When last we left our intrepid hero , Pythagoras was in the midst of manifesting a revolutionary bound in Western music – by plucking a single string at points along its length.

Let’s revisit the basics of the four string-lengths he was focusing his attention on. Pythagoras would be using a monochord, or single string of a particular length, and would pluck the whole string, or clamp the string at certain points along its length to pluck only a portion of it. Because he believed celestial harmony could be discovered through the integers 1, 2, 3, and 4, Pythagoras was plucking his string at its full length, 1/2 the length, 2/3 the length, and 3/4 the length.

The pitch created by plucking the full string is called the fundamental. The pitches generated by the other string lengths are called the octave, the dominant, and the subdominant, respectively. These string-lengths are shown below, and again you can hear the sound of these string-lengths, played relative to the fundamental.

Fundamental



Octave



Dominant



Subdominant

In music we talk about the ‘distance’ between pitches as ‘intervals.’ So, there would be a particular interval between, say, the pitch of the fundamental and the pitch of the octave, in our above example. When dealing with vibrating strings, the ratio of the length of the string being plucked gives us this musical interval. So, the interval between the fundamental and the octave would be 1/2.

Let’s jump for a moment from Pythagoras’s time to our own. Do you recall our of frequency of pitches? Recall that the ‘highness’ or ‘lowness’ of a sound is a result of how many vibrations are being generated per second, how many cycles of the sound wave are completed every second – which we describe as its number of Hertz. Digging into the physics of a vibrating string, we’ve discovered that decreasing the length of a particular string to be plucked causes an inverse change in frequency of pitch generated. So, if we pluck half the length of a string, we generate a pitch with a frequency twice as high as would have been created by plucking the whole string. If we pluck 2/3 the length of the string, we generate a pitch with a frequency 3/2 times as high. So for instance, if plucking the length of a whole string generated a pitch of 100 Hz, plucking 2/3 the string would generate a pitch of 100 Hz * (3/2), or 150 Hz.

how the relationship between a pitch and the same pitch one octave higher is doubling the frequency? Here we see the practical manifestation of that! Pluck half the string, and you’ll double the frequency of pitch and generate the octave. Now of course, Pythagoras didn’t have this depth of understanding sound wave dynamics when he was doing his work – but it can be helpful to have this understanding at one’s disposal, when approaching Pythagoras’s work from our time!

Now, back to Pythagoras. Pythagoras wanted to see what was created by the intervals of these four pitches. He looked at the dominant (plucking 2/3 of the string) and the subdominant (plucking 3/4 of the string) and wanted to determine what the interval was between them.


Pythagoras was one of the great mathematical minds of his time, and like him, we’ll use a little math to describe this interval.

In wanting to describe the ‘interval’ between the dominant and subdominant, we want to calculate the ratio of the length of the dominant to create the length of the subdominant. Another way of asking the same thing is: by what percent do we increase string-length to go from the dominant (2/3 of our total string length) to the subdominant (3/4 of our total string length).

Mathematically, we can set this up as:

(2/3) * x = (3/4)

where ‘x’ describes the interval we’re looking for. Solving this…

x = (3/4)*(3/2)
x = (9/8)

So, the interval between the dominant and subdominant is 9/8, meaning the length of string that generates the subdominant is 9/8ths the length of string that generates the dominant. This can also be described as increasing the string length by 12.5%. This interval came to be called a ‘whole tone.’

What about the interval between the subdominant and fundamental – could that be described in a series of ‘whole tone’ intervals?


As Pythagoras discovered: no. Let’s see that in action, shall we?

Recall, we’re trying to reach 1, or the full string-length, from 3/4 of the string length, with whole-tone intervals of 9/8.

Increasing by one whole tone we will have:

(3/4)*(9/8) = 27/32. Closer, but not at the full string length yet… let’s increase by another whole tone interval:

(27/32)*(9/8) = 243/256. Close, but 243/256 is still a fraction under 1, so we’re still not yet at the full string length. Let’s take another whole tone interval and see what happens:

(243/256)*(9/8) = 2187/2048. Our numbers are getting a little complicated, but what’s important to note here is that 2187 is higher than 2048, meaning we’ve gone higher than 1.

So, the subdominant and fundamental are separated by more than two whole-tone intervals, but less than three whole-tone intervals. Let’s figure out what that little smidge necessary is, shall we? We’ll set up a calculation like we did earlier, starting with the length of string after two whole-tone intervals:

(243/256)*x = 1

To reiterate (fractions and algebra can become confusing sometimes!), this is asking: what interval, x, is necessary to take us from a string length of 243/256 of the full string, to the entire full string, or 1?

Solving, we find:

x = 1*(256/243)
x = 256/243.

The interval of 256/243 can also be described as increasing the string length by about 5.3%. This interval came to be called a ‘semi tone.’

To recap, Pythagoras has described a ‘whole tone’ interval and a ‘semi tone’ interval. A ‘whole tone’ is a musical distance of 9/8, which can be physically observed by plucking a string 12.5% longer. A ‘semi tone’ is a musical distance of 256/243, which can be physically observed by plucking a string about 5.3% longer.

Using the same monochord example, let’s hear one whole-tone interval and one half-tone interval, from our same fundamental generated by the full string:

Fundamental and Whole-Tone Interval

Fundamental and Semi-Tone Interval

Through Pythagoras’s work, the Greeks began to only play notes separated by whole tones or semi tones. This tradition followed through into our time too. Let’s look at our musical alphabet of natural notes:

A B C D E F G

What are the musical distances between these notes? Those intervals are:

A (whole tone →) B (semi tone →) C (whole tone →) D (whole tone →) E (semi tone →) F (whole tone →) G

Let’s look at the rest of our musical alphabet, the sharps and flats:

Bb C# D# F# G#

How do these fit in with our whole tones and semi tones? The intervals for these are:

Bb (semi tone →) B
C (semi tone →) C#
D (semi tone →) D#
F (semi tone →) F#
G (semi tone →) G#

There you have it! Pythagoras’s work is alive and vibrant in our musical language today.

And yet, the question might arise… why are some natural notes separated by whole tones, and other separated by semi tones? Why that particular order or whole tones and semi tones among the natural notes? Why do the flats and sharps come in where they do?

Continue the journey with us in the next Music Monday – “Learning Our ABC’s”

Did you enjoy this exploration into our musical history? Did you enjoy it, or find it a bit difficult to swallow at first? Do you have your own insights into what our musical history has looked like?

Join the discussion – share your thoughts and ideas in the comments!

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