorian scale (Giordano, 2010).
1. Our scale will consist of a series of notes. The first note can be any note of frequency f, but the last one should be an octave higher, which has a frequency 2f.
2. Our scale should contain notes that make a "pleasing sound" when played together, which means the frequencies of the notes should be in simple ratios to each other.
We will follow the example given by Fauvel, Flood and Wilson (2003). Let us start with a tone of frequency f. We call this note C. Next we consider its second harmonic with frequency 2f. We name this note C', which is by definition an octave higher. The interval between notes C' and C in terms of the ratio of the frequencies is 2:1. If we choose a note with frequency 3f, the intervals between the notes are 3:2:1. Since we want to confine our scale within one octave, we get our third note by moving 3f one octave down, which is 3f/2 = 1.5f. We name this note G.
Now our scale looks like this:
Therefore we now have two new intervals. The interval between C and G is 3/2:1 = 3/2, which is also called a perfect fifth. And the interval between G and C' is 2:3/2 = 4/3, which is a perfect fourth in the western music convention.
With the two new intervals, we will generate more notes between C and C'. If we lower C' by a perfect fifth, which means 2f divided by 3/2, or 4f/3, we get note F. In the same way, if we lower G by a perfect fourth, we get D, which is (3f/2) / (4/3) = 9f/8. Now in an ascending order our scale looks like the following:
We have a new interval between F and G, which is (3f/2) / (4f/3) = 9/8. We call this interval a major second, or a whole tone. If we raise D and G by a whole tone, we get two new notes E and A, which is 81f/64 and 27f/16 respectively. In the same way, if we raise A by a whole tone, we get B, which is 243f/128. Again let us look at what we have in our scale.
Notice the intervals E to F and B to C': 256/243. We give it a name minor second or semitone.