Commas on the Monochord

Definitions:

The Pythagorean comma is the difference between 12 fifths and 7 octaves.

The Syntonic comma is the difference between two pure major seconds (two 8:9s, which is 64:81, otherwise known as a ditone) and a pure major third (4:5).

Meantone tunings (i.e. 1/4 comma meantone) refer to fractions of the syntonic comma.

Finding the Pythagorean comma on the monochord:

This comes from an early fifteenth century treatise, which is so beautifully simple and brief I will copy it here in its entirety (from Christian Meyer, Mensura Monochordi, p 145.)

4 4 2P 4 2P; 4 4 2P 4 2p; 4 2. et cetera.

Versus

Bis quatuor duo passum quatuor et duo passum

Replica hoc metrum superaddito quatuor duo.

Beautiful, isn’t it? My directions below seem awfully tedious in comparison. Essentially you are going up by pure fourths (3:4) twelve times, but to keep it within a range that will be useful on your monochord, you occasionally go down by a pure fifth (3:2). At the end, he divides each note length in half to find the octave above (“2. et cetera”) I will leave out that part, but add two final steps of my own: up one final fourth from Gb to find Cb, and up an octave from B to find the next B, thus illustrating the difference between Cb and B, or the Pythagorean comma.

*Directions on where to mark are written so as to best display the proportion. "Mark E at 3/4" means you have divided the length into four parts and your new note is three parts from the end of the stick (but one part from your starting point). This is the system Meyer adopted in his shorthand, and I find it works quite well.

Get the B from your Guido stick, and start there (Meyer started with B, but this series could be made starting from anywhere.)

Divide B to the end into fourths, and mark E at 3/4.*

Divide E to the end into fourths, and mark a at 3/4.

Divide a to the end into fourths, and go down (towards B) 6/4, marking D.

Divide D to the end into fourths, and mark G at 3/4.

Divide G to the end into fourths, and go down (towards B) 6/4, marking C.

Divide C to the end into fourths, and mark F at 3/4.

Divide F to the end into fourths, and mark b at 3/4.

Divide b to the end into fourths, and go down 6/4, marking Eb.

Divide Eb to the end into fourths, and mark ab at 3/4.

Divide ab to the end into fourths, and go down 6/4 to Db.

Divide Db to the end into fourths, and mark Gb at 3/4.

Divide Gb to the end into fourths, and mark Cb at 3/4.

Starting from B, divide the stick into eighths and mark h (b natural) at 4/8.

The difference between Cb (found by pure fourths and fifths) and h (found by going up one octave) is the Pythagorean comma.

Finding the Syntonic comma on the monochord:

This too can be started from anywhere, but for the best visual comparison to the tuning above, start at G, the halfway point of the stick.

Divide G to the end into ninths, and mark a at 8/9.

Divide a to the end into ninths, and mark h1 at 8/9.

Divide G to the end into fifths, and mark h2 at 4/5.

The ditone is h1, which is two pure whole steps up from G. The pure major third is h2, which is a pure third up from G. The difference between h1 and h2 is the syntonic comma.

Do the Pythagorean and Syntonic commas look alike? They should, but they are not identical. The Pythagorean comma is 23.46 cents, while the Syntonic comma is 21.51.