The Spiral of Fifths


I am preparing for an upcoming lecture, at NYU in Florence, about the "Evolution of Tuning Systems", so, I am trying to figure out a way to demonstrate some tuning techniques in a comprehensible and "entertaining" manner.

I am convinced nothing beats a good tutorial video and this is an example introducing the
Pythagorean tuning system and the, so called, spiral of fifths.

If the only way to build intervals is stacking perfect fifths, as in the Pythagorean system, this is what happens:

perfect fifth equals ratio 3/2 and measures 701.955 cents. It is "just" 1.955 cents wider than a tempered one.
This "micro" interval is below what is generally considered the threshold of
JND (just noticeable difference) set at around 5 cents, and can be perceived only under very special circumstances as a slow "beating" between the 2 frequencies, depending on many different factors such as timbre, loudness, register and duration.

Many would conclude that such a small interval is not relevant and unnoticeable but this experiment shows the opposite because small and "insignificant" differences quickly add up to major ones such as the famous "
Pythagorean comma".

circle of fifths, that all music students are required to learn (maybe), is "a geometrical representation of relationships" among the 12 notes of equal temperament system, but what happens when we try the same with "just" fifths? Every consecutive fifth introduces a difference of 1.955 cents and after 12 of them we do not get back to our starting point (as in equal temperament) but to a pitch that is 23.46 cents higher than that (this small interval is called Pythagorean comma) and clearly noticeable!

This is why we talk about a spiral of fifths, because, given a starting point (usually called root or fundamental note), pitches populate a ever expanding world of notes that get further and further away from its center (a mind boggling problem that has fascinated music theorists, musicians and instrument builders for thousands of years).

(click on the image to watch the movie)

This short video shows what happens:
We start on C and move along the spiral of justly tuned perfect fifths until we get to B#, that in equal temperament is
enharmonically the same of C but not here. It is easy to hear the difference between B# and C (B# is 23.46 cents sharper than C). This spiral of fifths is repeated twice and ends with both B# and C sounding together.

I hope you enjoyed it!!

Isn’t this explanation comprehensible and entertaining?

Let me know!

R. C. commented:
The epic of gilgamesh is based entirely on the fifth, in fact it is his name "two thirds"  
it applies directly to the zodiac, because Gilgamesh as a god traverses the cosmos "riding" on the fifth.  He goes for example from Gemini to Scorpio.  In fact he goes anticlockwise, 5 steps, a journey back in time in this context.  
By repeated journeys he marks out the square of the sun festivals, he "squares the earth".

A. L. commented:
Very good Carlo! Even I got that. Comprehensible and entertaining it was Happy